UNIVERSITY OF CALIFORNIA,
IRVINESEARCHES FOR EXOTIC BARYON NUMBER-VIOLATINGPROCESSES AT SUPER-KAMIOKANDEDISSERTATION

submitted in partial satisfaction of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
in Physics

byVolodymyr Takhistov

Dissertation Committee:
Professor Henry W. Sobel, Chair
Professor Mu-Chun Chen
Professor William R. Molzon

2016c 2016 Volodymyr Takhistov

TABLE OF CONTENTS
Page
LIST OF FIGURESv

LIST OF TABLES

viii

ACKNOWLEDGMENTS

ix

CURRICULUM VITAE

xi

ABSTRACT OF THE DISSERTATION

xiii

1 Introduction
1.1 Baryon Number Violation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.1 In the Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.2 In Unified Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.3 In Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Experimental Searches for Nucleon Decay . . . . . . . . . . . . . . . . . . .
1.3 Overview of the Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
1
3
7
15
18
20

2 The Super-Kamiokande Experiment
2.1 Cherenkov Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Detector Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Photomultiplier Tubes . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Water and Air Purification Systems . . . . . . . . . . . . . . . . . . .
2.2.4 Electronics and Data Acquisition (DAQ) System . . . . . . . . . . . .
2.3 Detector Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Relative PMT Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Relative PMT Timing . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 Absolute PMT Gain . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.4 Water Transparency . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.5 Energy Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Data Taking Periods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23
23
25
25
26
26
27
29
29
29
30
30
33
36

ii

3 Monte Carlo Simulations
3.1 Signal: Nucleon Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2 Nuclear Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.3 Generation of τ and X . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Background: Atmospheric Neutrinos . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Neutrino Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Neutrino Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.3 Neutrino Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37
37
37
38
41
41
42
43
48
48

4 Event Reconstruction
4.1 Vertex Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Vertex Point Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2 Ring Edge Finding . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.3 Refined Vertex Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Ring Counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Particle Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Momentum Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Ring Number Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6 Decay Electron Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49
50
50
50
51
51
52
53
54
55

5 Fully Contained Data Sample
5.1 FC Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1 First and Second Reductions . . . . . . . . . . . . . . . . . . . . . . .
5.1.2 Third Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.3 Fourth Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.4 Fifth Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.5 Final FC Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 FC Sample Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56
56
57
57
57
58
58
59

6 Nucleon Decay Analysis
6.1 Data/MC Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.1 Event Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.2 Analysis Momentum Distributions . . . . . . . . . . . . . . . . . . . .
6.2.3 Atmospheric Neutrino Background . . . . . . . . . . . . . . . . . . .
6.3 Spectral Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.1 Fitting Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.2 Systematic Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4 Fit Results and Lifetime Calculation . . . . . . . . . . . . . . . . . . . . . .

64
64
65
65
67
69
70
70
71
74

7 Summary and Future Prospects

78

Bibliography

80

iii

A Trilepton Decay Channels from Pati-Salam89

B Approximating Spectrum of 3-body Decays

91

C Additional Analysis Information
95
C.1 Final Momentum Distributions . . . . . . . . . . . . . . . . . . . . . . . . . 95
C.2 Sensitivity Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
C.3 Data Fit Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

iv

LIST OF FIGURES
Page
1.1

1.2

1.3

Evolution of the inverse of the three coupling constants in the SM [left] and in
the MSSM [right]. Only in the latter case unification is obtained. The SUSY
particles are assumed to contribute only above the effective SUSY scale MSU SY
of „TeV. From Ref. [64]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Nucleon decay p Ñ e` π 0 from X, Y gauge boson exchange, resulting in effective dim-6 contribution. This is typically the dominant nucleon decay channel
within non- SUSY GUTs. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Typical nucleon decay diagrams in SUSY GUTs. On the left is p Ñ e` π 0 ,coming from dim-4 terms and a squark exchange. On the right is p Ñ νK ` ,coming from dim-5 triplet Higgsino exchange with the superpartners dressedby a Wino or Higgsino loop. . . . . . . . . . . . . . . . . . . . . . . . . . . .

Schematic diagram for the emitted cone-like pattern of light resulting fromCherenkov radiation. The direction of the charged particle (red arrow) andthe emitted light (blue arrows) is shown. . . . . . . . . . . . . . . . . . . . .
2.2 Overview of the Super-Kamiokande experiment. From Ref. [22]. . . . . . . .
2.3 [left] Inner detector PMT quantum efficiency as a function of wavelength.
[right] Single photo-electron PMT analog-to-digital converter (ADC) response,with the peak around zero being dark noise. From Ref. [22]. . . . . . . . . .
2.4 Overview of the Super-Kamiokande water purification system. From Ref. [22].
2.5 Overview of the Super-Kamiokande air purification system. From Ref. [22]. .
2.6 Schematic of the data acquisition system for the inner detector (arrows show
data flow). From Ref. [22]. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7 Setup for the relative gain calibration system [left] and the relative gain distribution of the PMTs after recalibration [right]. From Ref. [22]. . . . . . . .
2.8 Setup for the relative timing calibration system [left] and a sample TQ map
for a single PMT [right]. From Ref. [22]. . . . . . . . . . . . . . . . . . . . .
2.9 Schematic view of the absolute gain calibration system setup [left] as well as
the charge distribution of a sample PMT [right]. From Ref. [22]. . . . . . . .
2.10 Setup for the water transparency measurements using diffuser ball and camera.
From Ref. [22]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.11 Schematic view of the water calibration laser injection setup [left] and themeasured attenuation length along with the theoretical predictions [right].
From Ref. [22]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

12

13

2.1

v

24
25

26
27
27
28
29
30
31
31

32

2.12 Results for logrQl{f pθqs as a function of l for sample data [left] and watertransparency variation over time [right]. From Ref. [22]. . . . . . . . . . . . .
2.13 Momentum distribution of the stopping muon decay electrons [left] and thereconstructed invariant π 0 mass [right] for SK-I. MC simulation is depictedby continuous line [left] and boxes [right]. From Ref. [22]. . . . . . . . . . . .
2.14 Results for SK-I energy calibration. From Ref. [22]. . . . . . . . . . . . . . .
3.1

3.2
3.3
3.4

3.5

3.6

4.1

4.2

Theoretical calculations (solid and dashed lines) [161] for the 1p3{2 proton
state (left) and 1s1{2 proton state (right) of 12 C, compared with experimental
data. From Ref. [162]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Proton invariant mass in 16 O, modified by nuclear binding energy. From
Ref. [164]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Schematic view of cosmic ray interactions in the atmosphere. From Ref. [174].
[left] Primary cosmic ray flux measurements compared with the Honda flux
model (solid line) for protons at solar minimum. The data are taken from
Webber et. al. [176] (crosses), LEAP [177] (upward triangles), WIZARDMASS [178] (open circles), CAPRICE-94 [179] (vertical diamonds), IMAX[180] (downward triangles), BESS [181] (closed circles), AMS [182] (squares),
Ryan et. al. [183] (horizontal diamonds), JACEE [184] (downward open triangles), Ivanenko et. al. [185] (upward open triangles), Kawamura et. al. [186](open squares) and RUNJOB [187] (open diamonds). [right] Atmospheric
neutrino flux at Super-Kamiokande as predicted by the Honda (solid line),
Fluka (dashed line), and Bartol (dotted line) flux models. From Ref. [175]. .
Comparison of NEUT MC simulations (solid line) and data for the ν and ν
cross- sections of the charged current quasi-elastic scattering. The data from
ANL [202], Gargamelle [203, 204, 205], BNL [206], Serpukhov [207] and SKAT
[208] are shown. From Ref. [209]. . . . . . . . . . . . . . . . . . . . . . . . .
Total neutrino (a) [left] and anti-neutrino (b) [right] cross sections as a function of energy. The calculated quasi-elastic scattering cross section is shownin the dashed line, that of single meson production appears in the dotted line,
and the dash-dotted line shows deep inelastic scattering. Data from several
experiments are overlaid. From Ref. [160]. . . . . . . . . . . . . . . . . . . .
The basic concept of the Hough transform applied to Cherenkov rings. The
charge in a PMT is distributed along a circle corresponding to a 42˝ Cherenkov
opening angle as seen from the event vertex. The circles of the hit PMTs fromthe same Cherenkov ring will then overlap in the center of the actual ring.
From Ref. [236]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Event display of single-ring electron [left] and single-ring muon [right] data
events, with reconstructed momenta of 492 MeV and 603 MeV as well as thetime scale widths of 130 ns and 162 ns, respectively. Color shows time ofarrival of light to PMTs. An e-like event gives diffused ring pattern, while aµ-like event has a sharp ring edge. From Ref. [237]. . . . . . . . . . . . . . .

vi

33

34
36

39
40
42

43

45

47

52

52

4.3

4.4
5.1
5.2
5.3
5.4
5.5
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8

PID likelihood distribution for data and atmospheric MC with a visible energy
below 1.33 GeV. Good separation between e-like and µ-like events is seen.
From Ref. [164]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
RTOT vs. momentum table for the electrons (blue crosses), muons (redcrosses) and the kaons (black crosses). From Ref. [239]. . . . . . . . . . . . .

54

Schematic representation of the fully contained, partially contained, upwardgoing stopping and through-going muon event classes at Super-K. . . . . . .
Data vs. atm.-ν MC comparison for the SK-I data set (1489.2 days). . . . .
Data vs. atm.-ν MC comparison for the SK-II data set (798.6 days). . . . . .
Data vs. atm.-ν MC comparison for the SK-III data set (518.1 days). . . . .
Data vs. atm.-ν MC comparison for the SK-IV data set (1294.7 days). . . .

56
60
61
62
63

Final e-like momentum distributions for p Ñ e` X. . . . . . . . . . . . . . . .
Final µ-like momentum distributions for p Ñ µ` X. . . . . . . . . . . . . . .
Contribution of individual atmospheric ν-channels to the background of the
e-like analysis spectra, normalized to SK-I livetime of 1489.2 days. . . . . . .
Contribution of the individual atmospheric ν-channels to the background of
the µ-like analysis spectra, normalized to SK-I livetime of 1489.2 days. . . .
Fermi momentum signal (spectral function) and background (relativistic Fermi
gas) model comparison for p Ñ µ` X, using SK-I sample. . . . . . . . . . . .
MC sensitivity fit for p Ñ e` X, resulting in β90CL “ 0.013 (at α “ 1) [right].
Data fit for p Ñ e` X, resulting in β90CL “ 0.013 [left]. . . . . . . . . . . . .
[top] Reconstructed momentum distribution for 273.4 kton¨years of combined
SK data (black dots) and the best-fit result for the atmospheric neutrino
background Monte Carlo (solid line). The corresponding residuals are shown
below, after fitted background subtraction from data. [bottom] The 90%
confidence level allowed nucleon decay signal multiplied by 10 (hatched histograms), from the signal and background MC fit to data. All modes areshown (overlaid), with e-like channels on the left and µ-like channels on theright. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

53

67
68
69
70
73
74
75

77

LIST OF TABLES
Page
1.1

Standard Model field content and the respective symmetry transformations.
Index i labels the generation. . . . . . . . . . . . . . . . . . . . . . . . . . .
Proton decay searches. We have denoted their operating principle: search
for fission fragments from radioactive ore (FI), liquid scintillator (SC), water
Cherenkov (WC) and fine-grained calorimeters (FGC). . . . . . . . . . . . .

19

3.1

Summary of nuclear de-excitation after nucleon decay [166]. . . . . . . . . .

41

5.1

Number of events/day after each reduction step for the SK-I FC sample. The
atmospheric Monte Carlo numbers and efficiencies down to the fifth reduction
are for events whose real vertex is in the fiducial volume, the number of OD
hits fewer than 10 and the visible energy larger than 30 MeV. In the last line,
the fitted vertex is used for both data and Monte Carlo [160]. . . . . . . . .
Event summary for the reference SK-I to SK-IV single-ring samples. . . . . .

58
59

1.2

5.2
6.1
6.2
6.3
6.4

6.5

7.1

Event selection criteria for nucleon decay channels with one e-like (showering)ring. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Event selection criteria for nucleon decay channels with one µ-like (non- showering) ring. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Nucleon decay signal detection efficiency, SK-I through SK-IV. . . . . . . . .
Systematic errors of the two representative spectral fits, with 1σ uncertainties
and resulting fit pull terms. Errors specific to signal and background are
denoted by S and B, while those that are common to both are denoted by SB.
Best fit pα, βq parameter values, best fit χ2 { d.o.f., no signal ∆χ2 , 90% C.L.
value of β parameter, allowed number of nucleon decay events in the full
273.4 kton¨years exposure and a partial lifetime limit for each decay mode at
90% C.L. The sensitivity and lifetime limit for dinucleon decay modes are per
16
O nucleus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Final analysis results (sensitivity and lifetime limits) for the eight studied
nucleon decay modes. Comparison to previous experimental results is shown.
The results for dinucleon channels are reported per nucleus. . . . . . . . . .

viii

3

66
66
68

72

76

78

ACKNOWLEDGMENTSFirst and foremost, I would like to express significant gratitude to Hank Sobel. I have been
tremendously lucky to have him as my advisor. He has believed in me and provided me with
unparalleled support, encouragement and freedom to pursue orthogonally-related projects
throughout my studies. I am not sure my “ventures” into both theoretical and experimental
physics would have been possible if I had another advisor. He was always available to consult
and assist me with any issues that might have occurred and I can’t thank him enough forthis. Additionally, he was willing to seriously entertain any random idea that I had whilst
walking to his office, even if it was nonsense. Despite having a long and successful career,
I have seen him many times discussing with great passion new developments in science andtechnology, even unrelated to his field. He is a prime example of a gentleman and a scientist,in the best connotations associated with these words. I know of many good things that have
happened because of his leadership and involvement (direct or indirect).
I would also like to express large appreciation to Mu-Chun Chen, who has acted as
my unofficial advisor for theoretical work. She has agreed to work with me while knowing
about my involvement with experiment and has been very understanding. I would like toespecially thank her for patience and bearing with my stubbornness in general. Thanks to
her support, I was able to attend various theoretical physics schools, which proved to be avery positive experience. She has also been very encouraging of me proceeding, unimpeded,in directions different to the original research themes that we were working on, catering to
my interests. I really appreciate this.
My best regards are extended to Ed Kearns, one of the conveners of the Super-Kamiokande
nucleon decay and atmospheric neutrino (high energy) group. Many times has he provided
me with mentorship and encouragement on work not only related to Super-Kamiokande, butalso on scientific ventures in general. A lot of the Super-Kamiokande analysis results would
have been drastically weaker (or not possible at all) without his input and general knowledgeof many areas of physics. He is one of the leading world experts on nucleon decay and it has
been a pleasure working with him.
Additionally, I would like to thank Michael Smy, Shunichi Mine and Roger Wendell,with whom I have closely collaborated on SK-related work. Their knowledge of SuperKamiokande and willingness to help was instrumental in my successful and speedy completionof analyses. I would also like to stress, that Michael Smy’s thorough and broad understandingof experimental techniques has been directly responsible for timely completion of more than
one thesis at UCI.
I would also like to acknowledge Mark Vagins and Michael Ratz for many enlightening
discussions. Collaborating with Michael Ratz, who commands vast knowledge of various
topics in theoretical high energy physics, has been a valuable learning experience. I would
also like to thank other faculty, students and researchers at UCI, including Bill Kropp, James
Rutledge, Pearce Weatherly and Muhammad Elnimr for being helpful in various situations.

ixSuper-Kamiokande is an international effort and would not have been possible without
vigorous and extensive labor of many people, especially researchers at the Kamioka Observatory (Yasunori Hayato, Makoto Miura and others). I would especially like to accentuate
Masato Shiozawa, the other convener of the SK high energy group and the person in chargeof the upcoming Hyper-K experiment, for his leadership and encouragement. The first SKp Ñ e` π 0 analysis, which he conducted for his 1998 PhD. work, has served as a useful guidefor multiple future theses, including this one. I must also express my gratitude to Masayuki
Nakahata (spokesperson of Super-K) and Yoichiro Suzuki (previous Super-K spokesperson),
who with tireless effort have been steering the collaboration in the right direction for many
years. Without the leadership of above individuals, it would not have been possible tomaintain the high level of quality and effort that permeates all areas of Super-K work.
Finally, I would like to thank numerous other people who have been un-named but who
also deserve gratitude and who have positively influenced me and my world view, including
various scientists and thinkers (of past and present), friends and family. I appreciate thatmembers of the thesis committee (Hank Sobel, Mu-Chun Chen and Bill Molzon) took thetime out of their busy schedules to deal with my defense and thesis.
I acknowledge support of the U.S. Department of Energy. The Super-Kamiokande
experiment was built and has been operated with funding from the Japanese Ministry of
Education, Culture, Sports, Science and Technology and the U.S. National Science Foundation. Special thanks goes to Kamioka Mining and Smelting Company for helping to host theexperiment.

x

CURRICULUM VITAE
Volodymyr Takhistov
EDUCATION
Doctor of Philosophy (PhD.), Physics
University of California, Irvine

2016
Irvine, CA, USA

Bachelor of Science (BSc.), Physics and Philosophy
Rutgers University

2010
New Brunswick, NJ, USA

SELECTED PUBLICATIONS
V. Takhistov, “Review of Nucleon Decay Searches at Super-Kamiokande”, Proc. of Moriond2016 (Electroweak Int. and Unified Theor.), (2016), arXiv:1605.03235 [hep-ex]
J. Huang, M.C. Chen and V. Takhistov, “Beyond Minimal Lepton Flavored Dark Matter”,
J. High Energy Phys. 1602, 060 (2016), arXiv:1510.04694 [hep-ph]V. Takhistov et. al. (Super-Kamiokande Collaboration), “Search for Nucleon and Dinucleon
Decays with an Invisible Particle and a Charged Lepton in the Final State at the SuperKamiokande Experiment”, Phys. Rev. Lett. 115, 121803 (2015), arXiv:1508.05530 [hep-ex]

M.C. Chen, M. Ratz and V. Takhistov, “R-parity violation from discrete R-symmetries”,
Nucl. Phys. B891, 322 (2015), arXiv:1410.3474 [hep-ph]V. Takhistov et. al. (Super-Kamiokande Collaboration), “Search for Trilepton Nucleon Decay via p Ñ e` νν and p Ñ µ` νν in the Super-Kamiokande Experiment”, Phys. Rev. Lett.
113, 101801 (2014), arXiv:1409.1947 [hep-ex]

M.C. Chen and V. Takhistov, “Charged Lepton Spectrum Approximation in a Three Body
Nucleon Decay”, Phys. Rev. D89, 095003 (2014), arXiv:1402.7360 [hep-ph]

OTHER PUBLICATIONS
K. Abe et. al. (Super-Kamiokande Collaboration), “Real-Time Supernova Neutrino Burst
Monitor at Super-Kamiokande”, Astropart. Phys. 81, 39 (2016), arXiv:1601.04778 [astroph.HE]

xi

E. Richard et. al. (Super-Kamiokande Collaboration), “Measurements of the atmospheric
neutrino flux by Super-Kamiokande: energy spectra, geomagnetic effects, and solar modulation”, Proc. of Sci. (Int. Cosmic Ray Conf. 2015) 1044, (2015) arXiv:1510.08127 [hep-ex]

Y. Zhang et. al. (Super-Kamiokande Collaboration), “First measurement of radioactive isotope production through cosmic-ray muon spallation in Super-Kamiokande IV”, Phys. Rev.
D93, 012004 (2016), arXiv:1509.08168 [hep-ex]

J. Gustafson et. al. (Super-Kamiokande Collaboration), “Search for dinucleon decay into
pions at Super-Kamiokande”, Phys. Rev. D91, 072009 (2015), arXiv:1504.01041 [hep-ex]

K. Abe et. al. (Super-Kamiokande Collaboration), “Test of Lorentz invariance with atmospheric neutrinos” Phys. Rev. D91, 052003 (2015), arXiv:1410.4267 [hep-ex]

K. Abe et. al. (Super-Kamiokande Collaboration) “Limits on sterile neutrino mixing using
atmospheric neutrinos in Super-Kamiokande”, Phys. Rev. D91, 052019 (2015), arXiv:1410.2008[hep-ex]

D.D. Stancil et. al. (MINERνA Collaboration), “Demonstration of Communication using
Neutrinos”, Mod. Phys. Lett. A27, 1250077 (2012), arXiv:1203.2847 [hep-ex]

xii

ABSTRACT OF THE DISSERTATIONSEARCHES FOR EXOTIC BARYON NUMBER-VIOLATINGPROCESSES AT SUPER-KAMIOKANDE
By
Volodymyr Takhistov
Doctor of Philosophy in Physics
University of California, Irvine, 2016
Professor Henry W. Sobel, Chair

Various theoretical considerations suggest that baryon number should be violated. Nucleon decay, which typically appears within the context of unified theories, would provide
a definitive signature of baryon number violation. In this dissertation, we report on the
search results for p Ñ e` X, p Ñ µ` (where X is an invisible, massless particle), n Ñ νγ,
p Ñ e` νν, p Ñ µ` νν, np Ñ e` ν, np Ñ µ` ν and np Ñ τ ` ν nucleon and dinucleon decays
at the Super-Kamiokande experiment. Some of these searches are novel. Using data froma combined exposure of 273.4 kton¨years and a χ2 spectral fitting technique, a search forthese decays yields a result consistent with no signal. Accordingly, lower limits on the partial lifetimes of τpÑe` X ą 7.9 ˆ 1032 years, τnÑνγ ą 5.5 ˆ 1032 years, τpÑµ` X ą 4.1 ˆ 1032
years, τpÑe` νν ą 1.7 ˆ 1032 years, τpÑµ` νν ą 2.2 ˆ 1032 years, τnpÑe` ν ą 2.6 ˆ 1032 years,τnpÑµ` ν ą 2.0 ˆ 1032 years and τnpÑτ ` ν ą 3.0 ˆ 1031 years at a 90% confidence level are
obtained. These results provide stringent test of new physics and also limit the parameter
space of models that allow for such processes.

xiiiChapter 1
Introduction

1.1

Baryon Number Violation

One of the primary goals of physics is to establish fundamental laws that govern Nature.
At the microscopic level, these laws can be stated in terms of symmetries and corresponding
conserved quantum numbers, which impose restrictions on particle interactions. An illustrative example of this is the baryon number pBq. The conservation of B was originallyproposed in 1929 by Weyl [1], to explain why protons, which make up matter and carry acharge of B “ 1, are stable. After discovery of the positron e` by Anderson in 1933 [2], thisissue became more pressing, since a natural question arises why the p Ñ e` γ decay is notobserved. Hence, the concept of baryon number conservation was further reinforced in 1939
by Stuckelberg [3] and in 1949 by Wigner [4].
In the 1960s, a coherent relativistic quantum field theory (QFT), which describes interactions coming from three of the four known forces, has been developed. This QFT, theStandard Model (SM) of particle physics, is based on electroweak-theory combined with thestrong force, but without gravity. The fundamental forces in QFTs are mathematically described by gauge (local) symmetries. In the SM, baryon number happens to be an accidental
global symmetry, which ensures that protons are stable within the theory at the classicallevel.
Since then, various theoretical arguments have been put forward that suggest that B isonly an approximate symmetry and should be violated. Already in 1966, Sakharov [5] haspointed out that to explain the observed matter-anti-matter asymmetry of the Universe [6]1one needs baryon number violating processes. In fact, even within the SM itself, the baryon
number is violated by a minute amount due to quantum non-perturbative effects [7]. Whenone ventures beyond the SM in search of a deeper underlying theory, the baryon number isalso typically violated. This is the case, for example, when QFTs based on supersymmetry
or Grand Unification are considered [8]. More so, in the more fundamental theories of quantum gravity, global symmetries are expected to be violated in general [9]. This issue is ofsome importance, since even a small amount of B-violation can have profound effect on theultimate fate of the Universe [10]. For a comprehensive review of baryon number violationin various contexts see Ref. [11].
Thus far, baryon number violation has not been experimentally established. An observation of B-violation would not be the first time that a “non-fundamental” symmetry isfound to be only approximate. Conservation of some quantities, such as the electric charge[12], CP T [13] and energy-momentum [14] follow directly from the fundamental principles
on which QFTs are built: gauge invariance, Lorentz invariance and unitarity. On the other
hand, several of the conservation laws which did not follow from any fundamental principles
have been later found to be violated. These include: parity pP q in 1956 [15], charge conjugation combined with parity pCP q in 1964 [16], lepton family symmetries (the muon lepton
symmetry pLµ q in 1998 [17] and the electron lepton family symmetry pLe q in 2001 [18]) as
well as time reversal pT q in 2012 [19]. Since 1998, it has been established that neutrinos
oscillate and thus are massive [17]. If neutrinos are found to possess Majorana mass [20],which along with the Dirac mass is one of the two options, this would indicate [21] that
lepton number L is violated also. Given the above, continuation of testing B-conservation
remains a high priority.
In this Thesis, we will describe searches for exotic-baryon number violating processes ata large underground water Cherenkov experiment, Super-Kamiokande (SK, Super-K) [22].
The Thesis is organized as follows. Throughout Chapter 1 we will review in more detailvarious theoretical motivations for baryon number violation as well as outline some of thepossible experimental signatures. We then describe and motivate our analysis. In Chapter 2,
we will describe the mechanism of the Cherenkov radiation and then provide an overview of
the Super-Kamiokande experiment, including the description of its design and calibration.
In Chapter 3, we discuss the Monte Carlo (MC) simulation for nucleon decay, SK detector
as well as the atmospheric neutrino background. Then, in Chapter 4, we will outline theevent reconstruction algorithms used in the experiment. Later, in Chapter 5, we describe thefully contained (FC) data reduction algorithm, used to select useful events for the physics
analyses. Chapter 6 contains the actual nucleon decay analysis, including the description2

of the utilized data and MC samples, the event selection, systematics as well as the fitting
technique and finally the results. Lastly, Chapter 7 provides a summary and gives an outlook
for the future.

1.1.1

In the Standard Model

Review of the Standard Model
We start with a brief overview of the Standard Model and refer the reader to standard
texts [23] for a more comprehensive description. The SM is a QFT based on the
SU p3qC ˆ SU p2qL ˆ U p1qY

(1.1)

gauge group. The SU p2qL ˆ U p1qY sector describes interactions of the electro-weak force,through the Glashow-Weinberg-Salam [24, 25, 26] theory. The SU p3qC sector describes thestrong force interactions, which allow for confinement and asymptotic freedom [27, 28]. The
field content, summarized in Table 1.1, consists of spin-0 (the Higgs boson), spin-1/2 (thematter quarks and leptons) and spin-1 (the gauge bosons) fields. The matter fields, which
Field
QiL
uiR
diR
LiL
eiR
H
GA
µ
Wµa
Bµ

Spin
1/2
1/2
1/2
1/2
1/2
0
1
1
1

SU p3qC
3
3
3
1
1
1
8
1
1

SU p2qL
2
1
1
2
1
2
1
3
1

U p1qY
1/6
2/3
-1/3
-1/2
-1
1/2
0
0
0

B
1/3
1/3
1/3
0
0
0
0
0
0

L
0
0
0
1
1
0
0
0
0

Table 1.1: Standard Model field content and the respective symmetry transformations.
Index i labels the generation.
come in three generations labeled by index i, are the left handed doublets QiL “ puiL , diL q and
LiL “ pνLi , eiL q as well as the right-handed singlets uiR , diR and eiR . The gauge fields mediateathe forces and are given by GAµ , Wµ and Bµ , where the indices are A “ 1, ..., 8 and a “ 1, 2, 3.
Finally, H is the Higgs boson. The discrepancy between the description of the left- and theright-handed fields signifies that SM is a chiral theory. The field transformations under the3

SM gauge groups, as well as the global baryon number B and the lepton number L, aredisplayed.
The Lagrangian description of the Standard Model is given by
LSM “ Lgauge ` Lkinetic ` LHiggs ` LYukawa ,(1.2)

where the gauge and Lorentz invariant components are
LgaugeLkinetic
LHiggs
LYukawa

1
1
1“ ´ TrpGµν Gµν q ´ TrpWµν W µν q ´ Bµν B µν
2
8
4
i
i
i
i { i
i
i
{
{
{ iR ` eiR iDe{ iR“ QL iDQL ` LL iDLL ` uR iDuR ` dR iDd´v 2 ¯2“ pDµ Hq: pDµ Hq ´ λ H : H ´
2
i
ij i
j
ij
˚ j
“ ´Yu QH uR ´ Yd QL HdR ´ Yeij LL HejR ` h.c.

Here, Lgauge is built from the field strength tensors Gµν , Wµν , Bµν which are functions of thegauge fields and describe their dynamics. The spinor and color indices have been neglected.
The Lkinetic component provides dynamics for the fermion matter fields. The covariant derivative Dµ , in case of the quark doublet QiL , is given byAa a
Dµ “ Bµ ` igGAµ T ` ig2 Wµ T ` ig1 Bµ Y ,(1.3)

where T A , T a , Y are the SU p3qC , SU p2qL , U p1qY generators and g, g2 , g3 are the corresponding couplings. The LHiggs sector is responsible for the Higgs kinetic and potential
terms. Finally, LYukawa describes the (generically complex) couplings of fermions with the
Higgs boson and contains the “flavor structure” of the SM. Due to the mismatch between themass and the weak eigenstates, the up and down quark mass matrices cannot be diagonalized
simultaneously. This results in the quark sector Cabbibo-Kobayashi-Maskawa (CKM) mixing [29, 30]. Similarly, the lepton sector has the Pontecorvo-Maki-Nakagawa-Sakata (PMNS)
mixing1 [31, 32, 33].
The SU p2qL ˆ U p1qY sector is spontaneously broken down to the electromagnetic (EM)U p1qEM , according do the “Higgs mechanism” [34, 35, 36]. This is implemented by having
the Higgs doublet acquire a vacuum expectation value (vev)
˜ ¸
˜ ¸
H`
1 0
.
H“
Ñ xHy “ ?
2 v
H0
1

(1.4)

The PMNS CP violating phases depend on whether the neutrinos are Majorana or Dirac fermions,which is still unknown.

4After spontaneous symmetry breaking, the interaction gauge fields are the photon γ (EM),the W ˘ and the Z 0 bosons (weak force)2 and the gluons g (strong force). The Higgs mechanism provides fermions as well as the W ˘ and the Z 0 bosons with masses, according toMW “

g22 v 2
4and MZ2 “ pg22 ` g12 q

2
v2
MW
“
,
4
cos2 θW(1.5)

where θW is the Weinberg angle defined assin2 θW “

2
g12
MW
“
1
´
.
g22 ` g12
MZ2

(1.6)

The electric coupling e is given by e “ g2 sin θW .
The SM has been extremely successful, as can be illustrated by the agreement between
the theoretical prediction and the experimental measurement of the anomalous magnetic
moment of electron being good to more than 10 significant digits [37]. That being said, manyobservations suggest that there exists a more fundamental underlying theory. For example,
the SM contains more than 15 unexplained parameters, does not account for the dark matterand the massive neutrinos and also does not provide an explanation for the observed electriccharge quantization. Moreover, a complete description of nature should also include gravity.

Baryon Number Violation in the Standard ModelAs already mentioned, the baryon number B and lepton number L are global accidental
symmetries of the SM and there are no gauge invariant renormalizable (dimension ď 4)
operators which could be written down that violate them. The lowest allowed operator in
the SM which violates B, is the non-renormalizable dimension-6 term
p6q

OB{ „

QQQL
,
Λ2

(1.7)

where Λ is a suppression scale associated with some high energy physics, coming from a more
fundamental theory. Similarly, for lepton number L, the lowest order L-violating term is thedimension-5 operator LHLH{Λ. In the above, we have suppressed the relevant indices forbrevity.
2

The photon and the weak bosons are formed from the linear combinations of Bµ and Wµa .

5

At the classical level, the Standard Model B-current Bµ jBµ is given by
Bµ jBµ “ Bµ

”1 ÿ
3

ı
i
i
pQL γ µ QiL ` uiR γ µ uiR ` `dR γ µ diR q “ 0 .

(1.8)

i

The description for the L-current Bµ jLµ is similar. At the quantum level, however, these global
symmetries are violated by the electro-weak anomalies [38, 39], resulting in
Bµ jBµ “¯
3 αβγδ ´ 2 a a
2

g
W
W
`
g
B
B
2
αβ
γδ
1 αβ γδ .
64π 2

(1.9)

Importantly, pB ´ Lq still remains conserved, since the equivalent B and L violating contributions can be subtracted. While anomalies associated with gauge symmetries signify
an inconsistency of the theory and violation of unitarity, this is not the case for anomalies
associated with global symmetries3 . Although one can attempt to gauge the baryon number
B [41], gauge anomaly cancellation will require that extra charged fields are introduced tothe theory, which is not desired.
For the SU p2qL , a non-Abelian gauge symmetry of the SM, there exist non-perturbative
field configurations (“instantons” [42]) which contribute to Bµ jBµ and violate the baryon number. Due to their non-perturbativate nature, these effects cannot be deduced from the standard perturbative Feynman diagram calculations. Instantons originate from non-uniquenessof the vacuum ground state in non-Abelian theories and represent the tunneling rate between
them. In analogy with regular quantum mechanics, these transitions, which violate baryon
2 2and lepton number separately, are exponentially suppressed and are proportional to e´8π {g2 ,in case of SU p2qL . Because the vacua are topologically distinct (can’t be morphed into one
another), they can be labeled by an integer (the Chern-Simons number), which in a certain
gauge is stated as
g22
nCS “
16π 2

ż

´
¯
g
a
a
a
b
c
d x
Bi Wj Wk ` abc Wi Wj Wk .
3
3

ijk

(1.10)

The field configurations at the top of the barriers, which need to be tunneled for a vacuumtransition, are the “sphalerons” [43]. In the electro-weak case, the barrier height (sphaleronenergy) is given by Esp „ 8πMW {g22 . It was found by ’t Hooft [44] that the lowest orderbaryon number violation in the SM due to instanton effects can be described by 12-fermion
operator
3 ´´ 1 ¯14¯ź´8π 2 {g22Osp „eαβγ QiαL QiαL QiαL LiL ,(1.11)MWi
3

The decay rate of π 0 Ñ γγ is explained in the SM through global axial anomalies [40].

6where α, β, γ represent color indices. This transition leads to ∆B “ ∆L “ 3. Its rate,
2 2
however, is suppressed by Γsp „ |e´8π {g2 |2 « 10´173 and is thus negligible. The above
calculations have assumed T “ 0 temperature, but for finite temperature the rate of thermal
fluctuations across a vacuum barrier is found to be [7, 45, 46]
Γsp “ T 4 e´Esp {T .

(1.12)

Hence, at high enough temperatures (e.g. the early Universe), these processes become significant and are important for baryogenesis, as we will discuss.
In the above discussion, we have followed the review of Dine [48]. A comprehensive
treatment of the subject can be found in Coleman’s textbook [47].

1.1.2

In Unified Theories

Review of Grand Unified Theories
Grand Unified Theories (GUTs) are some of the most well motivated theoretical concepts beyond the SM. In GUTs, the three gauge groups of the SM are combined together.
This allows to treat quarks and leptons similarly, explain charge quantization, gauge coupling unification and naturally include neutrino masses, among other things. Since leptons
can interact with quarks in GUTs, within this context, baryon number violation is inevitable.
The two simplest unifications accounts are the Georgi-Glashow SU p5q [49] model and
the Pati-Salam (PS) SU p4qC ˆ SU p2qL ˆ SU p2qR [50] theory4 . In the PS model, leptons areincluded as the “4th color”, by extending the QCD SU p3qC to SU p4qC . One family of particles
rpQL , LL q, pQR , LR qs transforms as rp4, 2, 1q, p4, 1, 2qs under PS, where QR “ pdR , uR q andLR “ peR , νR q under SU p2qR . Unlike the SM, in the PS model the left- and right-handed
fields are treated on the same footing. Additionally, it includes pB ´ Lq as subgroup andcontains a right handed neutrino νR , which is absent in the SM but desirable for neutrinomass generation. Since it is not built from a simple gauge group, strictly speaking it is not
The Georgi-Glashow SU p5q is the minimal simple group which fully contains the SM5
4Since PS is not based on a simple gauge group which would include the SM, strictly speaking it is not
a GUT, but rather a theory of “matter unification”. This is also the reason why in PS coupling unification
is not a prediction.
5
This can be seen from the fact that both groups have rank = 4.

7

and we will discuss it in more detail. The theory has one universal gauge coupling, αG ,defined at the grand unification scale of the three SM couplings, MG .
Within the SU p5q, quark and lepton fields reside in two irreducible representations:5 “ rdR , LL s and 10 “ rQR , uR , eR s. This can be visualized diagrammatically as
˛dc,1
˚ c,2 ‹
˚d ‹
‹
˚
c,3 ‹
5“˚
d
‹
˚
˚ ´‹
˝e ‚
´νe

¨

¨

and

0
uc,3 ´uc,2 u1
˚ c,3
˚´u
0
uc,1
u2
˚
c,2
10 “ ˚
´uc,1
0
u3
˚u
˚
0
˝ ´u1 ´u2 ´u3´d1 ´d2 ´d3 ´ec

˛
d1
‹
d2 ‹
‹
d3 ‹
‹ ,
c‹
e‚
0

(1.13)

where the numerical superscripts represent color indices and c stands for charge conjugation6 . The gauge sector contains 24 bosons that form a 24-plet7 , which can be seen diagrammatically as¨
2BG1 ´ ?
G13
G12
˚ 1
30
˚
2B
˚
G22 ´ ?
G23
G21
˚
˚
30
˚
2B
˚
3
3
G1
G2
G23 ´ ?
Vµ “ ˚
30
˚
˚
˚
X1
X2
X3
˚
˚
˝
Y1
Y2
Y3

˛
X1

Y1

‹
‹
‹
X2
Y2
‹
‹
‹
‹
X3
Y3
‹ ,
‹
3
‹
3B
W
‹
`
? `?
W
‹
2
30
‹
3
W
3B ‚
´
W
´? ` ?
230(1.14)

where Gji are the 8 gluons and the W a and the B combine to form the γ, Z 0 and W ˘ bosonsof the SM. The other 12 X and Y bosons are new. Identifying the lower right 2 ˆ 2 quadrantof the above 5 ˆ 5 matrix with SU p2qL rotations and the upper left 3 ˆ 3 quadrant with theSU p3qC rotations, it can be seen that the off-diagonal X and Y gauge bosons can contributeto both and thus give lepto-quark interactions. These interactions can mediate proton decay.
The SU p5q is spontaneously broken SU p5q Ñ SU p3qC ˆSU p2qL ˆU p1qY to the Standard
Model through the vev of the 24H Higgs multiplet, as depicted in Equation (1.15), which
is proportional to the hypercharge generator Y with ´3Y “ diagp2, 2, 2, ´3, ´3q. After the
breaking, the heavy X and Y gauge bosons decouple with mass MG .
6

Since gauge boson couplings observe helicity, left and right handed fields cannot be in the same multiplets. Hence, the right-handed fields are replaced by corresponding conjugated left-handed ones.
7
The adjoint 24-plet can be constructed from the SU p5q fundamental 5-plet according to 5 ˆ 5 “ 24 ` 1.

8

¨
2
˚
˚0
v ˚
x24H y Ñ ? ˚
0
30 ˚
˚
˝0
0

0
2
0
0
0

˛
0 0
0
‹
0 0
0‹
‹
2 0
0‹
‹ .
‹
0 ´3 0 ‚0 0 ´3(1.15)The electro-weak Higgs doublet H “ pH ` , H 0 q resides in the 5H of SU p5q, according
to 5H “ rT 1 , T 2 , T 3 , H ` , H 0 s. Here, T “ pT 1 , T 2 , T 3 q represents a new Higgs triplet. The
fermion Yukawa couplings are formed via 5 10 5H and 10 10 5H . Thus, the new Higgs triplet
will also couple to fermions and allow them to transform into one another. This leads to rapid
proton decay mediation, unless the T mass is ą 1011 GeV [51].
The charge operator Q “ T3 ` Y {2 is a linear combination of the SU p2qL and U p1qY
generators, which can be identified with the SU p5q generators. Since SU pN q generators aretraceless, this implies that Q eigenvalues add up to 0. Hence, this leads to one family offermions having a quantized charge, according to
Qpν e q ` Qpe` q ` 3Qpqq “ 0Ñ

1
Qpqq “ ´ e .
3(1.16)

Thus, charge quantization is explained in GUTs, which is not the case for the SM.
While SU p5q has many virtues, there are also issues. To achieve light Higgs doublet
but a heavy Higgs triplet, a tuning in the choice of vev that breaks 5H is required, resulting in the doublet-triplet splitting problem. Some proposed solutions include considering
more complicated representations and their combinations (e.g the “hidden partner mechanism”) [52], but this is not very satisfactory. Additionally, SU p5q gives prediction for the
Weinberg angle sin θW inconsistent with experiment [53], showing that coupling unification
doesn’t really work out. Finally, as we will discuss later, minimal SU p5q is ruled out by
proton decay limits.
Going to SOp10q [54] allows to unify all the matter fields in a single SOp10q multiplet
16 “ r10 ` 5 ` 1s, where the multiples in the parenthesis transform under SU p5q and
1 represents a right-handed neutrino singlet. Because the group rank of SOp10q is 5, its

9breaking proceeds to SM in two steps. The breaking patterns can be visualized as$’&
SOp10q Ñ’%SU p4qC ˆ SU p2qL ˆ SU p2qR
SU p3qC ˆ SU p2qL ˆ SU p2qR ˆ U p1qB´L
SU p5q ˆ U p1qX,/.
Ñ SU p3qC ˆ SU p2qL ˆ U p1qY ,/-among which we can identify the Pati-Salam and the SU p5q sub-groups. If QEM is partof the extra U p1qX , one obtains a “flipped” SU p5q model [55, 56] instead of the usual
SU p5q. Typically, the SOp10q breaking is done by rather large Higgs multiplets, residing
in 10H , 45H , 126H , 54H representations. The doublet-triplet problem of SU p5q also persists
in SOp10q. The Yukawa couplings originate from 10H 16 16 and the neutrino masses are
typically implemented via “see-saw” mechanism [57, 58, 59, 60, 61], since it requires a righthanded neutrino with mass around the GUT scale. To successfully account for the observed
fermion masses and mixing one can impose additional “flavor symmetries”8 .
Supersymmetry (SUSY) can be added to GUTs, allowing to make gauge coupling unification precise9 , as depicted in Figure 1.1. There are many reasons why SUSY is appealing
and we point the reader to a comprehensive review of Ref. [65] for details. It is the maximal
possible extension of the Poincare symmetry (Haag-Lopuszanski-Sohnius theorem [66]) for
4-D QFTs10 and allows to solve the gauge hierarchy problem of the SM (why is there such a
discrepancy between the electroweak and the Planck scales), more so, it is thought to be aconsistency requirement for string theory and is motivated phenomenologically. With SUSY,
the fermions and the bosons are related. Hence, in the minimal supersymmetric SM with
global supersymmetry (the MSSM) each field has an accompanying superpartner. The resolution of hierarchy problem requires superpartners to be around the TeV scale, which so far
have not been observed at the Large Hadron Collider (LHC) [67, 68]. At the cost of abandoning the hierarchy problem, one could raise the scale of SUSY breaking / superpartner masses
and thus alleviate the experimental constraints. When combined with GUTs, SUSY GUTs
can ease some of difficulties encountered within non-SUSY GUTs, such as precise coupling
unification. SUSY GUTs can have significantly different predictions for nucleon decay than
the non-SUSY GUTs, as we will discuss below.
To conclude the overview of unified theories, let us briefly mention ventures beyond
8

For an overview of discrete flavor symmetries and how they can be applied to GUTs see Ref. [62].
If one considers complicated enough breaking of SOp10q with appropriate intermediate scales, coupling
unification consistent with experiment can be achieved without SUSY [63]. Such models, however, are overly
contrived.
10
These combined Super-Poincare transformations are the reason why gauged (local) SUSY, the supergravity, automatically incorporates General Relativity.
910

Figure 1.1: Evolution of the inverse of the three coupling constants in the SM [left] and in
the MSSM [right]. Only in the latter case unification is obtained. The SUSY particles are
assumed to contribute only above the effective SUSY scale MSU SY of „TeV. From Ref. [64].
4-D GUTs and SUSY. String theory (ST) is the only known consistent theory of quantum
gravity. While there is a strong model building effort (e.g. [69, 70, 71, 72]) to get concrete
phenomenological predictions from ST, due to a large number of possibilities how the 10-D
theory can be “compactified” to the 4-D theory, this path is difficult. However, generic predictions (e.g. abundance of axions) do exist [73] and string-motivated constructions can beboth manageable and predictive. For example, some of the recent work has been focused on
GUTs in higher dimensions (5-D and 6-D). Such constructions allow to alter proton decayrate predictions [74] and explain various scale hierarchies, by considering a specific arrangement of fields in the higher dimensional space. This also allows to potentially address the
doublet-triplet splitting issue, outlined earlier. With this approach, however, one substitutes
the choice of a selecting a specific Higgs vev in 4-D with the choice of boundary conditions
in extra dimensions [75]. Similarly, motivated by E 8 ˆ E 8 symmetry of ST and the embedding E8 Ą E6 Ą SOp10q, one can consider even larger GUT groups with more complicatedbreaking patterns than those of SOp10q. However, the number of extra fields and parameters introduced is substantial and the models become overly contrived with not very generalpredictions. A comprehensive review of unified theories and supersymmetry can be found in
Mohapatra’s textbook [8].

11Baryon Number Violation in Unified Theories
We will now discuss several of the most popular nucleon decay signatures. In the context
of SU p5q, nucleon decay is mediated by the X and Y gauge boson exchange. This gives
effective dim-6 baryon number-violating operators. Figure 1.2 displays diagrams for p Ñe` π 0 mediation, which is typically the dominant decay channel in non-SUSY GUTs. The

Figure 1.2: Nucleon decay p Ñ e` π 0 from X, Y gauge boson exchange, resulting in effective
dim-6 contribution. This is typically the dominant nucleon decay channel within nonSUSY GUTs.
lifetime of the proton is then given byτpÑe` π0

„

MG4
,
m5p

(1.17)

where mp is the mass of the proton and the X and Y masses have been taken as MG .
Minimal Georgi-Glashow SU p5q [49] predicts the lifetime of p Ñ e` π 0 to be around 4.5 ˆ
1029˘1.7 years, which has been decisively ruled out by experimental limits of τpÑe` π0 ą 1033
years [77, 78, 79, 80].
Once SUSY is introduced, the unification scaled is pushed up from 1015 GeV to more
{ terms [81, 82, 83], whichthan 1016 GeV. This leads to significant suppression of the dim-6 B
2
are proportional to 1{MG , resulting in an increased proton lifetime of τ ą 1034´38 years.
On the other hand, SUSY also introduces new dim-4 and dim-5 operators that could also
mediate nucleon decay. The most dangerous are the dim-4 baryon number-violating U DD
and the lepton number-violating QLD, LLE terms11 . If both of these are allowed, they lead2
to very fast proton decay, with lifetime proportional to the squark mass τ „ 1{MSUSY .
By itself, this can be taken as supersymmetry predicting baryon number violation even
without any Grand Unification scheme. In order to suppress nucleon decay coming from{ or B{ terms independently (leading to R-parity
these operators, one can forbid either L
11Here, U , D, L, Q and E denote superfields and contain both the original fermion and a boson superpartner. Recall that fermions contribute with a mass dimension of 3/2 and bosons with 1.

12

{ and B{ terms simultaneously (e.g. with R-parity12 [85] or ZR
violation [84]), or forbid both L4[86]). If only the L-violating or the B-violating terms are forbidden independently, however,
the resulting scenario will not be compatible with SU p5q or SOp10q, where these terms areallowed or forbidden simultaneously. Although, in this case, Pati-Salam compatibility is still
an option [84].
Having forbidden SUSY dim-4 operators, the dominant contribution will typically comefrom the dim-5 terms QQQL and U U DE. Within supersymmetric GUTs, these terms arise
from the Yukawa couplings of the triplet T Higgs-superpartner (Higgsino) and they areproportional to 1{MT . As the final state sfermion superpartners are not observed, they need
to be converted to the regular SM fermions (“dressed”) by a gaugino (Wino or Higgsino)loop. Usually, these considerations lead to p Ñ νK ` being the dominant nucleon decay modein SUSY GUTs. In Figure 1.3 we depict typical diagrams originating from dim-4 and dim-5
baryon number-violating operators. The lifetime of the proton coming from the effective

Figure 1.3: Typical nucleon decay diagrams in SUSY GUTs. On the left is p Ñ e` π 0 ,coming from dim-4 terms and a squark exchange. On the right is p Ñ νK ` , coming from
dim-5 triplet Higgsino exchange with the superpartners dressed by a Wino or Higgsino loop.
dim-5 terms is then given byτpÑνK `

2
MG2 MSU
SY
,
m5p

„

(1.18)

where we have substituted MG for MT (assuming the triplet Higgsino is at the GUT scale).
In a full nucleon decay calculation (e.g. see Ref. [87]) one needs to also determine the loop
factor and the matrix element. The matrix element represents transition between vacuum
state and creation of the nucleon x0|qqq|N y. It can found via Chiral Lagrangian [88] or latticecalculations [89]. The final amplitude has a strong dependence on the fermion masses andmixing and the details of SUSY breaking / arrangement of the SUSY spectrum. The minimal
12

Matter parity is a discrete Z2 symmetry under which the matter superfields Q, U , D, L and D are odd{
and the Higgs superfields Hu and Hd are even. This forbids the unwanted dim-4 (R-parity violating) B{
and L terms. A combination of matter parity with a discrete subgroup of the Lorentz group gives R-parity,
under which all SM fields are even and superpartners odd.

13

SUSY SU p5q, with SUSY taken at the TeV scale as motivated by the hierarchy problem, isalso ruled out [90] by experimental constraints on τpÑνK ` ą 1033 years [91].
Thus far we have discussed nucleon decay in the simplest scenarios of minimal SUSY
and non-SUSY SU p5q. Going to larger groups such as SOp10q adds more variation in model
constructions and thus predictions. In the case of SUSY GUTs, there is a large theoreticaluncertainty coming from SUSY breaking sector and the resulting superpartner spectrum.
Additionally, because in GUTs many model sectors are related, there are typically multiple
nucleon decay channels which are predicted with various branching ratios. These predictions
have strong dependency on such things as the fermion mass and mixing spectrum. Overall,majority of models predict that the dominant channels will have τp ă 1036 years [92].
Aside from regular single nucleon decay, it is also worthwhile to mention other baryon
number violating processes. One such process is neutron - anti-neutron oscillation n ´ n (seeRef. [93] for review). It can be described by an effective 6-fermion operator
On´n

„

¯
1´
Q
Q
Q
Q
d
d
`
u
d
d
u
d
d
,
L
L
L
L
R
R
R
R
R
R
R
R
Λ5

(1.19)

where Λ is the suppression scale. The n ´ n lifetime can be calculated using the usual
quantum mechanical 2-state Hamiltonian approach. Taking the current experimental limitof τn´n ą 0.86 ˆ 108 s. [94] yields Λ of the order of „ 100 TeV [93]. If SUSY is introduced,
superpartner appearance in the above operator lowers its dimension, which in turn allows toraise the suppression scale. Even with SUSY, unlike regular proton decay, n ´ n allows totest scales far below the GUT scale. Inside matter, the neutron and anti-neutron potentials
are different and the n ´ n lifetime in matter is related to the lifetime in vacuum according
to
N uc.
τn´n
“ Rpτn´n q2 ,(1.20)
N uc.
where R „ 0.3 ˆ 1024 s´1 is a nuclear suppression factor. This gives τn´n
„ 1032 years,
comparable to the regular proton decay limits. An additional feature that separates thisprocess from the typical p Ñ e` π 0 and p Ñ νK ` channels, is that neutron oscillation is a|∆pB ´ Lq| “ 2 transition. It is possible to construct models [95] where this process arises
within the context of GUTs and is connected to baryogenesis and Majorana neutrino masses,
which have ∆L “ 2 and thus also break pB ´ Lq symmetry.

Another example of how baryon number violation can occur is through monopoles [96]catalyzing nucleon decay. When a breaking of a simple GUT gauge group leads to a residual
U p1q (i.e. G Ñ H ˆ U p1q), magnetic monopoles which carry a U p1q charge will appear14[97, 98]. With a mass of the order MG , they are expected to be abundant in the early
Universe. Since central core of a GUT monopole contains heavy gauge bosons, they canviolate baryon number. In fact, monopoles can catalyze proton decay through the CallanRubakov effect [99, 100], resulting inM ` p Ñ M ` e` ` mesons .

(1.21)The striking feature of this process is that the amplitude is not suppressed by 1{MG . The
problem lies in estimating the exact expected proton decay rate from this reaction. There
are other objects, such as supersymmetric Q-balls [101], which can be produced in the early
Universe and could catalyze nucleon decay [102].
Finally, effects of quantum gravity can also lead to baryon number violation. As an
example, consider proton decay catalyzed by virtual black holes. As already mentioned, it isthought that quantum gravity generally violates global symmetries (e.g. baryon number) [9].
Consider the process [76] where 2 quarks of a proton fall into a small black hole of mass
mBH „ MP l and that baryon number is violated (e.g. q ` q Ñ q ` l). Then, one can roughly
estimate the proton’s lifetime asτp „ 1036 ˆ

´ M
¯4
QG
yrs ,
1016 GeV(1.22)

which for MQG “ MP L gives τp „ 1045 yrs.
For an exhaustive review of possible baryon number-violating mechanisms and theirsignatures see Ref. [76].

1.1.3

In Cosmology

The Standard Model of Cosmology is Λ-CDM. It is described by the Big Bang cosmology, a cosmological constant Λ, associated with dark energy, and abundance of cold dark
matter (CDM). It is well supported by observation of light element abundance from Big
Bang nucleosynthesis (BBN), Hubble expansion and cosmic microwave background (CMB).
Fitting the astronomical data to parameters of Λ-CDM [103] gives a description of the
Universe that is composed of 68.5% dark energy, 26.5% dark matter and 4.9% of ordinary
matter (baryons). There is an inherent matter - anti-matter asymmetry observed in the Universe. From cosmic microwave background radiation (CMB), the measured ratio of baryon

15

nb to photon nγ density is found to be [104]η“nb“ p6.19 ˘ 0.14q ˆ 10´10 .
nγ(1.23)The lack of anti-baryon excess is reaffirmed by the astrophysical anti-proton flux [105], which
is found to be consistent with secondary cosmic ray production.
To explain the cosmologically observed baryon asymmetry, naively, one could expectthat there has always been a B asymmetry (i.e. B ą 0) starting with the Big Bang. However,such an initial asymmetry would be diluted to a negligible quantity by inflation [106, 107,108], a period of rapid expansion of the Universe. While multiple models of inflation exist (seeRef. [109] for review), the concept itself provides a simple resolution to several key problems
in cosmology and thus abandoning it would be undesirable. Namely, inflation addresses theissues of horizon (why the Universe is homogeneous and isotropic), flatness (why spatial
curvature of Universe is small) and the exotic-relic abundance (why such exotics as GUT
monopoles, produced during the early Universe, have not been observed). Another possibility
is that there is no global B asymmetry, and we are just located in a special separated regionof the Universe filled with matter and not anti-matter, but in other regions it is the opposite.
However, not only is this a non-minimal assumption, there is evidence that if large regionsof anti-matter exist, they are on cosmic distance scales away [110]. The final option, whichis the one that we shall focus on, is that initially B “ 0 and then baryon asymmetry was
generated via some mechanism of baryogenesis.
For a successful mechanism of baryogenesis, three conditions, as outlined by Sakharov[5], must be satisfied:
1. B-number violation processes exist: if all interactions conserve B, then B is globally conserved.
2. C and CP violation exist: C and CP symmetries interchange particles Ø antiparticles and particles Ø anti-particles followed by a parity flip, respectively. The
requirement of C and CP violation ensures that B-violating processes are not cancelledout by B-generating processes.
3. Period with B-violating processes out of equilibrium exists: CP T theorem
assures that masses of particles and anti-particles are equal and hence their thermal{ processes which violate C and CP
equilibrium densities are also same. Thus, even if B
exist, they must happen during thermal non-equilibrium to ensure that the thermal
16average of B is non-zero.
While a large number of specific baryogenesis mechanisms have been proposed, weoutline below some of the most popular categories and their respective problems, following
the subject review by Dine and Kusenko [111]:• Electroweak Baryogenesis: SM by itself satisfies all of the Sakharov’s conditions for{baryogenesis [7]. While B is a global symmetry in SM, as discussed in Section 1.1.1, B
is induced by sphalerons and at high enough temperatures the rates for these processes
can be significant. The CP violation already occurs in quark sector and out of equilibrium condition can happen during the electroweak phase transition. However, if one
considers SM alone, CP violation in the quark sector is too small and some difficulties
arise with getting the appropriate strength of the phase transition.
• Planck-scale (gravity-induced) Baryogenesis: as discussed in Section 1.1.2, global
symmetries are violated by quantum gravity effects and one would expect that Bviolating processes are present. These effects are suppressed by the Planck scale MP land can be significant in the early Universe. However, not only is it difficult to calculate{ will be diluted by inflation
things precisely during that epoch, any early generated B
later.
• GUT-scale Baryogenesis: as quarks and leptons transform in common GUT multiplets, baryon number-violating processes, like proton decay, naturally occur within
GUTs. It is typically assumed that above the GUT scale there is thermal equilibrium.
At a scale below the mass of the new GUT gauge bosons, dynamics of their interactions do not produce them sufficiently fast to maintain the equilibrium. Additionally,
their decays have new CP violating couplings. However, inflation again can dilute any
generated B number at GUT scale unless reheating temperature is very high, which isproblematic for cosmology.
• Leptogenesis [112]: as already mentioned in Section 1.1.1, the lowest order nonrenormalizable operator for the neutrino masses in the SM is a dimenstion-5 L-violatingterm LHLH{Λ. This results in Majorana neutrinos. Invoking the see-saw mechanism
to explain their mass scale requires introduction of the Dirac and the Majorana massterms for the additional heavy right-handed neutrinos Ni . In the early Universe, Ni
can decay to neutrinos and produce their overabundance, assuming that the new Ni
CP violating phases are appropriate. This L-violation can then be transformed by17

sphalerons into B-violation, since they conserve B ´ L. However, this mechanism isdifficult to test.
• Coherent Motion of Scalar Fields (Affleck-Dine mechanism [113]): assuming
supersymmetry, the scalar superpartners carry B and L. Their potentials typically haveflat directions which allow to easily offset the field from the minimum. A large classicalvalue of such field (“coherent field”) can carry a large amount of B and later decay toquarks, producing observed B-asymmetry. This scenario is naturally realized primarily
within the context of supersymmetric theories and is not very viable otherwise.

Finally, it is important to stress that baryon number-violating processes like proton
decay will affect the ultimate fate of the Universe [10]. As quantum gravity effects cancatalyze proton decay, the result will affect cosmology on the larger cosmological time scales,
during the epoch of t ě 1015´35 years. The reactions like p Ñ e` π 0 will power stars (e.g.
neutron stars, white dwarfs) through the processes likep ` e´ Ñ γ ` γ ` γ ` γ ,(1.24)

with one pair of γs coming from e` ` e´ annihilation and another from π 0 Ñ γ ` γ. The
precise determination of the cosmological era when such processes become significant depends
on the details of the relevant nucleon decay lifetime calculation.

1.2

Experimental Searches for Nucleon Decay

The experimental considerations of nucleon decay began with Goldhaber 13 in 1954 [114],
searching for fission products of thorium (Th232 ) and placing a limit on proton’s lifetimeof τp ą 1020 years. Here, one looks for isotope production due to nucleon decay. Thisis a very general search technique and does not assume any particular decay mode. At
the same time, direct searches for proton decay were conducted by Cowan, Goldhaberand Reines [115]. These were done using scintillator as a source and the detector. Suchsearches are less general and rely on the decay products coming from specific channels.
Many searches have followed, eventually leading up to large underground experiments based
13

Even before the actual experiments in 1950s, Goldhaber proposed a heuristic argument that proton
lifetime should be ą 1016 years, based on „ 1029 protons in human body and a human lifetime span of„ 50 ´ 100 years.

18

on iron calorimeters and water Cherenkov detectors. The iron calorimeters offer great position and energy resolution, but they are expensive and consequently suffer from limited
fiducial volume, which for nucleon decay is vital and that is made up in water Cherenkov
detectors.
Year
1954
1954
1958
1958
1960
1962
1965
1967
1974
1974
1977
1977
1981
1982
1986
1987
1989
1991
1999
2012

Reference
Goldhaber [114]Reines, Cowan and Goldhaber [115]Reines, Cowan and Kruse [116]Flerov et. al. [117]
Backenstoss et. al. [118]
Giamati and Reines [119]Kropp and Reines [120]Gurr et. al. [121]Bergamasco and Picchi [122]Reines and Crouch [123]Steinberg and Evans [124]Fireman [125]Cherry et. al. (Homestake) [126]Battistoni et. al. (NUSEX) [127]
Krishnaswamy et. al. (KGF) [128]Bartelt et. al. (Soudan) [129]Hirata et. al. (Kamiokande) [79]Berger et. al. (Frejus) [130]McGrew et. al. (IMB) [131]
Nishino et. al. (Super-Kamiokande) [132]Lifetime Limit2 ˆ 1020 yrs.
1 ˆ 1022 yrs.
4 ˆ 1023 yrs.
2 ˆ 1023 yrs.
3 ˆ 1026 yrs.
7 ˆ 1027 yrs.
4 ˆ 1028 yrs.
8 ˆ 1029 yrs.
1 ˆ 1029 yrs.
2 ˆ 1030 yrs.
2 ˆ 1025 yrs.
2 ˆ 1026 yrs.
3 ˆ 1030 yrs.
2 ˆ 1031 yrs.
4 ˆ 1031 yrs.
2 ˆ 1030 yrs.
3 ˆ 1032 yrs.
2 ˆ 1032 yrs.
8 ˆ 1032 yrs.
8 ˆ 1033 yrs.

TechniqueFI (Th232 )SCSC
FI (Th232 )
SC
SC
SC
SC
SC
SC
FI (Te130 )FI (K39 )WC
FGC (Fe26 )FGC (Fe26 )FGC (Fe26 )WC
FGC (Fe26 )WC
WC

Table 1.2: Proton decay searches. We have denoted their operating principle: search forfission fragments from radioactive ore (FI), liquid scintillator (SC), water Cherenkov (WC)and fine-grained calorimeters (FGC).

Many nucleon decay channels have been searched for, including n ´ n and dinucleon
np, nn, pp decays. However, no convincing evidence for nucleon decay has been obtained
throughout the experiments. We have summarized in Table 1.2 their results, including
the operating technique and the best obtained nucleon decay limit. Stringent experimental
constraints also exist on other processes which could lead to baryon number violation, suchas proton decay-catalysis from the Q-balls [133] and the GUT monopoles [134].

19

1.3

Overview of the Analysis

As already discussed, the simplest unification scenarios based on minimal non-SUSY
and (TeV-)SUSY SU p5q have been strongly constrained by the experimental lifetime limits
on p Ñ e` π 0 [77, 78, 79, 80] and p Ñ ν̄K ` [91], respectively. Additionally, no signs of(TeV-)SUSY have been observed at the Large Hadron Collider (LHC) [67, 68]. Hence, there
is reinvigorated interest in alternative possible signatures.
After SU p5q, the next logical GUT to study is SOp10q, which is very well theoreticallymotivated and arguably more appealing than SU p5q. However, due to a large number ofpossible constructions, there are not very many concrete universal predictions from SOp10q,
as we have discussed in Section 1.1.2. In the schemes of Ref. [135, 136], trilepton decay
channels p Ñ e` νν and p Ñ µ` νν could become significant within the Pati-Salam SU p4qC ˆ
SU p2qL ˆ SU p2qR [137] model coming from spontaneous breaking of SOp10q. Signatures ofthis breaking pattern have not been very well experimentally explored. Additionally, thesechannels present unusual decay patterns. We describe in more detail a possible setup andhow these channels can arise with significant branching ratios in Appendix A. Assuming
trilepton mediation, these scenarios predict proton lifetime of around 1030´33 years. On theother hand, the old experimental limits are 1.7 ˆ 1031 years from IMB [131] and 2.1 ˆ 1031
years from Frejus [138], for p Ñ e` νν and p Ñ µ` νν, respectively. Hence, searching forthese modes in the current state of the art-experiments would allow to restrict a significant
portion of the allowed parameter space for such models. Additionally, observation of thesechannels, coupled with non-observation of p Ñ e` π 0 , may allow to differentiate between PS
and its SOp10q embedding [135]. Depending on the combination of neutrinos/anti-neutrinos
that are present in the final state, the channels also have |∆pB ´ Lq| “ 2, unusual for single
nucleon decay. In Ref. [139] it has been suggested that observation of trilepton modes canhave favorable implications for baryogenesis. Interestingly, the proton decay through thesechannels was offered as a possible explanation [140, 141] of the observed atmospheric neutrino
flavor “anomaly” [142, 143], prior to the discovery of the neutrino oscillations [17].
In this work, we will describe a search for the trilepton decay channels using the best
available instrument, the largest underground water Cherenkov detector, Super-K [22]. Because neutrinos are invisible within Super-K, one cannot reconstruct the invariant mass andmomentum of the parent nucleon from the final state particles. Hence, the only signatureof the above two modes is a charged lepton, which produces a showering or a non-showering
single Cherenkov ring within the detector. Since these signatures are also common for a class20

of other channels, we will take this as an opportunity to do a broad search for multiple decay
modes, which is important due to a large theoretical uncertainty in predictions, as discussed
in Section 1.1.2. In addition to the trilepton channels, we will consider two general 2-bodydecays p Ñ e` X and p Ñ µ` X, where X is taken to be single unknown and invisible particle
that is assumed to be massless. Note, that our searches for p Ñ e` X and p Ñ µ` X are
distinct from the model-dependent inclusive analyses of Ref. [126, 144] that are listed in the
PDG [37] with the same name14 . We will also search for n Ñ νγ, with γ producing thesame signature as the e˘ in the detector. Though this radiative process is suppressed, it hasa clean signal and has been considered in the context of SUp5q [145], with some models [76]predicting a lifetime of 1038˘1 years.
In addition to the single nucleon decay channels, we will also look for dinucleon decay
modes which also produce the above signatures. While single nucleon ∆B “ 1 processes
have been in general well studied, dinucleon ∆B “ 2 channels also pose great interest.
These higher-dimensional processes can become significant in models that suppress proton
decay and could be connected to baryogenesis [146]. The disappearance ∆B “ 2 reactions,with invisible final state particles, have already been studied and no signal excess was observed [138, 147, 148]. We will thus focus on the dinucleon channels np Ñ e` ν, np Ñ µ` νand np Ñ τ ` ν, which violate baryon number by two units and violate lepton number byeither two or zero units, depending on neutrino/anti-neutrino final state. They can become
significant in models with an extended Higgs sector [146, 149], which could be also consideredin the context of GUTs [150]. It is worthwhile to note, that τ final states cannot occur insingle nucleon decay due to its large mass of mτ « 1.777 GeV. On the other hand, it canappear within dinucleon decay, as has been stressed in Ref. [151]. The process np Ñ τ ` ν has
in fact never been experimentally studied before and in addition to the electron and muonchannel searches we present the first search in the τ channel.
Since the signal signatures for all of the above channels are single Cherenkov rings, thesignal lies on top of a large amount of background coming from atmospheric neutrinos, as will
be discussed later. Previous analyses in older experiments performed the search by using acounting method within some selected signal region. On the other hand, for all our searches,
we will be using a spectral χ2 fit15 [154]. This will allow us to discriminate between thesignal and the background distribution shapes and thus significantly improve the sensitivityof the analyses. This is particularly relevant for trilepton and np Ñ τ ` ν channels, since their
14

Inclusive analyses were done by assuming specific branching ratios of SU p5q and then adding up all of
the contributions from the various decay channels, such as p Ñ e` π 0 , weighted by the branching ratios.
15
The fit is also used in the Super-Kamiokande neutrino oscillation [152] and dark matter analyses [153].

21

signal comes from 3-body decays and is thus spread out.
The main analyses presented within this Thesis accompany and describe in more detail the published results of Ref. [155] and Ref. [156]. The 3-body spectrum approximation
method discussed in Appendix B follows the published results of Ref. [157].

22

Chapter 2
The Super-Kamiokande Experiment
The Super-Kamiokande experiment operates by detecting the Cherenkov radiation photons produced within the water-based detector. The emitted light produces ring-like patternson the detector walls. The timing, amount of charge in the photomultiplier tubes (PMTs)of the detector and the shape of these patterns gives information about the kinds of particles that appear within the detector. Below, we describe the physics mechanism behind
Cherenkov radiation. Then, we provide an overview of the experimental apparatus. A comprehensive description of the detector and its calibration can be found in Ref. [22] andRef. [158], respectively.

2.1

Cherenkov Radiation

Cherenkov radiation [159] occurs when a charged particle traverses a dielectric medium
with a speed pvp q greater than the phase velocity of light pc{nq in the same medium, givenbycc ą vp ą .
(2.1)
nHere, c is the velocity of light in vacuum and n is the refraction index of the medium (n “ 1.33
for water at 20˝ and λ “ 580 nm). The electrically polarized medium then de-excites andcoherently radiates back, resulting in a cone-like pattern of emitted light with respect tothe particle’s trajectory. A schematic diagram for this effect is displayed in Figure 2.1. The

23

Figure 2.1: Schematic diagram for the emitted cone-like pattern of light resulting fromCherenkov radiation. The direction of the charged particle (red arrow) and the emitted
light (blue arrows) is shown.
cone’s opening angle θC is given bydcos θC “

1
1
“
nβ
n

1`

m2
,
p2

(2.2)

where β “ v{c and m, p are the mass and momentum of the charged particle, respectively.
From Equation (2.2), one can deduce the momentum threshold ppthr. q to emit the Cherenkov
light (at θC “ 0q in a medium of refraction index n asmpthr. “ ? 2
.
n ´1

(2.3)

Hence, in water, the Cherenkov radiation threshold of the electrons pe˘ q, muons pµ˘ q,charged pions pπ ˘ q and the kaons (K ˘ ) is 0.58 MeV/c, 121 MeV/c, 159 MeV/c and 563
MeV/c, respectively. For an ultra-relativistic particle (with β Ñ 1) traversing water, theCherenkov angle for the generated cone of light will approach θ „ 42˝ . The emitted spectra
has a wavelength dependence and is described by
´
d2 N
1 ¯1
“ 2πα 1 ´,
dxdλpnβq2 λ2(2.4)

where dN , α, dλ and dx correspond to the number of emitted photons, fine structure constant, differential unit of the wavelength and the distance, respectively.

24

2.2

Detector Description

2.2.1

DesignThe Super-Kamiokande experiment is located in a zinc mine at Kamioka, Gifu Prefecture, Japan. Being beneath a „ 1000 m rock overburden (2700 m water equivalent), allows forstrong shielding of the cosmic ray muon background, reducing its flux to 6ˆ10´8 cm´2 s´1 sr´1
(five orders of magnitude compared to the Earth’s surface). A schematic overview of theexperiment can be found in Figure 2.2. The Super-Kamiokande detector consists of a cylin-

Figure 2.2: Overview of the Super-Kamiokande experiment. From Ref. [22].
drical stainless steel tank (39.3 m in diameter and 41.4 m in height). It is separated into
three regions, the inner detector (ID), dead space and the outer detector (OD). The innerdetector is comprised of 32 kilotons of water and a size of 36.2 m in height and 33.8 m indiameter. The ID walls are covered with 20-inch (in diameter) 11,146 uniformly distributed
and inward-facing PMTs, giving detector a 40% photo-coverage. The dead space is a 0.55 mregion between the ID and the OD and contains support structures. The OD fully encloses
ID with „ 2 m thickness of water and contains 1,885 outward-facing 8-inch (in diameter)
PMTs. The inner detector walls are lined with an opaque covering to improve signal discrimination. For the OD, a reflective material is used to improve sensitivity, allowing tobetter reject noise/background. Additionally, Helmholtz coils are used to reduce the Earth’s
geomagnetic field effect, which can disturb the PMT electronics, from 450 Ñ 50 mG. The
purpose of the outer detector is to act as an active veto against background as well as ashield from radiation coming from the surrounding rocks.
25

2.2.2

Photomultiplier Tubes

The custom design of the SK PMTs maximizes their photo-sensitivity. The quantum
efficiency, overlaid with the wave-length dependent spectrum of the Cherenkov radiationin water, as well as the single photo-electron response of the ID PMTs can be found in
Figure 2.3. The custom PMTs have their quantum efficiency maximized around „ 400 nm,

Figure 2.3: [left] Inner detector PMT quantum efficiency as a function of wavelength.
[right] Single photo-electron PMT analog-to-digital converter (ADC) response, with thepeak around zero being dark noise. From Ref. [22].
which also corresponds to the peak light yield for the Cherenkov radiation spectrum in waterdescribed by Equation (2.4).

2.2.3

Water and Air Purification Systems

The SK detector water comes from abundant amounts of spring water in the mine. It ispassed through the water filtration system, depicted on Figure 2.4, which improves the purity
and thus ensures a high attenuation length of around 100 m at λ “ 420 nm, essential for highsensitivity of the physics analyses. Additionally, the filtration system removes radioactive
radon (Rn), which serves as a background for solar neutrino oscillation analysis („ MeV
energy range). The detector water is continuously circulated through the filtration system
at a rate of „ 35 ton/hour. The purification system allows to reduce the number of particles
larger than 0.2 µm to about 6 particles/cm3 . The resistivity of the water is also improved
from 11 MΩ ¨ cm Ñ 18.2 MΩ ¨ cm, approaching the chemical limit.
Additionally, to reduce radon concentration in the experimental area, air is continuously
pumped from outside the mine. This allows to keep the radon concentration around 20 ´ 30
mBq/m3 inside the top of the SK tank and „ 100 Bq/m3 at the SK dome outside the26

Figure 2.4: Overview of the Super-Kamiokande water purification system. From Ref. [22].
tank (compared to 100 ´ 3000 Bq/m3 in the mine). To ensure that the purified water in
the SK detector stays radon free, a separate air-filtration system pumps radon-purified air(Rn concentration 3 mBq/m3 ) right above the water within the detector. The air-filtration
system is depicted on Figure 2.5.

Figure 2.5: Overview of the Super-Kamiokande air purification system. From Ref. [22].

2.2.4Electronics and Data Acquisition (DAQ) System

A schematic for the ID data acquisition system (DAQ) is presented in Figure 2.6. Signals
from ID PMTs are collected by the analog-timing-modules (ATM), located inside Tristan
KEK Online (TKO) crates, which are responsible for recording and digitizing the arrival
time and the integrated charge for each PMT using an analog-to-digital converter (ADC).
The full timing and the integrated charge ranges are 1.6 µs (0.3 ns resolution) and 600 pC
27

« 300 photo-electrons (0.2 pC resolution), respectively. There are 12 PMTs per ATM and 20
ATMs per TKO crate. Eatch TKO crate also contains a go/no-go (GONG) trigger control
module and a super-control-header (SCH). The SCH sends information from the ATM to thesuper-memory-partner (SMP), located within a VME crate. The SMP acts as an information
storage buffer, which is to be read out and processed later.

Figure 2.6: Schematic of the data acquisition system for the inner detector (arrows show
data flow). From Ref. [22].
The electronics system for the outer detector consists of Charge-To-Timing Converters(QTCs), which read out the OD PMT output signal and generate a timing signal proportional
to the total charge. If a 0.25 photo-electron (p.e.) threshold is reached, the signal is thendigitized by the Timing-To-Digital Converter (TDC) and merged with the ID data. The
signal range of the digitized output is 16 µs (0.5 ns resolution).
Finally, the event hardware trigger system checks the combined total hits from ID ATMs
(or OD QTCs) within a 200 ns window. If the number of hits exceeds 29 (19) PMTs for ID(OD), a global trigger signal is issued to the Trigger Module (TRG). The TRG then records
the timing and type of the event, generating a global event trigger signal. This initiates thedata to be read out and processed.

28

2.3

Detector Calibration

2.3.1

Relative PMT Gain

To ensure that the PMTs produce a uniform response, their charge gain is measured
and each of the high voltages supplying the PMT is individually adjusted. The measurement
is performed using a xenon (Xe)-powered scintillator ball, adjusted to give the output at 440
nm, which is near the maximum of the PMT’s quantum efficiency. The schematic for thissetup is displayed on Figure 2.7. The relative gain Gi of the i´th PMT is obtained by

Figure 2.7: Setup for the relative gain calibration system [left] and the relative gain
distribution of the PMTs after recalibration [right]. From Ref. [22].

”l ı
Qi ¯
i
¨ li ¨ Exp,
Gi “
Q0 f pθq
L
´

(2.5)

where Qi is the observed PMT charge, Q0 is a constant, L is the effective light attenuation
length, li is the distance to the light source and f pθq is the relative PMT photo-sensitivity
as a function of the light incident angle.

2.3.2

Relative PMT Timing

Similarly, a uniform PMT timing response is also important. A diffuser ball lighted by
the nitrogen laser is used to calibrate the PMT timing response. The laser emits 3 ns lightpulses at 337 nm, which are shifted to 384 nm by a dye laser module. This is done for several
different positions. PMT timing vs. charge map (TQ map) is constructed for each PMT and29

is later used within MC and data reconstruction. The calibration setup as well as a sample
TQ map for a single PMT are shown in Figure 2.8.

Figure 2.8: Setup for the relative timing calibration system [left] and a sample TQ map fora single PMT [right]. From Ref. [22].

2.3.3

Absolute PMT Gain

To obtain the PMT charge corresponding to 1 p.e., a californium (Cf252 ) source surrounded by nickel (Ni) wire is used. Neutrons produced by spontaneous fission of Cf areabsorbed by Ni, which in turn re-emits γ-rays (mostly at 9 MeV). The observed EM shower
deposits no more than 1 p.e. per PMT. For each PMT, charge distribution corresponding to
1 p.e. is determined. The mean value of the charge distribution peaks is found to be 2.055 pC.
The calibration system setup as well as the charge distribution of a typical PMT are shownin Figure 2.9.

2.3.4

Water Transparency

Precise measurements of the water transparency are important for obtaining correct
MC simulations. The water transparency measurements at SK are performed with threedifferent techniques: using diffuser ball, nitrogen laser beam as well as cosmic ray muons.

30

Figure 2.9: Schematic view of the absolute gain calibration system setup [left] as well as
the charge distribution of a sample PMT [right]. From Ref. [22].
Diffuser Ball
The attenuation length can be defined as a sum of absorption and scattering coefficientsL“

1.
αabs ` αscat(2.6)

To measure it, a diffuser ball, powered by titanium-sapphire (Ti:Sa) laser pumped byneodymium-doped yttrium aluminium garnet (Nd:YAG) laser, is placed in the detector andthe emitted light that passes through water is monitored by a camera on top. The setup
is shown on Figure 2.10. After performing several measurements at varying depths andFigure 2.10: Setup for the water transparency measurements using diffuser ball andcamera. From Ref. [22].

31wavelengths, one can calculate the attenuation length as a function of wavelength using
log

”I

CCD

Ilaser

ı
“A´

ld
,
Lpλq

(2.7)

where A is a constant, ICCD is the intensity of light at the charged-coupled device (CCD)
camera, Ilaser is the intensity of light at the monitoring PMT and ld is the depth of the
diffuser ball.

Laser Beam
To measure the absorption and scattering coefficients separately, a nitrogen (N) laser
beam is injected from the top of the detector. Each laser, wavelength of 337 nm, 371 nm, 400
nm and 420 nm, fires every 6 seconds during normal data taking. Since the laser shines downward, the measurements of PMT response obtained from the top of the detector correspondto backward light scattering. The setup is depicted in Figure 2.11, along with the measured
attenuation coefficient overlaid with the theoretical scattering and absorption models usedwithin the SK MC.

Figure 2.11: Schematic view of the water calibration laser injection setup [left] and themeasured attenuation length along with the theoretical predictions [right]. From Ref. [22].

Cosmic Ray Muons
While the previous two methods were intrusive, cosmic ray muons provide continuous
detector monitoring throughout regular data-taking periods. We select only the downward
going muons. The energetic muons typically deposit nearly a constant 2 MeV/cm amount of32

energy along their track. The attenuation length can then be inferred from the PMT charge
response function
” lı
f pθq
¨ Exp ´,
(2.8)
Q “ Q0 ¨lLwhere Q0 is a constant, f pθq is the PMT acceptance, l is the photon path length and L isthe attenuation length. The results for sample data are shown in Figure 2.12, which yields
attenuation length of „ 100 m. Additionally, time variation of the attenuation length, whichis accounted for in the SK MC, is displayed.

Figure 2.12: Results for logrQl{f pθqs as a function of l for sample data [left] and watertransparency variation over time [right]. From Ref. [22].

2.3.5Energy Calibration

For the typical nucleon decay searches like p Ñ e` π 0 , the analysis is performed byreconstructing the invariant mass and the momentum of the parent particle. For our spectral
searches, we will be looking at the momentum distributions of the final state particles.
In either case, energy calibration of the detector provides one of the dominant systematic
uncertainties for nucleon decay studies. The energy calibration is performed in SK by four
independent methods, covering the energy range from „ 1 MeV/c to 10 GeV/c. These are:decay electrons from stopping muons, π 0 events and the low/high energy stopping muons
themselves. The energy calibration is continuously monitored and the uniformity throughout
the detector is also accounted for.

Decay Electrons
The energy distribution of the decay electrons produces a well known µ Ñ e` νν 3-bodydecay Michel spectrum up to „ 50 MeV (see also Appendix B). The mean of the Michel
33spectrum is compared between the data and the MC, as depicted on Figure 2.13 for the SK-I
phase of the detector. The decay electrons are selected by the following criteria:
1. time from stopping muon event is 2.0 ´ 8.0 µs
2. number of hit PMTs within 50 ns window is ą 60 (for SK-I)
3. goodness of vertex fit is ą 0.5
4. reconstructed vertex is ą 2 m from the ID wall
The p2q criterion rejects 6 MeV γ-rays from µ´ capture on the nucleons. In MC, measured
cosmic ray µ` {µ´ charged muon ratio of 1.37 is used and the µ´ capture is also accountedfor.

Figure 2.13: Momentum distribution of the stopping muon decay electrons [left] and thereconstructed invariant π 0 mass [right] for SK-I. MC simulation is depicted by continuousline [left] and boxes [right]. From Ref. [22].

Reconstructed Pions
Pions that are produced from the atmospheric neutrino interactions can also be used
for energy calibration. As π 0 immediately decays via π 0 Ñ γγ, the pion invariant mass Mπ0
can be reconstructed from momenta Pγ1 and Pγ2 of the two photons via
Mπ0 “b2Pγ1 Pγ2 p1 ´ cos θq ,where θ is the opening angle between γ1 and γ2 . These events are selected as follows:
34

(2.9)

1. 2 showering rings
2. 0 decay-electrons
3. reconstructed vertex is ą 2 m from the ID wall
where the (2) criterion rejects signal contamination from π ˘ π 0 and µ˘ π 0 events. The reconstructed pion mass is then compared with MC predictions, as depicted on Figure 2.13, witha clear peak visible around the true pion mass of 135 MeV/c2 .

Low/High Energy Stopping Muons
Recall from Equation (2.2) that the Cherenkov angle depends on the charge particle
momentum. One can use the Cherenkov angle of the downward going low-energy (ă 400
MeV/c) stopping muons to infer their momenta and then compare with the reconstructedvalue. The event selection for this study is the following:
1. number of p.e. is ă 1500 (for SK-I), i.e. the muon momentum is ă 380 MeV/c
2. one cluster of hit OD PMTs
3. entrance point is on the top wall
4. downward going (cos θzenith ą 0.9)
5. 1 decay-electron (muon event)The averaged momentum ratio Pp.e. {Pθ of the observed p.e. and the momentum inferredfrom Cherenkov angle are then compared between the data and the MC.
Since high energy stopping muons deposit about 2 MeV/cm of energy throughout theirpath, path length can be used to estimate the momentum from 1 to 10 GeV/c independentlyof the p.e.-based momentum measurement. The event selection for this study is the following:
1. entrance point is on the top wall
2. downward going (cos θzenith ą 0.94)
3. 1 decay-electron (muon event)35

4. reconstructed muon path length ą 7 m
To ensure that energy calibration is precisely accounted for, detector uniformity as
well as time variation of the uncertainty is also studied. The combined uncertainty fromenergy calibration for SK-I is shown in Figure 2.14. The total combined energy calibration
uncertainty for each SK phase, SK I-IV, is within 3% for high energies (Á 100 MeV) andă 1% for low energies („ 10 ´ 100 MeV).

Figure 2.14: Results for SK-I energy calibration. From Ref. [22].

2.4

Data Taking Periods

The Super-Kamiokande has collected data during four different experimental phases:
SK-I (May 1996-Jul. 2001, 1489.2 live days), SK-II (Jan. 2003-Oct. 2005, 798.6 live days),
SK-III (Sep. 2006-Aug. 2008, 518.1 live days) and the ongoing SK-IV experiment (Sep. 2008Oct. 2013, 1632.3 live days), corresponding to a combined exposure of 273.4 kton¨years. In
the SK-II phase of the detector the photo-coverage was only 19%, due to a PMT implosion
accident. The full photo-coverage was restored in SK-III and electronics were upgraded in
SK-IV, allowing for a better decay-electron detection efficiency.

36

Chapter 3
Monte Carlo Simulations

3.1

Signal: Nucleon Decay

3.1.1

Assumptions

Nucleon decay generation is typically done by randomly generating the final state particles of the decay in the parent nucleon’s center of mass frame, taking into account the 4momentum (energy and momentum) conservation. The final state particles are then Lorentz
boosted to the lab frame, which allows to account for the Fermi motion if the parent nucleons are bound inside the nucleus (i.e. within 16 O). Dinucleon decay is generated similarly,
but now one obtains a 2-body decaying system (nn, np, pp). The nucleon decay MC is spininsensitive.
The above approach is a general model independent technique for rare searches, which
does not depend on the specifics of how a decay is actually mediated and implicitly assumes
a flat phase-space distribution (see Appendix B). This has been a standard approach forthe past experimental nucleon decay searches (e.g. at IMB [131] and Frejus [138]). In thecase of the typical 2-body decay channel, such as p Ñ e` π 0 , the energy distribution of the
final state particles is fully determined by kinematics. This is so, since each of the resulting
particles takes „ mn {2 of energy (free nucleon decay) and the above approach works well.
On the other, for n ą 2 body decays, such as p Ñ e` νν or p Ñ µ` νν, the energy spectra is
not uniquely determined by the 4-momentum conservation and has a distribution from 0 to
mn {2 « 470 MeV and a mean of „ 313 MeV (free nucleon decay). In such a scenario, the37

energy dependence of the matrix element, which is determined by the mediating mechanism,
can play a role in n ą 2 decays. While we want to keep the searches as model independent as
possible, it is also important to understand the degree of approximation used in generating
the trilepton nucleon decays with a flat phase-space distribution, which experiments in thepast have just assumed.
Using effective 4-fermion interaction formalism of the Fermi theory and employing a relatively general set of assumptions, one can approximate [157] the decay spectra of p Ñ e` ννand p Ñ µ` νν. The approximation method is discussed in Appendix B. We have used thisresult to confirm that the e` and µ` energy distributions, which we will use in the nucleon
decay analysis and which are generated using the standard SK nucleon decay MC that assumes a flat phase-space distribution, are reasonable. In fact, the technique outlined in Ref.
[157] can be useful for other decay searches that satisfy the above assumptions, such asp Ñ e` e´ e` .
We note that in the nucleon decay analyses it is generally assumed that the bound andthe free nucleons have the same decay rate Γf ree « Γbound . The phase space for the two are
different, due to nuclear environment, which in principle affects the bound nucleon lifetimeτbound “ 1{Γbound . This effect, however, is taken to be negligible.

3.1.2

Nuclear Effects

Fermi Motion and Nucleon Decay Position
If a nucleon decay occurs inside oxygen 16 O, the bound parent nucleon has Fermi momentum. On the other hand, if the nucleon originates from hydrogen H, it is free (stationary).
All of the dinucleon decays originate from the bound nucleons of 16 O. In the SK nucleon
decay MC simulations, the Fermi momentum of nucleon decay is based on a spectral function
found by the e´12 C scattering experiments [162]. The S- and P -state momenta based on theoretical calculations and the experimental data are shown on Figure 3.1. On the other hand,atmospheric neutrino MC uses Fermi momentum calculated from the relativistic Fermi gas
model, as we will discuss later. The difference between the two models is used to estimate
the systematic uncertainty.
The position of the decaying nucleon is determined using the Woods-Saxon model [163],38

Figure 3.1: Theoretical calculations (solid and dashed lines) [161] for the 1p3{2 proton state(left) and 1s1{2 proton state (right) of 12 C, compared with experimental data. From
Ref. [162].
according to the following distribution
ρn prq “

ρp0q
Z
ˆ
,
A 1 ` Expr r ´ a s
b

(3.1)

where ρn prq is the nuclear density as a function of radial distance r from the nuclear center,ρ0 “ 0.48 m3π , a “ 2.69 fm is the maximal nuclear radius for oxygen, b “ 0.41 fm is the“surface thickness” of the oxygen nucleus and Z and A are the atomic and the mass numbers1 .
From the above, one can determine the Fermi surface momentum using
pF prq “

´3
2

¯1{3
π 2 ρn prq
.

(3.2)

The Fermi surface momentum is used for studying nuclear interactions of the mesons, which
are important in nucleon decay searches such as p Ñ l ` meson (see Ref. [132]). The
same technique is used for neutrino meson-production interactions in the atmospheric MC
simulations, as discussed later.

Binding Energy
The nuclear binding energy is taken into account by modifying the invariant mass ofthe nucleon according to Mn˚ “ Mn ´ Eb , where Mn˚ is the modified nucleon mass, Mn
is the rest nucleon mass and Eb is the binding energy. Binding energy Eb is assumed to1

For 16 O, Z “ 8 and A “ 16.

39have a random Gaussian distribution with a mean and the standard deviation of pµ, σq “
p39.0 MeV, 10.2 MeVq for the S-state and pµ, σq “ p15.5 MeV, 3.82 MeVq for the P -state.
The results for proton in 16 O are shown in Figure 3.2.

Figure 3.2: Proton invariant mass in 16 O, modified by nuclear binding energy. From
Ref. [164].

Correlated Decay
In the nuclear environment it is predicted that due to overlapping wave functions of
the nucleons, an additional “spectator” nucleon becomes involved in the decay and thus
modifies the invariant mass of the decaying system. Pictorially, this effect can be viewed
as several bound nucleons, which have Fermi motion, colliding and redistributing momenta.
In case this happens, a single nucleon decay effectively becomes a 2-body system decay anddinucleon decay becomes a 3-body system decay. Approximately 10% of nucleon decays are
predicted to experience this effect [165], which SK MC simulations take into account. We
consider a 100% systematic uncertainty for correlated decay, since it is not well studied.

Nuclear De-excitation
As the nucleon decays, the remaining nucleus (e.g. 16 O Ñ 15 N) can be left in an excited state, which then de-excites through a γ-ray or a nucleon emission. These effects were
calculated in Ref. [166] (more recently in Ref. [167]), as summarized in Table 3.1. They areincluded in SK MC simulations. For the nuclear decays to the states with a de-excitation
nucleon, no γ-ray is generated. The de-excitation γs are particularly important for somenucleon decay analyses, such as p Ñ νK ` [168] and n Ñ ννν [169].
40

StateEnergy of γp3{26.3 MeVp3{29.9 MeVs1{27.0 MeV
others
3.5 MeVp{n emission ground state -Probability
41%
3%
4%
16%
11%
25%

Table 3.1: Summary of nuclear de-excitation after nucleon decay [166].

3.1.3

Generation of τ and X

Since τ lepton decay is not included in the GEANT-3 [170] package that is used for the
SK detector simulation (see Section 3.3), we use the dedicated TAUOLA [171] package fordecaying the τ in the np Ñ τ ` ν search2 . For the np Ñ τ ` ν, we will be interested only inthe leptonic τ channels τ Ñ e` νν and τ Ñ µ` νν. We have confirmed that the e` and µ`
energy spectra from these decays, for both stationary and moving τ , agrees well with thetheoretical predictions of the pV ´ Aq SM theory [173], using the calculations of Ref. [157](see Appendix B).
For the p Ñ e` X and p Ñ µ` X decays, X is assumed to be a single unknown masslessand invisible particle, which by spin conservation is a boson. For the analysis, X is generated
as a neutrino, since SK nucleon decay MC is spin insensitive and the final state neutrino isalso effectively massless and invisible.

3.2

Background: Atmospheric Neutrinos

The main background for nucleon decay searches comes from interactions of atmospheric
neutrinos, which are produced by cosmic rays. The cosmic rays mainly consist of protons
and α-particles p„ 99%q, which strike the atmosphere. The isotropically distributed cosmic
rays interact with the air molecule nuclei, producing copious amounts of pions and kaons.
The decay of charged pions π ˘ leads to an eventual production of two muon neutrinos (νµor ν µ ) and one electron neutrino (νe or ν e ). These processes result in a specific neutrino flux
composition at the surface of the Earth. The whole reaction chain is shown schematicallyon Figure 3.3. While most of the neutrinos pass through Earth (travelling 10 - 10,000 km
2

The τ decay is treated in the same manner within the SK ντ -appearance analysis [172].

41

Figure 3.3: Schematic view of cosmic ray interactions in the atmosphere. From Ref. [174].
distance) without interaction, some interact at Super-Kamiokande and a neutrino event rateof about „ 8 events/day is observed within the detector.

3.2.1

Neutrino Flux

Atmospheric neutrino flux at Super-K is modeled using calculations of Honda et. al.
(Honda flux) [175], which parametrize and fit experimental cosmic ray data, as shown on
Figure 3.4. These 3-dimensional calculations propagate the cosmic rays through the atmosphere and take into account the effects of geomagnetic field and solar wind, which can affectthe cosmic ray flux by up to a factor of 2 at 1 GeV and „10% at 10 GeV energy range.
Interactions with the air nuclei are treated using NUCRIN [188] and DPMJET-III [189]models, with the resulting secondary particle profile for mesons and kaons used for neutrino
flux computation. Neutrino flux calculations of G. Battistoni et. al. (Fluka flux) [190] and
G.D. Barr et. al. (Bartol flux) [191] are used for comparison with Honda flux, as shown
on Figure 3.4, to determine the systematic uncertainty. Flux calculations cover the neutrinoenergy ranging from 30 MeV to 3 TeV.

42

Figure 3.4: [left] Primary cosmic ray flux measurements compared with the Honda flux
model (solid line) for protons at solar minimum. The data are taken from Webber et. al.
[176] (crosses), LEAP [177] (upward triangles), WIZARD-MASS [178] (open circles),
CAPRICE-94 [179] (vertical diamonds), IMAX [180] (downward triangles), BESS [181](closed circles), AMS [182] (squares), Ryan et. al. [183] (horizontal diamonds), JACEE[184] (downward open triangles), Ivanenko et. al. [185] (upward open triangles), Kawamura
et. al. [186] (open squares) and RUNJOB [187] (open diamonds). [right] Atmospheric
neutrino flux at Super-Kamiokande as predicted by the Honda (solid line), Fluka (dashedline), and Bartol (dotted line) flux models. From Ref. [175].

3.2.2

Neutrino Interactions

At the SK site and within the surrounding rock, neutrino interactions have been simulated using the NEUT [192] model and cross-checked with the NUANCE [193] model. The
following interactions are calculated:
CC/NC (quasi-)elastic scattering:
CC/NC single-meson production:
CC/NC coherent-pion production:
CC/NC deep inelastic scattering (DIS):ν ` N Ñ l ` N1
ν ` N Ñ l ` N 1 ` mesonν `16 O Ñ l `16 O ` π ˘,0ν ` N Ñ l ` N 1 ` hadrons .

Here, ν represents an incoming neutrino or anti-neutrino of some flavor (e or µ), N and N 1
are the original and the outgoing nucleons (p or n), respectively, and l is an outgoing lepton.
The outgoing lepton l can be a charged lepton, if there is a charged current (CC) interaction,
or a neutrino, for the neutral current (NC) interaction. Reactions with negligible crosssections, such as neutrino-electron scattering, are omitted. Below we provide an overview
of the above processes. Since CC quasi-elastic (QE) scattering and single-pion production
provide the predominant majority of the background for our analyses, they will be discussed
in more detail. Further information regarding the neutrino interactions and the associated43systematic uncertainties can be found in Ref. [160, 194].

Elastic and Quasi-Elastic Scattering
These events typically result in a single Cherenkov ring pattern and can sometimes giveoff a recoiling proton above threshold. For a free proton scattering, the cross-section of thecharged current quasi-elastic interaction is calculated from the pV ´ Aq theory and is given
by Llewellyn-Smith [195] as
2 ıdσ νpνq
M 2 G2F cos2 θC ”
2
2 s´u
2 ps ´ uq q
Apq
q
˘
Bpq
q
,
“
`
Cpq
q
dq 2
8πEν2
M2
M4

(3.3)

where Eν is the neutrino energy, M is the target nucleon mass, GF is the Fermi constant,
θC is the Cabbibo angle, q is the 4-momentum transferred to the lepton and s and u are theMandelstam variables. The form factors Apq 2 q, Bpq 2 q and Cpq 2 q can be re-written in termsof axial-vector and vector form factors FV1 pq 2 q, FV2 pq 2 q and FA pq 2 q as well as the electricand magnetic form factors GE pq 2 q and GM pq 2 q, which are typically the ones used in theexperiments. Two additional parameters, the vector mass MV and the axial-vector mass
MA , are needed to be specified in order to fully determine the form factor structure. They
are taken to be MV “ 0.84 GeV and MA “ 1.2 GeV (see K2K [196] measurement).
For bound nucleon scattering (16 O nuclei), the calculation of Smith and Moniz [197]is used. Nucleons are treated using relativistic Fermi gas model with a flat momentum
distribution up to 225 MeV/c (Fermi surface momentum). Models of Ref. [198, 199] better account for the intermediate energy ranges and are used to determine the systematic
uncertainty. Pauli exclusion principle is enforced by requiring that scattered nucleon has
momentum higher than the Fermi surface momentum.
The cross-sections for the neutral current elastic scattering are estimated using thefollowing relations [200, 201]:σpνp Ñ νpq “ 0.153 ˆ σpνp Ñ e´ pq
σpνp Ñ νpq “ 0.218 ˆ σpνp Ñ e` pq
σpνn Ñ νnq “ 0.150 ˆ σpνp Ñ νpq
σpνn Ñ νnq “ 1.000 ˆ σpνp Ñ νpq
Comparison between data and NEUT simulations for ν and ν quasi-elastic scattering isshown on Figure 3.5.
44

Figure 3.5: Comparison of NEUT MC simulations (solid line) and data for the ν and ν
cross- sections of the charged current quasi-elastic scattering. The data from ANL [202],
Gargamelle [203, 204, 205], BNL [206], Serpukhov [207] and SKAT [208] are shown. From
Ref. [209].
Single-Meson Production
Single-meson production of π, K and η is simulated using the model of Rein and
Sehgal [210, 211]. The production happens through baryonic resonance via two steps:ν ` N Ñ l ` N˚
N˚ Ñ m ` N1
where m is a meson, N and N 1 are nucleons and N ˚ is the baryon resonance. If the decay width of baryon resonance is negligible, the differential cross-section of single meson
production is given by
1
1 ÿ
d2 σ
“
ˆ
|T pνN Ñ lNj˚ q|2 δpW 2 ´ Mj2 q ,dq 2 dEν
32πM Eν2 2 j,spin(3.4)

where M is the target nucleon mass, Eν is the neutrino energy, W is the invariant massof the hadronic system (intermediate baryon resonance), Mj is the baryon resonance mass
and T pνN Ñ lNj˚ q is the resonance production amplitude, calculated from the FeynmanKislinger-Ravndal model [212]. If the resonance width is non-negligible, δ-function in Equation (3.4) is substituted by the usual Breit-Wigner formulaδpW 2 ´ Mj2 q Ñ

1Γ
.
2π pW ´ Mj q2 ` Γ2 {4

45

(3.5)

For the above calculations, W was restricted to be ă 2 GeV/c2 (region above 2 GeV/c2 istreated in DIS) and MA is taken to be 1.2 GeV as for QE scattering. A total of 28 resonances
are considered in SK simulations. Angular distribution of the final state pions is found fromthe ∆p1232q resonance, while angular distribution of pions coming from other resonances is
set to be isotropic. The simulated π ` angular distribution is found to be in agreement withexperimental data from νp Ñ µ´ π ` p scattering [213]. Pion-less ∆ decay [214], where 20% of
events only have lepton and a nucleon, is also simulated. As before, Pauli exclusion principle
is taken into account by requiring that the scattered nucleon momentum is greater than the
Fermi surface momentum.

Coherent-Pion Production
If the incoming neutrino energy is small, it effectively “sees” the whole 16 O nucleus.
Due to the large mass, the nucleus is not affected much in the interaction and the resulting
pion has a distinctive forward scattering in the angular distribution. This process is also
simulated using the Rein and Sehgal model [211]. However, the K2K data [215] agrees
better with the model of Kartavtsev et. al. [216], which is used to determine the systematic
uncertainty.

DIS
If the incoming neutrino energy is high, the neutrino can fragment the nucleon it collides with. The DIS is simulated using GRV98 (Glück-Reya-Vogt) [217] parton distributionfunction. Here, one restricts W ą 1.3 GeV/c2 . In the region 1.3 GeV/c2 ă W ă 2.0 GeV/c2
only pions are considered as the outgoing mesons. Their multiplicity is required to be larger
than 1, such that the process does not overlap with the single-pion production, with themean multiplicity estimated from the Fermilab bubble chamber experiment [218] and its
forward-backward asymmetry further studied by the BEBC experiment [219]. For region
W ą 2 GeV/c2 , hadronic system kinematics are determined using PYTHIA/JETSET [220]package, which allows to also treat the K, η and other mesons.
The neutral current DIS cross-section is determined from the DIS charged current crosssection through the means of empirically determined relations [221, 222], similar to thequasi-elastic scattering.

46

Nuclear Effects
As the mesons are created inside 16 O, they may hadronically interact within the nucleus
itself via charge exchange, absorption or inelastic scattering. This is particularly relevant
for pions, which have large cross-section around Eν „ 1 GeV. Their position within the
nucleus is calculated by the Woods-Saxon model [163], which is also used for nucleon decay
MC. Cascade model is then used to simulate the interaction, which is determined by thethe mean free path found from Ref. [223]. Fermi motion and Pauli exclusion principle aretaken into account as for the other interactions. The nuclear effect simulations are verified
by the experiments using the pion´12 C scattering, pion´16 O and the pion photo-production(γ `12 C Ñ π ´ ` X) [224, 225]. Other mesons are also considered.
Nucleon re-scattering is also treated with a cascade model. Scattering cross-sections aretaken from experiment [226] and the pion production from ∆s is taken into account using
the isobar production model [227].
Full MC simulation and data comparison of neutrino cross-section is shown on Figure 3.6. Simulations are seen to describe the data well.

Figure 3.6: Total neutrino (a) [left] and anti-neutrino (b) [right] cross sections as a functionof energy. The calculated quasi-elastic scattering cross section is shown in the dashed line,that of single meson production appears in the dotted line, and the dash-dotted line showsdeep inelastic scattering. Data from several experiments are overlaid. From Ref. [160].

47

3.2.3

Neutrino Oscillations

The phenomena of neutrino oscillations [32] originates from the mismatch of the neutrino flavor and mass eigenstates and signifies that neutrinos have mass. This was establishedby Super-Kamiokande [17] in 1998 for atmospheric νµ Ø ντ neutrino oscillation. Similarly,
the solar neutrino νe Ø νµ pντ q oscillation was established by SNO [18] in 2001.
To take into account neutrino oscillations, the generated SK atmospheric neutrino MC
events are reweighed accordingly. The following oscillation parameters have been assumed forthe atmospheric MC used in our analysis [152, 228]: normal hierarchy, ∆m232 “ 2.1ˆ10´3 eV2 ,∆m221 “ 7.6 ˆ 10´5 eV2 sin2 2θ23 “ 1.0, sin2 2θ12 “ 0.84 and δCP “ 0. The final backgroundrates for each SK data-taking period are normalized by the observed total sub-GeV event
rate (see Chapter 5 for sub-GeV sample characteristics).

3.3

Detector

After the initial intra-nuclear interactions, propagation of particles through the detector,
Cherenkov radiation, PMT response and the electronics (including the dark noise) are all
taken into account by the SK detector simulation software SKDETSIM, which based on the
GEANT-3 [170] package.
For hadrons, interactions above 500 MeV are treated with GCALOR [229, 230]. Interactions below 500 MeV are done with a custom package [231] based on the π´16 O [232]and the π ´ p scattering data [233]. For details regarding how KL0 Ñ KS0 regeneration andKaon-nucleon interactions are taken into account see Ref. [234].
Cherenkov radiation is included by generating photons according to Section 2.1. Rayleighscattering, Mie scattering and absorption are all taken into account as shown on Figure 2.11.
For production of the upward-going muons and interactions within the surroundingrock (assumed to be sillicon-oxide (SiO2 ), with density of 2.65 g/cm3 ) are simulated with
NEUT. These processes are not relevant for our analysis.

48

Chapter 4
Event Reconstruction
The event reconstruction algorithm is applied to fully contained events after data reduction (see Chapter 5). The same algorithm is applied to both, the data and the MC. The
reconstruction procedure contains the following steps:
1. Vertex Fitting:the vertex position point is determined from maximizing a likelihood function basedon the PMT timing distributions; Cherenkov cone direction and the ring edges are
estimated.
2. Ring Counting:search for additional rings is performed using a likelihood method and the vertex andposition of the dominant ring (as determined in the previous step); the total numberof rings is estimated.
3. Particle Identification (PID):
a likelihood function, based on ring pattern and opening angle, is formed to classify
each ring as showering or e-like (for e˘ , γ) and non-showering or µ-like (for µ˘ , π ˘ ).
4. Momentum Determination:
the total ring charge within a 70˝ cone is converted to ring momentum, using MC
simulation and detector calibration data.
5. Ring Number Correction:“fake” low energy rings that overlap with primary rings are removed.

49

6. Decay Electron Search:
decay electrons from the primary events are obtained.
In addition the above standard algorithm, alternative and more specialized tools also exist,
such as the multi-vertex fitter (MVFIT). However, they are not relevant for the studies
presented within this work and thus will be omitted. Below, we provide an overview of eachalgorithm step. For a more comprehensive description see Ref. [164].

4.1

Vertex Fitting

The vertex fitting algorithm consists of three steps: vertex point fit, ring edge finding
and a refined vertex fit (TDC-fit).

4.1.1

Vertex Point Fit

Point fit estimates the vertex position. The fit samples various test vertices and, assuming that all photons originated simultaneously from a single point, maximizes the “vertex
goodness” function Gver. . The function depends on the subtracted photon time of flight(TOF) recorded PMT timings and is given by
Gver. “”pti ´ t0 q2 ı
1 ÿ
,
Exp ´
N i
2p1.5 ˆ σq2(4.1)

where N is the number of hit PMTs, ti is the TOF-subtracted timing of the ith PMT, t0
is a free parameter chosen to maximize Gver. and σ is the PMT resolution (2.5 ns). Ring
direction is also roughly estimated in this step, by summing up the detected p.e. vector.

4.1.2

Ring Edge Finding

Using information from the previous fit portion, the Cherenkov opening angle and thering edge position are obtained. A distribution Qpθq is built from detected p.e.s as a functionof Cherenkov opening angle θ. The ring edge is then estimated where the second derivate ofQpθq vanishes.

50

Various directions and opening angles are tested to maximize the “ring direction goodness” function
żθC

Qpθqdθ
Gdir “0sin θCˆ Expr´pθC ´ θexp q2
s,
2σa2

(4.2)

where θC is the test opening angle, Qpθq is the charge distribution relative to the test
direction, θexp is the expected opening angle (θexp “ 42˝ for γ) of the ring and σa is its
resolution. The initial test direction is found by the previous step.

4.1.3

Refined Vertex Fit

The TDC-fit re-computes the vertex position by also utilizing directional and opening
angle information found above. Unlike the vertex point fit, which assumed that all photons
originated simultaneously at a single vertex point, TDC-fit takes into account that photons originated along the charged particle track length to get the residual PMT timing andmaximizes the corresponding likelihood function. Scattered light is also considered here.
An additional fit (MS-Fit), which uses particle ID (described later), further improveson the vertex position identification for the single-ring events.

4.2

Ring Counting

After the main ring has been identified, additional rings (up to 5) are searched for using
Hough transform [235] pattern recognition technique and the likelihood method. With theHough transform, a “virtual ring” pattern (42˝ Cherenkov angle) is considered around each
hit PMT. The “virtual rings” are weighted by the charges from the hit PMTs and a “real”
ring is identified from the peak of the overlapping “virtual” distributions. The concept is
illustrated in Figure 4.1. In practice, instead of virtual circles, the method is implemented byusing charge distribution function for each hit PMT and the a ring candidate is identified by
the peak in the overlapping distributions. A log-likelihood method is then used to determine
if a true ring has been found.

51

Figure 4.1: The basic concept of the Hough transform applied to Cherenkov rings. The
charge in a PMT is distributed along a circle corresponding to a 42˝ Cherenkov opening
angle as seen from the event vertex. The circles of the hit PMTs from the same Cherenkov
ring will then overlap in the center of the actual ring. From Ref. [236].

4.3

Particle Identification

Particle identification in SK classifies the observed rings as showering or e-like (fore˘ , γ) and non-showering or µ-like (for µ˘ , π ˘ ). The e-like rings produce a fuzzy pattern,
due to scattering and electromagnetic showers (bremsstrahlung, followed by the γ Ñ e` e´pair production). On the other hand, a clear pattern is observed for heavier particles which

Figure 4.2: Event display of single-ring electron [left] and single-ring muon [right] data
events, with reconstructed momenta of 492 MeV and 603 MeV as well as the time scale
widths of 130 ns and 162 ns, respectively. Color shows time of arrival of light to PMTs. An
e-like event gives diffused ring pattern, while a µ-like event has a sharp ring edge. From
Ref. [237].
52

don’t scatter much. The two types of rings are illustrated in Figure 4.2.
The reconstruction PID algorithm considers expected charge distribution functions forthe two types of rings, found from the MC simulations and the analytical considerations.
A likelihood test is then performed to determine which type of ring is seen. Figure 4.3
shows PID likelihood distribution for SK-I and the relevant contributing MC processes in
FC sample with a visible energy below 1.33 GeV. Quality of the PID algorithm has been

Figure 4.3: PID likelihood distribution for data and atmospheric MC with a visible energy
below 1.33 GeV. Good separation between e-like and µ-like events is seen. From Ref. [164].
checked using 1 kiloton water Cherenkov detector with electron and muon beams comingfrom the 12 GeV synchrotron at KEK [238].

4.4

Momentum Determination

Ring momentum is calculated from the integrated charge within 70˝ cone around ring’s
reconstructed direction. The integrated charge for ring n, RTn OT , is given by
RTn OT “

GMC ´
α
GData

ÿ
ă70˝

θi,n
´50 nsăti ă250 ns´¯
ÿri cos ωi ¯
Obs.
qi,n
Expr s´Si .
L f pωi qθ ă70˝i,nHere, the variables stand for:α ´ normalization factor
53

(4.3)

GData , GMC ´ relative PMT gain parameter for Data and MC
θi,n ´ opening angle between the nth ring and the ith PMT directions
Obs.
qi,n
´ observed charge in ith PMT from nth ringti ´ residual timing of the ith PMT
L ´ attenuation length in waterri ´ distance from vertex to the ith PMT
f pωi q ´ PMT acceptance correction as a function of incidence angle ωi
Si ´ expected amount of p.e.s from scattered photons for the ith PMT
The summation timing window is chosen to exclude spurious decay electrons. From RTn OT
the corresponding ring momentum is found using a correspondence table built from MC.
Figure 4.4 shows the correspondence of the two for e˘ , µ˘ and K ˘ .

Figure 4.4: RTOT vs. momentum table for the electrons (blue crosses), muons (redcrosses) and the kaons (black crosses). From Ref. [239].
The reconstructed momentum resolution for single ring events is estimated to be arounda˘p2.5{ P pGeVq ` 0.5q% for the electrons and ˘3% for the muons.

4.5

Ring Number Correction

Ring number correction is applied when multiple rings are found. Rings with low
momenta and which overlap other visible rings are rejected as “fake rings” (mis-fit).
54

4.6

Decay Electron Search

The decay electrons are categorized into three sets, those observed in a separate event(sub-event type), those observed with primary event (primary-event type) and those recorded
around the end of event timing window (split type). The only ones relevant for our analysis
are the primary-event type. They are obtained by the following selection criteria:
1. the time interval to the primary event is 1.2 µs ă ∆t ă 20 µs or 0.1 µs ă ∆t ă 0.8 µs
2. number of hit PMTs is ą 50
3. vertex is well reconstructed
4. total p.e.s ă 2000
5. number of hit PMTs in 50 ns window is ą 60
6. number of hit PMTs in 30 ns window is ą 40
The last two criteria reject gamma emission from µ´ capture on 16 O nuclei. The efficiency
for µ` is about „ 20% higher because it does not capture.

55

Chapter 5
Fully Contained Data Sample

5.1

FC Reduction

Approximately 106 events/day are collected by Super-K. The data relevant for physics
studies must be separated from the low-energy radioactivity (e.g. radon) and cosmic ray
muon-induced signals. Additionally, “flasher” events, caused by the PMT dynode discharge,
further contaminate the data and must be removed. Below, we provide an overview of theautomated “data reduction” algorithm for the fully contained (FC) data sample, whose
events show activity only within the ID volume. This is the sample relevant for the nucleon
decay analyses. Further details of the FC data reduction, as well as the reduction descriptions
for the upward-going muon (UPMU) and the partially contained (PC) data samples, can befound in Ref. [160, 240]. Figure 5.1 schematically shows different types of events in SK. For
brevity, we focus on SK-I, with other SK periods treated similarly.

Figure 5.1: Schematic representation of the fully contained, partially contained,
upward-going stopping and through-going muon event classes at Super-K.
56

5.1.1First and Second Reductions

The selection criteria for the 1st and the 2nd reductions are the following:
1. total ID charge within 300 ns window ą 200 p.e. (« 22 MeV/c, in case of electron)
2. ratio of (max p.e.s in any ID PMT)/(total number of p.e.s) is ă 0.5
3. number of hits in OD within 800 ns window is ă 25
4. time interval between events is ą 100 µs
Criterion (1) rejects the low energy radioactivity signals, (2) removes the flasher events, (3)ensures that events happen within the ID and (4) rejects the stopping muon decay-electrons.

5.1.2

Third Reduction

The selection criteria for the 3rd reduction is the following:
1. no clusters of ą 10 OD PMT hits within 8 m of the detector exit/entrance point of
the event2. number of ID hits in 50 ns residual window is ě 50
This reduction uses reconstruction tools to further reject the radioactivity and the stoppingmuon-related events.

5.1.3

Fourth Reduction

The 4th reduction focuses on further rejecting spurious events (e.g. flashers), using thefollowing selection criteria:
1. events with very broad timing distribution, typical of flasher events
2. events whose signature has a high correlation match with signal patterns expected fromflasher events
57

5.1.4

Fifth Reduction

The 5th reduction consists of:
1. removal of events with ą 10 OD PMT hits in 200 ns preceding the trigger time,
rejecting “invisible” cosmic ray muons which are below the Cherenkov threshold but
have an observed decay-electron
2. cosmic ray muon removal using a more precise fitter

5.1.5

Final FC Sample

The final reduction consists of selecting events according to:
1. vertex is in the ID fiducial volume (FV), defined as ą 2 m away from the wall
2. visible energy pEvis. q ą 30 MeV
After the final reduction, data sample contamination from undesired events is ă 1%.
The estimated efficiency of the reduction for nucleon decay and atmospheric MC is ą 95%.
The MC detection efficiency is defined as the fraction of events passing the selection criteria
compared to the total number of events generated within the true fiducial volume. Table 5.1
summarizes the event rates at each stage of the reduction process for SK-I.
Reduction Step
Trigger
1st Reduction
2st Reduction
3st Reduction
4st Reduction
5st Reduction
Final ReductionData
1269039.1
3083.7
202.7
44.9
18.1
16.1
8.2

Atm.-ν MC
9.4 (100.00%)
9.4 (99.95%)
9.4 (99.94%)
9.4 (99.85%)
9.3 (99.17%)
9.3 (99.15%)
9.2 (97.59%)

Table 5.1: Number of events/day after each reduction step for the SK-I FC sample. The
atmospheric Monte Carlo numbers and efficiencies down to the fifth reduction are for
events whose real vertex is in the fiducial volume, the number of OD hits fewer than 10 andthe visible energy larger than 30 MeV. In the last line, the fitted vertex is used for bothdata and Monte Carlo [160].

58

5.2

FC Sample Characteristics

The FC data set is made by applying the FC reduction to the data, atmospheric neutrino and nucleon decay MC (see Chapter 3). In Table 5.2, we show the data vs. atmospheric
MC reduction summary for the reference SK-I to SK-IV sub-GeV samples (SK-IV here isshown only up to 1294.7 live-days), which are most relevant for our analyses. For the atmospheric MC, 500 years worth of exposure-equivalent are generated for each of the SK periods.
Atmospheric MC is normalized by SK livetime for each period. Events are reweighed to take
into account the neutrino oscillations and the calculated flux, as described in Chapter 3.

Event Sample
FC Total
sub-GeV e-like
sub-GeV µ-like

SK-I(1489.2 days)
DataMC
8608.0 8314.5
3469.0 3312.6
3184.0 3064.7

SK-II(798.6 days)
DataMC
4554.0 4473.8
1856.0 1752.2
1684.0 1640.0

SK-III(518.1 days)
DataMC
3063.0 2941.8
1268.0 1171.2
1139.0 1090.4

SK-IV(1294.7 days)
DataMC
7318.0 7208.2
2885.0 2842.3
2837.0 2708.8

Table 5.2: Event summary for the reference SK-I to SK-IV single-ring samples.
We display on Figure 5.2 several key distributions for the SK-I to SK-IV FC sub-GeV
data samples. Agreement between data and MC is generally observed. The most important
distributions for our analysis are the e-like and the µ-like momenta.

59

(a) number of rings(b) particle identification(c) sub-GeV 1-ring e-like momentum(d) sub-GeV 1-ring µ-like momentum(e) sub-GeV 1-ring e-like decay electrons(f) sub-GeV 1-ring µ-like decay electrons

Figure 5.2: Data vs. atm.-ν MC comparison for the SK-I data set (1489.2 days).

60

(a) number of rings(b) particle identification(c) sub-GeV 1-ring e-like momentum(d) sub-GeV 1-ring µ-like momentum(e) sub-GeV 1-ring e-like decay electrons(f) sub-GeV 1-ring µ-like decay electrons

Figure 5.3: Data vs. atm.-ν MC comparison for the SK-II data set (798.6 days).

61

(a) number of rings(b) particle identification(c) sub-GeV 1-ring e-like momentum(d) sub-GeV 1-ring µ-like momentum(e) sub-GeV 1-ring e-like decay electrons(f) sub-GeV 1-ring µ-like decay electrons

Figure 5.4: Data vs. atm.-ν MC comparison for the SK-III data set (518.1 days).

62

(a) number of rings(b) particle identification(c) sub-GeV 1-ring e-like momentum(d) sub-GeV 1-ring µ-like momentum(e) sub-GeV 1-ring e-like decay electrons(f) sub-GeV 1-ring µ-like decay electrons

Figure 5.5: Data vs. atm.-ν MC comparison for the SK-IV data set (1294.7 days).

63

Chapter 6
Nucleon Decay Analysis
Below we present the analyses for the p Ñ e` X, p Ñ µ` X, n Ñ νγ, p Ñ e` νν,p Ñ µ` νν, np Ñ e` ν, np Ñ µ` ν and np Ñ τ ` ν nucleon decay searches, describing in moredetail the published results of Ref. [155] and Ref. [156].

6.1

Data/MC Set

For the analyses, we have employed FC data from the SK-I (1489.2 live-days), SK-II(796.8 live-days), SK-III (518.1 live-days) and SK-IV (1632.3 live-days) Super-K data taking
periods, corresponding to a combined exposure of 273.4 kton¨years.
The signal and background MC have been generated1 according to the methods outlined
in Chapter 3. For background, we use the same atmospheric-ν MC samples that are usedin the SK oscillation analysis [152]. For each SK period, 500 year exposure-equivalent of
atmospheric-ν MC is employed. Neutrino oscillations are taken into account by re-weighting
the atmospheric-ν MC events2 (see Chapter 3).
For the nucleon decay signal MC, events were generated 1 m away from the ID wall,
which allows to take into account possible event migration across the fiducial volume bound1

Both signal and background MC are simulated using 12p80 version of SKDETSIM (SK detector simulation software) and 11d version of Apfit (SK reconstruction software) for SK-I, SK-II and SK-III. For SK-IV,
13p80 version for SKDETSIM and 15a version for Apfit are used.
2
The atmospheric MC re-weighting scheme is based on Honda et. al. [241] 2006 and Honda et. al. [242]2011 neutrino flux calculations and is different for each SK period.

64ary (2 m away from the ID wall). In total, 5,000 signal events („ 4, 200 within FV) weregenerated for each SK period in the case of single nucleon decay channels, and 10,000 signal
events („ 8, 400 within FV) in the case of dinucleon decay channels.
The τ and X are generated according to the procedure outlined in Chapter 3. For thenp Ñ τ ` ν analysis only the leptonic τ decay channels τ ` Ñ e` νν and τ ` Ñ µ` νν are used,which have the branching ratios of 17.8% and 17.4%, respectively. This allows to treat thischannel in a similar manner as the other modes in the spectral fit and thus use a common
analysis framework. We have generated in total three different np Ñ τ ` ν MC samples, with
τ decaying to e` νν, µ` νν and with all τ decay channels. The third sample allows us to
study contamination from hadronic τ channels in the two selected leptonic channels, which
are used in the analysis.
The same FC reduction and reconstruction is applied to both MC and data.

6.2

Preliminaries

6.2.1

Event Selection

After the reduction and reconstruction, event selection is applied to obtain the final
analysis sample.
As already mentioned in the introduction, all of the decay modes considered within
this analysis contain a neutrino in the final state3 . Since they are not observable at SuperKamiokande, one cannot use the final state particles to reconstruct the invariant mass of theparent nucleon(s), which is done, for example, in the p Ñ e` π 0 analysis. The only observables
are those related to a single charged lepton (e` , µ` ) or a γ in the final state. The e` , µ` andγ constitute a single ring signature (showering for γ, e` and non-showering for µ` ). Since
the muon further decays into an electron, we also account for decay-electrons.
For a single nucleon, a two body decay results in a sharp peak with the energy near
„ mn {2 (free nucleon). Hence, we shall restrict ourselves to the sub-GeV (ă 1.33 GeV)
analysis sample. The trilepton 3-body decays, p Ñ e` νν and τ Ñ e` νν, have energy spectrathat spread out from 0 to mn {2. For the dinucleon decay, the total available invariant mass
3

Recall that X is generated as a neutrino.

65

is „ 2mn , with a peak around mn in a 2 Ñ 2 decay. For dinucleon np Ñ τ ` ν decay, after
the 2 Ñ 2 decay the τ further decays through trilepton channels, which results in a spreadout energy distribution from 0 to „ mτ {2. Due to the above considerations, we will enlarge
our analysis region to 1500 MeV for the dinucleon decay searches.
For the decay channels p Ñ e` X, n Ñ νγ, np Ñ e` ν, np Ñ τ ` νpτ ` Ñ e` ννq andp Ñ e` νν, the resulting visible energy is from e` or γ and thus the observed ring is e-like(showering). The selection criteria for the e-like single nucleon decays is shown in Tabe 6.1.
For the dinucleon decay channels, criterion (3) is omitted and (8) is extended to ă 1500
MeV/c.
#
1
2
3
4
5
6
7
8
a

Selectionevis ą 30 and nhitac ď 9awall ą 200
evis ă 1330
nring ““ 1ip[0] == 2
nmue == 0
amome[0] ą 100
amome[0] ă 1000

Description
FC
within FV, ă 2 m away from ID wall
visible energy ă 1.33 GeV, sub-GeV
1 ring
PID is e-like (showering)0 decay-electronse-like momentum ą 100 MeV/ce-like momentum ă 1000 MeV/c

This is for SK-I, in case of SK-II, SK-III, SK-IV criterion is nhitac ď 15.

Table 6.1: Event selection criteria for nucleon decay channels with one e-like (showering)
ring.

Similarly, for the decay channels p Ñ µ` X, np Ñ µ` ν, np Ñ τ ` νpτ ` Ñ µ` ννq theresulting visible energy is from µ` , producing a µ-like (non-showering) ring, with a decay
electron expected to be present. The selection criteria for the µ-like single nucleon decays
#
1
2
3
4
5
6
7
8
a

Selectionevis ą 30 and nhitac ď 9awall ą 200
evis ă 1330
nring ““ 1ip[0] == 3
nmue == 1
amomm[0] ą 200
amomm[0] ă 1000

Description
FC
within FV, ă 2 m away from ID wall
visible energy ă 1.33 GeV, sub-GeV
1 ring
PID is µ-like (showering)
1 decay-electronµ-like momentum ą 200 MeV/cµ-like momentum ă 1000 MeV/c

This is for SK-I, in case of SK-II, SK-III, SK-IV criterion is nhitac ď 15.

Table 6.2: Event selection criteria for nucleon decay channels with one µ-like (nonshowering) ring.
is shown in Tabe 6.2. For the dinucleon decay channels, criterion (3) is omitted and (8) is
extended to ă 1500 MeV/c.
66

After the event selection, the final single nucleon decay data samples with an e-like ring
contain around 8,500 events and 6,000 events for the case of a µ-like ring. The final samples
for the dinucleon decays contain around 9,500 events for the e-like channels and 6,500 events
for the µ-like ones.

6.2.2

Analysis Momentum Distributions

We can now obtain the final e-like and µ-like momentum distributions that will be usedin the spectral χ2 fit, discussed in the next section. The two representative e-like and µ-like
momentum spectra are displayed on Figure 6.1 and Figure 6.2, using p Ñ e` X and p Ñ µ` X
channels as examples, respectively. The final momentum distributions for the other 6 decay
modes can be found in Appendix C. For all our analysis distributions, which serve as input
to the fit, we have used 50 MeV momentum binning. Here, the signal has been normalized
to the background by area, which in turn is normalized to each SK-period’s livetime.

Figure 6.1: Final e-like momentum distributions for p Ñ e` X.
The signal detection efficiency is defined as before. In Table 6.3 we display the signal
detection efficiency for all decay channels. Channels with a µ-like ring have lower overall
67

Figure 6.2: Final µ-like momentum distributions for p Ñ µ` X.
signal efficiency, due to the efficiency of finding the decay-electron of the muon. The increasein efficiency observed in SK-IV for the µ-like channels comes from a 20% improvement in

Decay Mode

SK-I

`

92.7 ˘ 0.4
93.1 ˘ 0.4
77.2 ˘ 0.7
88.8 ˘ 0.5
64.4 ˘ 0.7
93.5 ˘ 0.3
73.6 ˘ 0.5
91.9 ˘ 0.3
74.6 ˘ 0.5

pÑe Xn Ñ νγp Ñ µ` Xp Ñ e` ννp Ñ µ` νν
np Ñ e` ν
np Ñ µ` ν
np Ñ τ ` pe` ννqν
np Ñ τ ` pµ` ννqν

Signal Efficiency (%)
SK-II
SK-III
95.1 ˘ 0.3
94.4 ˘ 0.4
77.7 ˘ 0.7
88.0 ˘ 0.5
65.0 ˘ 0.7
95.6 ˘ 0.2
74.3 ˘ 0.5
96.0 ˘ 0.3
76.9 ˘ 0.5

94.4 ˘ 0.4
94.3 ˘ 0.4
80.3 ˘ 0.6
89.2 ˘ 0.5
67.0 ˘ 0.7
94.2 ˘ 0.3
75.6 ˘ 0.5
94.5 ˘ 0.3
77.9 ˘ 0.5

SK-IV
93.7 ˘ 0.4
93.3 ˘ 0.4
94.5 ˘ 0.4
87.8 ˘ 0.5
78.4 ˘ 0.6
94.5 ˘ 0.3
89.4 ˘ 0.3
93.5 ˘ 0.3
91.2 ˘ 0.3

Table 6.3: Nucleon decay signal detection efficiency, SK-I through SK-IV.

68

the decay-electron detection after an upgrade of the SK detector electronics [158]. For thetrilepton p Ñ e` νν and p Ñ µ` νν channels, the average efficiency is slightly lower, due tothe spread in the momentum of the charged lepton coming from a 3-body decay combinedwith a cut at 100 MeV/c for the e-like and 200 MeV/c for the µ-like channels, respectively.

6.2.3

Atmospheric Neutrino Background

On Figure 6.3 and Figure 6.4 we show the remaining atmospheric neutrino backgroundfor the e-like and the µ-like single-nucleon decay channels after the event selection, alongwith the specific contribution of each respective neutrino-interaction channel. This allows us
to approximately identify which background systematic errors are expected to be significant.
For the e-like momentum distribution up to 1500 MeV/c, the dominant backgroundcontribution, composing about 75.8% of the events, comes from the νe charged-current (CC)quasi-elastic (QE) neutrino channel. The νe CC single-pion production constitutes around
13.0% of the background, while the νe CC coherent-pion, CC multi-pion and neutral-current(NC) single-pion productions contribute around 1.1%, 1.1% and 1.6%, respectively. About
3.5% and 1.1% of events come from νµ NC single-pion and coherent-pion production. For
the µ-like momentum spectrum up to 1500 MeV/c, the dominant contribution of around
78.6% comes from νµ CCQE. Similarly, νµ CC single-pion, CC coherent-pion and CC multipion as well as NC single-pion production contribute around 16.2%, 1.4%, 1.6% and 0.8%,
respectively.
ν-mode
CC QE
CC 1-πCC coh.-π
CC multi-π
NC 1-πNC coh.-π
NC multi-π
Total

νe (%)
75.83
13.07
1.13
1.09
1.55
0.49
0.21
93.37

νµ (%)
0.34
0.30
0.01
0.07
3.53
1.07
0.51
5.83

νe ` νµ (%)
76.17
13.37
1.14
1.16
5.08
1.56
0.72
99.20

Figure 6.3: Contribution of individual atmospheric ν-channels to the background of the
e-like analysis spectra, normalized to SK-I livetime of 1489.2 days.

69ν-modeνe (%)

νµ (%)

νe ` νµ (%)

CC QE

0.00

78.64

78.64

CC 1-π

0.05

16.23

16.28

CC coh.-π

0.01

1.40

1.40

CC multi-π

0.00

1.60

1.60

NC 1-π

0.37

0.81

1.19

NC coh.-π

0.00

0.00

0.00

NC multi-π

0.14

0.31

0.45

Total

0.57

98.99

99.56

Figure 6.4: Contribution of the individual atmospheric ν-channels to the background of theµ-like analysis spectra, normalized to SK-I livetime of 1489.2 days.

6.3

Spectral Fit

We now apply a spectral χ2 fit to the obtained final momentum distributions4 , allowing
us to take into account the information coming from the shape difference between the signal
and the background distributions.

6.3.1

Fitting Technique

The χ2 minimization fit is based on the Poisson distribution, with the systematic uncertainties accounted for by quadratic penalties (“pull terms”) [154]. The χ2 function usedin the analysis is
2

χ “ 2

nbins
ÿ ´

Niexp ` Niobs

i“1Niexp “”syserr
ı¯ Nÿ
j
Niobs
ln exp ´ 1 `
p q2
Ni
σj
j“1

Nsyserrı”ÿ j jsigback
α ¨ Ni` β ¨ Ni p1 `fi q ,
σj
j“1

(6.1)

where i labels the analysis bin. The terms Niobs , Nisig , Niback , Niexp are the numbers of observed
data, signal MC, background MC and the total (signal and background) MC events in each4

In this analysis, as in other SK searches, a custom SK software algorithm (Osc3++) which performs
the spectral fit is used.

70bin i. The index j labels the systematic errors, while j and fij correspond to the fit error
parameter and the fractional change in the Niexp bin due to 1-sigma error uncertainty σj ,respectively. The fit is performed for two parameters α and β, which denote the backgroundand signal normalizations, respectively. After the event selection, the signal MC distribution
is normalized to the background by the integral, which in turn is normalized to the SK
livetime. This allows us to identify the fit point pα, βq “ p1, 0q with the no-signal hypothesis.
Similarly, pα, βq “ p0, 1q signifies that the data is described by signal only, with the signal
amount equal to background MC normalized (pre-fit) to livetime. The χ2 minimization iscarried out over each α and β in the grid according to Bχ2 {Bj “ 0. The resulting global
minimum is defined as the best fit. For the np Ñ τ ` νν mode, after the appropriate event
selection is applied to both MC samples of τ Ñ e` νν and τ Ñ µ` νν, the samples are
combined for the fit, allowing us to obtain a single value for the permitted number of nucleon
decays at 90% CL. Further details regarding the fit and the treatment of systematic errors
can be found in Ref. [152, 153], where this technique is applied to the standard SK oscillation
analysis and the indirect dark matter search. This analysis technique was also applied to
the n Ñ νπ 0 and p Ñ νπ ` [243] nucleon decay searches.

6.3.2

Systematic Errors

The systematic errors in the analysis are considered within the fit through the fijcoefficients, as described above. For simplicity, it is assumed that the errors affect linearly
the bin content Ni0 of the analysis bin i. Then, the fij can be defined as a slope of the line
between the bin i content changed by ˘σ of the systematic error, which can be expressed as

fij ”

¯
´
`σ
´σ
Ni j ´ N ti j
2Ni0

.

(6.2)

For the error estimation in our analysis, unlike Ref. [154], the signal and error bins are split.
This ensures that signal systematic is applied to the signal bins and background systematicto the background bins, while a common systematic (such as energy calibration) is applied
to both. The two are then merged together for the χ2 minimization.
The systematics can be divided into signal specific (S), background-specific (B) as well
as detector and reconstruction errors, which are common to both signal and background(SB). The two signal specific systematics are from Fermi motion and nucleon-nucleon correlated decay. For background, many systematics, such as the neutrino flux normalization or71

the neutrino-interaction cross-section uncertainty, can contribute. In order to methodicallyselect the dominant systematics, we started from more than 150 errors employed in the SK
oscillation analysis [152] and chose those which affect the e-like and µ-like spectra analyses
bins by more than 5% (i.e. |fij ą 0.05|). Relaxing this criteria to 1% does not significantly
alter the results, but complicates the analysis [155]. Some of the most dominant contributions originate from the uncertainties related to the neutrino flux and the energy calibration(common to both signal and background). Including the two signal systematics, a total of
11 errors are considered and they are the same for all channels. In Table 6.4 we display
the complete list of systematics, their uncertainties and the fitted pull terms for the two
representative decay channels, p Ñ e` X and p Ñ µ` X.
Decay mode
Systematic error
Final state interactions (FSI)
Flux normalization (Eν ă 1 GeV)
Flux normalization (Eν ą 1 GeV)
MA in ν interactions
Single meson cross-section in ν interactions
Energy calibration of SK-I
Energy calibration of SK-II
Energy calibration of SK-III
Energy calibration of SK-IV
Fermi model comparison
Nucleon-nucleon correlated decay

1-σ uncertainty (%)
10
25 a
15 b
10
10
1.1
1.7
2.7
2.3
10 c
100

p Ñ e` X
Fit pull (σ)
0.10
-0.23
-1.44
0.69
-0.55
0.58
-0.91
0.48
0.38
-0.08
0.00

p Ñ µ` X
Fit pull (σ)
-0.60
-0.08
-0.50
0.23
-0.14
-0.54
-0.07
0.26
-0.14
0.70
0.06

a

B
B
B
B
B
SB
SB
SB
SB
S
S

Uncertainty linearly decreases with log Eν from 25% (0.1 GeV) to 7% (1 GeV).
Uncertainty is 7% up to 10 GeV, linearly increases with log Eν from 7% (10 GeV) to 12% (100 GeV) and then
20% (1 TeV).
c
Estimated from comparison of spectral function and Fermi gas model.

b

Table 6.4: Systematic errors of the two representative spectral fits, with 1σ uncertainties
and resulting fit pull terms. Errors specific to signal and background are denoted by S andB, while those that are common to both are denoted by SB.
To illustrate the above approach, we describe in more detail how some of the systematics
are estimated. For the correlated decay, there is an additional (wave-function correlated)nucleon in the final state (see Chapter 3). Assuming an uncertainty of σcorr “ 100%, thenumber of such events in the final sample is reweighed by a factor of 2 or 0 (correspondingto ˘ σcorr “ 100%) and then the event rate change in the e-like and the µ-like analysis binsprovides the respective fij .
For the case of Fermi momentum, the error estimation is related to the Fermi momentum
model comparison between the signal (uses spectral function) and the background (uses72Fermi gas model), as described in Chapter 3. As an example, we display in Figure 6.5 theFermi momentum distributions for both signal and background using the p Ñ µ` X SK-I
sample. This error is estimated by re-weighting every 5 MeV Fermi momentum signal bin to

Figure 6.5: Fermi momentum signal (spectral function) and background (relativistic Fermi
gas) model comparison for p Ñ µ` X, using SK-I sample.
match the shape of the background Fermi momentum distribution. Then, we compare thefinal resulting e-like (or µ-like) analysis spectra, with the bin rates given by Niold model andNinew model , respectively. The fij are then estimated via
fij ”

¯
´
old model
new model´ Ni
Ni
Niold model

,

(6.3)

where both distributions have been normalized by area.
For energy calibration, which is a common systematic error for both signal and background, we just shift the final energy spectra by ˘σ of the energy calibration uncertainty(within 3%) to obtain the new event count in each bin for the fij calculation.
The background systematics are treated similarly. For more details see the SK oscillation analysis [194, 244]. For example, the fij for the uncertainty in meson interaction
cross-section is found by generating MC samples with reweighed cross-sections by ˘σ andthen studying the shifts in the final analysis bins in the manner described above.
In our analysis the systematic errors due to reconstruction (ring counting, PID, etc.)
are negligible and are thus omitted.

73

6.4

Fit Results and Lifetime Calculation

The spectral fit determines the overall background and signal normalizations α and β.
The sensitivity reach and lower lifetime limit on the process (in case no excess is observed),
can then be computed from the 90% confidence level value of β (β90CL ), which translates into
the allowed amount of signal at 90% confidence level according to N90CL “ β90CL ˆ N signal ,where N signal is the total number of signal events. The partial lifetime limit is then calculated
from
řSK4
nucleonssk“SK1 λsk ˆ sk ˆ N,(6.4)
τ90CL {B “
N90CL
where B is the branching ratio of a process, sk and λsk are the signal efficiency and theexposure in kton¨years for each SK phase, N90CL is the amount of signal allowed at the 90%
confidence level and N nucleons is the number of nucleons per kiloton of water, corresponding
to 3.3 ˆ 1032 , 2.7 ˆ 1032 and 3.3 ˆ 1031 for the proton, the neutron and the dinucleon decay
searches, respectively.
To calculate the MC sensitivity, we use atmospheric MC as “fake data”. This corresponds to the scenario where no excess is observed in the data, which follows the background
distribution. Hence, in this case, the normalizations of background and signal are α “ 1and β “ 0, for background normalized to the SK-period’s livetime. As an example of the χ2
contour output, we display the sensitivity fit results for p Ñ e` X on Figure 6.6. In the χ2

Figure 6.6: MC sensitivity fit for p Ñ e` X, resulting in β90CL “ 0.013 (at α “ 1) [right].
contour plots, the red, blue and yellow lines correspond to 68%, 90% and 95% confidence level(C.L.) intervals, respectively. The sensitivity is obtained by finding the β90CL (red curve) at
the point of α “ 1 (right plot, α vs. β). In the given case, this value is β90CL “ 0.013, whichcorresponds to N90CL “ 108. Taking into account the signal efficiency (see Table 6.3), N90CL
74

and the live-time for each of the SK phases, one obtains the p Ñ e` X MC sensitivity reachof 7.9 ˆ 1032 years.
Proceeding similarly with the real data, we obtain χ2 distributions as shown on Figure 6.7. Here, the β90CL is obtained from the left figure above (χ2 vs. β) at the intersection

Figure 6.7: Data fit for p Ñ e` X, resulting in β90CL “ 0.013 [left].
with the red curve (the 90% CL interval). This corresponds to profiling out α as a nuisanceparameter. In our case, the β90CL happens to be 0.013 and the best fit point corresponds topα, βq “ p1.05, 0.002q. The similarity of the results between the sensitivity and the data fits
signify that data is described well by the background. Additionally, χ2 { d.o.f. “ 70.9{70 « 1,
as expected for a properly performed fit. For the data fit ∆χ2 “ χ2 ´ χ2min is 0.19 and iswithin 1σ of the background-only hypothesis. Hence, we establish that no significant excess
is observed and we proceed to calculate the lower lifetime limit on the process, which forp Ñ e` X is found to be 7.9 ¨ 1032 years at 90% CL.
We summarize the sensitivity and the data fit results for all 8 channels in Table 6.5.
The χ2 fit contours for other modes can be found in Appendix C.2. The results reported for
dinucleon decay channels are per 16 O nucleus5 . For the np Ñ τ ν mode we have combined theτ decay channels τ Ñ e` νν and τ Ñ µ` νν, weighted by their respective branching ratios.
This limit is then multiplied by 1.15 to account for roughly 85% sample purity of the tau
channels. All of the fits have χ2 {d.o.f. « 1, which is expected for correctly executed fits.
The fit outcomes show that ∆χ2 is within 1σ level of the background only hypothesis for
all channels, aside p Ñ µ` X, which is within 2σ. Overall, we conclude that no significant
excess has been observed in any of the studied channels.
To visualize the analysis results, we have shown the the fitted spectra for the 273.4
5This is consistent with other experiments (e.g. Frejus dinucleon results are reported per iron nucleus).

75

Decay modep Ñ e` Xn Ñ νγp Ñ µ` Xp Ñ e` ννp Ñ µ` νν
np Ñ e` ν
np Ñ µ` ν
np Ñ τ ` νBest fitpα, βq(1.050, 0.002)(1.045, 0.004)(0.960, 0.016)(1.050, 0.030)(0.990, 0.020)(0.955, 0.000)(0.910, 0.000)(0.910, 0.000)

Best fitχ2 {d.o.f.
70.9/7070.5/70
63.2/6265.6/70
66.1/62122.5/110
97.0/102224.6/214

No signal
∆χ2
0.19
0.43
3.43
1.50
0.50
0.00
0.00
0.00

Data
β90CL
0.013
0.015
0.032
0.060
0.050
0.004
0.005
0.006

Data
N90CL
108
125
187
459
286
33
36
96

Sensitivity(ˆ1031 yr.)
79
58
77
27
25
10
11
1

τ {B
(ˆ1031 yr.)
79
55
41
17
22
26
20
3

Table 6.5: Best fit pα, βq parameter values, best fit χ2 { d.o.f., no signal ∆χ2 , 90% C.L.
value of β parameter, allowed number of nucleon decay events in the full 273.4 kton¨years
exposure and a partial lifetime limit for each decay mode at 90% C.L. The sensitivity andlifetime limit for dinucleon decay modes are per 16 O nucleus.
kton¨years of combined SK data in Figure 6.8. The upper figures display best-fit result foratmospheric neutrino background (solid line) without signal fitted to data (black dots) andthe corresponding residuals after the fitted MC is subtracted from data. It is seen that thebackground MC describes the data well. The bottom figures display the 90% C.L. allowed
signal multiplied by 10 (hatched histogram), obtained from the fit of background with signal
to data, with all the e-like and µ-like spectra overlaid for all the modes.

76Figure 6.8: [top] Reconstructed momentum distribution for 273.4 kton¨years of combined
SK data (black dots) and the best-fit result for the atmospheric neutrino backgroundMonte Carlo (solid line). The corresponding residuals are shown below, after fitted
background subtraction from data. [bottom] The 90% confidence level allowed nucleon
decay signal multiplied by 10 (hatched histograms), from the signal and background MC fit
to data. All modes are shown (overlaid), with e-like channels on the left and µ-likechannels on the right.

77

Chapter 7
Summary and Future Prospects
While baryon number violation is expected from theoretical considerations in various
contexts (see Chapter 1), it remains to be seen experimentally. In this Thesis, we havedescribed searches for eight exotic baryon-number violating processes using 273.4 kton¨yearsof combined data from the Super-Kamiokande large underground water Cherenkov experiment. All of the searched channels produce single ring showering or non-showering signatures,
which are well described by atmospheric neutrinos, within the detector. We have studied
the single Cherenkov ring momentum spectra, including the effect of neutrino oscillation andsystematic uncertainties, up to 1500 MeV/c. No significant excess of signal over backgroundhas been observed in the data, allowing us to set the following lower lifetime limits on theprocesses, as shown in Table 7.1.
Decay modep Ñ e` Xn Ñ νγp Ñ µ` Xp Ñ e` ννp Ñ µ` νν
np Ñ e` ν
np Ñ µ` ν
np Ñ τ ` ν

Sensitivity(ˆ1031 yr.)
79
58
77
27
25
10
11
1

τ {B
(ˆ1031 yr.)
79
55
41
17
22
26
20
3

Previous τ {B(ˆ1031 yr.)
2.8 [131]
1.7 [131]
2.1 [138]
0.3 [138]
0.2 [138]-

Table 7.1: Final analysis results (sensitivity and lifetime limits) for the eight studied
nucleon decay modes. Comparison to previous experimental results is shown. The resultsfor dinucleon channels are reported per nucleus.
The obtained lifetime limits for the trilepton decay channels p Ñ e` νν and p Ñ µ` νν78are an order of magnitude improvement over the previous searches, conducted by IMB [131]and Frejus [138]. These results provide strong constraints to both the permitted parameter
space of models presented in Refs. [135, 139], which predict lifetimes of around 1030 ´ 1033
years, and the other GUT models that allow for similar processes. The analyses for thesechannels presented in this work are only weakly model dependent, due to the assumptionof a flat phase space in the signal generation. However, this assumption agrees well withalternative phase space considerations [157] (see Appendix B) in the context of vector- or
scalar-mediated proton decays, which are typical of GUT models [49, 54, 137]. The obtained
limits on the other six channels represent more than an order of magnitude improvement
over the previous analyses of n Ñ νγ at IMB [131] and two orders of magnitude for thenp Ñ e` ν and np Ñ µ` ν searches at Frejus [138]. The searches for p Ñ e` X, p Ñ µ` X
(where X is an invisible, massless particle) and np Ñ τ ` ν are novel. The results of all of theabove analyses provide a stringent test of new physics. The dinucleon decay limits restrict∆B “ 2 processes with L violated by either zero or two units.
In the future, the proposed Hyper-Kamiokande [245] large underground water Cherenkov
experiment with a half-megaton fiducial volume (20 times that of Super-K) is expected toallow for a factor of „ 5 ´ 101 improvement in the lifetime reach over the SK results withina „ 10 year running period. Additionally, the proposed 40 kiloton underground liquid argon
TPC (LArTPC) experiment DUNE/LBNF [246] is expected to also improve the nucleon
decay studies. The LArTPC technology will allow to see kaons, which are below thresholdat the water Cherenkov experiments for the p Ñ νK ` decays.

1

Rough sensitivity estimates have been calculated by the author.

79

Bibliography[1] H. Weyl, Z. Phys. 56, 330 (1929)[2] C.D. Anderson, Phys. Rev. 43, 491 (1933)[3] E.C.G. Stueckelberg, Helv. Phys. Acta 11, 299 (1938)[4] E.P. Wigner, Am. Phil. Soc. 93, 521 (1949)[5] A.D. Sakharov, Pisma Zh. Eksp. Teor. Fiz. 5, 32 (1967)[6] L. Canetti, M. Drewe and M. Shaposhnikov, New J. Phys. 14, 095012 (2012),
arXiv:1204.4186 [hep-ph][7] V.A. Kuzmin, V.A. Rubakov and M.E. Shaposhnikov, Phys. Lett. B155, 36 (1985)[8] R. Mohapatra, Unification and Supersymmetry: The Frontiers of Quark-Lepton Physics,
Springer (2012)[9] T. Banks and N. Seiberg, Phys. Rev. D83, 084019 (2011), arXiv:1011.5120 [hep-th][10] F.C. Adams and G. Laughlin, Rev. Mod. Phys. 69, 337 (1997), arXiv:astro-ph/9701131[11] K.S. Babu et. al., Snowmass 2013 Intensity Frontier Report (2013), arXiv:1310.6677[hep-ph][12] W. Pauli, Rev. Mod. Phys. 13, 203 (1941)[13] J.S. Schwinger, Phys. Rev. 82, 914 (1951)[14] E. Noether, Gott. Nachr. 1918, 235 (1918), arXiv:physics/0503066[15] T.D. Lee and C.N. Yang, Phys. Rev. 104, 254 (1956)[16] J.H. Christenson et. al., Phys. Rev. Lett. 13, 138 (1964)[17] Y. Fukuda et. al. (Super-Kamiokande Collaboration), Phys. Rev. Lett. 81, 1562 (1998),
arXiv:hep-ex/9807003[18] Q.R. Ahmad et. al. (SNO Collaboration), Phys. Rev. Lett. 87, 071301 (2001)[19] J.P. Lees et. al. (BaBar Collaboration), Phys. Rev. Lett. 109, 211801 (2012)arXiv:1207.5832 [hep-ex][20] E. Majorana, Nuovo Cim. 14, 171 (1937)[21] J.D. Vergados, H. Ejiri and F. Simkovic, Rept. Prog. Phys. 75, 106301 (2012),
arXiv:1205.0649 [hep-ph]80[22] Y. Fukuda et. al. (Super-Kamiokande Collaboration), Nucl. Instrum. Meth. A501, 418(2003)[23] P. Langacker, The Standard Model and Beyond, CRC Press (2009)[24] S.L. Glashow, Nucl. Phys. 22, 579 (1961)[25] S. Weinberg, Phys. Rev. Lett. 19, 1264 (1967)[26] A. Salam, Conf. Proc. C680519, 367 (1968)[27] H.D. Politzer, Phys. Rev. Lett. 30, 1346 (1973)[28] D.J. Gross and F. Wilczek, Phys. Rev. Lett. 30, 1343 (1973)[29] N. Cabibbo, Phys. Rev. Lett. 10, 531 (1963)[30] M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49, 652 (1973)[31] B. Pontecorvo, Sov. Phys. JETP 6, 429 (1957)[32] B. Pontecorvo, Sov. Phys. JETP 26, 984 (1968)[33] Z. Maki, M. Nakagawa and S. Sakata, Prog. Theor. Phys. 28, 870 (1962)[34] F. Englert and R. Brout, Phys. Rev. Lett. 13, 321 (1964)[35] G.S. Guralnik, C.R. Hagen and T.W.B. Kibble, Phys. Rev. Lett. 13, 585 (1964)[36] P.W. Anderson, Phys. Rev. 130, 439 (1963)[37] K.A. Olive et. al. (Particle Data Group), Chin. Phys. C38, 090001 (2014)[38] J.S. Bell and R. Jackiw, Nuovo Cim. A60, 47 (1969)[39] S.L. Adler, Phys. Rev. 177, 2426 (1969)[40] J. Steinberger, Phys. Rev. 76, 1180 (1949)[41] P.F. Perez and M. Wise, Phys. Rev. D82, 011901 (2010), arXiv:100.1754 [hep-ph][42] A.A. Belavin et. al., Phys. Lett. B59, 85 (1975)[43] N.S. Manton, Phys. Rev. D28, 2019 (1983)[44] G. t’ Hooft, Phys. Rev. Lett. 37, 8 (1976)[45] P.B. Arnold and L.D. McLerran, Phys. Rev. D37, 1020 (1988)[46] M. Dine et. al., Nucl. Phys. B342, 381 (1990)[47] S. Coleman, Aspects of Symmetry, Cambridge University Press (1989)[48] M. Dine, Proc. 18th SLAC Inst. Part. Phys. (1990)[49] H. Georgi and S.L. Glashow, Phys. Rev. Lett. 32, 438 (1974)[50] J.C. Pati and A. Salam, Phys. Rev. D8, 1240 (1973)[51] E. Golowich, Phys. Rev. D24, 2899 (1981)[52] S. Dimopoulos and F. Wilczek, Proc. Eric Sum. Sch. (1981)[53] P. Langacker and N. Polonsky, Phys. Rev. D47, 4028 (1993), arXiv:hep-ph/9210235[54] H. Fritzsch and P. Minkowski, Annals Phys. 93, 193 (1975)[55] S.M. Barr, Phys. Lett. B112, 219 (1982)[56] J.R. Ellis, D.V. Nanopoulos and J. Walker, Phys. Lett. B550, 99 (2002), arXiv:hep81

ph/0205336[57] P. Minkowski, Phys. Lett. B67, 421 (1977)[58] M. Gell-Mann, P. Ramond and R. Slansky, Proceedings to Supergravity Workshop (1979)[59] T. Yanagida, Prog. Theor. Phys. 64, 1103 (1980)[60] R.N. Mohapatra and G. Senjanovic, Phys. Rev. Lett. 44, 912 (1980)[61] J. Schechter and J.W.F. Valle, Phys. Rev. D22, 2227 (1980)[62] G. Altarelli and F. Feruglio, Rev. Mod. Phys. 82, 2701 (2010)[63] R.N. Mohapatra and M.K. Parida, Phys. Rev. D47, 264 (1993), hep-ph/9204234[64] D.I. Kazakov, Proc. of Eu. School of High En. Phys. 2000 (2000), arXiv:hep-ph/001228[65] S.P. Martin, Adv. Ser. Direct. High Energy Phys. 18, 1 (1998), arXiv:hep-ph/9709356[66] R. Haag, J.T. Lopuszanski and M. Sohnius, Nucl. Phys. B88, 257 (1975)[67] S. Chatrchyan et. al. (CMS Collaboration), Phys. Rev. D88, 0520217 (2013)[68] G. Aad et. al. (ATLAS Collaboration), Phys. Rev. Lett. 106, 131802 (2011),
arXiv:1102.2357 [hep-ex][69] L.B. Anderson, J. Gray and A. Lukas, Phys. Rev. D84, 106005 (2011), arXiv:1106.4804[hep-th][70] B. Volker et. al., J. High Energy Phys. 05, 043 (2006), arXiv:hep-th/0512177[71] W. Buchmuller et. al., Phys. Rev. Lett. 96, 121602 (2006), arXiv:hep-ph/0511035[72] W. Buchmuller et. al., Nucl. Phys. B785, 149 (2007), arXiv:hep-th/0606187[73] B.S. Acharya, G. Kane and P. Kuma, Int. J. Mod. Phys. A27, 1230012 (2012),
arXiv:1204.2795 [hep-ph][74] A. Hebecker and J. March-Russell, Phys. Lett. B539, 119 (2002), arXiv:hep-ph/0204037[75] Y. Kawamura, Prog. Theor. Phys. 105, 999 (2001), arXiv:hep-ph/0012125[76] P. Nath and P.F. Perez, Phys. Rept. 441, 191 (2007), arXiv:hep-ph/0601023[77] T. Haines et. al. (IMB Collaboration), Phys. Rev. Lett. 57, 1986 (1986)[78] W. Gajewski et. al. (IMB Collaboration), Phys. Rev. D42, 2974 (1990)[79] K.S. Hirata et. al. (Kamiokande Collaboration), Phys. Lett. B220, 308 (1989)[80] M. Shiozawa et. al. (Kamiokande Collaboration), Phys. Rev. Lett. D81, 3319 (1998),
arXiv:hep-ex/9806014[81] M.B. Einhorn and D.R.T. Jones, Nucl. Phys. B196, 475 (1982)[82] L.E. Ibanez and G.G. Ross, Phys. Lett. B105, 439 (1981)[83] S. Dimopoulos and H. Georgi, Nucl. Phys. B193, 150 (1981)[84] M.C. Chen, M. Ratz and V. Takhistov, Nucl. Phys. B891, 322 (2014), arXiv:1410.3474[hep-ph][85] G.R. Farrar and P. Fayet, Phys. Lett. B76, 575 (1978)[86] H.M. Lee et. al., Phys. Lett. B694, 491 (2011), arXiv:1009.0905 [hep-ph]82[87] J. Hisano, H. Murayama and T. Yanagida, Nucl. Phys. B402, 46 (1993), hepph/9207279[88] Y. Aoki et. al. (RBC-UKQCD Collaboration), Phys. Rev. D78, 054505 (2008),
arXiv:0806.1031 [hep-lat][89] Y. Aoki et. al., Phys. Rev. D75, 014507 (2007), arXiv:hep-lat/0607002[90] H. Murayama and A. Pierce, Phys. Rev. D65, 055009 (2002), arXiv:hep-ph/0108104[91] K. Kobayashi et. al. (Super-Kamiokande Collaboration), Phys. Rev. D72, 052007(2005), arXiv:hep-ex/0502026[92] J. Beringer et. al. (Particle Data Group), Phys. Rev. D86, 010001 (2012)[93] R.N. Mohapatra, J. Phys. G36, 104006 (2009), arXiv:0902.0834 [hep-ph][94] M. Baldo-Ceolin et. al., Z. Phys. C36, 409 (1994)[95] K.S. Babu and R.N. Mohapatra, Phys. Lett. B715, 328 (2012), arXiv:1206.570 [hep-ph][96] P.A.M. Dirac, Proc. Roy. Soc. Lond. A133, 60 (1931)[97] G. t’ Hooft, Nucl. Phys. B79, 276 (1974)[98] A.M. Polyakov, JETP Lett. 20, 194 (1974)[99] V.A. Rubakov, JETP Lett. 33, 644 (1981)[100] C.G. Callan, Nucl. Phys. B212, 391 (1983)[101] S. Coleman, Phys. Lett. B262, 263 (1985)[102] G.R. Dvali, A. Kusenko and M.E. Shaposhnikov, Phys. Lett. B417, 99 (1998),
arXiv:hep-ph/9707423[103] P.A.R. Ade et. al. (Planck Collaboration), Astron. Astrophys. 571, A16 (2014),
arXiv:1303.5076 [astro-ph.CO][104] C.L. Bennett et. al. (WMAP Collaboration), Astrophys. J. Suppl. 208, 20 (2013),
arXiv:1212.5225 [astro-ph.CO][105] O. Adriani et. al. (PAMELA Collaboration), Phys. Rev. Lett. 105, 121101 (2010),
arXiv:1007.0821 [astro-ph.HE][106] A.H. Guth, Phys. Rev. D23, 347 (1981)[107] A.D. Linde, Phys. Lett. B108, 389 (1982)[108] A. Albrecht and P.J. Steinhardt, Phys. Rev. Lett. 48, 1220 (1982)[109] J. Martin et. al. (WMAP Collaboration), JCAP 1403, 039 (2014), arXiv:1312.3529[astro-ph.CO][110] A.G. Cohen, A.D. Rujula and S.L. Glashow, Astrophys. J. 495, 539 (1998)[111] M. Dine and A. Kusenko, Rev. Mod. Phys. 76, 1 (2003)[112] M. Fukugita and T. Yanagida, Phys. Lett. B174, 45 (1986)[113] I. Affleck and M. Dine, Nucl. Phys. B249, 361 (1985)[114] M. Goldhaber, unpublished (1954)83

[115] F. Reines, C.L. Cowan and M. Goldhaber, Phys. Rev. 96, 1157 (1954)[116] F. Reines, C.L. Cowan and H.W. Kruse, Phys. Rev. 109, 609 (1958)[117] G.N. Flerov et. al., Sov. Phys. Dokl. 3, 79 (1958)[118] G.K. Backenstoss et. al., Nuovo Cim. 16, 749 (1960)[119] C.C. Giamati and F. Reines, Phys. Rev. 126, 2178 (1962)[120] W.R. Kropp and F. Reines, Phys. Rev. 137, B740 (1965)[121] H.S. Gurr et. al., Phys. Rev. 158, 1321 (1967)[122] L. Bergamasco and P. Picchi, Nuovo Cim. Lett. 11, 636 (1974)[123] F. Reines and M.F. Crouch, Phys. Rev. Lett. 32, 493 (1974)[124] R.I. Steinberg and J.C. Evans, Proceedings for 5th Neutrino Conference (1977)[125] E.L. Fireman, Proceedings for 16th International Cosmic Ray Conference (1977)[126] M.L. Cherry et. al. (Homestake Collaboration), Phys. Rev. Lett. 47, 1507 (1981)[127] G. Battistoni et. al. (NUSEX Collaboration), Phys. Lett. B118, 461 (1982)[128] M.R. Krishnaswamy et. al. (KGF Collaboration), Nuovo Cim. C9, 167 (1986)[129] J.E. Bartelt et. al. (Soudan Collaboration), Phys. Rev. D36, 1987 (1990)[130] C. Berger et. al. (Frejus Collaboration), Z. Phys. C50, 385 (1991)[131] C. McGrew et. al. (IMB Collaboration), Phys. Rev. D59, 052004 (1999)[132] H. Nishino et. al. (Super-Kamiokande Collaboration), Phys. Rev. D85, 112001 (2012),
arXiv:1203.4030 [hep-ex][133] Y. Takenaga et. al. (Super-Kamiokande Collaboration), Phys. Lett. B647, 18 (2007),
arXiv:hep-ex/0608057 [hep-ex][134] K. Ueno et. al. (Super-Kamiokande Collaboration), Astropart. Phys. 36, 131 (2012),
arXiv:1203.0940 [hep-ex][135] J.C. Pati, Phys. Rev. D29, 1549 (1984)[136] J.C. Pati, A. Salam and U. Sarkar, Phys. Lett. B133, 330 (1983)[137] J.C. Pati and A. Salam, Phys. Rev. D10, 275 (1974)[138] C. Berger et. al., Phys. Lett. B269, 227 (1991)[139] P.H. Gu and U. Sarkar, Mod. Phys. Lett. A28, 1350159 (2013), arXiv:1110.4581 [hepph][140] W.A. Mann, T. Kafka and W. Leeson, Phys. Lett. B291, 200 (1992)[141] P.J. O’Donnell and U. Sarkar, Phys. Lett. B316, 121 (1993), arXiv:hep-ph/9307254[142] Y. Fukuda et. al. (Kamiokande Collaboration), Phys. Lett. B335, 237 (1994)[143] R. Becker-Szenday et. al. (IMB Collaboration), Phys. Rev. D46, 3720 (1992)[144] J. Learned, F. Reines and A. Soni, Phys. Rev. Lett. 43, 907 (1979)[145] D. Silverman and A. Soni, Phys. Lett. B100, 131 (1981)[146] J.M. Arnold, B. Fornal and M. Wise, Phys. Rev. D88, 035009 (2013), arXiv:1304.6119
84

[hep-ph][147] M. Litos et. al. (Super-Kamiokande Collaboration), Phys. Rev. Lett. 13, 112 (2014)[148] T. Araki et. al. (KamLAND Collaboration), Phys. Rev. Lett. 96, 101802 (2006),
arXiv:hep-ex/0512059[149] R.N. Mohapatra and G. Senjanovic, Phys. Rev. Lett. 49, 7 (1982)[150] L. Arnellos and W.J. Marciano, Phys. Rev. Lett. 48, 1708 (1982)[151] D. Bryman, Phys. Lett. B733, 190 (2014), arXiv:1404.7776 [hep-ex][152] R. Wendell et. al. (Super-Kamiokande Collaboration), Phys. Rev. D81, 092004 (2010),
arXiv:1002.3471 [hep-ex][153] K. Choi et. al. (Super-Kamiokande Collaboration), Phys. Rev. Lett. 114, 141301(2015), arXiv:1503.04858 [hep-ex][154] G. Fogli et. al., Phys. Rev. D66, 053010 (2002), arXiv:hep-ph/0206162[155] V. Takhistov et. al. (Super-Kamiokande Collaboration), Phys. Rev. Lett. 113, 101801(2014), arXiv:1409.1947 [hep-ex][156] V. Takhistov et. al. (Super-Kamiokande Collaboration), Phys. Rev. Lett. 115, 12(2015), arXiv:1508.05530 [hep-ex][157] M.C. Chen and V. Takhistov, Phys. Rev. D89, 095003 (2014)[158] K. Abe et. al. (Super-Kamiokande Collaboration), Nucl. Instrum. Meth. A737, 418(2003), arXiv:1307.0162 [physics.ins-det][159] I.M. Frank and I. Tamm, C. R. Acad. Sci. USSR 14, 109 (1937)[160] Y. Ashie et. al. (Super-Kamiokande Collaboration), Phys. Rev. D71, 112005 (2005)[161] L.R.B. Elton and A. Swift, Nucl. Phys. A94, 52 (1967)[162] K. Nakamura et. al., Nucl. Phys. A268, 381 (1976)[163] R. Woods and D. Saxon Phys. Rev. 95, 577 (1954)[164] M. Shiozawa, Ph.D. Thesis, University of Tokyo (1999)[165] T. Yamazaki and Y. Akaishi, Phys. Lett. B453, 1 (1999)[166] H. Ejiri, Phys. Rev. C48, 1442 (1993)[167] Y.A Kamyshov and E. Kolbe, Phys. Rev. D67, 076007 (2003)[168] K. Abe et. al. (Super-Kamiokande Collaboration), Phys. Rev. D90, 072005 (2014),
1408.1195 [hep-ex][169] Y. Suzuki et. al. (Kamiokande Collaboration), Phys. Lett. B311, 357 (1993)[170] R. Brun, F. Carminati and S. Giani, CERN-W5013 (1994)[171] S. Jadach et. al., Comp. Phys. Comm. 76, 361 (1993)[172] K. Abe et. al. (Super-Kamiokande Collaboration), Phys. Rev. Lett. 110, 181802 (2013),
arXiv:1206.0328 [hep-ex][173] W. Fetscher, Phys. Rev. D42, 1544 (1990)85

[174] B. Louis et. al., Los Alamos Sci. 25, 16 (1997)[175] M. Honda et. al., Phys. Rev. D70, 4 (2004)[176] W.R. Webber, R.L. Golden and S.A. Stephens, Proceedings of 20th Int. Cosmic Ray
Conf. 1, 325 (1987)[177] W.S. Seo et. al. (LEAP Collaboration), Astrophys. J. 378, 763 (1987)[178] P. Pappini et. al. (WIZARD-MASS Collaboration), Proceedings of 23rd Int. Cosmic
Ray Conf. 1, 579 (1993)[179] G. Boezio et. al. (CAPRICE Collaboration), Astrophys. J. 518, 457 (1999)[180] W. Menn et. al. (IMAX Collaboration), Astrophys. J. 533, 281 (2000)[181] T. Sanuki et. al. (BESS Collaboration), Astrophys. J. 545, 1135 (2000)[182] S. Alcaraz et. al. (AMS Collaboration), Phys. Lett. B490, 27 (2000)[183] M. Ryan, J.F. Ormes and V.K. Balasubrahmanyan, Phys. Rev. Lett. 28, 985 (1972)[184] K. Asakimori et. al. (JACEE Collaboration), Astrophys. J. 502, 278 (1998)[185] I.P. Ivanenko et. al., Proceedings of 23rd Int. Cosmic Ray Conf. 2, 17 (1993)[186] Y. Kawamura et. al., Phys. Rev. D40, 729 (1989)[187] A.V. Apanasenko et. al. (RUNJOB Collaboration), Astropart. Phys. 16, 13 (2001)[188] K. Hänssgen and J. Ranft, Comp. Phys. Comm. 39, 1153 (1986)[189] S. Roesler, R. Engel and J. Ranft, Phys. Rev. D57, 2889 (1998)[190] G. Battistoni et. al., Astropart. Phys. 19, 269 (2003)[191] G.D. Barr et. al., Phys. Rev. D70, 023006 (2004)[192] Y. Hayato, (Proceeding Suppl.) Nucl. Phys. B112, 171 (2002)[193] D. Casper, (Proceeding Suppl.) Nucl. Phys. B112, 161 (2002)[194] R. Wendell, Ph.D. Thesis, University of North Carolina at Chapel Hill (2008)[195] C.H. Llewellyn-Smith, Phys. Rep. 3, 261 (1972)[196] R. Gran et. al., Phys. Rev. D74, 052002 (2006)[197] R.A. Smith and E.J. Moniz, Nucl. Phys. B43, 605 (2002)[198] J. Nieves et. al., Phys. Rev. C70, 055503 (2004)[199] H. Nakamura et. al., (Proceeding Suppl.) Nucl. Phys. B112, 197 (2002)[200] C.H. Albright et. al., Phys. Rev. D14, 1780 (1976)[201] K. Abe et. al., Phys. Rev. Lett. 56, 1107 (1986)[202] S.J. Barish et. al., Phys. Rev. D16, 3103 (1977)[203] S. Bonetti et. al. (Gargamelle Collaboration), Nuovo Cim. A38, 260 (1977)[204] M. Pohl et. al. (Gargamelle Collaboration), Lett. Nuovo Cim. 26, 332 (1979)[205] N. Armenise et. al. (Gargamelle Collaboration), Nucl. Phys. B152, 365 (1979)[206] A.S. Vovenko et. al., Sov. J. Nucl. Phys. 30, 528 (1979)[207] S.V. Belikov et. al., Z. Phys. A320, 625 (1985)86

[208] J. Brunner et. al. (SKAT Collaboration), Z. Phys. C45, 551 (1990)[209] Y. Hayato, (Proceedings) Acta Phys. Polon. B40, 2477 (2009)[210] D. Rein and L.M. Sehgal, Ann. Phys. 133, 79 (1981)[211] D. Rein, Z. Phys. C35, 43 (1987)[212] R.P. Feynman, M. Kislinger and F. Ravndal, Phys. Rev. D3, 2706 (1971)[213] T. Kitagaki et. al., Phys. Rev. D34, 2554 (1986)[214] S.K. Singh, M.J. Vicente-Vacas and E. Oset, Phys. Lett. B416, 23 (1998)[215] M. Hasegawa et. al. (K2K Collaboration), Phys. Rev. Lett. 95, 252301 (2005)[216] A. Kartavtsev, E.A. Paschos and G.J. Gounaris, Phys. Rev. D74, 054007 (2006)[217] M. Glück, E. Reya and A. Vogt, Eur. Phys. J. C5, 461 (1998)[218] S.J. Barish et. al., Phys. Rev. D17, 1 (1978)[219] S. Barlag et. al., Z. Phys. C11, 283 (1982)[220] T. Sjöstrand et. al., CERN-TH-7112-93 (1994)[221] P. Musset and J.P. Vialle, Phys. Rep. C39, 1 (1978)[222] J.E. Kim et. al., Rev. Mod. Phys. 53, 211 (1981)[223] L. Salcedo et. al., Nucl. Phys. A484, 577 (1988)[224] D. Ashery et. al., Phys. Rev. C23, 2173 (1981)[225] C. Ingram et. al., Phys. Rev. C27, 1578 (1983)[226] H.W. Bertinie et. al., Phys. Rev. C6, 631 (1972)[227] S.J. Lindenbaum et. al., Phys. Rev. 105, 1874 (1957)[228] K. Abe et. al. (Super-Kamiokande Collaboration), Phys. Rev. Lett. 107, 241801 (2011),
arXiv:1109.1621 [hep-ex][229] T. Gabriel, J. Brau and B. Bishop, Nucl. Sci. IEEE Trans. 36, 14 (1989)[230] C. Zeitnitz, T.A. Gabriel, Nucl. Inst. and Meth. in Phys. Res. A349, 106 (1994)[231] M. Nakahata et. al. (Kamiokande Collaboration), J. Phys. Soc. Jap. 55, 3786 (1986)[232] E. Bracci et. al., CERN-HERA-72-01 (1979)[233] A. Carroll et. al., Phys. Rev. C14, 635 (1976)[234] C. Regis, Ph.D. Thesis, University of California at Irvine (2010)[235] E. Davies, Machine Vision: Theory, Algorithms, Practicalities, Academic Press (1977)[236] H. Nishino, Ph.D. Thesis, University of Tokyo (2009)[237] Y. Itow, (Proceeding Supplements) Nucl. Phys. B112, 1 (2002)[238] S. Kasuga et. al. (Kamiokande Collaboration), Phys. Lett. B374, 238 (1996)[239] M. Litos, Ph.D. Thesis, Boston University (2010)[240] Y. Takenaga, Ph.D. Thesis, University of Tokyo (2008)[241] M. Honda et. al., Phys. Rev. D75, 043006 (2007), arXiv:astro-ph/0611418[242] M. Honda et. al., Phys. Rev. D83, 123001 (2011), arXiv:1102.2688 [astro-ph.HE]87[243] K. Abe et. al. (Super-Kamiokande Collaboration), Phys. Rev. Lett. 12, 113 (2014),
arXiv:1305.4391 [hep-ex][244] L.K. Pik, Ph.D. Thesis, University of Tokyo (2012)[245] K. Abe et. al. (Hyper-K Collaboration), (2011) arXiv:1109.3262 [hep-ex][246] J. Strait et. al. (DUNE Collaboration), (2016) arXiv:1601.05823 [physics.ins-det][247] J.C. Pati and A. Salam, Phys. Rev. Lett. 31, 661 (1973)[248] E. Commins and P. Bucksbaum, Weak Interactions of Leptons and Quarks, Cambridge
University Press (1983)[249] T. Kinoshita and A. Sirlin, Phys. Rev. 113, 1652 (1959)[250] L. Michel, Proc. Phys. Soc. A63, 514 (1950)[251] T. Kinoshita and A. Sirlin, Phys. Rev. 108, 844 (1957)[252] R. Behrends, R. Finkelstein and A. Sirlin, Phys. Rev. 101, 866 (1956)[253] W. Fetscher, Phys. Rev. D42, 1544 (1990)

88

Appendix A
Trilepton Decay Channels from
Pati-Salam
We provide an overview of the Pati-Salam model that could lead to sizable trilepton
decay channels, following Ref. [135]. The minimal Higgs content of the Pati-Salam theory
needed to break GP S Ñ GSM and explain the quark-lepton spectrum is the following [136]:∆R p10, 1, 3q, ∆L p10, 3, 1q, φp1, 2, 2q and ξp15, 2, 2q - where the SU p4q ˆ SU p2qR ˆ SU p2qL
gauge group transformations are noted in the parenthesis. Consistency with phenomenology
requires that the vevs are taken to be x∆R y " xφy „ xξy " x∆L y. The full symmetry
breaking chain is
SOp10qMGU TÓSU p4q ˆ SU p2qL ˆ SU p2qRx∆R y „ MIntermediate

Ó

(A.1)

SU p3qC ˆ SU p2qL ˆ U p1q
xφy „ xξy „ MEW

ÓSU p3qC ˆU p1qEM

where a choice of MU „ 1015 GeV as well as MR „ 1012 GeV is consistent with neutrino
related measurements and gives sin θW „ 0.23. The above can embedded into SOp10q, using
10 and 126 SOp10q multiplets, which contain all of the ∆R , ∆L , φ and ξ PS multiplets.
Additional 54, 210 can be used for SOp10q breaking. The explicit decomposition of the89relevant SOp10q multiplets under PS SU p4q ˆ SU p2qR ˆ SU p2qL is
54 “ p1, 1, 1q ` p6, 2, 2q ` p1, 3, 3q ` p20, 1, 1q ,126 “ p10, 1, 3q ` p10, 3, 1q ` p15, 2, 2q ` p6, 1, 1q ,(A.2)

10 “ p1, 2, 2q ` p6, 1, 1q .
In the scenario outlined above, the desired nucleon decay modes arise from invariant ∆2 ξ 2terms, which lead to diagrams on Figure A.1. Here, ξ3̄ and ξ8̄ are the triplet and the octet

Figure A.1: Diagrams for the p Ñ lll [left] and the p Ñ l ` meson [right] nucleon decays.
From Ref. [135].
components of ξ. The neutral component of ∆R p10, 1, 3q obtains a vev.
The respective amplitudes for the processes are given byhpλξ q2 λ∆ x∆R y
App Ñ l ` mesonq “,pmξ3̄ mξ8̄ m∆ q2

App Ñ l l ¯lq “(A.3a)

hpλξ q2 λ∆ x∆R y,pm2ξ3̄ m∆ q2

where λ’s denote the 3 Yukawa couplings of the lepto-quarks in the diagrams above, while m’s
are their masses and h is the ∆2 ξ 2 coupling. A priori, masses of mξ8̄ and mξ3̄ (the octet andthe triplet components of ξ) are of comparable magnitude and thus the above processes have
similar rate. To get the corresponding decay rates sizable, one can introduce an additionalξ 1 p15, 2, 2q (from 126 or 120 of SOp10q). This will allow to get the octet and the triplet lightusing a “Higgs see-saw”-like mixing between ξ and ξ 1 . More so, one can instead introduce
σp6, 2, 2q (from 54 of SO(10)), which contains the triplet but not the octet. This allows to
get the triplet light while keeping the octet heavy, which results in the tripleton modes being
more dominant than the p Ñ l ` meson channels.
90

Appendix B
Approximating Spectrum of 3-body
DecaysConsider n-body decays of particle b with a mass M . In the center of mass frame, thegeneral partial decay rate (dΓ) of b into n constituents, with a Lorentz invariant matrix
element M , is given by [92]
p2πq4
dΓ “
|M |2 dΦn ,(B.1)
2M
where dΦn is the is the n-body phase spacedΦn “ δ 4 pP ´

n
ÿ

pi q

i“1

n
ź

d3 pi
.
3 2E
p2πq
i
i“1

(B.2)

Here, P and pi represent the momenta of the original and the final state particles, whichhave energy Ei .
Assuming a 2-body decay of b, each of the final state particles will obtain energy equalto „ M {2. This uniquely determines the kinematics of the process, for which the partial
decay width dΓ2 is
1
|p |
dΓ2 “
|M |2 12 dΩ ,
(B.3)
2
32π
M
where p1 “ p2 label the resulting particle 1 and 2 momenta and dΩ is the solid angle ofparticle 1.

91

In the case of 3-body decay of b, the partial decay width dΓ3 is specified by
dΓ3 “

1
1
|M |2 dE1 dE2 dα dpcos βq dγ ,5p2πq 16M(B.4)

with dE1 , dE2 labeling energies of resulting particles 1 and 2 (with 3 being implicitly taken
into account) and pα, β, γq specifying the Euler angle orientation of momenta relative to theparent particle.
From the above, it can be seen that for 3-body decays, unlike for the 2-body decaycase, energy and momenta conservation are insufficient to fully determine the kinematics.
Possible energy dependence of the matrix element M , which encodes the model-dependent
mediation mechanism, can affect the energy distribution of the final state particles.
In the nucleon decay searches, one typically simulates only the final state particles,
taking into account the 4-momentum conservation. Thus, one implicitly assumes a flatdistributed phase space. For nucleon decay channels with ą 2 final state particles, the above
decay-rate dependence on the matrix element can be a potential issue. The trilepton p Ñe` νν and p Ñ µ` νν modes are of this type. We note that p Ñ e` νν has the same final
state particles as muon decay µ Ñ e` νν. It can be shown [157], that under a certain setof assumptions, we can employ the effective Fermi theory formalism for the muon decay toapproximate the decay spectra of the trilepton nucleon decay channels.
We will now outline the argument presented in Ref. [157]. Starting with the most general
4-fermion decay amplitude with arbitrary fermion couplings (of the vector and axial-vectorpV q, the scalar and pseudo-scalar pSq as well as the tensor pT q types) and assuming that
detector is insensitive to the spin of the charged lepton (e` ) and neutrinos are massless, thedecay rate of µ Ñ e` νν is given by [248]!
D G2F m5µ1 ` hpxqdΓ
2“ˆxdx d cos θ
32 192π 3
1 ` 4pme {mµ qη
”4
me p1 ´ xq ıˆ 12p1 ´ xq ` ρp8x ´ 6q ` 24
η
3
mµ x
”
4
α gpxq ı)˘ Pµ ξ cos θ 4p1 ´ xq ` δp8x ´ 6q `
,
3
2π x2

(B.5)

where GF , me , mµ , Ee , Pµ are the Fermi constant, electron mass, muon mass, electronenergy and muon polarization, respectively. Here, cos θ is the angle between the electron
momentum and muon spin, with x “ 2Ee {mµ . Functions gpxq and hpxq incorporate radiative
corrections [249], which in the case of muon decay give the usual peak shape in the electron’s92Michel spectrum. Parameters D, ρ, η, ξ, δ are the Michel parameters [250, 251]. At thispoint all the possible S, V and T couplings, are allowed. The information about the couplings
is encoded inside the Michel parameters, which are functions of the possible couplings. In theVcase of SM, only the left-left vector coupling gLL
is non-zero, corresponding to pV ´ Aq typecurrent, with the full set of parameters determined to be ρ “ ξδ “ 3{4, ξ “ 1, η “ 0 [92].
To employ Equation (B.5) for the p Ñ e` νν proton decay, we substitute mp instead ofmµ . To simplify this to the form that is useful, we note that we are only interested in theisotropic part of the spectrum, allowing us to integrate over the cos θ portion. Additionally,
since the energy spectrum (in case of free nucleon) is within 0 to mn {2 „ 470 MeV region
with a mean of around mn {3 „ 315 MeV, the low energy parameter η can be neglected.
Hence, neglecting the overall normalization and assuming that mass of the final statecharge lepton me is small with respect to mass of the initial particle mp , the approximate
isotropic spectrum for the trilepton nucleon decay can be stated as
ı)
!
”
4
dΓnuc
„ x̄2 p1 ` hpx̄qq ¨ 12p1 ´ x̄q ` ρp8x̄ ´ 6q ,
dx̄
3

(B.6)

where we have substituted proton mass into x̄ “ 2Ee {mp . Therefore, as seen from theabove, all the information about possible S, V, T couplings is encoded into a single Michelparameter ρ. It should be noted, that the radiative correction function hpx̄q has similar
distribution irrespective of considered couplings [252]. Thus, Eq. B.6 is considerably general.
0.5

dG  dE

0.4

0.3

0.2

0.1

0.0
0

100

200

300

400

500

Energy HMeVL

Figure B.1: Decay spectra of charge leptons e` (dotted line) and µ` (continuous line) inrespective p Ñ e` ν ν̄ and p Ñ µ` ν ν̄ decays. From Ref. [157].
Therefore, the whole isotropic energy spectra (up to normalization) for trilepton nucleon decays depends on a single parameter ρ that is a function of possible fermion S, V, T
couplings, depending on the theory. In SM ρ “ 3{4, which is responsible for the familiar Michel spectrum shape of the decay-electrons. This value is in fact quiet general and93

naturally appears if tensor as well as vector left-right couplings are neglected. This can beseen [253] in the SM τ Ñ e` νν decay spectrum calculations. Since the decay in question ismediated by the extra Higgs (scalars) of the Pati-Salam [135] and because there is usuallyno tensor mediated proton decay in GUTs (e.g. SU p5q [49]), the above assumption is valid
for our analysis.
We can now apply the given framework to the trilepton nucleon decays to obtain the
charged lepton spectra. Taking into account the radiative corrections as well as the charged
lepton and the initial particle masses of me “ 0.511 MeV and mp “ 938.2 MeV, the isotropic
spectrum, up to overall normalization, is shown in Fig. B.1 as a function of energy, forthe approximate e` spectrum in p Ñ e` ν ν̄ decay and the approximate µ` spectrum inp Ñ µ` ν ν̄. The µ` spectrum is also reasonably approximated since the condition that the
final state charged lepton mass mµ is significantly smaller than the original parent particle
mass mp still holds, given that mass of the muon is mµ “ 105.7 MeV.

94

Appendix C
Additional Analysis Information

C.1

Final Momentum Distributions

Figure C.1: Final e-like momentum distributions for n Ñ νγ .

95

Figure C.2: Final e-like momentum distributions for p Ñ e` νν .

Figure C.3: Final µ-like momentum distributions for p Ñ µ` νν .
96

Figure C.4: Final e-like momentum distributions for np Ñ e` ν .

Figure C.5: Final µ-like momentum distributions fornp Ñ µ` ν .

97

Figure C.6: Final e-like momentum distributions for np Ñ τ ` νpτ ` Ñ e` ννq.

Figure C.7: Final µ-like momentum distributions for np Ñ τ ` νpτ ` Ñ µ` ννq .
98

C.2

Sensitivity Results

Figure C.8: MC sensitivity fit for n Ñ νγ, resulting in β90CL “ 0.014 (at α “ 1) [right].

Figure C.9: MC sensitivity fit for p Ñ µ` X, resulting in β90CL “ 0.017 (at α “ 1) [right].

Figure C.10: MC sensitivity fit for p Ñ e` νν, resulting in β90CL “ 0.042 (at α “ 1) [right].
99

Figure C.11: MC sensitivity fit for p Ñ µ` νν, resulting in β90CL “ 0.051 (at α “ 1) [right].

Figure C.12: MC sensitivity fit for np Ñ e` ν, resulting in β90CL “ 0.009 (at α “ 1) [right].

Figure C.13: MC sensitivity fit for np Ñ µ` ν, resulting in β90CL “ 0.010 (at α “ 1) [right].
100

Figure C.14: MC sensitivity fit for np Ñ τ ` ν, resulting in β90CL “ 0.010 (at α “ 1) [right].

C.3

Data Fit Results

Figure C.15: Data fit for n Ñ νγ, resulting in β90CL “ 0.015 [left].

Figure C.16: Data fit for p Ñ µ` X, resulting in β90CL “ 0.032 [left].
101Figure C.17: Data fit for p Ñ e` νν, resulting in β90CL “ 0.050 [left].

Figure C.18: Data fit for p Ñ µ` νν, resulting in β90CL “ 0.060 [left].

Figure C.19: Data fit for np Ñ e` ν, resulting in β90CL “ 0.004 [left].
102

Figure C.20: Data fit for np Ñ µ` ν, resulting in β90CL “ 0.005 [left].

Figure C.21: Data fit for np Ñ τ ` ν, resulting in β90CL “ 0.006 [left].

103