Master Thesis
修士論文

Search for proton decay into charged antileptonand η meson in Super-Kamiokande（スーパーカミオカンデにおける反荷電レプトンとη 中間子に崩壊する陽子崩壊の探索）

January 2021
令和 3 年 1 月修士（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

Natsumi Ogawa
小川

夏実Abstract
There are four fundamental forces in nature: the electromagnetic, weak, strong, andgravitational forces. They are believed to be originated from a single uniﬁed force at thevery beginning of our universe. Grand Uniﬁed Theories explain the uniﬁcation of theelectromagnetic, weak, and strong forces and most of them predict protons to decay into
lighter particles. In this thesis, a search of proton decay into an antilepton and a η meson
has been performed. The cross sections of η nuclear eﬀect are revised from the latest
published paper, resulting in reducing their uncertainties around by a factor of two. We
analyze the data exposure of 55.7 kiloton·years (905.0 live days) of Super-Kamiokande. No
candidate event for the proton decay has been found. By combining the results of otherdata taking period from the previous research, the lifetime limit for p → e+ η (p → µ+ η)
has been set to be 1.33 × 1034 (6.1 × 1033 ) years with a total of 0.373 Megaton·years
exposure (3244.4 live days). This gives 28% (30%) longer lifetime limit than that of thelatest published result and the most stringent limit for these modes in the world.

v

Table of Contents

Abstractv

Table of Contentsvii

List of Figuresix

List of Tablesxiii

1 Underlying Physics
1
1.1 The Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Gauge Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Spontaneous Symmetry Breaking . . . . . . . . . . . . . . . . . . . . 3
1.1.3 Electroweak Uniﬁcation . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Grand Uniﬁed Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1 SU(5) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.2 SUSY SU(5) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.3 SO(10) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.4 Proton Decay into an Antilepton and a Meson . . . . . . . . . . . . 8
1.3 History of Nucleon Decay Research . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 The Super-Kamiokande Experiment
2.1 The Super-Kamiokande Detector . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Princeple of the Detector . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 Photodetection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.3 Water Puriﬁcation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Event Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Vertex Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Ring Counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Particle Identiﬁcation . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.4 Momentum Determination . . . . . . . . . . . . . . . . . . . . . . . .
2.2.5 Michel Electron Finding . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.6 Neutron Tagging for SK-IV . . . . . . . . . . . . . . . . . . . . . . .
2.3 Accumulated Data of Super-Kamiokande . . . . . . . . . . . . . . . . . . .

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3 Event Simulation
3.1 Proton Decay Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Fermi Motion of Nucleons . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2 Correlated Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.3 η Nuclear Eﬀect . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Background Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Atmospheric Neutrino Flux . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Neutrino Interactions . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.3 π Nuclear Eﬀect . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Detector Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Evaluation of η Meson Nuclear Eﬀect
4.1 Reaction Processes of η Mesons in Super-Kamiokande . . . . . . . . . . . .
4.1.1 Internal Nuclear Eﬀect . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2 Decay Modes of Survived η Mesons . . . . . . . . . . . . . . . . . . .
4.2 η Photoproduction Reaction . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Experiment at Mainz . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Simulation of η Photoproduction Reaction with η Nuclear Eﬀect . .
4.2.3 Systematic Uncertainties in the Previous Study . . . . . . . . . . . .
4.3 Interpolating the η Nuclear Eﬀect Cross Section . . . . . . . . . . . . . . .
4.3.1 Fitting Two Experimental Data with Least Chi-square Method . . .
4.3.2 Estimation of the Uncertainty of Fitting Parameters . . . . . . . . .
4.3.3 Reduced χ2 value and the goodness of ﬁt . . . . . . . . . . . . . . .
4.3.4 Fit with three diﬀerent functions . . . . . . . . . . . . . . . . . . . .
4.3.5 Goodness of Fittings . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.6 Uncertainty Estimation from Each Function . . . . . . . . . . . . . .
4.4 Estimation of Eﬀect on Proton Decay Search . . . . . . . . . . . . . . . . .

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5 Proton Decay Analysis
5.1 Proton Decay Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1 Selection Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.2 Signal Eﬃciencies and The Number of Background Events . . . . . .
5.1.3 Conﬁrmation of Data Agreement with MC . . . . . . . . . . . . . .
5.1.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Systematic Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Systematic Uncertainties on Signal Eﬃciency . . . . . . . . . . . . .
5.2.2 Systematic Uncertainties on the Number of Background Events . . .
5.3 Lifetime Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1 Lifetime Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.2 Final Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6 Discussion and Conclusion
6.1 Comparison with GUT models . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Future Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Acknowledgements

83

Bibliography

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viii

List of Figures

Shape of potential V (ϕ2 ) under µ2 < 0 and λ > 0 . . . . . . . . . . . . . . .
Feynman diagrams of proton decay via X and Y bosons . . . . . . . . . . .
Running coupling constants α1−1 , α2−1 and α3−1 as a function of energy scalein minimal SU(5) and minimal SUSY SU(5) model . . . . . . . . . . . . . .
Summary of lower limits of nucleon partial lifetime from several experiments

4
6Overview of the Super-Kamiokande detector . . . . . . . . . . . . . . . . . .
Outline drawing of Cherenkov light . . . . . . . . . . . . . . . . . . . . . . .
Schematic view of a 20-inch ID PMT . . . . . . . . . . . . . . . . . . . . . .
Quantum eﬃciency of the photocathode as a function of wavelength . . . .
Timing chart of ATM operation in SK-I to SK-III . . . . . . . . . . . . . .
Timing chart of QTC operation QBEE for SK-IV . . . . . . . . . . . . . . .
An observed charge distribution as a function of Cherenkov opening angle(top) and its second derivative (bottom) . . . . . . . . . . . . . . . . . . . .
2.8 An outline drawing of detecting additional rings . . . . . . . . . . . . . . .
2.9 Typical shower-like (e-like) ring pattern . . . . . . . . . . . . . . . . . . . .
2.10 Typical non shower-like (µ-like) ring pattern . . . . . . . . . . . . . . . . . .
2.11 Area where photons are emitted by a muon . . . . . . . . . . . . . . . . . .
2.12 Distributions of RTOT and true momentum up to 2000 MeV/c for bothe-like and µ-like rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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14
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16

1.1
1.2
1.3
1.4
2.1
2.2
2.3
2.4
2.5
2.6
2.7

3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9

Distributions for simulated momentum (left) and invariant mass (right)
after the proton decay events . . . . . . . . . . . . . . . . . . . . . . . . . .
Invariant mass distribution of the decayed proton in 16 O for p → e+ η MC .
Direction-averaged atmospheric neutrino ﬂux at SK calculated by the Honda
ﬂux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The simulated cross sections for ν (ν̄) interactions in NEUT with data . . .
Calculated cross sections of CCQE scattering with experimental data as thefunction of neutrino energy . . . . . . . . . . . . . . . . . . . . . . . . . . .
Calculated cross sections of ν + N → l + N ′ + π interactions with data as
the function of neutrino energy . . . . . . . . . . . . . . . . . . . . . . . . .
Total CC interactions of CCQE scattering, single meson production, anddeep inelastic scattering with data as the function of neutrino energy . . . .
Measered and NEUT simulated cross sections of π + − 12 C scattering (left)
and of π + − 12 C scattering (right) as a function of pion momentum . . . . .
Fraction of ﬁnal state of pion interaction as a function of pion momentum .

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23

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4.1

Measured diﬀerential cross sections of photoproduction of η mesons in thelaboratory system on a 12 C target identiﬁed by γ-rays with energies ofEγ = 735 − 765 MeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 Image of how the simulation of η photoproduction with a 12 C target proceeds 37
prev , from S (1535) resonance
4.3 Calculated cross section of η nuclear eﬀect, σnuc11
state by the Breit-Wigner formula . . . . . . . . . . . . . . . . . . . . . . . 38
4.4 Measured and simulated diﬀerential cross sections of photoproduction of η
mesons according to Breit-Wigner formula on a 12 C target with γ-rays withenergies of Eγ = 735 − 765 MeV . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.5 Distributions of the ﬁnal state of η mesons of the p → e+ η MC events . . . 40
4.6 Measured absorption cross section of η as a function of the laboratory momentum of η obtained by the experiment at Mainz . . . . . . . . . . . . . . 42
4.7 χ2 distribution when the cross section of the nuclear eﬀect is assumed as
σnuc = a [mb] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.8 Measured η absorption cross sections, σabs , and assumed cross sections ofη nuclear eﬀect in the η photoproduction simulation as σnuc = a [mb] . . . . 46
4.9 Measured and simulated diﬀerential cross sections of photoproduction of η
assuming the cross section of the η nuclear eﬀect as σnuc = a [mb] . . . . . . 46
4.10 χ2 distribution when the cross section of the nuclear eﬀect is assumed as
σnuc = a · exp (−b · pη ) [mb] . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.11 Measured η absorption cross sections, σabs , and assumed cross sectionsof η nuclear eﬀect in the η photoproduction simulation as σnuc = a ·exp (−b · pη ) [mb] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.12 Measured and simulated diﬀerential cross sections of photoproduction of η
assuming the cross section of the η nuclear eﬀect as σnuc = a·exp (−b · pη ) [mb] 48
4.13 χ2 distribution when the cross section of the nuclear eﬀect is assumed as
σnuc = a + b · exp (−c · pη ) [mb] . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.14 Measured η absorption cross sections, σabs , and assumed cross sectionsof η nuclear eﬀect in the η photoproduction simulation as σnuc = a ·exp (−b · pη ) [mb] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.15 Measured and simulated diﬀerential cross sections of photoproduction ofη assuming the cross section of the η nuclear eﬀect as σnuc = a + b ·exp (−c · pη ) [mb] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.16 Momentum distributions of η mesons from bound protons of p → e+ η (left)
and p → µ+ η (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.1
5.2
5.3
5.4
5.5
5.6
5.7

A typical PMT hit pattern of p → e+ η, η → 2γ events in SK . . . . . . . .
A typical PMT hit pattern of p → e+ η, η → 2γ events in SK . . . . . . . .
Distributions of the number of ring of the p → e+ η (left) and p → µ+ η(right) MC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Distributions of the number of ring depending on the decay modes of the η
meson from p → e+ η (left) and p → µ+ η (right) MC . . . . . . . . . . . . .
Distributions of reconstructed η invariant mass of the p → e+ η MC in
SK-IV after applying the selection criteria A and B . . . . . . . . . . . . . .
Distributions of reconstructed η invariant mass of the p → µ+ η MC in
SK-IV after applying the selection criteria A and B . . . . . . . . . . . . . .
Distributions of the number of Michel electrons of the p → e+ η (left) andp → µ+ η (right) MC after applying the selection criteria A-C . . . . . . . .

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Distributions of reconstructed invariant total mass of the p → e+ η (left)
and p → µ+ η (right) MC after applying the selection criteria A-D . . . . . .
5.9 Distributions of reconstructed total momentum of the p → e+ η (left) andp → µ+ η (right) MC after applying the selection criteria A-D . . . . . . . .
5.10 Distributions of the number of neutrons of the p → e+ η (left) and p → µ+ η(right) MC after applying the selection criteria A-D . . . . . . . . . . . . .
5.11 Scatter plots of total mass and total momentum after applying the selection
criteria A-D for p → e+ η, η → 2γ search . . . . . . . . . . . . . . . . . . . .
5.12 Scatter plots of total mass and total momentum after applying the selection
criteria A-D for p → µ+ η, η → 2γ search . . . . . . . . . . . . . . . . . . . .
5.13 Distributions of reconstructed total momenta outside the signal region after
applying the selection criteria A-D for p → e+ η (left) and p → µ+ η (right) .
5.14 Distributions of reconstructed total masses outside the signal region after
applying the selection criteria A-D for p → e+ η (left) and p → µ+ η (right) .
5.15 The signal eﬃciencies (upper) and the number of expected backgrounds(lower, red histogram) and data candidates (lower, black circles) of SK-IV
for p → e+ η, η → 2γ search on the left and p → µ+ η, η → 2γ search on theright . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.16 Reconstructed total invariant mass and momentum of proton decay MC(upper), atmospheric neutrino MC (middle), and data (lower) for p → e+ η,η → 2γ search on the left and p → µ+ η, η → 2γ search on the right . . . . .
5.17 Distributions of reconstructed total invariant momentum of the p → l+ ηMC (upper), and atmospheric neutrino MC and data (lower) for p → e+ η
search on the left and p → e+ η search on the right . . . . . . . . . . . . . .
5.18 The signal eﬃciencies (upper) and the number of expected backgrounds(lower, red histogram) and data candidates (lower, black circles) of from
SK-I to SK-IV for p → e+ η, η → 2γ search on the left and p → µ+ η,η → 2γ search on the right . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.19 Reconstructed total invariant mass and momentum of the data from SK-I
to SK-IV period for p → e+ η, η → 2γ search on the left and p → µ+ η,η → 2γ search on the right . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.8

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List of Tables

1.1

Predicted branching ratios for proton decay modes into a positron and ameson based on SU(5) and SO(10) models . . . . . . . . . . . . . . . . . . .

8

2.1
2.2
2.3

The summary of SK data taking periods . . . . . . . . . . . . . . . . . . . . 11
The summary of Cherenkov momentum thresholds in water . . . . . . . . . 12
Speciﬁcations of the 20-inch ID PMTs . . . . . . . . . . . . . . . . . . . . . 14

4.1
4.2
4.3

Decay mode and the branching ratio of the S11 (1535) resonance state . . .
Decay modes and the branching ratios of η mesons . . . . . . . . . . . . . .
Fraction of the ﬁnal states of η meson after the proton decay of p → l+ η in
16 O with the η nuclear eﬀect deduced from the Breit-Wigner formula . . . .
Fraction of the ﬁnal states of η meson from the proton decay of p → e+ ηin 16 O when the η nuclear eﬀect is assumed as σnuc = a [mb] . . . . . . . .
Fraction of the ﬁnal states of η meson from the proton decay of p → µ+ ηin 16 O when the η nuclear eﬀect is assumed as σnuc = a [mb] . . . . . . . .
Fraction of the ﬁnal states of η meson from the proton decay of p → e+ η in
16 O when the η nuclear eﬀect is assumed as σ
nuc = a · exp (−b · pη ) [mb] . .
Fraction of the ﬁnal states of η meson from the proton decay of p → µ+ ηin 16 O when the η nuclear eﬀect is assumed as σnuc = a · exp (−b · pη ) [mb] .
The no η nuclear interaction rate of η meson from the proton decay of
p → e+ η in 16 O when the η nuclear eﬀect is assumed as σnuc = a [mb] or
σnuc = a · exp (−b · pη ) [mb] . . . . . . . . . . . . . . . . . . . . . . . . . . .
The no η nuclear interaction rate of η meson from the proton decay ofp → µ+ η in 16 O when the η nuclear eﬀect is assumed as σnuc = a [mb] or
σnuc = a · exp (−b · pη ) [mb] . . . . . . . . . . . . . . . . . . . . . . . . . . .

35
36

Event selection criteria for p → l+ η, η → 2γ modes . . . . . . . . . . . . . .
Number of events satisfying each event selection criteria of p → l+ η, η → 2γ
mode within 10,000 of p → l+ η events in MC and 500 years of atmospheric
neutrino MC events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Signal eﬃciencies (ratios) for p → l+ η, η → 2γ mode against the number ofp → l+ η events in MC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Number of events and ratios satisfying each event selection criteria of p →l+ η, η → 2γ mode within 3615 (3553) of p → e+ η, η → 2γ (p → µ+ η,η → 2γ) events in MC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Breakdown (number of events contribution) of the neutrino interaction
modes of the background events in the signal boxes . . . . . . . . . . . . . .

57

4.4
4.5
4.6
4.7
4.8

4.9

5.1
5.2

5.3
5.4

5.5

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64

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5.6

Summary of the signal eﬃciencies, the number of expected background
events and the number of candidate events from 91.5, 49.1, 31.8, and 199.9
kton·years exposure during SK-I, SK-II, SK-III, and SK-IV periods . . . . .
5.7 Summary of the systematic uncertainties (percentage contribution) on signal eﬃciencies for each factors for SK-IV period . . . . . . . . . . . . . . . .
5.8 Summary of the systematic uncertainties (percentage contribution) on thenumber of background events for each factors for SK-IV period . . . . . . .
5.9 Impact of signal eﬃciencies and their systematic errors on lifetime limit . .
5.10 Summary of proton decay search of p → l+ η . . . . . . . . . . . . . . . . . .

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Chapter 1

Underlying Physics
In our universe, all the interactions, or forces, are known to be described by the four
fundamental forces: the electromagnetic, weak, strong, and gravitational forces. They are
believed to be fundamentally equivalent and to be diﬀerentiated from a uniﬁed force at thevery beginning of our universe. The elucidation of the uniﬁed description of the four forcesis one of the primary goals of particle physics. Many approaches both on experimental andtheoretical aspects have been made to verify the uniﬁcation. The Standard Model (SM),the theoretical framework of current particle physics, describes elementary particles withelectromagnetic, weak, and strong interactions. The Glashow-Weinberg-Salam theory in1967 and 1968 [1,2] successfully explained the uniﬁcation of the electromagnetic and weakinteractions as the electroweak force. Therefore, the next idea is to unify the three forcesof the electromagnetic, weak, and strong forces. Such theoretical hypotheses to merge
them are called Grand Uniﬁed Theories (GUT).

1.1

The Standard Model

The Standard Model is a mathematical model in particle physics which describes thethree fundamental forces: the electromagnetic, weak, and strong forces. The framework isbased on the gauge group SU(3) × SU(2) × U(1), where SU(2) × U(1) and SU(3) describe
the electroweak and the strong forces, respectively. In the model, the interactions aredescribed as exchanges of gauge bosons: photons, W, Z, and gluons.

1.1.1

Gauge Symmetry

Symmetry is the fundamental concept of the current particle physics. In particular, thegauge symmetry, which requires Lagrangians to be invariant under a group of local transformation, describes the dynamics of elementary particles in the Standard Model.
(1)

Quantum Electrodynamics (QED)

The ﬁrst idea of gauge symmetry can be found from classical electromagnetism. The
physical E and B ﬁelds, obtained from the scalar and vector potentials ϕ and A, do notchange under the gauge transformation of
ϕ → ϕ′ = ϕ −∂χand A → A′ = A+∇χ.
∂t

1

(1.1)

2

Chapter 1. Underlying Physics

This gauge transformation is written succinctly by Aµ = (ϕ, −A) and ∂µ = (∂0 , ∇) as
Aµ → A′µ = Aµ − ∂µ χ.

(1.2)

Suppose there is a fundamental symmetry requiring that physics is invariant in the universe
under local phase transformations deﬁned as
ψ(x) → ψ ′ (x) = Û (x)ψ(x) = eiqχ(x) ψ(x).

(1.3)

Here, the phase qχ(x) is a function of space-time x that can be diﬀerent at all points.
Under this local U(1) phase transformation, the free-particle Dirac equationiγ µ ∂µ ψ = mψ,(1.4)

becomesiγ µ ∂µ eiqχ(x)ψ = meiqχ(x)ψ ψ,(1.5)⇔ eiqχ(x) iγ µ [∂µ ψ + iq (∂µ χ) ψ] = eiqχ(x) mψ,

(1.6)

µ

iγ (∂µ + iq∂µ χ) ψ = mψ.

(1.7)

Here the additional term −qγ µ (∂µ χ)ψ appears compared to Eq. (1.4). Therefore, theinvariance under a U(1) local phase transformation is not satisﬁed for the free-particle
Dirac equation. To establish the required invariance, equation Eq. (1.4) needs to be
modiﬁed asiγ µ (∂µ + iqAµ )ψ − mψ = 0.

(1.8)

where Aµ is interpreted as the electromagnetic ﬁeld corresponding to a massless gaugeboson, photon. Then, Eq. (1.8) is kept invariant under the U(1) local gauge transformationof Eq. (1.3) with
Aµ → A′µ = Aµ − ∂µ χ.

(1.2)

In the form of Lagrangian L, Eq. (1.8) can be written as
L = ψ̄(iγµ Dµ − m)ψ,(1.9)

D = ∂ + ieA .

(1.10)µµ

They are invariant under the gauge transformation. The Lagrangian for the QED, LQED ,is described as below by adding the factors for the motions of photons
1
LQED = ψ̄(iγµ Dµ − m)ψ − f µν fµν ,4
f µν ≡ ∂ µ Aν − ∂ µ Aν .
(2)

(1.11)(1.12)

Quantum Chromodynamics (QCD)

The strong interaction is described by Quantum Chromodynamics (QCD) and requires tobe invariant under unitary transformations in color space with color charges of r, g, and b.

1.1. The Standard Model

3

This symmetry is preserved under the SU(3) local color phase transformation as described
below:
ψ(x) → ψ ′ (x) = exp [igS α(x) · T̂]ψ(x),(1.13)

where T̂ = {Ta } are the eight generators of the SU(3) symmetry group written in terms ofGell-Mann matrices and α(x) = α(x)a are the eight functions of the space-time coordinate
x. The Lagrangian for the strong interaction can be formulated as gauge invariant forSU(3) conversion as
1a
LQCD = ψ̄(iγµ Dµ − m)ψ − Faµν Fµν,
4Dµ = ∂ µ + igS (Gµa T a ),aFµν
= ∂µ Gaν − ∂ν Gaµ − gS fabc Gbµ Gcν ,(1.14)(1.15)(1.16)

where Gaµ are the eight gauge ﬁelds, fabc are the structure constants of the SU(3) groupwith commutation relations of [λa , λb ] = 2ifabc λc . Similar to QED, the phase transformations give rise to additional terms of gS fabc Gbµ Gcν due to the non-Abelian nature, whichcorresponds to the massless gauge bosons, gluons.
(3)

Weak Interaction

The weak interaction requires local gauge invariance under the transformation in the
weak isospin space. For the case of the ﬁrst generation, the wave function for the weakinteraction ϕ(x) can be expressed as weak isospin doublets of!

φ(x) =νe (x),e− (x)

!

u(x)
,
d(x)

φ(x) → φ′ (x) = exp[igW α(x) · T]φ(x).

(1.17)(1.18)The SU(2) gauge invariant Lagrangian for the weak interaction can be formulated as
L = iφ̄γµ (∂ µ + igW Wµ · T)φ,(1.19)

where W stands for the three gauge ﬁelds of massless gauge bosons.

1.1.2

Spontaneous Symmetry Breaking

While the Lagrangian indicates the massless gauge bosons for the weak interaction, theexperimental data indicate the non-zero masses of W and Z bosons. This fact breaks therequired gauge symmetry of the Standard Model. To solve this problem, the ideas of the
Higgs mechanism and the spontaneous symmetry breaking have been introduced [4, 5].
In general, the Lagrangian for a massless boson is described as
1
L = (∂µ ϕ)(∂ µ ϕ).
2(1.20)

Supposing the U(1) local gauge transformation ϕ(x) → ϕ′ (x) = eigχ(x) , the gauge invariance can be achieved by replacing with the corresponding covariant derivatives of∂µ → Dµ = ∂µ + igBµ .

(1.21)

4

Chapter 1. Underlying Physics

Fig. 1.1: Shape of potential V (ϕ2 ) under µ2 < 0 and λ > 0. From [3].

Considering a scalar ﬁeld ϕ with potential1
1
V (ϕ2 ) = µ2 ϕ2 + λϕ4 ,24(1.22)

1
1
L = (Dµ ϕ)∗ (Dµ ϕ) − V (ϕ2 ) − F µν Fµν ,24(1.23)

the Lagrangian for ϕ is given by

where F µν Fµν is the kinetic term for the new gauge ﬁeld B µ withF µν = ∂ µ B ν − ∂ ν B µ .

(1.24)

Here, the term (∂µ ϕ)(∂ µ ϕ)/2 in V (ϕ2 ) represents the kinetic energy of the scalar particle,µ2 /2 stands for the mass of the particle, and ϕ4 is related to the self-interactions of thescalar ﬁeld. While parameters λ and µ satisfy λ > 0 and µ2 < 0 respectively, the potential
V (ϕ2 ) is minimized ats

ϕ = ±v = ±

−µ2
,
λ

(1.25)as shown in Figure 1.1. The ﬁeld is expressed to have a non-zero vacuum expectation
value v. Since the vacuum state can take either of ϕ = ±v and breaks the symmetry of
the Lagrangian, this process is called spontaneous symmetry breaking. The LagrangianEq. (1.23) can be written with a function of the excitation of the ﬁeld η(x) = ϕ(x) − v as
1
1
1
L(η) = (∂µ η)(∂ µ η) − λv 2 η 2 − Fµν F µν + g 2 v 2 Bµ B µ + · · · .
2
42(1.26)

The former
and latter two terms represent the Lagrangian for η ﬁeld with the mass of√mη = 2λv and that for a gauge boson with the mass of mB = gv, respectively.

1.1. The Standard Model

1.1.3

5

Electroweak Uniﬁcation

The masses of W and Z gauge bosons are considered to be generated by the Higgs mechanism described in the previous section above. In the Glashow-Weinberg-Salam (GWS)
model [1], the mechanism is embedded in the U(1)×SU(2) local gauge symmetry of electroweak interaction. The GWS theory gives the uniﬁed description of the electromagnetic
and weak forces as electroweak interaction. To generate the masses of electroweak gaugebosons of W± and Z by the Higgs mechanism, the Higgs ﬁeld is required to have at least
two complex scalar ﬁelds:
ϕ=

ϕ+
ϕ0

!

!

1 ϕ1 + iϕ2.
=√
2 ϕ3 + iϕ4(1.27)The Lagrangian for this doublet of complex scalar ﬁelds is described as
L = (∂µ ϕ)† (∂ µ ϕ) − V (ϕ),
2 ††(1.28)

2

V (ϕ) = µ ϕ ϕ + λ(ϕ ϕ) .

(1.29)

In µ2 < 0 case, the potential takes the set of minima at
1v2µ2
ϕ† ϕ = (ϕ21 + ϕ22 + ϕ23 + ϕ24 ) =
=− .
2
2
2λ

(1.30)

When the set of ﬁelds is arbitrary chosen as ϕ1 = ϕ2 = ϕ4 = 0, the Higgs doublet isconverted after the symmetry breaking as follows ;
ϕ=

ϕ+
ϕ0

!

!

1
0
→√
.
v
+
h(x)
2

(1.31)The gauge symmetry has broken into SU(2)×U(1)→U(1). The Lagrangian after the symmetry breaking is described replacing the derivatives in Eq. (1.28) by∂µ → Dµ = ∂µ + igW T · Wµ + ig ′

Y
Bµ ,
2

(1.32)

where T are the three generators of the SU(2) symmetry and Y is the hypercharge. Then,the ﬁrst term in the Lagrangian can be written as1
1 2(Dµ ϕ)† (Dµ ϕ) = (∂µ h)(∂ µ h) + gW
(Wµ(1) + iWµ(2) )(W (1)µ − iW (2)µ )(v + h)22
81+ (gW Wµ(3) − g ′ Bµ )(gW W (3)µ − g ′ B µ )(v + h)2 .
8(1)(1.33)(1.34)(2)

Here, the mass terms for the W (1) and W (2) appear as m2W Wµ W (1)µ /2 and m2W Wµ W (2)µ /2
respectively, and therefore the mass of W boson is1
mW = gW v.
2(1.35)

From the remaining factors in Eq. (1.34) of W (3) and B, these two ﬁelds should satisfy∂µ ∂µW (3)µ
Bµ

!

v2
=
8

2
gW
−gW g ′
′
−gW g
g ′2

!

!

W (3)µ.
Bµ(1.36)

6

Chapter 1. Underlying Physics

Fig. 1.2: Feynman diagrams of proton decay via X and Y bosons. From [6].

The mass eigenstates and the mass can be obtained by solving this equation A = cos θ B + sin θ Wµ(3) : m = 0,
µ
W µ
W
A
q
 Zµ = − sin θW Bµ + cos θW Wµ(3) : mZ = 1 v g 2 + g ′2 ,W
2(1.37)

where θW is the Weinberg angle which satisﬁes
mW= cos θW .
mZ

(1.38)

Therefore, the GWS theory provide a uniﬁed description of electromagnetic and weakinteractions and agree well with the experimental results.

1.2

Grand Uniﬁed Theories

After the success of electroweak uniﬁcation, Grand Uniﬁed Theories have been proposed
to explain the merging of the electroweak and strong interactions. The main idea of theGrand Uniﬁed Theories is that the gauge symmetries of SU(2)×U(1) from the electroweak
interactions and SU(3) from the strong interaction are incorporated into a global symmetry
at high energy scale. In this section, some representative GUT models are introduced.

1.2.1

SU(5) Model

The ﬁrst GUT with the most simple model, minimal SU(5) model, was proposed by Georgi
and Glashow in 1974 [7], where SU(5) is the minimum group covering SU(3)×SU(2)×U(1).
24 independent matrices are deﬁning SU(5), which corresponds to 24 gauge bosons of 1
G1 − √2B
30

G21


3

VSU(5) =  G1


X1


Y1

G12
G22 − √2B
30
G32

G13
G23
G33 − √2B
30

X2

X3

Y2

Y3

X̄1
X̄2
X̄3

Y¯1
Y¯2
Y¯3

3
W
3B
√ + √
2
30
W−

W+
3

3B
√ + √
−W
2
30






,




(1.39)

where Gji is the eight gluons. W ± , W 3 , and B are the gauge bosons for electroweak
interactions. X and Y are the new gauge bosons with electric charges of 4/3 and 1/3,

1.2. Grand Uniﬁed Theories

7

Fig. 1.3: Running coupling constants α1−1 , α2−1 and α3−1 as a function of energy scale inminimal SU(5) and minimal SUSY SU(5) model. From [12].

respectively. The fermions are assigned to the 5̄ + 10 dimensional representations
 c d1
 dc 
 2 5̄ =  dc3  ,


 e 

−νe



0
uc3 −uc2 u1
−uc
0
uc1
u2
 3
 c
c
0
u3
10 =  u2 −u1
−u1 −u2 −u3 u1−d1 −d2 −d3 −ec



d1
d2 


d3  .
d1 
0(1.40)

Since the new bosons X and Y can intermediate quarks and leptons, the baryon and lepton
numbers are no longer conserved in SU(5) model. This indicates new interactions whereleptons and quarks transform into each other and violating the baryon number. The
most distinct phenomenon predicted by GUTs is the nucleon decays with considerably
long lifetimes. For example, two quarks in a proton can transform into a lepton andan antiquark mediated by X and Y bosons, and therefore a proton decays into a lepton
and a meson as shown in Figure 1.2. The p → e+ π 0 is the dominant decay mode in thisminimal SU(5) model with a lifetime of 1031±1 years [8]. However, this model was excluded
by several experiments [9–11]. The previous experimental approaches are summarized in
section 1.3.

1.2.2

SUSY SU(5) Model

One feasible extension of the SU(5) model is made by introducing the Supersymmetry(SUSY) hypothesis. SUSY proposes a new symmetry between fermions and bosons andthe existence of the SUSY partner for every particle, all of which have yet to be observed
so far. The clear diﬀerence between SU(5) and SUSY SU(5) can be seen in the threerunning coupling constants of electroweak and strong interactions as shown in Fig. 1.3:
the three couplings do not meet exactly in a point in SU(5), whereas they completely
match at around 2 × 1016 GeV in SUSY SU(5) [13–15]. The dominant proton decay modein SUSY SU(5) is p → ν̄K + . The lifetime limits are expected to be 2.9 × 1030 years for
p → ν̄K + , and 4.1 × 1033 years for p → e+ π 0 [16]. This model had also been excluded by
the results from Super-Kamiokande experiment [17].

8Chapter 1. Underlying Physics
Tab. 1.1: Predicted branching ratios for proton decay modes into a positron and a meson
based on SU(5) and SO(10) models.

Referencep → e+ π 0
p → e+ η
p → e+ ρ0
p → e+ ωothers1.2.3

Branching ratio [%]SU(5)SO(10)[22] [23] [24]
[25]
33
9
35
30
12
3
15
13
17
21
2
2
22
56
17
14
17
11
31
31

SO(10) Model

Another extension of GUT is to introduce a larger symmetry group containing the SU(3)×SU(2)×U(1)from the Standard Model. One of the plausible models is based on the SO(10) symmetry.
In this model, the right-hand neutrino is introduced and all the fermions are embedded
into a single representation and the neutrino masses are predicted [18]. The SO(10) symmetry breaks into SU(3)×SU(2)×U(1) intermediated by several symmetries, for example,
SU(4)×SU(2)L × SU(2)R known as the Pati-Salam GUT model [19]. The dominant decay
mode id p → e+ π 0 with its lifetime ranging from 1032 to 1039 years [20, 21].

1.2.4

Proton Decay into an Antilepton and a Meson

Although the branching ratio of p → e+ π 0 mode dominates in many models, those of theother modes are not negligible. The decay rates of modes with heavy non-strange mesons(η, ρ, ω) are comparable to the modes with pions [22–25]. The branching ratios of thenucleon decay modes into an antilepton and a meson based on SU(5) or SO(10) models
are summarized in Tab. 1.1. In addition to p → e+ (meson) modes, p → µ+ (meson) modes
are predicted to occur by the mixing eﬀects between families. In the SU(5) GUT models,
the decay rates of modes accompanied by muons are estimated to be [26]
Γ(N → µ+ (meson))sin θc 2 cos θc 2=
= 0.010,Γ(N → e+ (meson))(1 + sin θc 2 )2 + 1(1.41)

where θc is the Cabbibo angle. The nucleon decays are supposed to occur with such an
extremely low probability that the number of expected observation event is one or less fora single channel. Therefore, it is crucial to survey comprehensively across multiple decay
modes to enhance the chance of observing proton decay events.

1.3

History of Nucleon Decay Research

The ﬁrst attempt of proton decay search took place by Goldhaber [30] in 1954. The search
was conducted by seeking for the radioactive decay or ﬁssion products from the excitedstate of 232 Th after proton decay. The lifetime was concluded to be more than ∼ 1020
years. This indirect proton decay search had been conducted for several radiochemicalmaterials [31].
At the same time, the searches for nucleon decay by the direct detection of decay
products had started. Reins, Cowan and Goldhaber [32] had measured charged particles

1.3. History of Nucleon Decay Research

Fig. 1.4: Summary of lower limits of nucleon partial lifetime from several experiments. The
experimental results are taken from [9–11, 27, 28]. From [29].

910Chapter 1. Underlying Physics

with kinetic energies above 100 MeV from proton decay in 300 liters of liquid scintillator.
This experiment set a limit for proton lifetime of ∼ 1022 years. The experiment was
improved in 1958 and updated the limit to ∼ 1023 years [32].
After the publication of the ﬁrst GUT in 1974 [7], many nucleon decay experiments
were proposed motivated by the prediction of the minimal SU(5) GUT. One of the major
experimental ideas was the ﬁne-grained iron calorimeters with alternating layers of ironplates and tracking detectors: the Kolar Gold Field [33], NUSEX [34], Soudan [35], andFrejus [11] experiments. While these detectors showed good performance in energy resolution and particle identiﬁcation, the volume had limited to a few ktons due to the highcost.
To assure large detector volume with feasible cost, water Cherenkov experiments were
contrived. These detectors perform high momentum and direction resolution by detecting
the patterns of the Cherenkov ring images. The IMB [10], KAMIOKANDE [9], and SuperKamiokande (SK) experiment [17, 28, 29] have large ﬁducial masses of 3.3 kton, 1.0 kton,and 22.5 kton, respectively, compared to the iron calorimeters.
So far many proton decay modes have been searched by several experiments and noevidence has been found. The summary of the proton into an antilepton plus a mesonmodes are shown as Fig. 1.4.

1.4

Thesis Overview

As described in section 1.2.4, many GUTs predict protons to decay into heavy nonstrange mesons (η, ρ, ω) with comparatively large branching ratios to the dominantp → e+ π 0 mode. Although these modes were analyzed in the previous study [29] by
Super-Kamiokande, there is still room for improvement of proton decay simulation, especially for the modes decay into eta mesons, with additional data exposure of 55.7 kton·years(904.4 live days).
Therefore, this thesis studies the proton decay modes of p → e+ η, η → 2γ andp → µ+ η, η → 2γ by using 0.37 Mton·years exposure of data from Super-Kamiokande.
In chapter 2, the overview and event reconstruction algorithm of Super-Kamiokande aredescribed. Then, the Monte Carlo (MC) simulations for proton decay and background
events are described in chapter 3. chapter 4 shows the estimation of η nuclear eﬀects inthe proton decay simulation. The event selection criteria and the proton decay analysis
with ﬁnal results are represented in chapter 5. Finally, the discussion and conclusion aresummarized in chapter 6.
In contrast to the latest publication [29], the cross sections and their uncertainty of
nuclear eﬀect of η mesons have been updated. This revise has been applied to the searches
where η mesons decay with η → 2γ mode of a certain data taking period (SK-IV). The
analysis of the other data taking periods (SK-I, II, and III) of η → 2γ and other decay
mode of η mesons are yet to be updated. Their results are taken from the previous study.
The limits of proton lifetime have been calculated by combining the results of revised
study of p → l+ η, η → 2γ for SK-IV period, and the others from the latest publication.

Chapter 2

The Super-Kamiokande
Experiment
2.1

The Super-Kamiokande Detector

Super-Kamiokande (SK) is a large water Cherenkov detector whose main purpose is tosearch for the proton decay. The detector comprises a cylindrical stainless-steel tankwith 39.3 m diameter and 41.4 m height, 50 ktons of ultrapure water, and around 13,000
photomultiplier tubes (PMTs) as shown in Fig. 2.1. The detector is placed at 1,000 m
underground of Mt. Ikenoyama, in the Kamioka-mine, Hida-city, Gifu Prefecture, Japan.
This corresponds to the depth of 2700 m.w.e. and reduces the background cosmic muon
ﬂux to 6 × 10−8 cm−2 s−1 sr−1 , a reduction of ﬁve orders magnitude compared to the ﬂux
on the surface of the Earth. The observed rate of cosmic muon at SK is around 2 Hz.
The detector consists of two sections: the inner detector (ID) and the outer detector(OD) as illustrated in Fig. 2.1. ID is the main detector containing 32 ktons of water andcovered by 11,146 of 20-inch PMTs facing inwards, corresponding to around 40% coverage.
OD has been mainly used for rejecting cosmic muon events as a veto detector and also asa shield against γ-rays from the surrounding rocks. The outer surface is covered by 1,885
of 8-inch PMTs facing outwards. ID and OD are separated by PET black sheets.
The data taking periods of SK can be divided into four until 2018; SK-I, II, III, and
IV. The operation had started on April 1st 1996 and stopped in July 2001 to replace bad
PMTs (SK-I). Due to the accident that one of the ID PMTs collapsed to make a shock
wave in November 1996, 6,777 ID PMTs and 1,100 OD PMTs were destroyed. Ever since
this accident, ID PMTs have been protected by ﬁber reinforced plastics (FRPs) and acrylic
cases. The operation had resumed with 5,182 ID PMTs and 1,885 OD PMTs from OctoberTab. 2.1: The summary of SK data taking periods.

Operation
Livetime (days)
Number of PMTs
Photo coverage

Start
End
ID
OD

SK-I
Apr. 1996
Jul. 2001
1489.2
11,146
1,885
40%

11

SK-II
Oct. 2002
Oct. 2005
798.6
5,182
1,885
19%

SK-III
Jun. 2006
Sep. 2008
518.1
11,129
1,885
40%

SK-IV
Sep. 2008
May. 20183244.4
11,129
1,885
40%

12

Chapter 2. The Super-Kamiokande ExperimentFig. 2.1: Overview of the Super-Kamiokande detector. From [36].
Tab. 2.2: The summary of Cherenkov momentum thresholds in water.

pthresh [MeV/c]

e
0.57

µ
118

p
1052

2002 to October 2005 (SK-II). After the replenishment of the rest of the PMTs, the datataking had started with 11,129 ID PMTs from June 2006 to September 2008 (SK-III). In
September 2008, the readout electronics and the data acquisition system were upgraded.
The SK-IV period had begun at that time and stopped in May 2018, which is the longest
data taking period. The features of each period are summarized in Tab. 2.1.

2.1.1

Princeple of the Detector

Super-Kamiokande is capable of observing charged particles by detecting the emitted
Cherenkov light with the PMTs. When a charged particle passes through a dielectric
medium at a speed greater than that of light, Cherenkov light is emitted in a cone aroundthe direction of the particle with an opening angle ofcos θc =

1
,
nβ

(2.1)

where θc is the opening angle, n is the refractive index of the medium, and β = v/c is the
speed of the particle v divided by the propagation speed of light c. The schematic view of
Cherenkov radiation is shown in Fig. 2.2. Here, the charged particle emits the light in the
direction of θc until it reduces its speed to below the speed of light. Therefore, the lightis projected on the wall as a ring, which is called the Cherenkov ring. The momentum
threshold for the Cherenkov radiation is
mcβthresh
pthresh = q,2
1 − βthresh

withβthresh =

1
,
n

(2.2)

and the thresholds for various particles in water (n=1.34) are summarized in Tab. 2.2.
The opening angle is around 42◦ for particles travelling at β ∼ 1 in water.

2.1. The Super-Kamiokande Detector

13

Cherenkov
light𝜽𝒄Chargedparticle
Wall

Fig. 2.2: Outline drawing of Cherenkov light.

The spectrum of Cherenkov light as the number of emitted photons N per wavelength
λ per unit travel length x is given by the following formula




d2 N
2πα
1
= 2 1− 2 2 ,
dλdx
λ
n β

(2.3)

where α is the ﬁne structure constant. When 300 nm ≤ λ ≤ 600 nm, around 340 photons
are emitted per unit cm by a charged particle with β ∼ 1 in water.

2.1.2
(1)

Photodetection
Photosensors

20-inch PMTs with 11-stage Venetian blind type dynodes (R3600) manufactured by Hamamatsu Photonics K. K. are used for ID detectors. Figure 2.3 shows the structure of a
20-inch ID PMT. The bialkali (Sb-K-Cs) photocathodes are adopted and the quantum
eﬃciency peaks at 22% at the wavelength of 360 nm as shown in Fig. 2.4. The typical
speciﬁcations of ID PMTs are summarized in Tab. 2.3.
8-inch PMTs (R1408) are used for OD detectors. The details of this PMT are describedin [37].
(2)

Electronics and Data Acquisition

Due to the upgrade of SK in 2008, the electronics and data acquisition systems of SK-IV
are diﬀerent from those of SK-I, II, and III. The electronic circuit modules used for SK-I

14

Chapter 2. The Super-Kamiokande ExperimentFig. 2.3: Schematic view of a 20-inch ID PMT.

Fig. 2.4: Quantum eﬃciency of the photocathode as a function of wavelength.

Tab. 2.3: Speciﬁcations of the 20-inch ID PMTs.

Photocathode material
Quantum eﬃciency
Dynode structure
Operation high voltage
Gain
Dark current
Dark rate
Transit time
Transit time spread

Bialkali (Sb-K-Bs)
22% at λ = 360 nm
11-stage Venetian blind type
1700 - 2000 V
107 at 2000 V
200 nA at 107 gain
3 kHz at 107 gain with 0.25 p.e. threshold
90 ns at 107 gain
2.2 ns (1σ)

2.1. The Super-Kamiokande Detector

15

Fig. 2.5: Timing chart of ATM operation in SK-I to SK-III. From [38].

to SK-III period are called ATM for ID and OD QTC for OD with hardware triggers. For
SK-IV, QBEE circuits are used for both ID and OD with software trigger.
Systems for SK-I to SK-III The timing chart of ID trigger for SK-I to SK-III is shownin Fig. 2.5. The ATM (Analog Timing Module) records and digitizes the integrated charge
and the arrival time information for each PMT signal with Analog-to-Digital Converter(ADC) and Time-to-Digital Converter (TDC). One ATM module processes 12 ID PMT
signals. The input PMT signals of each ID PMTs are divided into four by a current
splitter. One of the four signals is input into a discriminator and a rectangular pulse
would be output if the signal is a hit. Then, the event trigger (global trigger) is issued by
the sum of the rectangular pulse (HITSUM) from all of the ID PMTs. In this system, three
types of event triggers are used depending on the energy; the super low energy (SLE), lowenergy (LE), and high energy (HE) triggers. Once the event trigger happens, each ATM
starts to digitize and record the information in internal memory buﬀers.
For OD PMTs, the signals are processed by charge-to-time converter modules in asimilar way to the ATM system. The details of electronics and data acquisition for SK-I
to SK-III periods are described in [36].
System for SK-IV To improve the problems of dead time and cable reﬂection of ATMs,
a new electronic module QBEE [39] was developed and installed for ID and OD PMTs

16

Chapter 2. The Super-Kamiokande ExperimentFig. 2.6: Timing chart of QTC operation QBEE for SK-IV. From [39].

since SK-IV period. One QBEE consists of 24 channels. The charge of a PMT signal isintegrated by a Charge-to-Time Converter (QTC), and then the QTC output is converted
into a digitized time and charge by a TDC in each channel. Figure 2.6 shows the timing
chart of QTC. One distinct feature compared to the previous system of SK-I, II, and III is
the acquisition of timing information. In QBEE modules, the charge integrated by QTC isdischarged as constant current. By measuring the discharging time, QTC output signals
contain both charge and timing information. Thanks to this new method, QBEE enables
to record events with no dead time. Another modiﬁcation had been made in the trigger
system. Unlike the hardware trigger in ATM, a software trigger is issued when the numberof PMT hits exceeds the threshold. In addition to the SLE, LE, and HE triggers, special
high energy (SHE) and after trigger (AFT) triggers were newly installed for a neutron
tagging algorithm, which is described in section 2.2.6.

2.1.3

Water Puriﬁcation

To achieve high detection eﬃciency of Cherenkov light, it is essential to keep high transparency of the water to avoid the light being absorbed or scatted. In addition, removing
radioactive materials, especially radon from bedrocks, is crucial to reduce the background
events. For these reasons, 50 ktons of water in the tank is constantly reprocessed by awater puriﬁcation system with the ﬂow rate of around 35 ton/hour.

2.2. Event Reconstruction

2.2

17

Event Reconstruction

The event reconstruction processes are applied to data sets to derive information of physics
quantities, such as the number of Cherenkov rings, momentum, particle identiﬁcation,
charge, and time information of PMTs. The APﬁt reconstruction scheme is used in thisanalysis and applied for both SK data and MC events. Here, the overview of the APﬁt is
described in this section and the details can be found in [40].
The reconstruction algorithm proceeds as the following steps.
1. Vertex ﬁtting
2. Ring Counting
3. Particle Identiﬁcation
4. Momentum Reconstruction
5. Michel Electron Finding
6. Neutron Tagging for SK-IV
The neutron tagging is applied only for data and MC events for SK-IV because it requires
the updated electronics.

2.2.1

Vertex Fitting

The reconstruction process starts from determining the vertex position of the observed
event. This process consists of three steps: point ﬁt, ring edge search, and TDC ﬁt.
Point Fit Firstly, the vertex position is roughly estimated by assuming that the Cherenkov
light has been emitted from a single point. The ﬁtting is proceeded on the basis that thetiming residual ((photon arrival time)-(time of ﬁght)) distribution should peak at the true
vertex position. The goodness of ﬁt G is therefore deﬁned as!

1 X(ti − t0 )2
G=
exp −
,
N i
2(1.5σ)2(2.4)

where N is the number of hit PMTs, ti is the time of ﬂight subtracted timing of i-th
PMT, t0 is the interaction time chosen to maximize the goodness, and σ is the PMT
timing resolution of 2.5 ns. The factor 1.5 is chosen to optimize the ﬁtting performance.
Ring Edge Search Secondly, the algorithm estimates the direction and the edge of themost energetic Cherenkov ring. The opening angle of the ring edge θedge has been deﬁned
as the angle satisfying
d2 PE(θ)
= 0,
(2.5)
d2 θ
where PE(θ) is the angular distribution of the observed charge as a function of opening
angle θ. Figure 2.7 shows the typical PE(θ) distribution and the θedge . The direction isevaluated by the estimator Q(θedge ) deﬁned as
R θedgeQ(θedge ) =0PE(θ) dθ×sin θedgedPE(θ)dθ
θ=θedge!2!

(θedge − θexp )2
× exp −
,
2σθ2

(2.6)

18

Chapter 2. The Super-Kamiokande ExperimentFig. 2.7: An observed charge distribution as a function of Cherenkov opening angle (top) andits second derivative (bottom). From [39].

2.2. Event Reconstruction

19

Fig. 2.8: An outline drawing of detecting additional rings. The shaded and dashed circles
stand for the Cherenkov ring and rings with 42◦ half angles. From [39].

where θexp is the expected Cherenkov opening angle calculated from total charge insideθedge and σθ is its resolution. Qedge is calculated by changing the particle direction aroundthe initial direction derived from point ﬁt and the direction that maximizes Qedge is chosen.
TDC Fit Finally, the precise vertex position is determined from the information of the
track of a charged particle and the scattered Cherenkov photons. The track length isestimated by the total charge with an expectation of 3 MeV/cm energy deposit. The
residual of observed and expected photon arrival timings are calculated diﬀerently for
PMTs inside and outside the ring edge. PMTs inside the edge mostly detect the photons
directly, whereas PMTs outside observe scattered photons or reﬂected light from the wall.
Therefore, the timing residuals for PMTs inside the ring edge are calculated by consideringthat Cherenkov light is emitted with the same angle along the track. For PMTs outside
the edge, on the other hand, the timing was modulated by the time of ﬂight of Cherenkov
light from the vertex with scattering parameters. By evaluating the goodness for PMTs
inside and outside respectively by Eq. (2.4), the vertex position is re-estimated precisely.

2.2.2

Ring Counting

After determining the vertex position and the most energetic Cherenkov ring, additional
Cherenkov rings are investigated by ring counting algorithm.
The Hough transformation [41], a feature extraction technique, is used to search for
the Cherenkov ring candidates. Figure 2.8 shows the basic concept of ﬁnding additionalrings. The shaded ring in shows the Cherenkov ring projected to the detector wall. The
dashed circles represent virtual circles with 42◦ half angles centered at the hit PMTs. The
center of the Cherenkov ring is deﬁned as the intersection point of these dashed circles andthereby the direction can be estimated. Then, a log likelihood method judges whether this20

Chapter 2. The Super-Kamiokande ExperimentFig. 2.9: Typical shower-like (e-like) ring pattern.

candidate ring is true or fake. This likelihood is calculated by the expected and observed
charge inside the tested ring.

2.2.3

Particle Identiﬁcation

Next, the particle identiﬁcation (PID) process runs to classify the Cherenkov rings into two
types: a shower-like (e-like) or a non-shower-like (µ-like). Electrons and γ-rays produce
diﬀused and blurred Cherenkov ring patterns as shown in Fig. 2.9 because of the eﬀects
of electromagnetic showering and multiple scattering. These rings are identiﬁed as e-like.
On the other hand, muons and charged pions create µ-like rings with sharper ring edges asshown in Fig. 2.10. In addition, the Cherenkov rings of electrons and γ-rays have opening
angles of ∼ 42 ◦ , while those of µ-like rings vary depending on the momenta and energy
losses. By using these diﬀerences, the PID algorithm identiﬁes whether the ring is e-like orµ-like. The PID consists of two processes: expected charge distribution and determinationof particle types.

Expected Charge Distribution Firstly, the expected charge in each PMT is calculated. The expected charge distribution in the i-th PMT from an electron qiexp (e), and2.2. Event Reconstruction

Fig. 2.10: Typical non shower-like (µ-like) ring pattern.

Fig. 2.11: Area where photons are emitted by a muon. From [42].

21

22

Chapter 2. The Super-Kamiokande Experimentfrom a muon qiexp (µ) are estimated by
qiexp (e) = αe Qexp (pe , θi )

qiexp (µ) = 


R
ri

3/2

e−ri /L f (Θi ) + qiscat ,αµ sin2 θxidθri sin θxi + ri dx

(2.7)





x=xi

+ q knock −r /L
 e i f (Θi ) + q scat ,
i


(2.8)

where
αe , αµ : normalization factor
Qexp (pe , θi ): expected charge distribution depending on the electron momentum and theangle θi for the i-th PMT
R: virtual sphere radius (16.9 m)
ri : distance from the vertex to the i-th PMT
L: light attenuation length in waterf (Θi ): angular acceptance as a function of the photon injection angle Θi for the i-th
PMT
qiscat : expected charge due to scattering light for the i-th PMT
θxi : Cherenkov angle of a muon track length at x
qiknock : expected charge due to knock-on electrons for the i-th PMT
Qexp (pe , θi ) distribution has been obtained by MC simulation. The term sin2 θxi in
Eq. (2.8) comes from the angular dependence of Cherenkov light intensity and the termr(sin θ + r(dθ/dx)) arises from the area where photons are emitted as shown in Fig. 2.11.
Determination of Particle Types The likelihood for the n-th ring is deﬁned asYLn (e or µ) =exp(e or µ) +prob qiobs , qi,nθi <(1.5×θc )


X exp

n′ ̸=n

qi,n′

,

(2.9)

exp
where qi,nis the expected charge distribution from the n-th ring assuming its particle as
an electron (Eq. (2.7)) or a muon (Eq. (2.8)) and qiobs is the observed charge in the i-th
PMT. Here, the function prob gives the probability to detect qiobs when qiexp is estimatedand deﬁned as!

prob(q

obs

,q

exp

1
(q obs − q exp )2
)= √
exp −
,
2σ 2
2πσ

(2.10)

where σ is the resolution for qexp . The likelihood Ln is optimized by changing the direction
and the opening angle of the n-th ring to take the maximum value. Then, Ln is converted
into the form of a probability to determine the particle type from the ring pattern as!

{χ2 (e or µ) − min[χ2n (e), χ2n (µ)]}
Pnpattern (e or µ) = exp − n,2σχ2 2(2.11)nχ2n (e or µ) = −2 log Ln (e or µ) + constant,(2.12)

2.2. Event Reconstruction

23

Fig. 2.12: Distributions of RTOT and true momentum up to 2000 MeV/c for both e-like andµ-like rings. From [43].

√where the resolution of χ2 distribution is given as σχ2n = 2N and N is the number of
PMTs used in the calculation. The probability coming from the Cherenkov opening angle
is deﬁned as!

(θobs − θnexp (e or µ))2
Pnangle (e or µ) = exp − n,2(δθn )2

(2.13)

where θnobs and δθn are the reconstructed Cherenkov opening angle of the n-th ring and its
ﬁtting error, respectively, and θnexp (e or µ) is the expected opening angle estimated by thereconstructed momentum assuming the particle as an electron or a muon. A probability
function was thus calculated by taking into account both the probabilities of the ringpattern and the opening angle,
P (e, µ) = P pattern (e, µ) × P angle (e, µ).

(2.14)At last, the type of the particle is determined by
PPID ≡q− log P (µ) −q− log P (e).

(2.15)The Cherenkov ring is classiﬁed as e-like when PPID < 0 and as µ-like when PPID > 0.

2.2.4

Momentum Determination

The momentum of each ring is evaluated by the observed charge inside the 70◦ half-angle
cone towards the ring direction. With the modiﬁcation of the light attenuation in waterand the angular acceptance of the PMTs, the total charge from the n-th ring RTOTn is
calculated as






X
GMC 
ri,n
RTOTn =
α×
qi,n × exp
GdataLθ <70◦
i,n

where



cos Θi×f (Θi )

−X
θi,n <70◦

Si  ,(2.16)

24

Chapter 2. The Super-Kamiokande Experiment

α: normalization factor
Gdata , GMC : relative PMT gain parameter for cosmic muon data and MC
θi,n : opening angle between the i-th PMT the n-th ring direction
ti : TOF subtracted hit timing of the i-th PMT
Si : expected charge of the i-th PMT due to scattering light
Figure 2.12 shows the linear relationship between RTOT and true momentum. RTOT
is converted to the corresponding momentum depending on the PID by conversion table
made from MC.

2.2.5

Michel Electron Finding

The Michel electrons, decayed electrons from muons, are a signiﬁcant indicator of theexistence of muons. These electrons are tagged by searching hit clusters after the primary
Cherenkov ring event around ∼ 1 µs. As around 20% of µ− are absorbed by the nucleus
and do not emit decay electrons, the Michel electron tagging eﬃciency of µ− is lower thanthat of µ+ . Moreover, the eﬃciency is higher for SK-IV than SK-I, II, and III periods
because of the elimination of dead time in the electronics. The tagging eﬃciency is ∼ 80%
for µ+ and ∼ 65% for µ− for SK-I to SK-III, and ∼ 95% for µ+ and ∼ 80% for µ− for
SK-IV.

2.2.6

Neutron Tagging for SK-IV

Atmospheric neutrino interactions often produce neutrons, whereas nucleus after proton
decay and hydrogen atoms seldom emit neutrons. Therefore, the neutron tagging is signiﬁcant for distinguishing proton decay events from the background.
Neutrons travelling through water get captured by hydrogen nucleus and emit deexcitation γ-rays of 2.2 MeV with the probability of 100% by the reactionn + p → d + γ (2.2 MeV).

(2.17)The mean travelling time was measured to be 204.87 µs. The produced neutron can betagged by detecting this 2.2 MeV γ-ray.
First, PMT hits from dark noise is removed. Due to the scintillation light from radioactive isotopes in the PMT glass, dark noise appears in PMTs with a time constantof around 10 µs. Therefore, the dark noise events are removed by rejecting the PMT
hits which had recorded another hit within 10 µs. Then, the candidate hit cluster from
2.2 MeV γ-ray is sought by setting conditions to the number of PMT hits within a 10 ns
sliding window after the prompt neutrino interaction. Finally, 16 variables that represent
the characteristics of the candidate cluster are calculated and input into a neural network
to classify into the real and fake clusters. The neutron tagging eﬃciency is 25.2% with a
mistagging rate of 1.8%. The details of neutron tag technology can be found in [44, 45]

2.3

Accumulated Data of Super-Kamiokande

In this study, only the events with their reconstructed vertices and all the visible particles
inside the Fiducial Volume (FV) are used for the proton decay analysis. These events arecalled Fully Contained (FC) events, and the FV is deﬁned as the region 2 m away from the2.3. Accumulated Data of Super-Kamiokande

25

inner detector wall with 22.5 ktons of water. Besides FC events, there are Partially Contained (PC) and Upward Going Muon (UPMU) events. The former corresponds to eventswith their vertices inside FV but charged particles go outside, and the latter derives fromhigh energy muon generated by upward-going neutrinos interacting with the bedrocks.
Most of the FC events are produced from atmospheric neutrinos. So as to extract the
FC data set, events which are not induced by atmospheric neutrinos, for example, cosmic
muons, radioactivities from materials around the detector, electronic noise, and ﬂashing
PMTs, are excluded by imposing conditions on the number of hits, observed charges of ID
and OD PMTs, and so on. Out of around 106 events accumulated in SK in a single day,∼ 8 events are extracted as FC events.

Chapter 3

Event Simulation
In order to evaluate the signal detection eﬃciency for the p → l+ η proton decay events,
a dedicated simulation has been developed. Firstly, in section 3.1, the processes taken inplace in SK followed by the proton decay events are discussed. Besides, the atmospheric
neutrino simulation has been contrived to estimate the expected number of background
events in section 3.2. As a summary, in section 3.3, the overall detector simulation scheme
is described.

3.1

Proton Decay Simulation

In the SK, proton decay events are assumed to occur both in hydrogen nuclei as free
protons and in oxygen nuclei as bound protons with a ratio at one to four in the H2 O
molecule system. The probabilities of the proton decays are considered to be the same
regardless of their initial state. The p → l+ η proton decay events are treated as two body
decays. While in the proton decay events from free protons a positron (an antimuon)
and an η meson are emitted back-to-back direction with a monochromatic momentumof 310.0 MeV/c (297.7 MeV/c) in the p → e+ η (p → µ+ η) decay, in bound protons themomentum of the decay products have distributions due to the Fermi motion, and η mesons
can react with residual nucleons in the originating nuclei. In this work, these eﬀects fromoxygen isotopes were taken into account in the simulation as the steps described in theFig. 3.1: Distributions for simulated momentum (left) and invariant mass (right) after the
proton decay events. The cyan, black, green, red, and purple histograms correspond to free,bound, s-state, p-state, and correlated decay protons, respectively [46].

27Chapter 3. Event Simulation

events [arbitrary]

28

102

10

1
0

200

400

600
800
1000 1200
total invariant mass [MeV/c2]

Fig. 3.2: Invariant mass distribution of the decayed proton in 16 O for p → e+ η MC. The
peaks around 900 MeV/c2 and 940 MeV/c2 corresponds to s-state and p-state, and the tail inthe low momentum region stands for peaks from correlated decays.

following paragraphs.

3.1.1

Fermi Motion of Nucleons

Firstly, the Fermi motion and the nuclear binding energy in 16 O should be evaluated forbound protons. The proton momentum in 16 O are simulated based on the measured andcalculated nucleon momentum and nuclear binding energy in 12 C nucleus from an electron
scattering experiment on 12 C [47]. The simulated proton momentum distributions forboth the s-state and p-state of 16 O are shown in Fig. 3.1. The nuclear binding energies are
39.0 MeV for the s-state and 15.5 MeV for the p-state. This eﬀect is taken into accountby subtracting the binding energy from the proton rest mass.

3.1.2

Correlated Decay

Second, the decaying protons can be aﬀected by the other nucleons in the same 16 O nucleus
by their wave function overlapping. In this case, the proton decays as a three body decay,
and its momentum changes. This event is called the correlated decay, predicted to occuer
with a probability of ∼ 10% from [48]. Fig. 3.2 shows the invariant mass distribution of
decaying protons from p → e+ η MC in 16 O. The lower tail corresponds to correlated decays
where the invariant masses are smaller than proton mass due to the invisible momentumof the recoiling nucleon.

3.2. Background Simulation

29

Fig. 3.3: Direction-averaged atmospheric neutrino ﬂux at SK calculated by the Honda ﬂux.
The ﬂux averaged over all direction is on the left and the ﬂux ratio is on the right. From [50].

3.1.3η Nuclear Eﬀect

Finally, the η mesons generated by bound protons in 16 O nucleus can interact with nucleons until escaping from the nucleus (η nuclear eﬀect). The decaying proton position in
16 O nucleus is given by the Woods-Saxon nuclear density model [49] asρ(r) =

ρ(0),1 + exp ( r−a
b )

(3.1)

where ρ(0) is the average density of nuclei, a and b are the the maximum radius and thesurface thickness of 16 O nucleus, respectively. The η mesons are considered to be emitted
from this position. The details of the η nuclear eﬀect are discussed in section 4.

3.2

Background Simulation

The background for proton decay search in SK is the events derived from the atmospheric
neutrino interactions since the signals can be similar to those of proton decay events. The
simulation events for atmospheric neutrino are created, and the number of background
events is calculated by simulating atmospheric neutrino ﬂux and the neutrino interactions.

3.2.1

Atmospheric Neutrino Flux

The atmospheric neutrino ﬂux in the event simulation is calculated by the Honda ﬂux from[50, 51]. The Honda ﬂux has been estimated based on the measured primary cosmic ray
ﬂuxes in AMS [52] and BESS [53] experiments. The eﬀects of solar wind and geomagnetic
ﬁeld are taken into account. Interactions of the primary cosmic ray with air nuclei are
simulated based on JAM [54] for energies of the primary cosmic rays below 32 GeV andDPMJET-III [55] for above 32 GeV. Finally, the ﬂux of atmospheric neutrinos is acquired
from the decays of pions, kaons, and the secondary particles from air nuclei.

30

Chapter 3. Event Simulation

Fig. 3.4: The simulated cross sections for ν (ν̄) interactions in NEUT with data. The
red, blue, green, and black solid lines correspond to (quasi-)elastic scattering, single π, K, η, γ
production, deep inelastic scattering, and the total of three interactions. From [44].

Figure 3.3 shows the atmospheric neutrino ﬂux averaged over all direction at SK by
the Honda ﬂux together with other ﬂux models from FLUKA [56] and the Bartol [57].
The dominant background events for proton decay search are neutrinos with 1 GeV orderenergy.

3.2.2

Neutrino Interactions

The atmospheric neutrinos interact with the nucleons in the SK tank. These interactions
are simulated by NEUT [58]. The interactions considered in the atmospheric neutrino MC
events are as follows for both Charged Current (CC) and Neutral Current (NC):
CC/NC elastic and quasi-elastic scattering ν + N → l + N ′CC/NC single π production ν + N → l + N ′ + π
CC/NC single K production ν + N → l + N ′ + K
CC/NC single η production ν + N → l + N ′ + η
CC/NC single γ production ν + N → l + N ′ + γ
CC/NC deep inelastic scattering ν + N → l + N ′ +hadrons
CC/NC coherent pion production ν + 16 O → l + 16 O + π
where ν is a neutrino, N and N ′ are the initial and ﬁnal nucleons, and l is a lepton (neutrino
for NC and charged lepton for CC interaction). The distributions for the simulated crosssections of neutrino interactions in NEUT are shown in Fig. 3.4.
(1)

Elastic and Quasi-Elastic Scattering

For NC elastic scattering interaction, a neutrino is scattered by a nucleon without generating any particle. In CC quasi-elastic scattering interaction (CCQE), on the other hand,a neutrino converts into a corresponding charged lepton. The CCQE cross sections forfree protons are calculated by the model in [60] in the simulation. For the interactions in3.2. Background Simulation

31

Fig. 3.5: Calculated cross sections of CCQE scattering with experimental data as the functionof neutrino energy. The solid and dashed lines show the calculated cross sections for free andbound targets. From [59].

bound nucleons in 16 O, the eﬀects of the Fermi motion and the Pauli exclusion principle
are evaluated by [61]. The cross section of NC elastic scattering are estimated by adopting
the relations as follows [62, 63]:σ(νp → νp) = 0.153 × σ(νn → e− p)(3.2)

σ(ν̄p → ν̄p) = 0.218 × σ(ν̄p → e n)

(3.3)

σ(νn → νn) = 1.5 × σ(νp → νp)(3.4)σ(ν̄n → ν̄n) = 1.0 × σ(ν̄p → ν̄p)

(3.5)

+

Figure 3.5 shows the calculated cross sections of CCQE scattering with experimental data.
(2)

Single Meson and Gamma Production

Single meson of π, K, and η and a γ-ray can be produced through the nucleon resonancestate N ∗ :ν + N → l + N∗∗′

N → N + (π, K, η, γ)

(3.6)
(3.7)

Single pion production has been simulated by the Rein-Sehgal Model [64]. The crosssection is obtained by the amplitudes of each resonance production and the rate of eachresonance decaying into a pion and a nucleon. The cross sections for the other production
interactions have been calculated by changing the decay rates corresponding to the generated particle. The estimated cross sections of ν + N → l + N ′ + π interactions are shownin Fig. 3.6.
(3)

Deep Inelastic Scattering

In the energy region of over 10 GeV, deep inelastic scattering becomes the dominant neutrino interaction. Neutrinos with such high energies can interact with a quark comprising
a nucleon and produce multiple hadrons. The cross section of deep inelastic scattering has
been calculated by following the theory from [65]. In the region of W < 2.0 GeV/c2 , only
the pions are considered as produced hadrons, and its multiplicity is estimated from the32

Chapter 3. Event Simulation

Fig. 3.6: Calculated cross sections of ν +N → l+N ′ +π interactions with data as the functionof neutrino energy. From [59].

Fig. 3.7: Total CC interactions of CCQE scattering, single meson production, and deep
inelastic scattering with data as the function of neutrino energy. The upper (lower) lines
corresponds to νµ (ν̄µ ). From [59].

3.2. Background Simulation

33

Fig. 3.8: Measered and NEUT simulated cross sections of π + − 12 C scattering (left) and of
π + − 12 C scattering (right) as a function of pion momentum. From [69].

Fig. 3.9: Fraction of ﬁnal state of pion interaction as a function of pion momentum. From [28].

results of bubble chamber experiments [66, 67]. The calculated cross sections for total CC
interactions of the CCQE scattering, single meson production, and deep inelastic scattering with data are shown in Fig. 3.7. The productions of K and η mesons are treated in
the region W > 2.0 GeV/c2 by PHYTIA/JETSET simulation package [68].
(4)

Coherent Pion Production

Neutrinos can interact with oxygen nuclei followed by a charged pion production. The
pion is emitted in the forward direction as little momentum is transferred to the oxygen
nucleus. This interaction is called the coherent pion production, and has been simulated
by Rein-Sehgal [64].

3.2.3

π Nuclear Eﬀect

The pions induced by atmospheric neutrino interactions are the dominant backgroundsource. These pions can interact with nucleons in 16 O nucleus in a similar way to the η34

Chapter 3. Event Simulation

nuclear eﬀect: pions can be scattered, absorbed, or charge exchanged by π nuclear eﬀect.
The cross sections for each interaction are calculated by the cascade model of [70] by
NEUT [58] with data from various π − (p, n) and π − N scattering experiment [69]. Dueto the Pauli exclusion principle, the nucleon momentum after π nuclear eﬀect needs to belarger than the Fermi surface momentum ofpF (r) =3 2
π ρ(r)
2

1

3

,

(3.8)

where ρ(r) is given by Eq. (3.1). Figure 3.8 shows the measured and NEUT simulated
cross sections of π ± -12 C scattering. The cross sections of π ± -16 O scattering, based on
the measured and simulated cross sections of π ± -12 C scattering, are adopted for generating simulation events for atmospheric neutrinos. The fraction of the ﬁnal state of pion
interaction is shown in Fig. 3.9.

3.3

Detector Simulation

The particles generated through the proton decay and atmospheric neutrino events are
simulated by a GEANT3 based simulation package SKDETSIM. The particle propagation in water, Cherenkov photon emission, and the response of PMTs and electronics are
simulated by SKDETSIM.
The Cherenkov photon propagation has been simulated by considering the Rayleigh/Miescattering and absorption by H2 O molecule. The Rayleigh scattering, caused by small particles (r ≪ λ, where r is the radius of a particle), dominates when photons have relatively
short wavelengths of (λ ≤ 450 nm). The scattering coeﬃcient dependence of λ−4 makephotons scatter symmetrically in the forward and backward directions. For longer wavelength (λ ≥ 450 nm), the dominant process is the absorption by H2 O molecule. Besides,
the Mie scattering is caused by large particles (r ≫ λ) to scatter photons more intensely
in the forward than in the backward. These scattering eﬀects tuned by calibration data
have been evaluated in the simulation package.

Chapter 4

Evaluation of η Meson Nuclear
Eﬀect
The process of η mesons absorbed immediately after proton decay events brought thebiggest uncertainty to determine the lifetime of the protons decaying into p → l+ η modes.
In this work, the reaction cross section between η mesons and nuclei has been updated
from the previous research [29]. The cross section had been studied from an experimental
work deducing the photoproduction cross sections of η mesons. In section 4.1, the elementary interaction processes and the detection schemes of η mesons in Super-Kamiokande
are described. To estimate the uncertainty of the cross section, the η photoproduction
reactions are discussed in section 4.2. From the obtained diﬀerential cross sections of η
photoproduction reactions and the measured η absorption cross sections, the cross sectionsof the η nuclear eﬀect have been evaluated by a minimum chi-square method as discussed
in section 4.3.

4.1

Reaction Processes of η Mesons in Super-Kamiokandeη mesons produced from oxygen atoms through the proton decay modes p → l+ η occasionally interact while travelling within the nucleus to escape from it. Since this nuclear
eﬀect plays a crucial role in proton decay search in that aﬀecting the number of observableη mesons, it is necessary to get a reliable estimation of the ηN reaction cross sections.

4.1.1

Internal Nuclear Eﬀect

η mesons interact with nucleons and can form several baryon resonance states. The
S11 (1535) resonance state is studied exclusively due to the property that it exists slightly
Tab. 4.1: Decay mode and the branching ratio of the S11 (1535) resonance state. Numbers
are taken from [71].

Decay mode
NπNηN ρ → N ππ
∆π → N ππ

Branching ratio[%]
42
42
13
3

35

36

Chapter 4. Evaluation of η Meson Nuclear Eﬀect

Tab. 4.2: Decay modes and the branching ratios of η mesons. Numbers are taken from [71].

Decay modes
2γ
3π 0
+
π π−π0π+π−γBranching ratios[%]
39.41 ± 0.20
32.68 ± 0.23
22.92 ± 0.28
4.22 ± 0.08

above the ηN threshold providing a large branching ratio of S11 (1535) → N + η [72, 73].
Therefore, the η nuclear eﬀect is evaluated through S11 (1535) resonance state in this study.
η + N −→ S11 (1535),
S11 (1535) −→ N + meson(η, π, ππ).

(4.1)
(4.2)

The decay modes of S11 (1535) state in Eq. (4.2) and their branching ratios from [71] arelisted in Tab. 4.1.

4.1.2

Decay Modes of Survived η Mesons

The main decay modes of η mesons and their branching ratios are shown in Tab. 4.2. In
this thesis, only the proton decay sequence into the p → l+ η, η → 2γ modes are studied
due to the largest decaying branching ratio of η mesons and their highest signal eﬃciencies,
as discussed in section 5.1.1, above all the decay modes of η mesons.

4.2

η Photoproduction Reaction

In order to conﬁrm the cross section of η nuclear eﬀect, σnuc , and its uncertainty, thephotoproduction reaction of η meson is utilized. The measured diﬀerential cross sectionsof η photoproduction reaction dσηphoto /dp from an experiment conducted at Mainz [73]are compared with the simulated values from an phenomenological ﬁtting of σnuc to verify
the assumption.

4.2.1

Experiment at Mainz

The experiment conducted by MAMI accelerator facility at Mainz Germany in 1996
[73] measured the diﬀerential cross sections of photoproduction reaction of η mesons,dσηphoto /dp, by injecting γ-rays with energies of Eγ = 735 − 765 MeV into a 12 C target.
The produced η mesons were identiﬁed by reconstructing the invariant masses from thetwo γ-rays emerging from the η → 2γ decay. Figure 4.1 shows the measured dσηphoto /dp
along with the laboratory momentum of η mesons of p = 0 − 600 MeV/c. The total systematic error for each dσηphoto /dp plot was estimated to be 6%, mainly coming from theuncertainty of detection eﬃciency.

4.2.2

Simulation of η Photoproduction Reaction with η Nuclear Eﬀect

The overview of the process of η photoproduction with a 12 C target is illustrated inFig. 4.2. Firstly, the elementary process of this reaction is the η photoproduction reactions

dσ/dp [ µ b×(MeV/c)-1]4.2. η Photoproduction Reaction

37

0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0

100

200

300

400

500
600
p [MeV/c]

Fig. 4.1: Measured diﬀerential cross sections of photoproduction of η mesons in the laboratory
system on a 12 C target identiﬁed by γ-rays with energies of Eγ = 735 − 765 MeV. Data istaken from [73].

Fig. 4.2: Image of how the simulation of η photoproduction with a 12 C target proceeds. The
red, blue, and yellow circles stand for η mesons, nucleons, and 12 C nucleus, respectively. Thisﬁgure illustrates when an η meson is scattered by nuclear eﬀect while propagating through thenucleus.

38

Chapter 4. Evaluation of η Meson Nuclear Eﬀect

prev, from S11 (1535) resonance stateFig. 4.3: Calculated cross section of η nuclear eﬀect, σnucby the Breit-Wigner formula. From [6].

at nucleons, creating an η meson with a γ ray striking a nucleon in the nucleus:
γp → ηp,(4.3)

γn → ηn.

(4.4)

The diﬀerential cross sections of the η photoproduction on a proton target (Eq. (4.3)) arecalculated by the Scattering Analysis Interactive Dial-in (SAID) program [74]. The SAID
provides data and the solutions of partial wave analysis of hadron elastic and chargeexchange scatterings. The contribution of the neutrons in the photoproduction crosssections is assumed to be 2/3 of that with a proton target from the results of η photoproduction measurement experiment with a deuteron target [75]. Secondly, η mesons
generated by these reactions propagate through the 12 C nucleus, and this propagation
is simulated by the cross section of η nuclear eﬀect, σnuc , within a simulation package,
NEUT [58]. As only the η mesons which can escape from nucleons without being absorbed
are observed, the cross section of the η photoproduction reaction on a 12 C target can be
estimated by the number of these escaped η mesons.
In the previous study [6,29], σnuc and its uncertainty were estimated by simulating theη photoproduction cross section in a 12 C nucleus by σnuc in this way and comparing to
the measured cross section in [73].

4.2.3

Systematic Uncertainties in the Previous Study

prev , was calculated
The cross section of the η nuclear eﬀect in the previous study [6, 29], σnucby the Breit-Wigner Formula:
prev
σnuc
=

π
ΓηN (Γtotal − ΓηN ),k 2 (ECMS − Mres )2 + Γ2total /4(4.5)

where ECMS is the energy of η-N in the center of mass frame, Mres is the mass of theresonance state S11 (1535), and Γtotal is the total width of S11 (1535) resonance taken

4.2. η Photoproduction Reaction

39

Fig. 4.4: Measured and simulated diﬀerential cross sections of photoproduction of η mesons
according to Breit-Wigner formula on a 12 C target with γ-rays with energies of Eγ = 735 −765 MeV. The experimental data from Mainz experiment [73] are shown with black circles.
The dashed line indicates the cross-section without η nuclear eﬀect, and the black, blue, andred solid lines show the simulated cross sections when they were multiplied by 100%, 50%, and200%, respectively. From [6].

Tab. 4.3: Fraction of the ﬁnal states of η meson after the proton decay of p → l+ η in 16 O
with the η nuclear eﬀect deduced from the Breit-Wigner formula. From [6].

η nuclear eﬀect in 16 O nucleus
no interaction
scatteredabsorbedprev
default σnuc
56%
6%
38%

prev × 0.5
σnuc
73%
4%
23%

prev × 2
σnuc
43%
5%
53%

40

Chapter 4. Evaluation of η Meson Nuclear Eﬀect

Fig. 4.5: Distributions of the ﬁnal state of η mesons of the p → e+ η MC events. The black
shaded histogram shows the contribution from free proton decay events only. The blue andcyan histograms show the fractions of events that can be reconstructed and identiﬁed as proton
decays originating from all protons and free protons, respectively. The ﬁnal interaction statesin the x-axis are described the top right table. The number of events in the bins of the blue
histogram from left to right are 1757, 26, 0, 0, 0, and 0.

4.3. Interpolating the η Nuclear Eﬀect Cross Section

41

from [71], ΓηN = Γtotal × BR(ηN ) is the partial width of S11 (1535) → ηN estimatedas the product of Γtotal and the branching ratio of S11 (1535) → ηN , and k is the wave
prev according to Eq. (4.5) as a function of thenumber. Figure 4.3 shows the calculated σnucinitial momentum of η meson in the center of mass frame.
Figure 4.4 illustrates the comparison between the measured cross section of η photoproduction from Mainz experiment [73] and the simulated cross section by Eq. (4.5) inthe manner described in section 4.2.2. The dashed line in Fig. 4.4 stands for the crosssection without any η nuclear eﬀect, which corresponds to the cross section of elementary
processes in Eq. (4.3) and Eq. (4.4) multiplied by the number of nucleons in the 12 C nucleus. By comparing the measured data plots and the simulated line, the uncertainty ofprev was estimated from −50% to +100% in the previous study.
σnuc
The analysis of the proton decay search in SK is conducted by reconstructing momenta
and masses of the decay particles from a proton or the decayed proton itself from detected
Cherenkov lights. If the decay particles are absorbed or scattered by the nuclear eﬀect,
the momenta and masses cannot be reconstructed correctly, and thus fail in identifying
the initial particles. As a result, only the proton decay candidates of p → l+ η without
experiencing any η nuclear eﬀect can be identiﬁed by SK. Figure 4.5 shows more than
98.5% of the events that can be reconstructed as proton decays in SK experienced nonuclear interaction. Against this background, the systematic error arising from η nuclear
eﬀect was estimated by the uncertainty of the no nuclear interaction rate in a 16 O nucleus in
the previous research [29]. The no nuclear interaction events are deﬁned as those of proton
decay MC where the η meson decayed before being scattered or absorbed (transferred into
other particles) by nucleons. Table 4.3 shows the fractions of the ﬁnal states of η mesons in
the proton decay MC events deduced in [29]. From this table, the no nuclear interaction
prev varies from 43% to 73% with arate with taking into account the uncertainty of σnucnominal value of 56%. Hence, the systematic error attributable to the nuclear eﬀect of
p → l+ η was estimated as

1
εprev
nuc =
2

4.3



(|73 − 56| |43 − 56|)
+
56
56



∼ 27%.

(4.6)

Interpolating the η Nuclear Eﬀect Cross Section

The experimental η absorption cross sections, σabs , were also measured in the Mainz
experiment [73] in Fig. 4.6 as a function of the momentum of η meson in the laboratory
prev , does not match
frame. As shown in Fig. 4.3, the cross sections of η nucleus eﬀect, σnucwith the measured η absorption cross sections. This discrepancy could be interpreted as
the former only consider a single excite state of S11 (1535) resonance state without taking
into account the eﬀect of interference between other resonance states, such as P11 (1440)
and D13 (1520).
In this study, the cross section of η nuclear eﬀect, σnuc , and its uncertainty are estimated by adopting the least χ2 method for both the measured η photoproduction crosssection, dσηphoto /dp, and η absorption cross section, σabs , from [73].

Chapter 4. Evaluation of η Meson Nuclear Eﬀect

σabs [mb]

42

160
140
120
100
80
60
40
20
0
0

100

200

300

400

500

600
pη [MeV]Fig. 4.6: Measured absorption cross section of η as a function of the laboratory momentumof η obtained by the experiment at Mainz. Data is taken from [73].

4.3.1

Fitting Two Experimental Data with Least Chi-square Method

The cross section σnuc is assessed with three types of functions by minimizing the χ-squarevalues:χ2 = χ2ηphoto + χ2abs ,(4.7)

where χ2ηphoto and χ2abs are given asnηphoto "χ2ηphoto =

X

1

i=1

εηphoto,iχ2abs =

n
abs
X
i=1



dσηphoto,i−dp

dσηphoto,idp

σabs,i − σnuc (pη )εabs,i



# 2

,

(4.8)

sim

!2

,

(4.9)

where (dσηphoto /dp)sim is the simulated dσηphoto /dp, σnuc (pη ) is the assumed cross sectionof nuclear eﬀect, εηphoto and εabs are the errors of measured (dσηphoto /dp)sim and σabs ,respectively, and nηphoto and nabs are the numbers of data points. Here, σabs ∼ σnuc isassumed as the diﬀerence between these two cross sections is the inclusion of the eﬀects
of the elastic scattering and multistep interactions, which can be assumed to play a minor
role of up to a few percentage as discussed in [73]. The estimation of the errors of ﬁtting
parameters are described in section 4.3.2.
While the uncertainties reported in the η photoproduction experiment [73] had been
contributed only by the systematic errors of 6% for the data points of the cross sections,the additional statistical errors are evaluated especially for the high momentum region,
where the dσηphoto /dp values converge to 0 µb × (MeV/c)−1 . By the ﬂuctuation of the4.3. Interpolating the η Nuclear Eﬀect Cross Section

43

points at these values from the experiment of Eγ = 634 − 705 MeV with a 12 C target,
the statistical error of 0.004 [mb] is additionally included in this study. Thus, The total
uncertainty εηphoto,i is given as follows:s

0.06 ×εηphoto,i =

4.3.2

dσηphoto,idp

2+ (0.004)2 mb.

(4.10)

Estimation of the Uncertainty of Fitting Parameters

The ﬁt of n-tuple data sets of (x1 , y1 ), · · · , (xn , yn ) with an arbitrary function f (x; a1 , · · · , am )to determine the parameters a1 , · · · , am , the least χ2 method is employed. In this method,
the set of the parameters satisfying the χ2 deﬁned below to be the least are adopted as
the best ﬁtn Xyi − y(xi ) 2

χ2 ≡

σi

i=1

.

(4.11)The uncertainty of yi given as σi is assumed to follow a Gaussian distributed random error.
When the true parameter sets are supposed as (a1 , · · · , am ), the likelihood of taking then-tuple data sets (x1 , y1 ), · · · , (xn , yn ) is given as
P (a1 , · · · , an ) ==nYi=1 σi
n
Y

1
√

"

#

(yi − f (xi ; a1 , · · · , am ))2· exp −
,
2σi2
2π

1
√

i=1 σi 2π!

(4.12)!

χ2,· exp −
2

(4.13)

where in the second line, the likelihood can be written as a function of χ2 in Eq. (4.13)
with the deﬁnition of χ2 (Eq. (4.11)). The likelihood P (a1 , · · · , an ) reaches the maximum
value when χ2 takes its minimum χ2min ofχ2min =

n Xyi − f (xi ; a˜1 , · · · , a˜m ) 2

σi

i=1

,

(4.14)

where a˜1 , · · · , a˜p are the parameters of f (xi ; a˜1 , · · · , a˜m ) with χ2min .
To estimate the uncertainties of the ﬁtting parameters, consider the residual of χ2 ,2
∆χ = χ2 − χ2min . From Eq. (4.11) and Eq. (4.14),
∆χ2 =
=

n
X
(yi − f (xi ; a1 , · · · , am ))2 − (yi − f (xi ; ã1 , · · · , ãm ))2
i=1
n
X

σi2

,

(4.15)(f (xi ; a1 , · · · , am ) + f (xi ; ã1 , · · · , ãm ) − 2yi ) (f (xi ; a1 , · · · , am ) − f (xi ; ã1 , · · · , ãm ))
.
2
σ
i
i=1
(4.16)

By taking the second order of the diﬀerential coeﬃcients,
∆χ ≈
2



,ãm )n X
m
2f (xi ; ã1 , · · · , ãm ) − 2yi + ∂f (xi ;ã∂a1 ,···
·
∆a
X
j
j
i=1 j=0

σi2!

·∂f (xi ; ã1 , · · · , ãm )· ∆aj ,
∂aj
(4.17)

44

Chapter 4. Evaluation of η Meson Nuclear Eﬀect

where ∆aj is deﬁned as ∆aj = aj − ãj . By the deﬁnition of minimum χ2 value, Eq. (4.14)
can be transformed as:∂χ2 (ã1 , · · · , ãm )
= 0,
∂aj
⇔

n
X
f (xi , ã1 , · · · , ãm ) − yi

σi2

i=0

(4.18)∂f (xi , ã1 , · · · , ãm )
· 2
∂aj

!

= 0.

(4.19)

Therefore, in the case of a ≈ ã, Eq. (4.17) can be approximated as:
∆χ2 ≈

m
X

Aj (aj − ãj )2 .

(4.20)

j=1

Equation (4.13) can be transformed as bellow.
P (a1 , · · · , an ) =∝∝nYi=1 σimY

1√

j=1 σimY
j=1 σi

1· exp − (∆χ2 + χ2min ) ,2
2π(4.21)Aj (aj − a˜j )2· exp −,
22π

(4.22)(aj − a˜j )2,· exp −
2δj2
2π(4.23)

1
√
1
√

"
"

#

#

p

where in the third line, Aj = 1/δj is adopted. From Eq. (4.23), it can be approximatedthat ∆χ2 would follow the χ2 distribution with the number of freedom of m, which is thenumber of ﬁtting parameters. From the χ2 distribution, the value of ∆χ2 when it is within
the 1σ = 68.3% region is deduced to satisfy ∆χ2 ≤ 1.00 when m = 1, ∆χ2 ≤ 2.30 whenm = 2, and ∆χ2 ≤ 3.53 when m = 3. Accordingly, the errors of the ﬁtting parameters are
estimated as the uncertainties within the χ2min + 1 region for m = 1.00 ﬁtting function,the χ2min + 2.30 region for m = 2 ﬁtting function, and the χ2min + 3.53 region for m = 3
ﬁtting function.

4.3.3

Reduced χ2 value and the goodness of ﬁt

The χ2 distribution is known to obey a χ2 distribution when the ﬁt function is exactly
same as the “true” function. This is based on the assumption that each data points should
distribute with a Gaussian of the standard deviation of the error bars. As the minimum
χ2 value after a certain ﬁt to n-point data with a k-parameter function is expected tofollow the χ2 distribution with the degree of freedom of n − k, the reduced χ2 , deﬁned asχ2ν,min =

χ2min
n−k

(4.24)

is often used as an indicator of the goodness of ﬁt. As the mean value of a χ2 distribution
with the n − k degree of freedom is n − k, it can be denoted that the ﬁt is reasonable
when χ2ν,min ∼ 1. While in the case of χ2ν,min > 1, it is interpreted as less ﬁt, the caseof χ2ν,min < 1 shows the function is over-ﬁtting. Here below, the result of three diﬀerent
functions ﬁt with χ2 minimisation are discussed comparing with their goodness of ﬁt.

χ2

4.3. Interpolating the η Nuclear Eﬀect Cross Section

45

54
52
50
48
46
44
42
40
38
36
20

22

24

26

28

30
a [mb]

Fig. 4.7: χ2 distribution when the cross section of the nuclear eﬀect is assumed as σnuc =a [mb]. The orange line shows the ﬁt results of χ2 distribution with a quartic function.

4.3.4

Fit with three diﬀerent functions

In this study, three types of functions listed below are compared to obtain the best estimation of σnuc function:
•
•
•
(1)

Case1: σnuc = a
Case2: σnuc = a · exp (−b · pη )Case3: σnuc = a + b · exp (−c · pη )Case1: σnuc = a

First, from the distribution of measured σabs in Fig. 4.6, σnuc is considered as a function
independent of pη :σnuc = a [mb].

(4.25)dσηphoto /dp is simulated by σnuc , and χ2 deﬁned in Eq. (4.7) is calculated. Figure 4.7 shows
the distribution of χ2 and parameter a. From the ﬁtting curve of Fig. 4.7, χ2 represents
the minimum value of χ2min = 38.2, and χ2ν,min = 2.12 at a = 24.3 mb, explained in
section 4.3.3. Furthermore, χ2 = χ2min + 1.00 is satisﬁed when a = 23.0 or 25.8 mb.
Therefore, as discussed in section 4.3.2, σnuc and its uncertainty is estimated as
σnuc = 24.3+1.5−1.3 mb.

(4.26)The measured and assumed absorption cross sections σabs and the measured and simulated
η photoproduction cross sections are shown in Fig. 4.8 and Fig. 4.9, respectively.
(2)

Case2: σnuc = a · exp (−b · pη )Figure 4.9 demonstrates the simulated diﬀerential cross sections of η photoproduction are
underestimated in the high momentum region compared to measured values. This means

Chapter 4. Evaluation of η Meson Nuclear Eﬀect

σabs (σnuc) [mb]

46

160
140

MAINZ(1996)
χ2 (a=24.3)

120

χ2 +1 (a=23.0)

minχ2 +1 (a=25.8)
min
min

100
80
60
40
20
0
0

100

200

300

400

500

600
pη [MeV]dσ/dp [ µ b×(MeV/c)-1]

Fig. 4.8: Measured η absorption cross sections, σabs , and assumed cross sections of η nuclear
eﬀect in the η photoproduction simulation as σnuc = a [mb]. The experimental data of σabs
from Mainz experiment [73] are shown as the black circles. The black, red, and blue solidlines show the nuclear eﬀect cross sections, σnuc , with parameters of a = 24.3, a = 25.8, anda = 23.0, respectively.

0.25

MAINZ(1996)
No nuclear effectσnuc (a=24.3)
σnuc (a=23.0)

0.2

σnuc (a=25.8)

0.15

0.1

0.05

0

100

200

300

400

500

600
p [MeV/c]

Fig. 4.9: Measured and simulated diﬀerential cross sections of photoproduction of η assuming
the cross section of the η nuclear eﬀect as σnuc = a [mb]. The experimental data from Mainz
experiment [73] are shown as the black circles. The dashed line shows the cross-section without
η nuclear eﬀect, and the black, red, and blue solid lines show the simulated cross sections withparameters of a = 24.3, a = 25.8, and a = 23.0, respectively.

4730
0.0024{χ}^2

b [/(MeV/c)]

4.3. Interpolating the η Nuclear Eﬀect Cross Section

29

0.00220.002

280.0018270.00160.001426

0.0012250.001
240.0008
30

35

40

45

50

55

a [mb]

σabs (σnuc) [mb]Fig. 4.10: χ2 distribution when the cross section of the nuclear eﬀect is assumed as σnuc =a · exp (−b · pη ) [mb]. The red circle in the center shows the parameter set whose χ2 showed
the minimum value χ2min . The parameter sets in the region colored with red satisfy χ2 > 30.0.

160
140

MAINZ(1996)
χ2 (a=42.0, b=0.0015)

120

χ2 +1 (a=52.0, b=0.0021)minχ2 +1 (a=34.0, b=0.0009)
min
min

100
80
60
40
20
0
0

100

200

300

400

500

600
pη [MeV]Fig. 4.11: Measured η absorption cross sections, σabs , and assumed cross sections of η nuclear
eﬀect in the η photoproduction simulation as σnuc = a · exp (−b · pη ) [mb]. The experimental
data of σabs from Mainz experiment [73] are shown as the black circles. The black, red,and blue solid lines show the nuclear eﬀect cross sections, σnuc , with parameters of (a, b) =(42.0, 0.0015) = (34.0, 0.0009), and (52.0, 0.0021), respectively. The parameter sets of (a, b) =(34.0, 0.0009) and (52.0, 0.0021) are chosen as representative points that fulﬁll χ = χmin + 2.30.

Chapter 4. Evaluation of η Meson Nuclear Eﬀect

dσ/dp [ µ b×(MeV/c)-1]

48

0.25

MAINZ(1996)
No nuclear effectσnuc (a=42.0, b=0.0015)
σnuc (a=34.0, b=0.0009)

0.2

σnuc (a=52.0, b=0.0021)

0.15

0.1

0.05

0

100

200

300

400

500

600
p [MeV/c]

Fig. 4.12: Measured and simulated diﬀerential cross sections of photoproduction of η assuming the cross section of the η nuclear eﬀect as σnuc = a · exp (−b · pη ) [mb]. The experimental data from Mainz experiment [73] are shown as the black circles. The dashed line
shows the cross-section without η nuclear eﬀect, and the black, red, and blue solid lines show
the simulated cross sections with parameters of (a, b) = (42.0, 0.0015) = (34.0, 0.0009), and(52.0, 0.0021), respectively. The parameter sets of (a, b) = (34.0, 0.0009), (52.0, 0.0021) arechosen as representative points that fulﬁll χ = χmin + 2.30.

the η mesons with high momenta have been much absorbed or scattered in the nuclei thanactually they are, indicating the overestimation of nuclear eﬀect cross section. Therefore,
so as to mitigate this inconsistency, an exponential function decreasing with momentum
is tested as:
σnuc = a · exp (−b · pη ) [mb].

(4.27)dσηphoto /dp is simulated by σnuc , and χ2 deﬁned in Eq. (4.7) is calculated. Figure 4.10
shows the distribution of χ2 , parameter a, and b. χ2 represents the minimum value ofχ2min = 23.5, χ2ν,min = 1.39 when (a, b) = (42.0, 0.0015). Thus, the most plausible σnuc asa function of exponential is deduced to be
σnuc = 42.0 · exp (−0.0015 · pη ) mb.

(4.28)The region satisfying χ2 = χ2min + 2.30 = 25.8 corresponds to the 68.3% conﬁdence levelof parameter set (a, b). The measured and assumed absorption cross sections σabs areshown in Fig. 4.11, and the measured and simulated η photoproduction cross sections aredescribed in Fig. 4.12.
(3)

Case3: σnuc = a + b · exp (−c · pη )Finally, the χ2 minimization is tested as a combined function of Eq. (4.25) and Eq. (4.27)
as:σnuc = a + b · exp (−c · pη ) [mb].

(4.29)

c [/(MeV/c)]

4.3. Interpolating the η Nuclear Eﬀect Cross Section

49

0.00160.00140.00120.001
0.00080.00060.0004130b [ 120
mb 110
]
100
90
80
70
60
50

-90 -80-70-40-60 -50-30 -20-10a [mb]

σabs (σnuc) [mb]Fig. 4.13: χ2 distribution when the cross section of the nuclear eﬀect is assumed as σnuc =a + b · exp (−c · pη ) [mb]. The red circle stands out for the parameter set whose χ2 showed theminimum value χ2min . Although the 1σ conﬁdence level of the ﬁtting parameters with threeparameter function is given by χ2 ≤ χ2min + 3.53, the black circles show the parameter sets
which satisﬁed χ2 ≤ χ2min + 1.

160
MAINZ(1996)
χ2 (a=-75, b=115, c=0.0004)

140

min

120
100
80
60
40
20
0
0

100

200

300

400

500

600
pη [MeV]Fig. 4.14: Measured η absorption cross sections, σabs , and assumed cross sections of η nuclear
eﬀect in the η photoproduction simulation as σnuc = a · exp (−b · pη ) [mb]. The experimental
data of σabs from Mainz experiment [73] are shown as the black circles. The red solid line
show the nuclear eﬀect cross section, σnuc , with parameters of (a, b, c) = (−75.0, 115, 0.0015)
with χ2min .

Chapter 4. Evaluation of η Meson Nuclear Eﬀect

dσ/dp [ µ b×(MeV/c)-1]

50

0.25

MAINZ(1996)
No nuclear effect
σnuc

0.2

0.15

0.1

0.05

0

100

200

300

400

500

600
p [MeV/c]

Fig. 4.15: Measured and simulated diﬀerential cross sections of photoproduction of η assuming
the cross section of the η nuclear eﬀect as σnuc = a + b · exp (−c · pη ) [mb]. The dashed line
shows the cross-section without η nuclear eﬀect, and the black solid line show the simulated
cross section with parameters of (a, b, c) = (−75.0, 115, 0.0015) with χ2min .

dσηphoto /dp is simulated by σnuc and χ2 deﬁned in Eq. (4.7) is calculated. Figure 4.13
indicates the three parameters, a, b, and c, satisfying the condition of χ2 ≤ χ2min + 1.
Here, χ2min = 22.3, χ2ν,min = 1.39 are achieved at (a, b, c) = (−75, 115, 0.0004) shownas the red circle in the ﬁgure. Thus, the most plausible σnuc written by this function is
deduced to be
σnuc = −75 + 115 · exp (−0.0004 · pη ) mb.

(4.30)The measured and assumed absorption cross sections σabs are shown in Fig. 4.14, andthe measured and simulated η photoproduction cross sections are described in Fig. 4.15.
From the the chart illustrated in Fig. 4.13, it can be seen that there is a correlation
between parameters a and b. The uncertainties of ﬁtting parameters (section 4.3.2) arenot estimated for this function since the goodness of ﬁt does not improved from Eq. (4.27)
as described in section 4.3.5.

4.3.5

Goodness of Fittings

To summarize the χ2 minimization with three types of functions, the achieved χ2ν,min andthe degrees of freedom for respective functions, σnuc = a, σnuc = a · exp (−b · pη ), andσnuc = a + b · exp (−c · pη ), are 2.12 with 18, 1.39 with 17, and 1.39 with 16. The pvalues are 0.004, 0.134, and 0.134, enumerated in the same order as above. While slight
improvement could be seen from the ﬁrst to the second ﬁtting, no diﬀerence appeared
between the second and the third ones. Thus, the third function is not dealt with in thisstudy. From this result, it is suggested that the simple exponential function Eq. (4.28)
σnuc = 42.0 · exp (−0.0015 · pη ) mb,(4.28)

4.3. Interpolating the η Nuclear Eﬀect Cross Section

51
p → µ+ ηevents [arbitrary]

events [arbitrary]p → e+ η
300
250
200

300
250
200

150

150

100

100

50

50

0
0

100

200

300

400
500
600
momentum [MeV/c]

0
0

100

200

300

400
500
600
momentum [MeV/c]

Fig. 4.16: Momentum distributions of η mesons from bound protons of p → e+ η (left) andp → µ+ η (right).
Tab. 4.4: Fraction of the ﬁnal states of η meson from the proton decay of p → e+ η in 16 O
when the η nuclear eﬀect is assumed as σnuc = a [mb].

Parameter aη nuclear eﬀect in 16 O nucleus
no interaction
scatteredabsorbeda = 24.3
χ2min
60%
6%
38%

a = 25.8
χ2min + 1.00
59%
4%
23%

a = 23.0
χ2min + 1.00
63%
5%
53%

would be the best among the three.
It is also noticeable that there is a discrepancy about two times of the standard deviation at the (245, 295) and (425, 460) MeV/c bins in every ﬁts shown in Fig. 4.9, Fig. 4.12,
and Fig. 4.15 could not be improved in any attempt. The momentum after the proton
decay from a bound proton is simulated to distribute around 300 MeV/c as indicatedin Fig. 4.16. Although the deﬁciency at the low momentum side and the excess at the
opposite side of η photoproduction cross sections can be found, the consistency of ﬁt functions with the measured σabs around 300 MeV/c region can support the validity of theestimation.

4.3.6

Uncertainty Estimation from Each Function

By applying the estimated cross section of η nuclear eﬀect as a function of σnuc = a(σnuc = a · exp (−b · pη )) to proton decay MC simulations, the fraction of the ﬁnal states ofη meson for p → e+ η and p → µ+ η are be obtained as Tab. 4.4 and Tab. 4.5 (Tab. 4.6 andTab. 4.7). The no interaction rate varies around 5% (10%) relative to the nominal value
in the column second to the left within the uncertainty of the parameter a (parameter setsof a and b) when the function is assumed as a constant value (an exponential).

52

Chapter 4. Evaluation of η Meson Nuclear Eﬀect

Tab. 4.5: Fraction of the ﬁnal states of η meson from the proton decay of p → µ+ η in 16 O
when the η nuclear eﬀect is assumed as σnuc = a [mb].

Parameter aη nuclear eﬀect in 16 O nucleus
no interaction
scatteredabsorbeda = 24.3
χ2min
62%
6%
38%

a = 25.8
χ2min + 1.00
60%
4%
23%

a = 23.0
χ2min + 1.00
65%
5%
53%

Tab. 4.6: Fraction of the ﬁnal states of η meson from the proton decay of p → e+ η in 16 O
when the η nuclear eﬀect is assumed as σnuc = a · exp (−b · pη ) [mb].

Parameters (a, b)
η nuclear eﬀect in 16 O nucleus
no interaction
scatteredabsorbed(42.0, 0.0015)
χ2min
65%
6%
38%

(52.0, 0.0021)
χ2min + 2.30
60%
4%
23%

(34.0, 0.0009)
χ2min + 2.30
67%
5%
53%

Tab. 4.7: Fraction of the ﬁnal states of η meson from the proton decay of p → µ+ η in 16 O
when the η nuclear eﬀect is assumed as σnuc = a · exp (−b · pη ) [mb].

Parameters (a, b)
η nuclear eﬀect in 16 O nucleus
no interaction
scatteredabsorbed(42.0, 0.0015)
χ2min
64%
6%
38%

(52.0, 0.0021)
χ2min + 2.30
60%
4%
23%

(34.0, 0.0009)
χ2min + 2.30
70%
5%
53%

4.4. Estimation of Eﬀect on Proton Decay Search

53

Tab. 4.8: The no η nuclear interaction rate of η meson from the proton decay of p → e+ η in16
O when the η nuclear eﬀect is assumed as σnuc = a [mb] or σnuc = a · exp (−b · pη ) [mb].

Function of σnuc
σnuc = a [mb]σnuc = a · exp (−b · pη ) [mb]AdoptedNo η nuclear interaction rate
Nominal σnuc Minimum Maximum
60%
59%
63%
65%
60%
67%
65%
59%
67%

Tab. 4.9: The no η nuclear interaction rate of η meson from the proton decay of p → µ+ η in16
O when the η nuclear eﬀect is assumed as σnuc = a [mb] or σnuc = a · exp (−b · pη ) [mb].

Function of σnuc
σnuc = a [mb]σnuc = a · exp (−b · pη ) [mb]Adopted4.4

No η nuclear interaction rate
Nominal σnuc Minimum Maximum
62%
60%
65%
64%
60%
70%
64%
60%
70%

Estimation of Eﬀect on Proton Decay Search

The no nuclear interaction rate of η meson of proton decay MC events for p → e+ η andp → µ+ η are individually summarized in Tab. 4.8 and Tab. 4.9. The nominal σnuc has
been determined as the σnuc with parameters minimizing χ2 . The right two columns in
Tab. 4.8 and Tab. 4.9 show the minimum and maximum ratio above all the proton decay
MC events generated with set of parameters within χ2min +1.00 or χ2min +2.30 for each case.
The adopted minimum and maximum values have been decided so that the uncertainty
would take the widest range. By taking the relative variations of no interaction rates,
the systematic errors arose from η nuclear eﬀect of p → e+ η and p → µ+ η modes are
estimated as|59 − 65|
∼ 10%,
65|70 − 64|
εnuc, p→µ+ η =∼ 10%.
64
εnuc, p→e+ η =(4.31)(4.32)

Chapter 5

Proton Decay Analysis
The proton decay into p → e+ η and p → µ+ η, followed by η → 2γ decay are searched byanalyzing the SK-IV data obtained from October 21st 2015 to May 31st 2018, which havenot been analyzed in the previous research [29]. The amount of the statistics corresponds
to 904.99 days of live time and 55.7 kton·years of exposure. The proton decay candidates
are extracted from data by the event selection criteria described in Tab. 5.1. The signal
eﬃciencies are evaluated by the number of events survived the selection criteria in the proton decay MC. The background events are estimated by applying the criteria to 500 yearsof the atmospheric neutrino MC and then normalizing by live time. Finally, the partial
lifetime limits of proton for the two decay modes are calculated from the newly analyzed
data in this study and the one scrutinized in the previous study [29].

5.1

Proton Decay Search

The proton decays into an antilepton and an η meson back-to-back with a monochromatic
momentum of 310.0 MeV/c for p → e+ η and 297.7 MeV/c for p → µ+ η in the proton
rest frame. Figure 5.1 shows the signals of a typical proton decay MC event of p → e+ η.
Three shower-like (e-like) Cherenkov rings are clearly seen; one of the rings is produced
by a positron and the rests are from η. On the other hand, Fig. 5.2 shows the signals ofa typical event of p → µ+ η. The ring in the right with a sharp outline corresponds to theantimuon, and the other two are derived from γ-rays.

5.1.1

Selection Criteria

In order to search the data sets of SK for proton decay events, it is essential to sort out
the events by the features of proton decay events and extract the candidates. Here, theinformation listed below are utilized for this event selection.
1. The number of the Cherenkov rings
2. The pattern of the Cherenkov rings
3. The reconstructed invariant η mass, mη
4. The number of Michel electrons
5. The reconstructed total mass, mtot

55

56

Chapter 5. Proton Decay Analysis

Fig. 5.1: A typical PMT hit pattern of p → e+ η, η → 2γ events in SK.

Fig. 5.2: A typical PMT hit pattern of p → e+ η, η → 2γ events in SK.

5.1. Proton Decay Search

57

Tab. 5.1: Event selection criteria for p → l+ η, η → 2γ modes.

A

Event cut
The number of ringsp → e+ η, η → 2γ
3

B

PID

all shower-likeC
D

Invariant η mass (mη )The number of Michel electrons
The total momentum (ptot ),the total invariant mass (mtot )The total momentum (ptot )The number of neutrons
The total momentum (ptot )The number of neutrons

480 ≤ mη ≤ 620
0
ptot ≤ 250
800 ≤ mtot ≤ 1050
100 < ptot ≤ 250
0ptot ≤ 100
0

E
F
H
G
I

p → µ+ η, η → 2γ
3
one non-shower-like,
two shower-like
480 ≤ mη ≤ 620
1
ptot ≤ 250
800 ≤ mtot ≤ 1050
100 < ptot ≤ 250
0ptot ≤ 100
0

6. The reconstructed total momentum, ptot
ptot , mtot and the total energy Etot are given bynpar

ptot =

X

pi ,

i=1
npar qEtot =

X

m2i + p2i ,

(5.1)
(5.2)

i=1

q

mtot =

2 − p2 ,
Etot
tot

(5.3)

where mi is the mass of i-th particle, pi is the momentum of i-th particle (ring), and npar
is the number of decay particles. For p → e+ η (p → µ+ η) mode, the decay particles a
positron (antimuon) and an η meson, and the number of decay particles equals two. mη iscalculated by the same manner as Eq. (5.1)-Eq. (5.3), with decay particles of γ-rays andnpar = 2.
All the conditions for proton decay events described above are summarized in Tab. 5.1.
The criteria H and I are applied for events passed F and G, respectively. Fully Contained
Fiducial Volume (FCFV) cut, which requires the distance between the reconstructed vertex
and the wall to be larger than 200 cm, has been applied to all the events in the proton decay
search. The deﬁnitions and details of FC events are described in section 2.3. Hereinafter,
the region of reconstructed total mass and momenta satisfying the criterion H (I) are called
as upper (lower) signal box.
A: Number of Rings In the proton decay modes studied in this thesis, a proton decays
into an antilepton and an η meson back-to-back in the proton rest frame, and the η meson
decays into two γ-rays. Therefore, all the proton decay events produce there rings: one
from the antilepton and the other two from the γ-rays. From these characteristics, proton
decay events should be detected with three Cherenkov rings, which is indicated as criteria
A in Tab. 5.1.
The number of rings distributions from all decay modes of η mesons are shown in
Fig. 5.3 for each p → e+ η and p → µ+ η MC events. It is obvious that 3-ring events
dominate for the proton decay MC events. The fractions of 3-ring events are 49% (48%)

58

Chapter 5. Proton Decay Analysis

p → µ+ ηevents [arbitrary]

events [arbitrary]p → e+ η

4500
4000
3500
3000

4500
4000
3500
3000

2500

2500

2000

2000

1500

1500

1000

1000

500

500

0

1

2

3

4

0

5
6
number of rings

1

2

3

4

5
6
number of rings

Fig. 5.3: Distributions of the number of ring of the p → e+ η (left) and p → µ+ η (right) MC.
All of the decay modes of η mesons are considered. The black and cyan shaded histogramshows the distribution from all and free protons, respectively. The red dashed line shows thenumber of rings distribution of the atmospheric neutrino MC in SK-IV normalized by the
maximum bin values. The green solid lines indicate the selection criterion A in Tab. 5.1

p → µ+ ηevents [arbitrary]

events [arbitrary]p → e+ η

4500
4000
3500
3000

4500
4000
3500
3000

2500

2500

2000

2000

1500

1500

1000

1000

500

500

0

1

2

3

4

5
6
number of rings

0

1

2

3

4

5
6
number of rings

Fig. 5.4: Distributions of the number of ring depending on the decay modes of the η mesonfrom p → e+ η (left) and p → µ+ η (right) MC. The red shaded histogram, the blue andgreen histograms shows the distribution of the η → 2γ, η → 3π 0 , and η → π + π − π 0 modes,
respectively. The orange histogram represent the distribution for events where the η mesons
are absorbed or scattered by the nuclear eﬀect.

59

events [arbitrary]

events [arbitrary]

5.1. Proton Decay Search

700
600
500

700
600
500

400

400

300

300

200

200

100

100

0
0

100 200 300 400 500 600 700 800 900 1000
η mass [MeV/c2]

00100 200 300 400 500 600 700 800 900 1000
η mass [MeV/c2]

Fig. 5.5: Distributions of reconstructed η invariant mass of the p → e+ η MC in SK-IV after
applying the selection criteria A and B. The black solid lines show all the decay branches ofthe η meson. Left ﬁgure: The cyan shaded histogram shows the distribution for free protons
and the red dashed line shows the the distribution of the atmospheric neutrino MC in SK-IV
normalized by the maximum bin values. The green solid lines indicate the selection criteria C
in Tab. 5.1. Right ﬁgure: The red shaded histogram, the blue and green histograms shows the
distribution of the η → 2γ, η → 3π 0 , and η → π + π − π 0 modes, respectively.

for p → e+ η (p → µ+ η) from all protons and 60% (58%) for p → e+ η (p → µ+ η) fromfree protons. The number of rings distributions for each η meson decay modes are shownin Fig. 5.4. Here, the 1, 2, and 4-ring events of η → 2γ mainly attribute to the overlaps
of Cherenkov rings to miscount the number of rings. It is clear that the majority of 4
and 5-ring events derives from η → 3π 0 . In addition, most of the 1-ring events arise fromevents with η mesons being absorbed by the nuclear eﬀect. On the other hand, 1-ring
events dominate for the background atmospheric neutrino events from a charged lepton.
B: Particle Identiﬁcation Another characteristic of the rings is their patterns. Positronsand γ-rays would produce shower-like (e-like) Cherenkov rings, whereas antimuons would
produce non-shower-like (µ-like) rings while propagating through the water. From thesecharacteristics, proton decay events should be detected with three Cherenkov rings, andall of the rings should be shower-like for p → e+ η and one of the rings should be nonon-shower-like for p → µ+ η events (criterion B).
C: Reconstructed η Mass If the two shower-like rings derived from γ-rays are detected, the invariant mass of the η meson can be reconstructed by the energies and momenta of these two rings. In the p → e+ η event, however, all three rings show shower-likepattern and the ring of a positron cannot be distinguished from those of γ-rays. To solve
this problem, mη is reconstructed by taking all the possible combinations of two Cherenkov
rings. The two rings that can reconstruct the closest mass to the rest mass of η meson aredetermined as rings derived from γ-rays from η meson decay.
Figure 5.5 shows the distributions of reconstructed η mass after selection criteria A
and B for p → e+ η proton decay MC and atmospheric neutrino MC on the left and foreach η decay modes on the right. The same distributions for p → µ+ η MC are shown inChapter 5. Proton Decay Analysis

events [arbitrary]

events [arbitrary]

60

500

400

500

400

300

300

200

200

100

100

0
0

100 200 300 400 500 600 700 800 900 1000
η mass [MeV/c2]

00100 200 300 400 500 600 700 800 900 1000
η mass [MeV/c2]

Fig. 5.6: Distributions of reconstructed η invariant mass of the p → µ+ η MC in SK-IV after
applying the selection criteria A and B. The black solid lines show all the decay branches ofthe η meson. Left ﬁgure: the cyan shaded histogram shows the distribution for free protons
and the red dashed line shows the the distribution of the atmospheric neutrino MC in SK-IV
normalized by the maximum bin values. The green solid lines indicate the selection criteria C
in Tab. 5.1. Right ﬁgure: the red shaded histogram, the blue and green histograms shows the
distribution of the η → 2γ, η → 3π 0 , and η → π + π − π 0 modes, respectively.

Fig. 5.6. From the ﬁgure on the left in Fig. 5.5, the distribution of proton decay MC events
has a peak at ∼ 550 MeV/c2 while the rest mass of η meson is 547.8 MeV/c2 . In contrast,
the atmospheric neutrino MC shows peaks around ∼ 140 MeV/c2 and ∼ 540 MeV/c2 .
Thus, events satisfying 480 ≤ mη ≤ 620 MeV/c2 are selected as candidates. Focusing on
the right, it can be seen that most of the events composing the 547.8 MeV/c2 peak arederived from η → 2γ. On the other hand, from the left ﬁgure of Fig. 5.6, p → µ+ η MC
makes another peak at around ∼ 140 MeV/c2 besides ∼ 550 MeV/c2 from η rest mass.
The right ﬁgure shows that this low momentum peak is mainly composed of η → π + π − π 0events.
The majority of the background events passing criterion A and B consists of an antilepton and single or multiple pions produced events, where a neutral pion decaying into
two γ-rays. Muons are generated in CC pion production events induced by muon neutrinos, and the observed three rings can be classiﬁed into one µ-like ring and two e-likerings from γ-rays. Therefore, pion including background events for p → µ+ η, η → 2γ canreconstruct masses from two γ-rays around the true pion mass of 135.0 MeV/c2 as seenin Fig. 5.6. CC pion production events from electron neutrino interactions, on the other
hand, include electrons and make three e-like rings. In this case, since the rings originatingfrom two γ-rays cannot be identiﬁed explicitly, the pion masses can be mis-reconstructed.
Thus, the reconstructed η mass in background events for p → e+ η, η → 2γ shows a broad
distribution around 550 MeV/c2 in addition to the peak at ∼140 MeV/c2 in Fig. 5.5. Furthermore, single η meson production contributed as the second most background events
for both events.
D: Number of Michel Electrons The number of Michel electrons also helps to constrain the number of background events. Since no muon and a single muon is generated

5.1. Proton Decay Search

61

2400

p → µ+ ηevents [arbitrary]

events [arbitrary]p → e+ η

2200
2000
1800
1600

1600
1400
1200

1400

1000

1200

800

1000
600

800
600

400

400

200

200
0

0

1

2

3
4
5
6
number of Michel electrons

0

0

1

2

3
4
5
6
number of Michel electronsFig. 5.7: Distributions of the number of Michel electrons of the p → e+ η (left) and p → µ+ η(right) MC after applying the selection criteria A-C. All of the decay modes of η mesons are
considered. The cyan shaded histogram shows the distribution for free protons and the reddashed line shows the distribution of the atmospheric neutrino MC in SK-IV normalized by
the maximum bin values. The green solid lines indicate the selection criterion D in Tab. 5.1

in p → e+ η and p → µ+ η decay respectively, the number of Michel electrons should bezero and one for each mode (criterion D). The number of Michel electrons considering all
the decay modes of η mesons after criteria A-C are shown in Fig. 5.7. More than 98%(96%) of the proton decay events from p → e+ η (p → µ+ η) are classiﬁed as having no(one) Michel electron.
E, F & G: Reconstructed Total Momentum and Mass The background eventsof atmospheric neutrino usually do not have isotropic signals because most of the neutrino
induced particles are emitted in the direction of movement. This results in background
events showing higher total momenta and lower invariant masses than those from true
proton decay signals. Hence, the reconstructed momenta and masses are signiﬁcant clues
to distinguish proton decay signals from backgrounds.
The reconstructed total mass and momentum are shown in Fig. 5.8 and Fig. 5.9 respectively for p → e+ η and p → µ+ η for all η meson decay modes. The peaks in the
total mass distributions around 930 MeV/c2 corresponds to the rest mass of a proton of
938.3 MeV/c2 . From Fig. 5.8, it can be found that event selection with ptot ≤ 100 MeV/cworks especially successful, where more than 99% of the events passed this are proton
decay events and most of them derive from free protons.
Against this background, the selection cut of ptot has been set to 250 MeV/c considering
the Fermi momentum of bound protons in 16 O nucleus (criterion H). In speciﬁc, events
that satisﬁes ptot ≤ 100 MeV/c are considered as signals mainly from free protons andhave high signiﬁcancy as proton decay candidates (criterion I). Besides, only the events
fulﬁlling 800 ≤ mtot ≤ 1050 MeV/c2 are selected as candidates (criterion E).
While pion production events comprise the majority of background events of p → e+ η,η → 2γ after criterion A-D, those for p → µ+ η, η → 2γ include η mesons decaying into
two γ-rays. This is because of diﬀerence in the distribution of reconstructed neutral pion

62

Chapter 5. Proton Decay Analysis

500

p → µ+ ηevents [arbitrary]

events [arbitrary]p → e+ η

400

300

400
350
300
250
200

200

150
100

100

50
0
0

200

400

0
0

600
800
1000 1200
total invariant mass [MeV/c2]

200

400

600
800
1000 1200
total invariant mass [MeV/c2]

Fig. 5.8: Distributions of reconstructed invariant total mass of the p → e+ η (left) andp → µ+ η (right) MC after applying the selection criteria A-D. All of the decay modes of η
mesons are considered. The cyan shaded histogram shows the distribution for free protons
and the red dashed line shows the distribution of the atmospheric neutrino MC in SK-IV
normalized by the maximum bin values. The green solid lines indicate the selection criterion
E in Tab. 5.1

p → µ+ ηevents [arbitrary]

events [arbitrary]p → e+ η

220
200
180
160
140

180
160
140
120

120

100

100

80

80

60

60

40

40

20

20
0
0

100 200 300 400 500 600 700 800 900 1000
total momentum [MeV/c]00

100 200 300 400 500 600 700 800 900 1000
total momentum [MeV/c]

Fig. 5.9: Distributions of reconstructed total momentum of the p → e+ η (left) and p → µ+ η(right) MC after applying the selection criteria A-D. All of the decay modes of η mesons are
considered. The cyan shaded histogram shows the distribution for free protons and the reddashed line shows the distribution of the atmospheric neutrino MC in SK-IV normalized by
the maximum bin values. The green solid lines indicate the selection criterion E in Tab. 5.1

5.1. Proton Decay Search

63
p → µ+ ηevents [arbitrary]

events [arbitrary]p → e+ η

2200
2000
1800
1600
1400
1200

1600
1400
1200
1000
800

1000
600

800
600

400

400

200

200
0

0

1

2

3

4
5
6
number of neutrons

0

0

1

2

3

4
5
6
number of neutronsFig. 5.10: Distributions of the number of neutrons of the p → e+ η (left) and p → µ+ η(right) MC after applying the selection criteria A-D. All of the decay modes of η mesons are
considered. The cyan shaded histogram shows the distribution for free protons and the reddashed line shows the distribution of the atmospheric neutrino MC in SK-IV normalized by the
maximum bin values. The green solid lines indicate the selection criteria H and I in Tab. 5.1.

mass as explained in section 5.1.1. Therefore, more background events are reconstructedwith total invariant masses with values close to the true proton mass in the analysis of
p → µ+ η, η → 2γ than that of p → e+ η, η → 2γ as seen in Fig. 5.8.
H & I: Number of Neutrons The last notable characteristic of proton decay eventsis the number of neutrons. Neutrons produced in the SK water tank are captured byhydrogens after a few microseconds of travel, followed by emission of a 2.2 MeV prompt
γ-ray from the de-excitation of hydrogen. Thus the number of neutrons can be identiﬁed
by detecting this prompt γ-rays. While neutrino interaction can produce neutrons, oxygen
nucleus emits neutrons with a low probability and hydrogen emits no neutron. For thesereasons, the information of the number of neutrons is used to select the events; no neutrons
should be found in proton decay candidates (criterion H and I). The number of neutrons
after criteria A-D are shown in Fig. 5.10. More than 98% (98%) of the proton decay events
for p → e+ η (p → µ+ η) are classiﬁed as including no neutrons.

5.1.2

Signal Eﬃciencies and The Number of Background Events

The signal eﬃciencies are deﬁned as(Signal eﬃciency of a certain criterion) =Number of events passed the criterion,
Number of p → l+ η events after FCFV cut(5.4)

and the number of background events have been estimated by normalizing the numberof events passed all the event selection criteria in 500 years of atmospheric neutrino MC
events to the data live time after the correction of neutrino oscillation eﬀect. Here, theneutrino oscillation probability has been calculated with ∆m2 = 2.5 × 10−3 eV2 andsin2 (2θ) = 1.0.

64

Chapter 5. Proton Decay Analysis

Tab. 5.2: Number of events satisfying each event selection criteria of p → l+ η, η → 2γ mode
within 10,000 of p → l+ η events in MC and 500 years of atmospheric neutrino MC events.

FCFV
A
B
C
D
E
F
G
H
I

p → e+ η, η → 2γ
proton decay MC atmospheric ν MC
8023
17719204038
145764
3635
75490
2345
17763
2301
8828
1816
22
1065
22
751
0
1046
7
738
0

p → µ+ η, η → 2γ
proton decay MC atmospheric ν MC
8038
17719203830
145764
3067
62600
1650
5344
1585
2627
1432
7
798
6
634
1
782
6
625
0

Tab. 5.3: Signal eﬃciencies (ratios) for p → l+ η, η → 2γ mode against the number of p → l+ η
events in MC.

A
B
C
D
E
F
G
H
I

p → e+ η, η → 2γ
48.8 ± 0.8%
43.9 ± 0.7%
28.3 ± 0.6%
27.8 ± 0.6%
21.9 ± 0.5%
12.9 ± 0.4%
9.1 ± 0.3%
12.6 ± 0.4%
8.9 ± 0.3%

p → µ+ η, η → 2γ
46.2 ± 0.7%
37.0 ± 0.7%
19.9 ± 0.5%
19.1 ± 0.5%
17.3 ± 0.5%
9.6 ± 0.3%
7.7 ± 0.3%
9.4 ± 0.3%
7.5 ± 0.3%

Tab. 5.4: Number of events and ratios satisfying each event selection criteria of p → l+ η,η → 2γ mode within 3615 (3553) of p → e+ η, η → 2γ (p → µ+ η, η → 2γ) events in MC.

FCFV
A
B
C
D
E
F
G
H
I

p → e+ η, η → 2γ
number of events
ratio[%]
3136
100
2618
83.5 ± 1.6
2396
76.4 ± 1.6
2056
65.6 ± 1.4
2054
65.5 ± 1.4
1674
53.8 ± 1.3
973
31.0 ± 1.0
70122.4 ± 0.895630.5 ± 1.0
68821.9 ± 0.8

p → µ+ η, η → 2γ
number of events
ratio[%]
2984
100
2389
80.1 ± 1.6
2078
69.6 ± 1.5
1643
55.1 ± 1.4
1579
52.9 ± 1.3
142747.8 ± 1.3
79626.7 ± 0.963121.1 ± 0.878026.1 ± 0.962220.8 ± 0.8

5.1. Proton Decay Search

65

Tab. 5.5: Breakdown (number of events contribution) of the neutrino interaction modes ofthe background events in the signal boxes.

Interaction
CC single π production
CC multiple π production
NC multiple π production
CC single η production
CC quasi-elastic scatteringp → e+ η, η → 2γ
4
1
0
1
1

p → µ+ η, η → 2γ
1
1
1
3
0

The numbers of events that survived each criterion are listed in Tab. 5.2 for 10,000
p → l+ η proton decay MC events and background MC events of 500 years for each decay
modes. Table 5.3 shows the eﬃciencies for each criterion deﬁned above together withstatistical errors. The number of background events with statistical errors arose fromlimited number of MC events are estimated to be 0.13 ± 0.05 (0.08 ± 0.04) for the full
SK-IV period for p → e+ η (p → µ+ η). All the MC events are produced by SK-IV
period simulation. Selection criteria H and I correspond to the ﬁnal signal region, andthe numbers in the row H and I of Tab. 5.2 and Tab. 5.3 corresponds to the number ofproton decay candidates and the ﬁnal signal eﬃciencies, respectively. Table 5.4 shows thenumber of events and ratios extracting only the events decaying in η → 2γ after eachcriterion. It has been shown that around 52% (47%) from η → 2γ for p → e+ η (p → µ+ η)MC events can be detected by SK. These ineﬃciencies are mainly caused by the η nuclear
eﬀect explained in section 4. The lower signal eﬃciency of p → µ+ η is mainly due to thehigher mis-PID rate.
The breakdown of the seven (six) events selected as p → e+ η, η → 2γ (p → µ+ η,η → 2γ) candidates from background MC are listed in Tab. 5.5. The majority of the events
are accompanied by one or two pions. In the events derived from CC single π production
interaction, a charged lepton and the decaying two γ-rays from π 0 generates three ringsand create similar signals to those of proton decays. A charged and a neutral pion in themultiple pions production events induced by deep inelastic scattering can also generate
three rings. Events of CC single η production interaction with an η meson decaying into
two γ-rays also survives the selection criteria. In the rest CCQE induced event, a single
electron generates two Cherenkov rings by changing its direction by scattering. These two
shower-like rings overlap to be mis-identiﬁed as three-ring event.

5.1.3

Conﬁrmation of Data Agreement with MC

The agreement between data and atmospheric neutrino MC events are conﬁrmed before
conducting the proton decay search with data. Only the data 100 MeV/c (MeV/c2 ) outside
the signal region are used so that the ﬁnal results cannot be seen. Figure 5.11 (Fig. 5.12)
shows the scatter diagram of total mass and total momentum, and the distributions of
total momentum and total mass are shown as Fig. 5.13 (Fig. 5.14) for p → e+ η, η → 2γ
(p → µ+ η, η → 2γ) respectively. Criterion A and B are applied and the events inside theblack box are not used. No signiﬁcant diﬀerence between data and atmospheric neutrino
MC is seen.

66

Chapter 5. Proton Decay Analysis

Atmospheric ν MC

SK-IV Data (904.4 days)

900

1000

500

total momentum [MeV/c]

900

total momentum [MeV/c]

1000

400

400

300

300

200

200

100

100

800

800

700

700

600

0
0

600
500

200

400

0
0

600
800
1000 1200
total invariant mass [MeV/c2]

200

400

600
800
1000 1200
total invariant mass [MeV/c2]

Fig. 5.11: Scatter plots of total mass and total momentum after applying the selection criteria
A-D for p → e+ η, η → 2γ search. The diagram on the left stand for atmospheric neutrino MC
events and the right for data from SK-IV period of 905.0 days.

Atmospheric ν MC

SK-IV Data (904.4 days)

900

1000

500

total momentum [MeV/c]

900

total momentum [MeV/c]

1000

400

400

300

300

200

200

100

100

800

800

700

700

600

0
0

600
500

200

400

600
800
1000 1200
total invariant mass [MeV/c2]

0
0

200

400

600
800
1000 1200
total invariant mass [MeV/c2]

Fig. 5.12: Scatter plots of total mass and total momentum after applying the selection criteria
A-D for p → µ+ η, η → 2γ search. The diagram on the left stand for atmospheric neutrino MC
events and the right for data from SK-IV period of 905.0 days.

67

eventsevents

5.1. Proton Decay Search

4.5
4

2
1.8
1.6

3.5

1.4

3

1.2

2.5

1

2

0.8

1.5

0.6

1

0.4

0.5

0.2

0
0

0
0

100 200 300 400 500 600 700 800 900 1000
total momentum [MeV/c]

100 200 300 400 500 600 700 800 900 1000
total momentum [MeV/c]

eventseventsFig. 5.13: Distributions of reconstructed total momenta outside the signal region after applying the selection criteria A-D for p → e+ η (left) and p → µ+ η (right). The black circles andred histograms stand for SK-IV data of 905.0 days and for atmospheric neutrino MC events,
respectively. The green boxes corresponds to the signal boxes (criterion H and I) and the black
box shows the 100 MeV/c (MeV/c2 ) outside region from the signal boxes.

7

3.5
3

6
2.5

5

2

4

1.5

3
2

1

1

0.5

0
0

200

400

600
800
1000 1200
total invariant mass [MeV/c2]

0
0

200

400

600
800
1000 1200
total invariant mass [MeV/c2]

Fig. 5.14: Distributions of reconstructed total masses outside the signal region after applying
the selection criteria A-D for p → e+ η (left) and p → µ+ η (right). The black circles andred histograms stand for SK-IV data of 905.0 days and for atmospheric neutrino MC events,
respectively. The green boxes corresponds to the signal boxes (criterion H and I) and the black
box shows the 100 MeV/c (MeV/c2 ) outside region from the signal boxes.

68

Chapter 5. Proton Decay Analysis

Fig. 5.15: The signal eﬃciencies (upper) and the number of expected backgrounds (lower,
red histogram) and data candidates (lower, black circles) of SK-IV for p → e+ η, η → 2γ searchon the left and p → µ+ η, η → 2γ search on the right. The atmospheric neutrino MC arenormalized by live time and the data corresponds to the 55.7 kt·years of exposure in SK-IV.
The event selection criteria are deﬁned in section 5.1.1.

5.1. Proton Decay Search
1000

total momentum [MeV/c]

total momentum [MeV/c]

1000
900
800
700
600
500

800
700
600
500
400

300

300

200

200

100

100
200

400

0
0

600
800
1000 1200
total invariant mass [MeV/c2]

900
800
700
600
500

600
800
1000 1200
total invariant mass [MeV/c2]

200

400

600
800
1000 1200
total invariant mass [MeV/c2]

200

400

600
800
1000 1200
total invariant mass [MeV/c2]

800
700
600
500
400

300

300

200

200

100

100
200

400

0
0

600
800
1000 1200
total invariant mass [MeV/c2]

1000

total momentum [MeV/c]

1000
900
800
700
600
500

900
800
700
600
500

400

400

300

300

200

200

100

100

0
0

400

900

400

0
0

200

1000

total momentum [MeV/c]

1000

total momentum [MeV/c]

900

400

0
0

total momentum [MeV/c]

69

200

400

600
800
1000 1200
total invariant mass [MeV/c2]

00Fig. 5.16: Reconstructed total invariant mass and momentum of proton decay MC (upper),atmospheric neutrino MC (middle), and data (lower) for p → e+ η, η → 2γ search on the left
and p → µ+ η, η → 2γ search on the right. The blue and cyan circles in the upper ﬁgures
show the distribution from all the decay branches of the η meson, and for free proton decay
events only, respectively. The red circles in the middle and the black circles in the lower eachcorrespond to the 500 years of atmospheric neutrino MC and the data for 55.7 kt·years of
SK-IV period. The green solid and dashed lines indicate the selection criteria E, F, and G in
section 5.1.1. All the event selections except E, F and G are applied.

Chapter 5. Proton Decay Analysis

events [arbitrary]

events [arbitrary]

70

450
400
350
300

350
300
250
200

250
150

200
150

100

100
50

50

0
0

100 200 300 400 500 600 700 800 900 1000
total momentum [MeV/c]

eventsevents003.5
3100 200 300 400 500 600 700 800 900 1000
total momentum [MeV/c]

2.2
2
1.8
1.6

2.5

1.4
2

1.2
1

1.5

0.8

1

0.6
0.4

0.5

0.2
0
0

100 200 300 400 500 600 700 800 900 1000
total momentum [MeV/c]00

100 200 300 400 500 600 700 800 900 1000
total momentum [MeV/c]

Fig. 5.17: Distributions of reconstructed total invariant momentum of the p → l+ η MC
(upper), and atmospheric neutrino MC and data (lower) for p → e+ η search on the left andp → e+ η search on the right. The blue open and shaded cyan histograms in the upper ﬁgures
show the distribution from all the decay branches of the η meson, and for free proton decay
events only, respectively. The red histograms and the black circles in the lower correspondto the atmospheric neutrino MC and the data for 55.7 kt·years of SK-IV period. The green
solid and dashed lines indicate the selection criteria F and G in section 5.1.1. All the event
selections except E, F, and G are applied.

5.1. Proton Decay Search

71

Fig. 5.18: The signal eﬃciencies (upper) and the number of expected backgrounds (lower,
red histogram) and data candidates (lower, black circles) of from SK-I to SK-IV for p → e+ η,η → 2γ search on the left and p → µ+ η, η → 2γ search on the right. All the results from
SK-I to SK-IV with 0.373 Mt·years of exposure are combined. The event selection criteria aredeﬁned in section 5.1.1.

72

Chapter 5. Proton Decay Analysis

p → e+ η
1000

total momentum [MeV/c]

total momentum [MeV/c]

1000

p → µ+ η

900
800
700
600
500

900
800
700
600
500

400

400

300

300

200

200

100

100

0
0

200

400

600
800
1000 1200
total invariant mass [MeV/c2]

0
0

200

400

600
800
1000 1200
total invariant mass [MeV/c2]

Fig. 5.19: Reconstructed total invariant mass and momentum of the data from SK-I to SK-IV
period for p → e+ η, η → 2γ search on the left and p → µ+ η, η → 2γ search on the right. The
black and red circles correspond to the results from SK-I to SK-IV in the previous study [29]with 0.316 Mt·years of exposure, and the newly analyzed SK-IV data with 55.7 kt·years of
exposure in this study, respectively. All the event selections except E, F, and G are applied.

Modesp → e+ η(2γ, upper)(2γ, lower)(3π 0 )
p → µ+ η(2γ, upper)(2γ, lower)(3π 0 )
10.9
6.7
8.2
6.5
5.6
6.2

7.3
5.8
6.9

7.2
6.0
6.9

10.7
8.2
7.6
9.4
7.5
7.9

12.6
8.9
7.7

Eﬃciency [%]
II
III
IV

11.0
7.9
8.0

I

0.05(2)0.000(6)0.34(5)0.17(4)0.01(1)0.15(3)I

0.02(1)0.000(4)0.13(2)0.10(2)0.01(1)0.06(2)0.01(1)0.000(3)0.12(2)0.05(1)0.003(3)0.06(1)

Background [events]
IIIII

0.08(4)0.000(10)0.22(6)0.13(5)0.000(10)0.04(3)

IV

0
0
0

0
0
0

0
0
1

0
0
0

0
0
0

0
0
0

0
0
1

0
0
0

Candidate [events]
I II IIIIV

Tab. 5.6: Summary of the signal eﬃciencies, the number of expected background events and the number of candidate events from 91.5, 49.1,
31.8, and 199.9 kton·years exposure during SK-I, SK-II, SK-III, and SK-IV periods. The numbers except for those of p → l+ η, η → 2γ searches
for SK -IV are cited from [29]. The errors of the background events are statistical errors from 500 years of atmospheric neutrino MC for each SK
period. The ”upper” and ”lower” in the η → 2γ searches stand for 100 < ptot < 250 and ptot < 100, respectively.

5.1. Proton Decay Search73

74

Chapter 5. Proton Decay Analysis

Tab. 5.7: Summary of the systematic uncertainties (percentage contribution) on signal efﬁciencies for each factors for SK-IV period. The numbers except for the η nuclear eﬀect of
p → l+ η, η → 2γ searches for SK-IV are cited from [29]. The ”upper” and ”lower” in theη → 2γ searches stand for 100 < ptot < 250 and ptot < 100, respectively.

Modes
p → e+ η(2γ, upper)(2γ, lower)(3π 0 )
p → µ+ η(2γ, upper)(2γ, lower)(3π 0 )

5.1.4

η nucleareﬀect

N-N correlated
decay

Fermi
momentum

Detector
performances

Total

10
4
12

8
3
4

9
13
15

2
2
4

16
14
20

10
4
17

9
3
6

10
12
2

3
3
5

17
13
19

Results

The proton decay events for p → e+ η, η → 2γ and p → µ+ η, η → 2γ are searched for55.7 kton · years of SK-IV data with the selection criteria in Tab. 5.1. Figure 5.15 shows
the signal eﬃciencies, the number of background events, and the number of candidates
in data. The distribution of the data candidates and that of expected background events
agree well. The total invariant mass and total momentum distributions are shown in
Fig. 5.16 and Fig. 5.17 for proton decay MC, atmospheric neutrino MC, and SK-IV data.
The number of background events in the signal boxes for 55.7 kton · years exposure are
expected to be 0.034 (0.000) events for the upper and 0.022 (0.000) events for the lower
box for p → e+ η, η → 2γ (p → µ+ η, η → 2γ). No event is found in the signal boxes; No
proton decay candidate is found.
The results for p → l+ η, η → 2γ modes from SK-I to SK-IV combined from the previous
study [29] are shown in Fig. 5.18 and Fig. 5.19. No candidate is found through SK-I to
SK-IV period. All the signal eﬃciency, the number of expected background events, andthe number of data candidate events for each SK period are summarized in Tab. 5.6. The
numbers of η → 2γ except for SK-IV period and those of η → 3π 0 are cited from [29]. Twocandidates were found for p → µ+ η, η → 3π 0 search in the previous study. The probability
to observe more than two candidates was calculated by the expected background eventsof Poisson statistics without considering systematic errors. The Poisson probability ofobserving two events for p → µ+ η, η → 3π 0 was deduced to be 20.9%.

5.2

Systematic Uncertainties

The systematic uncertainties on signal eﬃciency and the number of expected background
events are described in this section. The uncertainties discussed in section 4 are applied
as the systematic errors coming from η nuclear eﬀect for SK-IV period. The other uncertainties are cited from the latest research [29]. Each error sources are summarized in
Tab. 5.7 and Tab. 5.8.

5.2. Systematic Uncertainties

5.2.1
(1)

75

Systematic Uncertainties on Signal Eﬃciency
η Nuclear Eﬀect

The systematic uncertainties arising from η nuclear eﬀect of SK-IV period are estimatedby the relative variations of the number of events without experiencing nuclear eﬀect in
16 O nucleus as described in section 4. These estimated values in Eq. (4.31) and Eq. (4.32)
are adopted for the systematic uncertainties of upper signal boxes, whereas those of lower
signal boxes are estimated to be 0.35 times the error of upper ones. This is because thefractions of bound proton in the lower boxes are around 35%. For SK-I to SK-III period,
uncertainties are taken from [29].
(2)

N-N Correlated Decay

Around 10% [48] of the bound protons in 16 O nucleus are aﬀected by other nucleons inthe same 16 O nucleus as described in section 3.1. This eﬀect can make another nucleon
to recoil together with the decaying proton. Thus, this decay is considered as three body
decay with low signal eﬃciency. The systematic uncertainty is estimated by the ﬂuctuationof the number of events in the signal box with 100% uncertainty of the eﬀect of correlated
decay.
(3)

Fermi Momentum

The proton momentum is simulated by a spectral function from electron scattering experiment in 12 C [47]. The proton momentum distribution in atmospheric neutrino MC,on the other hand, is calculated based on the Fermi gas model. Therefore, the diﬀerence
between the number of events in the signal box based on the spectral function and the
Fermi gas model is considered as uncertainties.
(4)

Detector Performances

The uncertainties related to event reconstruction are combined as the systematic errors
from detector performances. A total of six error sources is included: vertex position, thenumber of Cherenkov rings, particle identiﬁcation (ring pattern), the number of Michelelectrons, and the energy scale. The detail of this information is described in section 2.2.

5.2.2

Systematic Uncertainties on the Number of Background Events

The uncertainties from atmospheric neutrino ﬂux, neutrino cross section, meson nuclear
eﬀect, and impact of hadron propagation in water are considered for the backgroundestimations. The systematic uncertainty due to detector performances is also evaluatedas well as that of signal eﬃciencies. The background estimation errors are considered tobe the same for both lower and upper signal boxes for η → 2γ decay. All of the values ofeach error source have been taken from [29].
(1)

Meson Nuclear Eﬀect

Pions and η mesons generated by neutrino interaction in 16 O can interact with 16 O nucleus.
Due to this nuclear eﬀect, the mesons get absorbed or scattered by the nucleon. In
this study, the systematic uncertainty of the π nuclear eﬀect and η nuclear eﬀect areevaluated. The former is taken from reference [29] evaluated by tuning the parameters

76

Chapter 5. Proton Decay Analysisof the interaction model of π-scattering experiment data [76] within 1σ uncertainty. The
uncertainty is adopted as 12% (14%) for p → e+ η, η → 2γ (p → µ+ η, η → 2γ) mode.
These parameters are the interaction probabilities of quasi-elastic scattering and charge
exchange, inelastic scattering, and absorption. Around 60% of the background events are
simulated to originate from pion productions as indicated in Tab. 5.5. The latter estimatedfrom Eq. (4.31) and Eq. (4.31) are about 10% for p → l+ η, η → 2γ decays, which arecomparable to the ones of pion nuclear eﬀect. About 30% of the background events include
single η meson. Since more than 90% of the background events are accompanied by mesons,
12% (14%) is adopted as a error source of meson nuclear eﬀect for all background eventsof (p → µ+ η, η → 2γ) search.
(2)

Hadron Propagation in Water

Charged pions can be generated by neutrino interactions and strongly interact with nucleons. This eﬀect of hadron propagation in water is evaluated by both the simulated andmeasured cross section of charged pions in water from [77, 78] with 100% uncertainty.
(3)

Neutrino Flux

The systematic uncertainty of neutrino ﬂux is evaluated by the atmospheric neutrino data
analyses in SK described in [79, 80]. Several error sources listed below are taken into
account.
•
•
•
•
•
•
•

Energy dependent normalization
Neutrino ﬂavor ratioν̄/ν ratio
Up/down asymmetry
Horizontal/vertical ratio
K/π production ratio
Neutrino ﬂight length

The dominant factor for the neutrino ﬂux error is the energy-dependent normalization.
(4)

Neutrino Cross Section

The systematic uncertainty of neutrino cross section is also evaluated by atmospheric
neutrino data analyses in SK [79, 80]. Each of the error sources is shown below.
•
•
•
•
•
•
•
•
(5)

MA (axial vector mass) in quasi-elastic scattering and single-meson production
Quasi-elastic scattering for bound nucleons (total cross section)
Quasi-elastic scattering for bound nucleons (ν̄/ν)Quasi-elastic scattering for bound nucleons (ﬂavor ratio)
Single(multi)-meson production (total cross section)
Single(multi)-meson production (model dependence)
Coherent pion production
NC/CC ratio
Detector Performances

The uncertainties related to event reconstruction are all combined as the systematic errors
from detector performances. A total of seven error sources is included: ﬁducial volume,Modesp → e+ η(2γ, upper)(2γ, lower)(3π 0 )
p → µ+ η(2γ, upper)(2γ, lower)(3π 0 )

Neutrino
cross section
15
15
12
15
15
12

Neutrino
ﬂux
8
8
8
9
9
8

14
14
9

12
12
34

Meson nuclear
eﬀect

8
8
13

5
5
17

Hadron propagationin water

28
28
12

21
21
30

Detector
performances

37
37
25

30
30
51

Total

Tab. 5.8: Summary of the systematic uncertainties (percentage contribution) on the number of background events for each factors for SK-IV
period. All the numbers are cited from [29]. The ”upper” and ”lower” in the η → 2γ searches stand for 100 < ptot < 250 and ptot < 100,
respectively.

5.2. Systematic Uncertainties
77

78

Chapter 5. Proton Decay Analysis

detector non-uniformity, energy scale, particle identiﬁcation (ring pattern), the number of
Cherenkov rings, the number of Michel electrons, and neutron tagging. The detail of thisinformation is described in section 2.2.

5.3

Lifetime Limits

The lower limits of the proton’s partial lifetime are calculated by the Bayesian method.
In this study, the results from 12 independent searches are conducted for p → l+ η decay:
the lower and upper box analysis for p → l + η, η → 2γ for SK-I to SK-IV, and thep → l + η, η → 3π 0 for SK-I to SK-IV.

5.3.1

Lifetime Estimation

The probability of detecting n events of proton decay is given as Poisson statistics:
P(n|Γλϵb) =e−(Γλϵ+b) (Γλϵ + b)n
,
n!

(5.5)

where Γ is the decay rate, λ is the exposure, ϵ is the signal eﬃciency, and b is the numberof background event. From the Baye’s theorem, Eq. (5.5) can be transformed as1
P(n|Γλϵb) · P(Γλϵb),P(n)1=P(n|Γλϵb) · P(Γ) · P(λ) · P(ϵ) · P(b),P(n)

P(Γλϵb|n) =

(5.6)
(5.7)

where the second line is derived from the assumption that Γ, λ, ϵ and b are all independent.
Here, the probability density function of Γ is
ZZZ

P(Γ|n) =1=A

P(Γλϵb|n) dϵdλdb,
ZZZ

(5.8)

e−(Γλϵ+b) (Γλϵ + b)nP(Γ) · P(λ) · P(ϵ) · P(b) dϵdλdb,
n!

(5.9)

where A is the normalization constant deﬁned as
Z ∞A=

P(Γ|n) dΓ.

(5.10)

0

The probabilities P(λ) and P(ϵ) are assumed to be Gaussians; exp − (λ−λ0 )2(λ > 0)22σλP(λ) ∝(λ ≤ 0)0(P(ϵ) ∝

0)
exp − (ϵ−ϵ
2σϵ2
0

2

(5.11)

(ϵ > 0)(ϵ ≤ 0)(5.12)

where λ0 (σλ ), ϵ0 (σϵ ), b0 (σb ) are the estimations (systematic errors) for exposure, signal
eﬃciency, and the number of background, respectively. The systematic errors of exposure
are set to be 0.01 kton·years. When the number of background b is low (less than around5.3. Lifetime Limits

7910), a convolution of Poisson and Gaussian distribution should be adopted for the probability of the number of background to take the statistical error into account. Therefore,P(b) is expressed asP(b) ∝

Z ∞ −Be (B)nbnb !

0(bC − B)2
· exp −
2σb2

!

dB.

(5.13)

Here nb is the number of events in 500 years of atmospheric neutrino MC, B is the numberof true background events in 500 years of atmospheric neutrino MC, C is the normalization
constant for MC live time to the data live time, and σb is the systematic error of the numberof background.
The relationship between the lower limit of the decay rate, Γlimit and the conﬁdence
level C.L. is
Z Γlimit

C.L. =

P(Γ|n) dΓ,(5.14)

0and the the lower limit of lifetime is deﬁned as;τ /B =

1Γlimit.

(5.15)The probability to detect ni events in the i-th independent proton decay search is given
as the product of each Poisson probabilities
P(n1 · · · ns |Γλ1 ϵ1 b1 · · · Γλs ϵs bs ) ==sYP(ni |Γλi ϵi bi )i=1e−(Γλ1 ϵ1 +b1 ) (Γλ1 ϵ1 + b1 )n1
n1 !

(5.16)
× ··· ×e−(Γλs ϵs +bs (Γλs ϵs + bs )ns,
ns !
(5.17)

where s is the number of independent searches, λi , ϵi and bi are the exposure, signal
eﬃciency and the number of background in SK-i period. Therefore, the lower limit ofdecay rate for the combined period searches is
0.9 =Z Γlimit Y
s
0

5.3.2

P(Γ|ni ) dΓ.

(5.18)

i=1

Final Results

The lifetime limits are calculated by combining the three diﬀerent search methods for four
diﬀerent SK periods. For each p → l+ η search, there are three methods: two for the lower
and upper box analysis for η → 2γ and one for η → 3π 0 , with four SK periods. Therefore,
12 independent searches are combined to estimate the limit for p → l+ η. The searches
for p → l+ η, η → 2γ lower and upper box analysis for SK-IV period are updated in thisstudy with new data, signal eﬃciencies, and systematic errors. The results of the other20 searches are cited from [29].
The lifetime limit for p → l+ η at 90% conﬁdence level are summarized in Tab. 5.9.
The updated factors compared to the previous study in this thesis are•

Extend exposure from 0.317 Mton·years→0.373 Mton·years for p → l+ η, η → 2γ

80

Chapter 5. Proton Decay Analysis

Tab. 5.9: Impact of signal eﬃciencies and their systematic errors on lifetime limit. The
numbers in the second row are cited from [29].

ModesPrevious study [29]Extended SK-IV datawith updated systematic errors
with updated signal eﬃciencies

Lifetime limit at 90% CL[×1033 years]
p → e+ η
p → µ+ η
10.4
4.7
11.6
5.4
12.1
5.8
13.3
6.1

Tab. 5.10: Summary of proton decay search of p → l+ η.

Modes
p → e+ η
p → µ+ η••

Background[events]
0.84 ± 0.32
0.93 ± 0.25

Candidate[events]
02

Probability
[%]
···
20.9

Lifetime limit
at 90% CL
13.3 × 1033 years
6.1 × 1033 years

Reduced the systematic errors on signal eﬃciencies by a factor of two for p → l+ η,η → 2γ in SK-IV period
Increased the signal eﬃciencies by 2.8% (1.0%) for the upper box analysis and 1.4%
(0.5%) for the lower box analysis of p → e+ η, η → 2γ (p → µ+ η, η → 2γ) in SK-IV
periodAs shown in Tab. 5.9, the extended exposure, updated systematic errors, and the updated signal eﬃciencies extend the lifetimes by around 11%(15%), 4%(7%), and 10%(5%),
respectively for p → e+ η (p → µ+ η). The summary of the proton decay search is shownin Tab. 5.10. As overall, the lower limit of partial proton lifetime for p → e+ η (p → µ+ η)
have increased by 28% (30%) with lifetimes ofτ /Bp→e+ η = 13.3 × 1033 years at 90% C.L.,(5.19)

τ /Bp→µ+ η = 6.1 × 10

(5.20)

33

years at 90% C.L..

Chapter 6

Discussion and Conclusion
6.1

Comparison with GUT models

As discussed in section 1.2.4, several GUT models suggest that the proton decay p → l+ η
occurs with a substantial branching ratio. The lower limits on the lifetime for the p → e+ ηmode estimated by four GUT models [22–25] are:∼ 3 × 1032 years(6.1)

SU(5) [23]:∼ 3 × 10

years

(6.2)

SU(5) [24]:

∼ 7 × 1032 years

(6.3)

:

∼ 1 × 10

(6.4)

SU(5) [22]SO(10) [25]3235

years.

The above lifetimes are estimated by the expected lifetime of the proton divided by thebranching ratio of p → e+ η mode. By contrast, the lower limit on the lifetime of the
p → e+ η mode estimated in this work isτ /Bp→e+ η = 13.3 × 1033 years at 90% C.L..

(6.5)

This represents the most stringent experimental limit reported so far, and excludes theSU(5) models of [22–24].

6.2

Conclusion

The proton decay search into a charged antilepton plus an eta meson and the eta meson
into two gamma-rays have been updated to include improved nuclear eﬀect estimation,
signal eﬃciencies, systematic errors for SK-IV period, and an additional 55.7 kton·years
exposure of SK-IV data. As for the newly analyzed 55.7 kton·years data, no evidence ofp → l+ η, η → 2γ is found in this study. The partial lifetimes of protons via p → l+ ηare estimated based on Bayes’ theorem by using total of 0.373 Mton·years exposure from
SK-I to SK-IV including the results from [29]. The lower limits at 90% conﬁdence levelare calculated as below.
τ /Bp→e+ η = 13.3 × 1033 years at 90% C.L.,
τ /Bp→µ+ η = 6.1 × 1033 years at 90% C.L..
The limits of p → e+ η and p → µ+ η have extended by 33% and 29% respectively fromthe previous study [29]. These results set the most stringent limits in the world.

81

82

6.3

Chapter 6. Discussion and Conclusion

Future Prospects

In this study, due to the re-estimation of the cross section of η nuclear eﬀect, improvementof the signal eﬃciencies and the systematic errors of p → l+ η, η → 2γ are demonstrated
for SK-IV data. This re-estimation will also be applied to both the rest of SK-I, II and III
data for p → l+ η, η → 2γ search, and the p → l+ η, η → 3π 0 search for SK-I to SK-IV datawith additional data. With all of these updates, the lifetime limit can roughly estimatedto be ∼ 20 × 1033 years (∼ 10 × 1033 ) for p → e+ η (p → µ+ η) mode.
SK is now running as a new phase, SK-Gd, with Gadolinium (Gd) dissolved in water.
Gd increases the capture eﬃciency of neutrons, and thus the neutron tagging technology
has improved. This phase has started from August 2020 and the atmospheric neutrino
background events are expected to decrease, which can contribute to the better sensitivityof proton decay searches. In addition, the Hyper-Kamiokande project (HK) [81], thesuccessor to SK, have started its construction planning to operate in 2027. The magnitudeis planned to be 10 times larger than of SK consisting of a cylindrical water tank of 74 m
diameter and 60 m height, 40,000 PMTs and 260 kton of pure water. HK is expected tocontribute to proton decay analysis with such high statistics.

Acknowledgements
My profound gratitude goes ﬁrst to my supervisor, Professor Masashi Yokoyama, for all
the continuous support during my Master s course. His enthusiasm towards physics had
led me to join the Super-Kamiokande experiment and his invaluable advice allowed me tocomplete this thesis.
I am deeply indebted to Makoto Miura and Shunichi Mine who generously gave metheir support. They taught me how to carry out research without hesitation. My analysis
would never have been accomplished without their insightful comments and suggestions.
I am grateful to all of the members in Super-Kamiokande collaboration, especially tothe following people: Yoshinari Hayato, Roger Wendell, Takuya Tashiro, Masaki Ishitsuka,
Hidekazu Tanaka, Yasuhiro Nishimura, Akira Takenaka and Ryo Matsumoto.
My appreciation also extends to all the laboratory colleagues. Thanks go to Konosuke
Iwamoto and Kota Nakagiri for being supportive not only for this study but also forbuilding skills as a physicist. My friends, Aoi Eguchi, Haruto Kikutani, Kohei Matsushita,
Yoshimi Yoshimoto had encouraged each other and made my time at The University ofTokyo enjoyable.
Finally, sincere thanks to my family for their considerations, encouragements and emotional support for years, and my beloved dog, Chiﬀon, for always having been with me,
which are the most essential to accomplish this big goal.

83

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89