BOSTON UNIVERSITYGRADUATE SCHOOL OF ARTS AND SCIENCES

Dissertation

A SEARCH FOR BARYON NUMBER VIOLATION BY TWO UNITSAT THE SUPER-KAMIOKANDE DETECTOR

byJeffrey D. Gustafson
B.A., Macalester College, 2009

Submitted in partial fulfillment of therequirements for the degree ofDoctor of Philosophy
2016

c

Copyright by
JEFFREY GUSTAFSON
2016

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

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

Acknowledgements
There are many, many people to whom I owe thanks for the successful completionof my Ph.D, and by extension, a completion of my life up to this point. I’ll try tolimit my scope to people who are in tangible danger of actually reading this.
My advisor, Professor Ed Kearns, cannot be thanked enough. Ed took me on
knowing I had very little high energy physics experience, and was always extremely
gracious and patient as I struggled to become a useful graduate student. His unparalleled mastery of neutrino physics, motivating attitude, generously-shared wisdom,
and infectious sense of humor were instrumental in helping me graduate. Ed knew
when to push, but he also taught me the valuable lesson of when to “declare victory”,
for which this occasion is ideal.
It has been an enormous honor and privilege to work at the Super-Kamiokande
experiment, which after 20 years (and a Nobel Prize!) remains one of the beacons ofmodern high-energy physics. The hard work and dedication of all collaborators pastand present, those I have met and those I haven’t, has been a continuous source ofinspiration. In particular, the leadership of Spokesmen Yoichiro Suzuki and Masayuke
Nakahata during my time on the experiment, and the leadership of the Atmospheric
Neutrino and Proton Decay group by Professor Kearns and Masato Shiozawa, has
ensured that Super-K continues to produce world-class physics at this late stage.
Within Super-K, I have been privileged to work closely with a number of greativ

physicists. With no pretense of exhaustion, I’d particularly like to thank Doctors
Makoto Miura and Shunichi Mine for their proton decay expertise and suggestions,and setting outstanding examples with their highly methodical approaches; Dr. Jennifer Raaf, for patiently showing me the ropes on all things UPMU; Dr. Yoshinari
Hayato, for promptly and kindly answering all of my dumb questions, particularly
related to NEUT; and Dr. Roger Wendell, for a list of things that would probably
reach the length of this thesis to properly do justice.
It has been a pleasure to work at the Boston University Department of Physics
at all levels of the graduate school experience. I’d particularly like to thank eachmember of my committee — Professor Rob Carey for being my second reader andpromptly providing helpful comments, Professor Shyam Erramilli for his friendly firstyear lessons in electromagnetism (as well as the “conservation of pain”), Professor
Tulika Bose for her valuable introduction to particle physics in PY551 that I somewhat
belatedly took, and Professor Andy Cohen for kindly agreeing to be on my committee
on short notice. I’d also like to thank my erstwhile committee member Professor
Martin Schmaltz for his helpful comments on the theory of my analysis. At theadministrative level, I’d like to thank Winna Somers for patiently dealing with mytravel forms, which somehow confused me anew each time I went to Japan, and Mirtha
Cabello for everything else. I’ve always felt good coming into work with office-matesand BU Neutrino group members like Chris Kachulis, Ryan Linehan, Flor de Maria
Blaszczyk, and Dan Gastler.
The friends I’ve made at BU and elsewhere during my time in graduate school
have been invaluable in making the experience great. In particular, the friendship ofAlex Long helped keep me sane, especially in the first two years. I’ve had great times
with Colin Howard over our shared love of music. Michael Chernicoff is, as far as I
can tell, incapable of holding a dull conversation. While I did not see him nearly asv

much as I would have liked, Euan Richard was always a good laugh.
My family has been a fundamental source of emotional, moral, and financial support since the day I was born. I am extremely fortunate to maintain a close relationship with my parents, Jim and Lori Gustafson, without whom I’d be nowhere. WhileI don’t talk to her enough, my sister Katie has always been there for me. The broad
wisdom and compassion of my aunt Aline has continually amazed me. Finally, I am
proud to say I am the second physics Ph.D in the family, after my wonderful uncle
Dr. Steve Gustafson, although now that we share the same title it will be somewhat
less of an event when I beat him in chess.

vi

A SEARCH FOR BARYON NUMBER VIOLATION BY TWO UNITSAT THE SUPER-KAMIOKANDE DETECTOR
JEFFREY D. GUSTAFSON
Boston University Graduate School of Arts and Sciences, 2016
Major Professor: Edward T. Kearns, Professor of Physics
ABSTRACTBaryon number B appears to be a conserved quantity in the Standard Model ofparticle physics, though there are compelling theoretical reasons to believe it isn’t.
This thesis describes searches for processes that violate B by two units with theSuper-Kamiokande experiment. Two types of searches are performed. One is thesimultaneous decay of two bound nucleons to pions (dinucleon decay), encompassing
the modes 16 O(pp) → 14 C π + π + , 16 O(pn) → 14 N π + π 0 , and 16 O(nn) → 14 O π 0 π 0 .
The second is a search for a neutron transforming into an antineutron, or neutronantineutron oscillation n → n̄.
This thesis uses the full dataset across four Super-Kamiokande detector periods
from April 1996 to March 2015, comprising 4972.4 livetime days (307 kiloton-years).
Monte Carlo simulations of the signal processes and their background from atmospheric neutrino interactions are used to estimate signal efficiency, expected background, and their associated uncertainties. Both multivariate analyses and simple
kinematical cuts are applied in these searches. This is the first search for dinucleon
decay to pions in oxygen. The search for n → n̄ entails some important updates froma previous search at Super-Kamiokande, mainly increased exposure and an improved
model of pion interactions.
In each mode investigated, no signal excess was observed, and all data are consistent with atmospheric neutrino background. In the absence of evidence for anyviisignal process, lower lifetimes are set at the 90% confidence level. For dinucleon decay, the limits are τpp→π+ π+ > 7.2 × 1031 years, τpn→π+ π0 > 1.7 × 1032 years, andτnn→π0 π0 > 4.0 × 1032 years. These limits are about two orders of magnitude more
stringent than those set by previous searches for dinucleon decay in iron. For n → n̄,
the limit for bound neutrons is τbound > 2.4 × 1032 years, corresponding to a freeneutron lifetime τfree > 3.8 × 108 s. This is comparable with previous results fromSuper-Kamiokande and other experiments.

viii

Contents

Acknowledgementsiv

Abstractvii

Table of Contentsix

List of Tables

xiv

List of Figures

xxi

List of Abbreviations

xxxvii

1 Introduction

1

2 Motivation and Theory4

2.1

Baryon Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

2.2

The Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2.3

Baryogenesis and Baryon Asymmetry . . . . . . . . . . . . . . . . . .

7

2.4

Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

2.5

Phenomenology and previous results . . . . . . . . . . . . . . . . . .

13

2.5.1

nn̄ oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

2.5.2

Dinucleon Decay . . . . . . . . . . . . . . . . . . . . . . . . .

19ix

3 The Super-Kamiokande Detector

22

3.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

3.2

Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

3.2.1

Inner Detector (ID) . . . . . . . . . . . . . . . . . . . . . . . .

24

3.2.2

Outer Detector (OD) . . . . . . . . . . . . . . . . . . . . . . .

25

3.3

Cherenkov Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

3.4

Photomultiplier tubes . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

3.4.1

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

27

3.4.2

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

28

Electronics and Data Acquisition (DAQ) . . . . . . . . . . . . . . . .

29

3.5.1

Inner Detector Electronics and DAQ (SK I-III) . . . . . . . .

30

3.5.2

Outer Detector Electronics and DAQ (SK-I-III) . . . . . . . .

31

3.5.3

Trigger (SK-I-III) . . . . . . . . . . . . . . . . . . . . . . . . .

32

3.5.4

SK-IV Electronics and DAQ Upgrade . . . . . . . . . . . . . .

33

3.6

Water Purification System . . . . . . . . . . . . . . . . . . . . . . . .

35

3.7

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

35

3.7.1

Water Calibration . . . . . . . . . . . . . . . . . . . . . . . . .

36

3.7.2

Charge calibration . . . . . . . . . . . . . . . . . . . . . . . .

38

3.7.3

Timing Calibration . . . . . . . . . . . . . . . . . . . . . . . .

42

3.7.4

Energy Scale . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

3.5

4 Atmospheric Neutrino Simulation

48

4.1

Neutrino Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

4.2

Neutrino interactions . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

4.2.1

(Quasi)-elastic scattering . . . . . . . . . . . . . . . . . . . . .

52

4.2.2

Single meson production . . . . . . . . . . . . . . . . . . . . .

52

x

4.3

4.2.3

Deep inelastic scattering . . . . . . . . . . . . . . . . . . . . .

53

4.2.4

Nuclear effects . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

Detector Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

5 Signal Simulation

61

5.1

Dinucleon Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

5.2

Neutron-antineutron oscillation . . . . . . . . . . . . . . . . . . . . .

63

6 Reduction and Reconstruction
6.1

6.2

67

Fully Contained Reduction Steps . . . . . . . . . . . . . . . . . . . .

68

6.1.1

First Reduction . . . . . . . . . . . . . . . . . . . . . . . . . .

68

6.1.2

Second Reduction . . . . . . . . . . . . . . . . . . . . . . . . .

68

6.1.3

Third Reduction . . . . . . . . . . . . . . . . . . . . . . . . .

70

6.1.4

Fourth Reduction . . . . . . . . . . . . . . . . . . . . . . . . .

75

6.1.5

Fifth reduction . . . . . . . . . . . . . . . . . . . . . . . . . .

76

6.1.6

Final FC sample . . . . . . . . . . . . . . . . . . . . . . . . .

78

Event Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

6.2.1

Vertex fitting . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

6.2.2

Ring counting . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

6.2.3

Particle identification . . . . . . . . . . . . . . . . . . . . . . .

84

6.2.4

Momentum reconstruction . . . . . . . . . . . . . . . . . . . .

88

6.2.5

Decay electrons . . . . . . . . . . . . . . . . . . . . . . . . . .

89

6.2.6

π 0 reconstruction . . . . . . . . . . . . . . . . . . . . . . . . .

90

7 Dinucleon Decay Search

91

7.1

Boosted Decision Trees . . . . . . . . . . . . . . . . . . . . . . . . . .

92

7.2

pp → π + π + . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

xi

7.3

7.4

7.5

7.2.1

Pre-selection for pp → π + π +. . . . . . . . . . . . . . . . . .

97

7.2.2

Multivariate analysis for pp → π + π + . . . . . . . . . . . . . .

98

7.2.3

Final sample for pp → π + π +pn → π + π 0. . . . . . . . . . . . . . . . . . 101. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

7.3.1

Pre-selection for pn → π + π 0 . . . . . . . . . . . . . . . . . . . 108

7.3.2

Multivariate analysis for pn → π + π 0

7.3.3

Final sample for pn → π + π 0 . . . . . . . . . . . . . . . . . . . 112

nn → π 0 π 0. . . . . . . . . . . . . . 111. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

7.4.1

Selection criteria for nn → π 0 π 0

7.4.2

Final sample for nn → π 0 π 0 . . . . . . . . . . . . . . . . . . . 119. . . . . . . . . . . . . . . . 118

Systematic Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . 122
7.5.1

Simulation Uncertainties . . . . . . . . . . . . . . . . . . . . . 123

7.5.2

Reconstruction and BDT uncertainties . . . . . . . . . . . . . 126

8 Neutron-Antineutron Oscillation Search

132

8.1

Pre-selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

8.2

Multivariate analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

8.3

8.2.1

Standard variables . . . . . . . . . . . . . . . . . . . . . . . . 134

8.2.2

Non-standard variables . . . . . . . . . . . . . . . . . . . . . . 136

Final selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
8.3.1

Systematic uncertainties . . . . . . . . . . . . . . . . . . . . . 143

8.3.2

Final cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

8.3.3

Data discrepancy . . . . . . . . . . . . . . . . . . . . . . . . . 145

9 Results and discussion
9.1

153

Lifetime limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
9.1.1

Dinucleon decay . . . . . . . . . . . . . . . . . . . . . . . . . . 154
xii

9.1.2
9.2

Neutron-antineutron oscillation . . . . . . . . . . . . . . . . . 154

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
9.2.1

Global BNV searches . . . . . . . . . . . . . . . . . . . . . . . 155

9.2.2

Comparison with previous SK n → n̄ analysis . . . . . . . . . 156

Appendices161

A n → n̄ Branching Fractions

162

B Plots of Multivariate Input Variables for SK-I-III

166

C Limit Calculations

182

Bibliography

186

Curriculum Vitae

193

xiii

List of Tables
2.1

Limits on bound neutron oscillation lifetimes from Irvine-MichiganBrookhaven ([31]), Kamiokande([32]), Frejus ([33]), Soudan-II ([34]),
Sudbury Neutrino Observatory([35]), and Super-Kamiokande ([23]).
Limits are converted to a free oscillation time via Equation 2.16. See
the text for a comparison to the free neutron experiment from [29].
The values of R for [31–35] are calculated in [36], while the value of
R used for [23] incorporates a more recent calculation from [27]. The
uncertainty in R is also incorporated in the conversion to the free lifetime in [23]. The study in [35] used a different search method than theother analyses listed, such that efficiency, background rate, and data
candidates are not illuminating figures. . . . . . . . . . . . . . . . . .

2.2

20

Summary of results for N N → ππ from Frejus [39], and pp → K + K + [40]and pp → e+ e+ [41] from Super-Kamiokande. The Frejus searches
were performed with 2.2 × 1031 iron-years of exposure, and the SuperKamiokande searches with 3.06 × 1032 oxygen-years. . . . . . . . . . .

21

3.1

Specifications for the ID PMTs. . . . . . . . . . . . . . . . . . . . . .

30

3.2

Specifications for the ATM. . . . . . . . . . . . . . . . . . . . . . . .

31

xiv

4.1

Interactions modeled for each particle type. Cherenkov radiation ismodeled for e± , µ± , and charged hadrons. . . . . . . . . . . . . . . .

5.1

60

Number of pions for pp → π + π + (top), pn → π + π 0 (middle), andnn → π 0 π 0 (bottom) after nuclear effects. Percentages indicate thepercentage of events with the specified number of π + ’s (columns) andnumber of π 0 ’s (rows). Entries with a “—” indicate a small contribution (<0.1%). π + ’s are required to be above Cherenkov threshold. The
number of pions prior to nuclear effects is boldened. . . . . . . . . . .

5.2

64

Percentage of pions undergoing different interactions for the analysis
in [23], and this analysis. From left to right, the interactions are absorption, quasi-elastic scattering, charge exchange, π production, andno interaction. The ∗ indicates that QE scattering and charge exchange
are combined. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.1

A set of common parameters used for the BDTs in the pp → π + π + andpn → π + π 0 searches. . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.2

66

96

Independent MC sample sizes used for each stage of the BDT for boththe pp → π + π + and pn → π + π 0 searches, prior to any cuts. Dinucleon
decay sample sizes show only the number of events, while the atmospheric neutrino sample sizes show the approximate number of eventsand the equivalent number of years. The same statistics are used for
SKI-IV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xv

96

7.3

Efficiency, expected background, and data events after each precut forthe pp → π + π + search. The FCFV cut is described in Section 6.1.6,
and cuts (A1)-(A4) are described in the text. Expected backgroundis weighted for oscillation effects, and scaled to the appropriate SK
detector livetime (from the 200 year analysis sample). . . . . . . . . .

7.4

Relative importance of each variable in the pp → π + π + search, averaged across SK periods.

7.5

99

. . . . . . . . . . . . . . . . . . . . . . . . . 101

Efficiency, expected background events, and candidate data events for
SKI-IV pp → π + π + search. Background is quoted both for the appropriate SK livetime, and per megaton-year. . . . . . . . . . . . . . . . 105

7.6

Atmospheric neutrino interaction modes in the remaining backgroundfor the pp → π + π + search by percentage, as estimated by atmospheric
neutrino MC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

7.7

Candidate variable values, and mean variable values for signal andbackground distributions, for each of the variables used in the BDT
for the pp → π + π + search. The description of each variable is in the
text. Values are given for SK-II and SK-IV, for which single candidate
events were found. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

7.8

Efficiency, expected background, and data events after each precut forthe pn → π + π 0 search. Cuts (B1)-(B5) are described in the text.
Expected background is weighted for oscillation effects, and scaled tothe appropriate SK detector livetime (from the 200 year analysis sample).110

7.9

Relative importance of each variable in the pn → π + π 0 search, averaged across SK periods.

. . . . . . . . . . . . . . . . . . . . . . . . . 112

xvi

7.10 Final BDT cut value, efficiency, expected background, and candidate
data events for the SKI-IV pn → π + π 0 search. Background rates arequoted for the appropriate SK livetime, and per megaton-year. . . . . 115
7.11 Neutrino interaction mode percentages for remaining background in
the pn → π + π 0 search for SKI-IV. . . . . . . . . . . . . . . . . . . . . 117
7.12 Candidate variable values, and mean variable values for signal andbackground distributions, for each of the variables used in the BDT
for the pn → π + π 0 search. Letters for each variable are assigned in the
text. Values are given for SK-I, for which a single candidate event was
found. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
7.13 Efficiency, expected background, and data events for the SKI-IV nn →π 0 π 0 search, for each analysis cut. The cuts are described in the text.

120

7.14 Neutrino interaction mode percentages for the remaining backgroundin the nn → π 0 π 0 search for SKI-IV. . . . . . . . . . . . . . . . . . . 122
7.15 Variations used to estimate systematic uncertainties from ion interactions for N N → ππ . The coefficients correspond to quasielastic scattering (QE), inelastic scattering including hadron production (INEL),
absorption (ABS), and charge exchange (CX). Both QE and CX are
considered in low- and high-energy contexts, with an “H” added forthe latter. Because hadron production is only considered for high momenta (pπ > 500 MeV/c), variations that change hadron production
are considered high-energy variations, which are of the most relevancefor N N → ππ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
7.16 Dominant interaction uncertainties that contribute to the overall atm.ν uncertainty for the N N → ππ searches. . . . . . . . . . . . . . . . . 125

xvii

7.17 Ring separation and PID errors, from the analysis in [61], for relevantsamples in the N N → ππ analyses. All samples are multi-ring. . . . . 126
7.18 Values for the absolute energy scale uncertainty in each SK period. . 127
7.19 Systematic uncertainties in the pp → π + π + search.

. . . . . . . . . . 129

7.20 Systematic uncertainties in the pn → π + π 0 search. . . . . . . . . . . . 130
7.21 Systematic uncertainties in the nn → π 0 π 0 search. Note that theenergy scale uncertainty is dominant for the background, unlike othersearches in which the simulation uncertainties are dominant. . . . . . 131
8.1

Efficiency, expected background, and data events after each precut for
the n → n̄ search. The FCFV cut is described in Section 6.1.6, and cuts(D1)-(D3) are described in the text. Expected background is weighted
by oscillation factors determined from the analysis in [86], and scaled
to the appropriate SK detector livetime (from the 200 year analysis
sample). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

8.2

Relative importance of each variable in the n → n̄ search, averaged
across SK periods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

8.3

Values for the DIS interaction uncertainty for Eν < 10 GeV. . . . . . 144

8.4

Efficiency, expected background events, and candidate data events for
the SKI-IV n → n̄ search. Background is quoted both for the appropriate SK livetime, and per megaton-year. . . . . . . . . . . . . . . . 146

8.5

Atmospheric neutrino interaction modes in the remaining backgroundfor the n → n̄ search by percentage, as estimated by atmospheric
neutrino MC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

8.6

Systematic uncertainties in the n → n̄ search. . . . . . . . . . . . . . 148

xviii

8.7

Values of | (Meandata − Meanatm.-ν ) /Meanatm.-ν | for each variable in
the n → n̄ analysis, following selections (D1)-(D3). . . . . . . . . . . 149

8.8

Values of the χ2 test statistic between data and atm.-ν MC, whose
value in [0, 1] gives the probability that differences in distributions are
due to statistical fluctuations, for each variable in the n → n̄ analysis,
following selections (D1)-(D3). . . . . . . . . . . . . . . . . . . . . . . 150

8.9

Shift value in data and atm.-ν MC total momentum, and associatedsignal and background uncertainties in the n → n̄ search. . . . . . . . 152

8.10 Summary of the adjustments made in SK-I/II/III to account for thediscrepancy seen between data and atm.-ν MC, with original and adjusted efficiency and background rates at the final cut value, and corresponding signal and background uncertainties. The number of variables removed starts from variables with the largest values of the meandifference between data and atm.-ν MC (Table 8.7). Background rates
show the nominal values ± 1σ systematic uncertainty, calculated withall the sources in Table 8.6. . . . . . . . . . . . . . . . . . . . . . . . 152
9.1

Lower lifetimes on N N → ππ modes from this analysis, compared to
the results of [39]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

9.2

Bound n → n̄ limit τ90 and τ90 (ncand = nexp. bkg. ) for the SK-I analysis
in [23], and this analysis. Note that τ90 (ncand = nexp. bkg. ) for this analysis does not include the additional systematic uncertainties describedin Table 8.10, while τ90 with the actual number of candidates does. . 154

9.3

Free n → n̄ lifetime limits, as measured in [29], and as calculated in [23]and this thesis. The calculations incorporate theoretical uncertainty on
the nuclear suppression R as calculated in [27]. . . . . . . . . . . . . . 156
xix

9.4

Comparison of efficiencies, background rates, systematic uncertainties,
and data candidates from the SK-I analysis in [23], the new analysis
MC using the same cut method, and the new analysis as described in
Chapter 8. The background rates are shown with ± 1-σ systematic
uncertainties to better compare to the number of data events. . . . . 159

A.1 Branching fractions used in the SK-I n → n̄ analysis in [23]. The values
are taken from bubble chamber experiments [75–77] . . . . . . . . . . 164
A.2 Purely pionic branching fractions used in this analysis. The data forn̄n is from data in [78]. Some of the data for n̄p has been updated fromresults in the OBELIX experiment, though most is replicated from [23]. 164
A.3 Branching fractions with one or more intermediate resonances thisanalysis. Data for n̄n annihiliaton is taken from [78] and [79]. . . . . 165
A.4 Branching fractions with one or more strange particles in this analysis
for n̄n annihilation, from data in [78]. . . . . . . . . . . . . . . . . . . 165
A.5 Efficiency after each of the n → n̄ selection pre-cuts, for three different
MC sets: MC from the old SK simulation, MC with the new pionmodel but same branching fractions as the old SK simulation, and MC
with both updated pion model and branching fractions (this analysis). 165

xx

List of Figures
2.1

Feynman diagram for n → n̄ (top) and N N → ππ (bottom) from oneof the models discussed in [22]. Spectator quarks q1 , q2 ∈ (u, d) are
required for N N → ππ . The X1 and X2 scalar particles belong to(6̄, 1, − 13 ) and (6̄, 1, 23 ) representations of the Standard Model SU (3) ⊗SU (2) ⊗ U (1) gauge group. . . . . . . . . . . . . . . . . . . . . . . .

2.2

13

The reduced n̄ radial wavefunction (top) and the n̄ nuclear potential
for a p1/2 -state n̄ (bottom), from [27]. . . . . . . . . . . . . . . . . . .

17

3.1

The SK detector [43]. . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

3.2

Relative spectrum of Cherenkov light in pure water, with quantum
efficiency of the ID PMT overlaid [43]. . . . . . . . . . . . . . . . . .

27

3.3

A schematic of the 20-inch PMT used in the ID [42]. . . . . . . . . .

28

3.4

Left: pulse height distribution for a single p.e. in a 20-inch PMT. The
peak near zero is from dark current. Right: relative transit time for a
20-inch PMT illuminated with light of 410 nm wavelength, for single
p.e. intensity. Both from [42]. . . . . . . . . . . . . . . . . . . . . . .

29

3.5

A schematic of the ID DAQ for SK-I-III [42]. . . . . . . . . . . . . . .

32

3.6

A schematic of the OD DAQ for SK-I-III [43]. . . . . . . . . . . . . .

33

3.7

A schematic of the SK-I-III trigger system [43]. . . . . . . . . . . . .

34

xxi

3.8

The SK water purification system [42]. . . . . . . . . . . . . . . . . .

3.9Attenuation measurements and fitted curves using the nitrogen laser [50].

Measured values for SK-IV are similar. . . . . . . . . . . . . . . . . .
vs. ` (in cm) for a typical run (top); variation of the
3.10 Plot of log Q(`)·`
f (θ)

36

38

attenuation length in the SK-I-III periods (bottom). Both from [50].
The time variation of the length is sensitive to water quality, and timedependent attenuation lengths are used in data processing. . . . . . .

39

3.11 A plot of the time vs. charge for a typical readout channel [49]. The
time of each hit is corrected for time of flight. . . . . . . . . . . . . .

43

3.12 Left: double ratio RM C /Rdata , where R is the ratio of measured momentum to momentum inferred from the Cherenkov angle, for low momentum stopping muons. Right: ratio of momentum to range for highmomentum stopping muons. Both from [42] . . . . . . . . . . . . . .

45

3.13 Momentum of decay electrons for data (crosses) and MC (solid histogram) [42].

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

3.14 Invariant mass of the π 0 sample (two showering rings), between data(crosses) and MC (boxes) [42]. . . . . . . . . . . . . . . . . . . . . . .

46

3.15 Energy scale uncertainty vs. momentum range used for calibration
measurements in SK-I [42], in order of increasing energy: decay electrons (solid circles), single π 0 ’s (solid triangles), low-energy stoppingmuons (open squares), and high-energy stopping muons (solid squares). 47

xxii

4.1

The primary cosmic ray flux for protons (at solar minimum), as calculated in [54]. Overlaid data points (see [54] for references) comeWebber (crosses), MASS (open circles), LEAP (closed upward triangles), IMAX (closed downward triangles), CAPRICE-94 (closed vertical diamonds), CAPRICE-98 (open vertical diamonds), BESS (closed
circles), AMS (closed horizontal diamonds), BESS-TeV (open upward
triangles), Ryan et al (closed horizontal diamonds), JACEE (large open
downward triangles), RUNJOB (large open diamonds), Ivanenko etal. (large open upward triangles), and Kawamura et al. (large open
squares). Dashed lines indicate calculations from a fitting procedure,
and the solid line includes additional corrections explained in [54]. . .

4.2

50

The atmospheric neutrino flux (left) and neutrino flavor ratios (right)
up to 50 GeV. The solid line indicates the calculation of [53], thedot-dashed line an earlier calculation from Honda et al. [55], the long
dashed line the BARTOL calculation [57], and the short dashed line
the FLUKA calculation [56]. . . . . . . . . . . . . . . . . . . . . . . .

4.3

Cross-sections for various CC1π interactions, with solid lines showing
the fit used in NEUT [58]. See [58] for details on the data points. . .

4.4

51

54

Total charged-current cross-section for neutrinos (upper lines) and antineutrinos (lower lines). The lines show a comparison of different parton distribution functions, with and without corrections from Bodek
and Young [64]. See [58] for details on the data points. . . . . . . . .

4.5

56

Pion interaction cross sections as a function of lab momentum, for π +(left) and π − (right) [68]. The “TunedFSI” plots are used in the mostrecent MC simulation. See [68] for details on the data points. . . . . .

xxiii

57

5.1

Fermi momentum predictions (solid line) and data (crosses) from [73].
The left shows a p-state nucleon, and the right an s-state in 12 C. . . .

5.2

62

Invariant mass for pp → π + π + . The modes pn → π + π 0 and nn → π 0 π 0are essentially identical. The dotted histogram shows events with acorrelated decay. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3

63

The integrand in the numerator of Equation 2.14 for an s-state andp-state neutron in 16 O. The solid and dotted lines show calculations
for two different approximations. . . . . . . . . . . . . . . . . . . . .

5.4

65

Charged, neutral, and total pion multiplicity (top left, top right, andbottom left), and average charged pion momentum (bottom right) forn → n̄ MC. The blue histogram shows the simulation of [23], the redhistogram shows the simulation using the same branching fractions
as [23] but the updated pion-nucleon model described in Section 4.2.4,
and the green histogram shows the simulation using new branching
fractions and the updated pion model. . . . . . . . . . . . . . . . . .

6.1

66

The second reduction cut 2a) for data (top), MC (middle), and MC normalized to data events after the selection (bottom) in SK-IV. The small
number of events for which NHITA800 > 25 have PEtotal > 100,000.
Taken from [61]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.2

69

The goodness parameter (Equation 6.1) for thrugoing (top) and stopping muons (bottom) in SK-IV, normalized to data taken over twodays (red crosses). The blue histogram shows data over the previous

6.3

two months. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

A schematic of one of the cable holes with a veto counter.

73

xxiv

. . . . . .

6.4

Typical distributions of NMIN100 for a flasher (top) and an atmospheric
neutrino MC event (bottom). The arrows indicate the region that isremoved with the third reduction flasher cut. Taken from [61]. . . . .

6.5

74

A plot of the event rate across each reduction stage in SK-IV, with first(black), second (red), third (cyan), fourth (dark blue), fifth (yellow),and FCFV (magenta). There is a slight gain increase with time mostapparent in the first two reductions. This is due to degrading PMT
quality, and is accounted for in event reconstruction. . . . . . . . . .

6.6

79

The Hough transform method. The grey area corresponds to hit PMTs,
and the dashed circles are drawn around each hit PMT with a 42-degree
opening angle. The ring center is determined by the point at which
the most circles overlap. . . . . . . . . . . . . . . . . . . . . . . . . .

6.7

82

Ring-counting likelihod in SK-IV, for the sub-GeV (Evis < 1.3 GeV)
sample. The MC is normalized to data events (crosses) and separated
by neutrino interaction type. . . . . . . . . . . . . . . . . . . . . . . .

6.8

83

A simulated µ+ (top) and e+ (bottom), both with momentum 800
MeV/c. The charge pattern is notably sharper for the µ+ and morediffuse for the e+ . The intensity scale indicates the amount of charge
deposited on the PMTs. The main display shows the ID, while thesmall display on the top right shows the OD, in which there is negligible activity. Both displays show the detector “unrolled” from its
cylindrical shape. The white +’s indicate the horizontal and verticallocation of the reconstructed vertex, projected onto the detector wall.

6.9

85

Distributons of RTOT vs. momentum up to 2000 MeV/c for bothe-like and µ-like rings. . . . . . . . . . . . . . . . . . . . . . . . . . .

xxv

89

7.1

An example tree for the pn → π + π 0 analysis. The tree has a depth offour. The variables used are the reconstructed π 0 momentum (“pi0ptot”),
the visible energy (“evis”), the π 0 − π + opening angle (“opang”), andthe ratio of charge carried in the highest-energy ring (“qratio”), all ofwhich are described in the text. Nodes that split to the left are moresignal-like, and nodes that split to the right more background-like. . .

7.2

93

Selection criteria (A1) (top left), (A3) (top right), and (A4) (bottom
left) for pp → π + π + MC (blue histogram), atm.-ν background (red histogram), and data (crosses) for SK-IV. All plots are area-normalized tothe data. The green line indicates the placement of the cuts describedin the text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.3

98

The angle between µ-like rings, the strongest discriminatory for the
pp → π + π + search, for signal MC (solid histogram), atm.-ν MC (dashedhistogram), and data (crosses). Distributions are shown for SK-IV andnormalized to the number of events in the data. . . . . . . . . . . . . 102

7.4

The ratio of charge carried by the leading ring (top left), the visibleenergy (top right), the maximum distance between the primary vertex
and a decay electron vertex (middle left), the largest angle between aµ-like ring and decay electron vertex (middle right), the magnitude of
the vector sum of RTOT (bottom left), and the number of Cherenkov
rings (bottom right) for the pp → π + π + search, for signal MC (solid
histogram), atm.-ν MC (dashed histogram), and data (crosses). Distributions are for SK-IV, and are normalized to the number of eventsin the data. Details of each variable are described in the text. . . . . 103

xxvi

7.5

The number of decay electrons (left) and the number of µ-like rings(right) for the pp → π + π + search, for signal MC (solid histogram),atm.-ν MC (dashed histogram), and data (crosses). Distributions areshown for SK-IV, and are normalized to the number of events in thedata. Details of each variable are described in the text. . . . . . . . . 104

7.6

BDT output for pp → π + π + signal (solid), atm.-ν background (dashed),
and data (crosses). The vertical green line indicates the BDT cut value,
and the arrow indicates that only events to the right of the cut are kept.104

7.7

A pp → π + π + MC event. The two solid orange rings correspond totrue π + ’s, and have an angle of 164 degrees between them. The dashedcyan ring is fit as e-like, but is actually a hard scatter of one of the π + ’s.106

7.8

Surviving atmospheric neutrino MC event for the pp → π + π + search.
The interaction type is νµ charged-current single π + production. The
uppermost µ-like ring corresponds to a µ− , while the smaller µ-like ringis a π + . The e-like ring is mis-fit, as it comes from a hard scatter ofthe π + . The angle between the two µ-like rings is 170 degrees, thereby
mimicking the main event signature. The momentum of the muon µlike ring is 402 MeV/c, and the momentum of the π + µ-like ring is301.3 MeV/c

7.9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

pp → π + π + candidate event in SK-II. The two µ-like rings are strikingly back-to-back, at almost exactly 180 degrees to each other. The
e-like ring could be due to a hard scatter. There is one Michel electron,
not shown. The µ-like rings have momenta of 399.0 MeV/c (solid) and
280.5 MeV/c (dashed), and the e-like ring has momentum 42.0 MeV/c.
The visible energy is 387.0 MeV, the total momentum 124.2 MeV/c,and the invariant mass 710.9 MeV/c2 . . . . . . . . . . . . . . . . . . . 107
xxvii

7.10 pp → π + π + candidate event in SK-IV. The two µ-like rings are at anangle of 151 degrees to each other, with momenta of 541.4 and 389.5
MeV/c. The dashed e-like ring could be due to a hard scatter. Thereare 2 Michel electrons, not shown. The visible energy is 742.8 MeV,
the total momentum is 445.8 MeV/c, and the invariant mass is 1022.3
MeV/c2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.11 Selection criteria (B1) (top left), (B3) (top right), (B4) (bottom left)and (B5) (bottom right) for pn → π + π 0 MC (blue histogram), atm.-ν
background (red histogram), and data (crosses) for SK-IV. All plots arearea-normalized to the data. The green line indicates the placement of
the cuts described in the text. . . . . . . . . . . . . . . . . . . . . . . 109
7.12 The total momentum of the π 0 candidate in SK-IV, the best discriminating variable for the pn → π + π 0 search, for signal MC (solid histogram), atm.-ν MC (dashed histogram), and data (crosses). Distributions are normalized to the number of events in the data. . . . . . 113
7.13 The opening angle between the π 0 and π + candidate rings (top left),the momentum of the π + candidate (top right), the invariant massof the π 0 candidate (middle left), the ratio of charge carried by themost energetic ring (middle right), the visible energy (bottom left),
and the number of decay electrons (bottom right) for the pn → π + π 0
search in SK-IV, for signal MC (solid histogram), atm.-ν MC (dashedhistogram), and data (crosses). Distributions are normalized to thenumber of events in the data. Details of each variable are described in
the text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

xxviii7.14 BDT output for pn → π + π 0 signal (solid), atmospheric neutrino background (dashed), and data (crosses). The green vertical line indicatesthe BDT cut value, and the arrow indicates that only events to theright of the cut are kept. One candidate event can be seen in the SK-I
data distribution, just to the right of the cut at 0.19. . . . . . . . . . 115
7.15 A pn → π + π 0 MC event. The µ-like ring corresponds to a true π + .
The e-like ring contains two overlapping γ’s from a π 0 . The angle
between the µ- and e-like rings is 165 degrees. The e-like ring has
a reconstructed momentum of 893 MeV/c, while the original π 0 had
momentum 876 MeV/c. . . . . . . . . . . . . . . . . . . . . . . . . . 116
7.16 A surviving atmospheric neutrino MC event for the pn → π + π 0 search.
The interaction type is CC1π 0 , νµ n → µ− π 0 p. The two overlapping
e-like rings are photons from a π 0 , with total momentum 939 MeV/c.
The µ-like ring corresponds to the µ− , with momentum 410 MeV/c.
The angle between them is 151 degrees, thereby appearing back-to-back.116
7.17 The pn → π + π 0 candidate in SK-I. The e-like and µ-like rings are atan angle of 140 degrees. The e-like ring has a high momentum of 987
MeV/c, but its fit π 0 mass is low, at 10 MeV/c2 . The µ-like ring has amomentum of 460.0 MeV/c. The total momentum and invariant mass
are 700.1 MeV/c and 1266.4 MeV/c2 , respectively. . . . . . . . . . . . 117
7.18 Total momentum (left) and invariant mass (right) for data (crosses),nn → π 0 π 0 MC (solid histogram), and atmospheric neutrino MC(dashed histogram). Cuts (C1)-(C3) are applied to both plots. The
invariant mass cut (C5) has been applied to the total momentum, andthe total momentum cut (C4) to the invariant mass. The thin green
vertical lines indicate the locations of the final cuts for each variable.
xxix

119

7.19 The total momentum and invariant mass distributions for nn → π 0 π 0MC (left), atmospheric neutrino MC (center), and data (right) combined for SK I-IV. Events have passed cuts (C1)-(C3). The solid box
indicates the total momentum and invariant mass cuts ((C4) and (C5)in the text). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.20 A nn → π 0 π 0 MC event. The two overlapping e-like rings on the top
left correspond to two γ’s from a π 0 decay, and the other ring also
corresponds to a π 0 . The true momentum of the π 0 corresponding to
the two-ring fit is 872.7 MeV/c, and the true momentum of the π 0 corresponding to the one-ring fit is 936.7 MeV/c. The total reconstructed
momentum of this event is 219 MeV/c, and the reconstructed invariant
mass is 1793 MeV/c2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.21 Surviving atmospheric neutrino MC event for the nn → π 0 π 0 search.
The interaction is νµ neutral current deep-inelastic scattering. In particular, the interaction produced a π 0 and an η meson, from which four
γ’s are produced. All but the left-most ring correspond to true photons from the π 0 and η meson. Many additional charged pions are also
produced, concentrated near the center of the display, thereby complicating reconstruction. The left-most ring corresponds to a charged
pion. There is a ring near the center from multiple, closely overlapping charged pion tracks, which is not found by the reconstruction
program. The total reconstructed momentum is 553 MeV/c, and thereconstructed invariant mass is 1763 MeV/c2 . . . . . . . . . . . . . . 122

xxx

7.22 The BDT output for pn → π + π 0 signal, for standard (solid histogram)and FSI-varied MC (dashed histogram). The varied MC output isfrom the decreased hadron production set of variations. This decreasein hadron production leads to a clearer signal. . . . . . . . . . . . . . 125
7.23 The invariant mass distribution of the nn → π 0 π 0 atm.-ν backgroundin SK-IV, after selections (C1)-(C4), with a function of the form exp (p0 + p1 x)
fit to the region 600 MeV/c2 < Minv < 2000 MeV/c2 . The uncertainty
was estimated by integrating the function between the invariant mass
cut values, and finding the difference in the integral when the cut values
were shifted. A similar procedure was done for the total momentum,
and the larger value was taken for the uncertainty. . . . . . . . . . . . 128
8.1

Selection criteria (D1)-(D3) for n → n̄ MC (blue histogram), atm.-ν
background (red histogram), and data (crosses) in SK-IV. All plots arearea-normalized to the data. The green line indicates the placement of
the cuts described in the text. . . . . . . . . . . . . . . . . . . . . . . 134

8.2

A comparison of the atm.-ν weight using two-flavor oscillation andsmall corrections from the flux in [53] (blue curve), and the atm.-ν
weight using the full oscillation analysis in [86] (red curve), for all
atm.-ν events passing the selection pre-cuts (D1)-(D3). . . . . . . . . 136

xxxi

8.3

The total momentum (top left), invariant mass (top right), (Etot Eprimary )/Etot (middle left), PID likelihood difference of the leading
ring (middle right), number of decay electrons (bottom left), and maximum distance between the primary vertex and a decay electron vertex(bottom right) for the n → n̄ search in SK-IV, for signal MC (solid
histogram), atm.-ν MC (dashed histogram), and data (crosses). Distributions are normalized to the number of events in the data. Detailsof each variable are described in the text. . . . . . . . . . . . . . . . . 139

8.4

The sphericity (top left), number of potential rings (top right), first
Fox-Wolfram moment (bottom left), and second Fox-Wolfram moment(bottom right) for the n → n̄ search in SK-IV, for signal MC (solid
histogram), atm.-ν MC (dashed histogram), and data (crosses). Distributions are normalized to the number of events in the data. Detailsof each variable are described in the text. . . . . . . . . . . . . . . . . 140

8.5

BDT output for n → n̄ signal and atm.-ν background (top), and theremaining signal efficiency and scaled background in SK-I as a functionof BDT cut (bottom). The BDT outputs are area-normalized

8.6. . . . 141

atm.-ν BDT output for SK-I MC, categorized by neutrino interaction
mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

8.7

Lifetime limit vs. BDT cut assuming the number of candidates is equal
to the number of background events. The value is maximized for a cutof 0.35. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

8.8

BDT output for n → n̄ signal (blue), atm.-ν background (red), anddata (crosses). All histograms are normalized to the number of data
entries. The vertical green line indicates the BDT cut value, and thearrow indicates that only events to the right of the cut are kept. . . . 146
xxxii

8.9

Remaining events as a function of the BDT cut, for data (crosses) andatm.-ν MC (red histogram). The atm.-ν MC is normalized by theappropriate livetime. The cyan curves show the one-sigma systematic
uncertainties (Table 8.6). . . . . . . . . . . . . . . . . . . . . . . . . . 149

8.10 Total momentum in SK-I, for data (crosses), atm.-ν MC (red), andatm.-ν MC shifted to match the data (green). . . . . . . . . . . . . . 151
8.11 Remaining events as a function of the BDT cut, for data (crosses) andatm.-ν MC (red histogram). The atm.-ν MC is normalized by theappropriate livetime. The cyan curves show the one-sigma systematic
uncertainties (Table 8.6). . . . . . . . . . . . . . . . . . . . . . . . . . 151
9.1

Total momentum vs. invariant mass following selections A) and B)
for n → n̄ signal (top), atm.-ν background (middle), and SK-I data(bottom) for the analysis in [23], with boxes indicating selections C)
and D). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

B.1 The angle between µ-like rings (left) and the ratio of charge carriedby the most-energetic ring (right) for the pp → π + π + search, for signal MC (solid histogram), atm.-ν MC (dashed histogram), and data(crosses). Distributions are normalized to the number of events in thedata. Details of each variable are described in the text, in Section 7.2.2.167
B.2 The visible energy (left) and the maximum distance between the primary vertex and a decay electron vertex (right) for the pp → π + π +search, for signal MC (solid histogram), atm.-ν MC (dashed histogram),and data (crosses). Distributions are normalized to the number of
events in the data. Details of each variable are described in the text,
in Section 7.2.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
xxxiiiB.3 The largest angle between a µ-like ring direction and a decay electron
vertex (left) and the magnitude of the vector sum of RTOT (right)
for the pp → π + π + search, for signal MC (solid histogram), atm.-ν
MC (dashed histogram), and data (crosses). Distributions are normalized to the number of events in the data. Details of each variable aredescribed in the text, in Section 7.2.2. . . . . . . . . . . . . . . . . . . 169
B.4 The number of Cherenkov rings (left) and the number of decay electrons (right) for the pp → π + π + search, for signal MC (solid histogram), atm.-ν MC (dashed histogram), and data (crosses). Distributions are normalized to the number of events in the data. Details ofeach variable are described in the text, in Section 7.2.2. . . . . . . . . 170
B.5 The number of µ-like rings for the pp → π + π + search, for signal MC(solid histogram), atm.-ν MC (dashed histogram), and data (crosses).
Distributions are normalized to the number of events in the data. Details are described in the text, in Section 7.2.2. . . . . . . . . . . . . . 171
B.6 The π 0 candidate momentum (left) and the angle between the π 0 andπ + candidate rings (right) for the pn → π + π 0 search, for signal MC(solid histogram), atm.-ν MC (dashed histogram), and data (crosses).
Distributions are normalized to the number of events in the data. Details of each variable are described in the text, in Section 7.3.2. . . . . 172
B.7 The π + candidate momentum (left) and the invariant mass of the π 0candidate (right) for the pn → π + π 0 search, for signal MC (solid histogram), atm.-ν MC (dashed histogram), and data (crosses). Distributions are normalized to the number of events in the data. Details ofeach variable are described in the text, in Section 7.3.2. . . . . . . . . 173

xxxiv

B.8 The ratio of charge carried by the most energetic ring (left) and thevisible energy (right) for the pn → π + π 0 search, for signal MC (solid
histogram), atm.-ν MC (dashed histogram), and data (crosses). Distributions are normalized to the number of events in the data. Detailsof each variable are described in the text, in Section 7.3.2. . . . . . . 174
B.9 The number of decay electrons for the pn → π + π 0 search, for signal MC (solid histogram), atm.-ν MC (dashed histogram), and data(crosses). Distributions are normalized to the number of events in thedata. Details of each variable are described in the text, in Section 7.3.2.175
B.10 Total momentum (left) and invariant mass (right) for data (crosses),nn → π 0 π 0 MC (solid histogram), and atmospheric neutrino MC(dashed histogram). Cuts (C1)-(C3) are applied to both plots. The
invariant mass cut (C5) has been applied to the total momentum, andthe total momentum cut (C4) to the invariant mass. The thin green
vertical lines indicate the locations of the final cuts for each variable.
Details of the selection criteria can be found in Section 7.4.1. . . . . . 176
B.11 The total momentum (left) and invariant mass (right) for the n →n̄ search, for signal MC (solid histogram), atm.-ν MC (dashed histogram), and data (crosses). Distributions are normalized to the number of events in the data. Details of each variable are described in thetext, in Section 8.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
B.12 (Etotal -Eprimary )/Etotal (left) and the PID likelihood difference of theleading ring (right) for the n → n̄ search, for signal MC (solid histogram), atm.-ν MC (dashed histogram), and data (crosses). Distributions are normalized to the number of events in the data. Details ofeach variable are described in the text, in Section 8.2. . . . . . . . . . 178
xxxv

B.13 The number of decay electrons (left) and the maximum distance fromthe primary vertex to a decay electron vertex (right) for the n →n̄ search, for signal MC (solid histogram), atm.-ν MC (dashed histogram), and data (crosses). Distributions are normalized to the number of events in the data. Details of each variable are described in thetext, in Section 8.2. The distance is only shown for events with one ormore decay electron. . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
B.14 The sphericity (left) and the number of potential rings (right) for then → n̄ search, for signal MC (solid histogram), atm.-ν MC (dashedhistogram), and data (crosses). Distributions are normalized to thenumber of events in the data. Details of each variable are described in
the text, in Section 8.2. . . . . . . . . . . . . . . . . . . . . . . . . . . 180
B.15 The first and second Fox-Wolfram moments (left and right, respectively) for the n → n̄ search, for signal MC (solid histogram), atm.-ν
MC (dashed histogram), and data (crosses). Distributions are normalized to the number of events in the data. Details of each variable aredescribed in the text, in Section 8.2. . . . . . . . . . . . . . . . . . . . 181

xxxvi

List of Abbreviations
1π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Single Pion Production
ADC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analog to Digital Converter
ATM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analog Timing Module
B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Baryon Number
BDT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Boosted Decision Tree
BNV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Baryon Number Violation
CC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Charged Current
DAQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Data Acquisition
DIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deep Inelastic Scattering
FC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fully contained
FSI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Final State Interaction
FV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Fiducial volume
GUT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Grand Unified Theory
ID . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Inner Detector
L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lepton Number
MC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Monte Carlo
NC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Neutral Currentn → n̄ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Neutron-antineutron oscillation
N N → ππ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dinucleon decay to pions
OD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Outer Detector
p.e. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . photoelectron
PID . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . particle identification
PMT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Photomultiplier TubeQBEE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . QTC Based Electronics with Ethernet

xxxviiQE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quasi-elastic
QTC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Charge to Time Converter
SK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Super-Kamiokande
SM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Standard Model
TDC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Time to Digital Converter
TKO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tristan KEK Online

xxxviii

Chapter 1
Introduction
The basic stability of matter is an intuitive necessity. In the language of particle
physics, everyday matter is composed of baryons, protons and neutrons that fill atomic
nuclei. While certain unstable atoms are known to decay, the decay of individual
baryons would make life as we know it impossible.
Thankfully, matter appears to be stable for all practical purposes. More specifically, baryon number appears to be conserved — that is, the total number of baryons
before and after an interaction, such as a decay or a high-energy collision, is constant.
Some physicists have suggested that baryon number is absolutely conserved, in thesame manner as properties like energy, momentum, and electric charge. Even those
who have predicted that baryons are ultimately unstable have predicted that baryons
decay with lifetimes of 1029 years or higher — roughly 1019 times longer than the
known universe.
The apparent stability of matter does not imply that it is absolutely stable, andthus baryon number is not necessarily an absolutely conserved quantity. There arecompelling theoretical reasons to believe that baryon number is not exactly conserved,
and the attempt to observe a baryon number non-conserving process has been a1

2
continuous experimental goal for decades. The discovery of such a process would bea spectacular confirmation of new physics, while non-observation places important
constraints on the theories that predict such processes. The goal of this thesis is toprobe a subset of these processes (“B” indicates the baryon number):• Dinucleon decay to pions, the simultaneous decay of two bound nucleons (B =2) into pions (B = 0), encompassing the decay channels 16 O(pp) → 14 C(π + π + ),16O(pn) → 14 N(π + π 0 ), and 16 O(nn) → 14 O(π 0 π 0 );• Neutron-antineutron oscillation, the transformation of a neutron (B = 1) into
its antiparticle (B = −1).
Each of these processes have been sought by experiments before. They have been
constrained to occur on timescales on the order of 1030 years or longer, depending on
the specific channel.
The search in this thesis follows the same basic logic as previous searches. Since
an experiment cannot reasonably run for longer than a few decades, it must consistof as large a sample of matter as possible. The potential to discover these processes(or further constrain them) scales with the product of the number of nucleons andthe running time of the experiment. In this thesis, data from the Super-Kamiokande
water Cherenkov detector is used. With a tank volume containing 7.5×1032 watermolecules and a running time of nearly two decades, Super-Kamiokande has by far
the largest exposure of any comparable experiment. The full dataset is used for thesesearches.
This thesis is organized as follows. In Chapter 2, I discuss the underlying theory and motivation of these searches, as well as relevant results from previous experiments. In Chapter 3, I describe the Super-Kamiokande detector, including its
basic structure, data acquisition system, and calibration sources. In Chapter 4, I

3describe the simulation of atmospheric neutrino Monte Carlo, a simulation used by
all analyses that use Super-Kamiokande data, and the main source of background tothese searches. In Chapter 5, I describe the simulation of both dinucleon decay topions and neutron-antineutron oscillation. Chapter 6 describes the data reductionand event reconstruction algorithms common for all data and Monte Carlo samples.
Chapter 7 describes the dinucleon decay searches; the features of each search are distinct, and prominently include multivariate analysis techniques. Chapter 8 discusses
the neutron-antineutron oscillation search. In Chapter 9, I present and discuss theresults.

Chapter 2
Motivation and Theory
Observation of neutron-antineutron oscillation or the related process of dinucleon
decay to pions can illuminate profound issues inexplicable within our current understanding of the Standard Model of particle physics. Here I describe the underlying
motivation and theory of these searches, as well as relevant previous experimental
results.

2.1

Baryon Number

Baryon number is defined in terms of the quark content of a system of particles:B=1(nq − nq̄ ) ,
3

(2.1)

where q and q̄ refer to quarks and antiquarks, respectively. Thus individual quarks(antiquarks) have B = 1/3 (B = −1/3); baryons, made of three quarks, have B =1; antibaryons have B = −1; mesons, composed of a quark and antiquark, have
B = 0; and leptons, which have no quarks or antiquarks, have B = 0. Though notfundamental to this thesis, lepton number is defined similarly:4

5

L = n` − n`¯,

(2.2)

where ` and `¯ refer to leptons and antileptons, respectively.
The conservation of baryon number was first suggested by Weyl in 1929 [1]. It
was framed as a conservation of both positive and negative charges; given the known
particles of the time, this meant the conservation of proton number and electron
number. The extension to baryons was made by Stueckelberg [2] and Wigner [3].
The necessarily long minimum lifetime of baryonic matter was known, in a roughquantitative sense, before any dedicated experiment. For instance, Goldhaber deduced as far back as the 1950s that the proton lifetime had to be at least 1016 years,
because if it were any shorter than this, radiation due to energetic particles from decaying protons in the ∼1027 nucleons in the human body would exceed the maximum
dosage for radiation workers [4]. The first experiment probing nucleon stability was
indirect, and looked for the spontaneous fission of 232 Th resulting from excitationfrom the decay of one of its nucleons. As no such fission was observed, this method
resulted in a lower nucleon lifetime of 2 × 1023 years [5]. A detailed history of nucleon
decay experiments can be found in [6].

2.2

The Standard Model

The Standard Model of particle physics has been spectacularly successful in explaining the subatomic universe as we know it. Like any physical theory, the Standard
Model has various conserved quantities. The conservation of some of these quantities
is exact, such as energy, momentum, and angular momentum. Noether’s theorem
dictates that any conserved quantity is associated with an underlying symmetry —6more formally, if the action is invariant under a group of symmetry transformations,
there is a corresponding conserved current. For the absolute quantities above, thecorresponding symmetries are translation in time, translation in space, and rotationin space, respectively.
Aside from these absolutes, quantities conserved within the Standard Model areassociated with its group structure:

GSM ⊃ SU (3)C × SU (2)L × U (1)Y .

(2.3)

Here SU (3)C encodes the color conservation of the strongly-interacting quarks andtheir messenger particles, gluons. The group SU (2)L characterizes the weak force,with the L indicating that only left-handed particles (or right-handed antiparticles)interact with this force. The conserved quantity here is weak isospin, associated withthe three SU (2) gauge bosons. Finally, U (1)Y conserves the weak hypercharge, which
is defined in terms of both the weak isospin and the more familiar electromagnetic
charge QEM . It has been well-known since the pioneering work of Glashow, Weinberg,
and Salam [7–9] that the subgroup SU (2)L × U (1)Y spontaneously breaks to a single
U (1)EM that conserves the electromagnetic charge, while the three weak SU (2) bosons
and single U (1) hypercharge boson mix to create the massive charged W bosons, themassive neutral Z boson, and the massless neutral photon. The Higgs mechanism bywhich the bosons obtain mass was recently vindicated with the thrilling discovery byboth the ATLAS [10] and CMS [11] detectors of a neutral particle consistent with theStandard Model Higgs boson.
The Standard Model gauge group reveals some interesting features about thenature of conserved quantities. For instance, even though parity was once thought tobe conserved in fundamental interactions, the fact that the SU (2)L weak force only

7
interacts with left-handed particles means that it violates parity (or spatial inversion)
maximally. It is natural to question whether other quantities may be non-conserved,
even if they appear conserved in all observations thus far. One such quantity is baryon
number.
Although no baryon number violating process has been observed, baryon number
is not strictly conserved within the Standard Model1 . Whether such processes occur
at an observable level is still an interesting question, and both neutron-antineutron
oscillation and dinucleon decay to pions (hereafter “n → n̄ ” and “N N → ππ ”)
violate baryon number by two units.

2.3

Baryogenesis and Baryon Asymmetry

Despite its success, there are several unresolved issues within the Standard Model,
as well as explanations that are incomplete or unsatisfactory. One such issue is theobserved excess of matter to antimatter. Indeed, the level of antimatter in the universeis negligible. In the Standard Model, processes which produce particles and theirassociated antiparticles should occur at equal rates, thus leading to no net excess.
Baryogenesis - a set of processes and conditions that produced the observed excess inbaryons to antibaryons - is thus difficult to explain.
Sakharov [12] developed three necessary conditions for generating this imbalance,
often called the Baryon Asymmetry of the Universe (BAU):
I. There must be one or more baryon number violating (BNV) processes.
II. There must be charge and charge-parity violation. If there weren’t, any baryon
number violating process would be matched by its charge and charge-parity
1

If it were, Equation 2.3 would have an additional U(1) symmetry for baryon number, analogous
to weak hypercharge.

8
conjugate processes, thereby leading to no net (anti-)baryon excess.
III. There must be interactions outside of thermal equilibrium. At thermal equilibrium, the energy difference between baryons and anti-baryons is zero, andthus the Boltzmann distribution indicates that there should be equal amountsof matter and anti-matter.
BNV processes have been studied extensively. In particular, single nucleon decay
has been a topic of strong experimental interest for decades, with the most stringent
results coming from Super-Kamiokande [13–17]. Modes such as p → e+ π 0 , in addition
to having a clear experimental signature, are predicted to occur at an observable rate
in Grand Unified Theories (GUTs). Such GUTs predict that the three subatomic
forces of the Standard Model - strong, weak, and electromagnetic - should have equal
strength at an unobservably high energy of ∼ 1014 −1016 GeV. This scale is determinedfrom an extrapolation of the observed change with energy of the strong, weak, andelectromagnetic coupling constants, to the point where they become approximately
equal. For many physicists, these theories provide satisfying closure to the seemingly
arbitrary differences present in the three forces at observable energy scales. Though
the current lower lifetime on p → e+ π 0 of 1.2 × 1034 years [18] has ruled out thesimplest GUTs, such as those based on the gauge group SU (5) [19], an unstable
proton (or bound neutron) is the only observable aspect of many GUTs, and thus its
discovery would be a profound revelation.
In one sense, then, the search for processes like n → n̄ and N N → ππ are complementary to searches for single nucleon decay. Given the non-observation of single
nucleon decay despite many years of searches, any additional avenues are welcome.
It should also be noted that most single proton decay modes of interest violate both

9baryon number B and lepton number L by one unit, but conserve B − L2 . Bothn → n̄ and N N → ππ violate B by two units and do not involve L, so they also
violate B − L by two units. This is an important distinction. As mentioned, no BNVprocess has been observed, but such processes occur in the Standard Model. In theelectroweak component of the Standard Model, B and L are violated by anomalous“sphaleron” processes that convert three baryons to three antileptons, and vice-versa.
Such processes conserve B − L, however, and thus the difference B − L is a conserved
quantity of the Standard Model. In the current universe, these interactions are highly
suppressed. In the early universe, however, when the temperature was at or above
the electroweak scale T ∼ 100 GeV and below the GUT scale T ∼ 1014 − 1016 GeV,
such processes occurred rapidly, such that any previously generated baryon asymmetry that conserved B − L would have been washed out [20]. Thus, while proton
decay is extremely important as a signifier of GUT-scale physics, its conservation of
the quantity B − L makes it unlikely to contribute to the BAU. Neutron-antineutron
oscillation and dinucleon decay to pions remain good candidate processes.
There are other options for generating the BAU, apart from the ∆B = 2 processes analyzed in this thesis. One such option is baryogenesis via leptogenesis, firstsuggested in [20]. Leptogenesis is just the analogue to baryogenesis for leptons. This
option makes use of L-violating processes above the scale in which sphalerons are active, which are then converted to the BAU via the sphaleron mechanism of converting
antileptons to baryons. This scenario is attractive particularly in light of the strong
theoretical suspicion that neutrinos are Majorana particles that acquire mass via a∆L = 2 operator, a possibility that is being probed in neutrinoless double beta decay
experiments. The same logic to ∆B = 2 proceses is relevant, in that the Standard
2

An exception to this is a trilepton decay mode like p → e+ νν [14], which violates B − L by twounits assuming two normal neutrinos. In general, a single nucleon decay must have an odd numberof leptons in the final state, to conserve spin.

10
Model-conserved difference B − L is violated by two units.

2.4

ModelsFor a ∆B = 2 process, the Lagrangian requires an effective six-quark operator of
the form:O∆B=2 =

1
O6Q ,
Mn

(2.4)

where O6Q is an operator with two up-type and four down-type quarks, and M is amass scale that suppresses the operator. For all first-generation quarks, this operator
leads to n → n̄ and N N → ππ . The power of the mass scale n depends on additional
details of the operator. If all other parameters are dimensionless, it follows fromdimensional analysis that n = 5, since quark operators have dimension 3/2.
A crude estimation of the n → n̄ rate can be obtained from this operator, byabsorbing additional details of the operator into the suppression scale M . The (free)n → n̄ transition rate δm isδm =1
hn̄|U DDU DD|ni.
M5

(2.5)

The matrix element hn̄|U DDU DD|ni is difficult to calculate. Though there is considerable uncertainty, theorists typically estimate it to be Λ6 ≈ 1 × 10−4 GeV6 , or
Λ ≈ 200 MeV, based on a calculation in [21]. Thus the n → n̄ lifetime τnn̄ = 1/δm
depends only on the suppression scale M and a calculable, if uncertain, hadronicmatrix element:τnn̄ ≈

M5
.
Λ6

(2.6)

11
Again roughly, the current lower limit on the n → n̄ lifetime of about 108 s implies thatM > 106 GeV. This is an interesting scale - it is above the ∼10 TeV scale achievable
at the Large Hadron Collider, yet is well below the GUT scale that motivates searches
for many other BNV processes.
It should be noted that the operator in Equation 2.5 is not renormalizable. In
fact, no operator that causes BNV is renormalizable, and since the Standard Model
Lagrangian only has renormalizable operators, B is sometimes called an “accidental”
symmetry of the Standard Model. However, because the suppression scale M is muchhigher than the ∼100 MeV scale of nuclear interactions, the operator is valid in an
effective field theory context.
There are several broad classes of models that feature n → n̄ , some of which
explicitly forbid single proton decay [22]. For example, one model described in [22]includes the Lagrangian terms0
L = −g1ab X1αβ QaLα QbLβ − g2ab X2αβ daRα dbRβ − g1ab X1αβ uaRα dbRβ000

+ λX1αα X1ββ X2γγ αβγ α0 β 0 γ 0 .

(2.7)

Here Q is an SU (2)L quark doublet; uR and dR are right-handed SU (2) up- and down0

type quark singlets; g1 , g2 , and g1 are dimensionless couplings; and λ is a couplingwith dimension one. Roman indices indicate quark generation, and Greek indicesindicate color. The X1 and X2 particles are heavy scalars introduced by the modelthat couple to pairs of quarks, with X1 ∈ (6̄, 1, − 31 ) and X2 ∈ (6̄, 1, 23 ) representationsof the Standard Model SU (3) × SU (2) × U (1) gauge group. These terms lead toan effective operator shown in Figure 2.1. Note that the diagrams for n → n̄ andN N → ππ are the same under crossing symmetry; the latter simply requires spectator

12quarks of appropriate charge.
Following [22], the mixing element δm = 1/τn→n̄ , where τn→n̄ is the n → n̄ lifetime,
is calculated withδm = hn̄|Hef f |ni(2.8)(g 011 ) g 11 λHef f = − 1 4 22 dR dR uR dR uR dR × Ncontraction
4M1 M2

(2.9)

where
2

and Ncontraction is a factor accounting for all spin and color contractions. The mixing
element is estimated to be2

δm ≈ 2λΛ6(g1011 ) g211
3M14 M22.

(2.10)

Here Λ is the same hadronic matrix element discussed above. To get a rough senseof the lower bounds on the mass scale, consider the simple scenario g1011 = g211 = 1,
and M1 = M2 = λ = M . The upper limit on δm, is 1/τlim = 2.4 × 10−33 GeV, whereτlim = 2.7 × 108 s from [23]. Plugging this into 2.10, we get M > 600 TeV.
Some models that predict ∆B = 2 processes are driven specifically by the BAU.
One recent model is called “Post-Sphaleron Baryogenesis” [24–26]. As the nameimplies, the model provides a BNV mechanism that occurs after sphaleron processes
have been suppressed, when the universal temperature has cooled below 100 GeV or
so. The mechanism involves the direct decay of a scalar particle to six quarks, and aset of scalar particles like the Xi bosons from the generic model of [22] with masses
that cannot be much higher than the TeV scale. This model has a number of testablefeatures. It predicts an upper limit on the n → n̄ oscillation time τn→n̄ ≤ 5× 1010

13
d¯

d
X2
u

ū
X1

X1

d

d¯

q1

q1
d¯

u
d
d
u
q2

X1
X2
X1

d¯
q2

Figure 2.1: Feynman diagram for n → n̄ (top) and N N → ππ (bottom) from one of
the models discussed in [22]. Spectator quarks q1 , q2 ∈ (u, d) are required for
N N → ππ . The X1 and X2 scalar particles belong to (6̄, 1, − 13 ) and (6̄, 1, 23 )representations of the Standard Model SU (3) ⊗ SU (2) ⊗ U (1) gauge group.

s [26], which is about two orders of magnitude beyond the current limit. It is also
sensitive to the neutrino mass hierarchy, and is only valid if the so-called inverted
hierarchy is discovered by neutrino experiments [25]. Finally, the TeV-scale scalarparticles can be sought directly at the Large Hadron Collider.

2.5

Phenomenology and previous results

Here I discuss the phenomenology of n → n̄ and N N → ππ . Though the context
for this study is bound neutrons in 16 O, the phenomenology of free neutron oscillations
is instructive, and so I discuss it also.

14

2.5.1

nn̄ oscillations

Free Neutrons
The probability of a free neutron oscillating to an antineutron can be parametrizedwith a simple 2 × 2 Hamiltonian, as in a textbook example of a two-state system:

i









d  n   En δm   n 
 =
 
dt
n̄
δm En̄
n̄(2.11)

where n and n̄ denote the neutron and antineutron wavefunctions, En and En̄ theirenergies, δm = 1/τnn̄ a mixing element that provides the oscillation, and τnn̄ is theneutron oscillation time. From this, we obtain the probability for nn̄ oscillations:Pn→n̄ (t) =

√

δm2
2
2 + δm2 t
sin∆E∆E 2 + δm2(2.12)

where ∆E = En − En̄ . If the energy levels are degenerate, as is the case in past andproposed experimental contexts, the probability simplifies to Pn→n̄ (t) ≈ (δmt)2 =(t/τnn̄ )2 . Thus the probability scales simply with both the time of observation andthe neutron oscillation time.

Bound neutrons
To study the possibility of n → n̄ in nuclei, we must account for the different
nuclear potentials felt by neutrons and antineutrons. Here we follow the work ofFriedman and Gal [27].
The radial wavefunctions of the neutron and antineutron satisfy a coupled set ofdifferential equations:

15

−

~2 d2
~2 ` (` + 1)u(r)+uν`j (r) − Un (r)uν`j (r)ν`j
2µ dr22µr2−Eν`j uν`j (r) + δmwν`j (r) =
−

0

~2 d2
~2 ` (` + 1)w
(r)+wν`j (r)ν`j
2µ dr22µr2− (Un̄ (r) + iWn̄ ) wν`j (r) − Eν`j wν`j (r) + δmuν`j (r) = 0.
Here u and w are the radial wavefunctions for the neutron and antineutron, respectively; `, ν, and j are the principal, orbital, and magnetic quantum numbers; µ is
the reduced nuclear mass; δm is the mixing element mentioned in the free neutroncase; and Eν`j are the energy eigenvalues. The nuclear potentials corresponding to
the neutron and antineutron wavefunctions are real and complex, respectively, with
the imaginary part of the antineutron potential Wn̄ responsible for inducing oscillations. Though data on antineutron scattering is sparse, the identical interactions of(anti)-protons and (anti)-neutrons at the nuclear level enable Un̄ (r) to be fit with recent antiproton scattering data. The energy eigenvalues Eν`j are also complex, withEν`j = −Bν`j − iΓν`j /2, where Γ is the oscillation width. Equations 2.13 can bealgebraically manipulated and integrated over r from zero to infinity to produce anexpression for the width,Γν`j =−2δmR


Im wν`j (r)u∗ν`j (r) drR|uν`j (r)|2 dr(2.13)

Assuming the initial condition of a pure neutron state, the ratio |w/u| ∼ δm, andtherefore Γν`j ∼ δm2 . Equation 2.13 can be re-arranged, dropping higher order terms,
and to a good approximation Equation 2.13 is16Γν`j ≈

2δm2

R

W (r) |vν`j (r)|2 dr
R n̄2uν`j (r)dr(2.14)

where vν`j = wν`j /δm is the reduced antineutron wavefunction, and uν`j is approximated as real. Thus the disappearance width is proportional to the overlap of theneutron and antineutron wavefunctions (Equation 2.13), and the imaginary part of
the antineutron potential (Equation 2.14). Equation 2.14, or rather Γν`j /δm2 , can becalculated numerically for a given atomic state. Plots of the reduced wavefunction
and n̄ nuclear potential can be seen in Figure 2.2.
The average disappearance width per neutron is then

Γd =

1 X
nν`j Γν`jN ν`j(2.15)

where N is the number of neutrons in the atom and nν`j is the occupancy of stateν`j. For oxygen, N = 8, and there are four p3/2 , two p1/2 , and two s1/2 states. Thenthe nuclear disappearance time Td is
1
=
Td =
Γd



1
2
Γd τnn̄



2
2
τnn̄
= Rτnn̄.

(2.16)

R is typically called the nuclear suppression factor. The numerical value of R is
calculated in [27] for oxygen as 0.517 × 1023 s−1 .
One can arrive at the same approximate result as the careful analysis of [27] using
a simpler argument. Neutrons bound in nuclei are quasi-free for a time tf ∼ 1/Eb ,where Eb ≈ 30 MeV is a typical nuclear binding energy. During this time, theprobability of oscillation is given by Equation 2.12. This quasi-free condition occurs
1/tf times per second, and the nuclear oscillation probability per second Pnuc is then

17

−

16

1p1/2 n in O
0.005v(r) (fm−1/2MeV−1)

Real−0.005Imag.

−0.015

0

2

4

6

8

10

r (fm)
FIG. 1: Antineutron 1p1/ 2 reduced radial wavefunction v in 16 O.
0.020

0

−

16

n in− O162___ 2W |v|−−− −2 u Im(v)n in O
1p1/2

0.015

potentials (MeV)

MeV−1 fm−1

−50

Real

0.010

1s1/2

−100

Imag.

0 005

−150

−200

0

2

4

6

r (fm)
FIG. 3: T he antineutron optical potential in 16 O.

Figure 2.2: The reduced n̄ radial wavefunction (top) and the n̄ nuclear potential for a0
p1/2 -state n̄ (bottom),−
56from [27].
n in Fes (MeV)−50

Pnuc (n → n̄) =

1
Tnuc


≈

tf
τnn̄

2


×

1
tf


.

(2.17)

2. Thus, the nuclear nn̄ transition time Tnuc can be estimated as Tnuc = Rτnn̄
, withR = 1/tf = Eb /~ ≈ 0.45 × 1023 s−1 .

18Experimental issues - free vs. boundBoth free and bound n → n̄ searches have been conducted in the past. There are
relative advantages and drawbacks to each, which I describe briefly here.
Free and bound n → n̄ searches rely on the same basic experimental signature. A
neutron that has oscillated will quickly annihilate with a nearby nucleon, resulting in
multiple pions. For bound neutrons this is just a nucleon in the surrounding atom,and for free neutrons this is a nucleon contained in a target material.
A typical free neutron experimental setup consists of a beam of non-relativistic
neutrons traveling through a de-magnetized vacuum chamber3 , with an annihilation
target material at the end. The sensitivity of such a setup depends on the flight timet and the number of neutrons observed N , with a figure of merit N t2 , where thesquare dependence on t follows from Equation 2.12. The most sensitive free neutron
experiment to date was performed at Grenoble [29], for which the flight time wast ∼ 0.1 s and the flux of neutrons was Φ ∼ 1011 neutrons/second. The experiment
ran for about a year, thus producing a total of N ∼ 3 × 1018 neutrons. With nocandidates seen, the lower lifetime for n → n̄ was set at τn→n̄ > 0.86 × 108 s.
Free neutron experiments have the extremely attractive feature that they are
essentially background-free, but they are limited by the flux of free neutrons thatcan be feasibly produced. By contrast, bound neutron experiments such as SuperKamiokande have abundant neutrons at any given time. Super-Kamiokande’s fiducial
volume, for instance, contains ≈ 6 × 1033 neutrons bound in oxygen — more thansufficient to account for the suppression factor that bound neutrons suffer. Bound
3

This is because the Earth’s magnetic field will cause an energy splitting in the neutron andantineutron of µn B ≈ 3 × 10−21 GeV, where µn is the neutron magnetic moment. This splitting
would suppress the n → n̄ probability (Equation 2.12), although it has been recently argued in [28]that a proper tuning of the magnetic field would not suppress the n → n̄ transition rate, if spin istaken into account.

19
neutron experiments are limited by constant atmospheric neutrino background, however. In particular, it is difficult to separate the multi-pion signal of n → n̄ from thefinal state particles in a deep inelastic scattering neutrino interaction, which produces
multiple hadrons. Also, the density of the material in bound neutron experiments
can significantly distort the n → n̄ signal, due to pion-nucleon interactions.
Despite being very different experiments, the null results from free and boundneutron searches have resulted in lower limits on the n → n̄ oscillation time within
the same order of magnitude. Table 2.1 shows the limits set by bound neutron experiments. The background limitation of such neutron experiments is evident; notice,for instance, that even though Super-Kamiokande has a larger neutron exposure than
Soudan-II by more than a factor of ten, the Super-Kamiokande limit is less thana factor of three more stringent than that of Soudan-II. Future free neutron experiments, in particular the proposed NNbarX experiment [30], intend to incorporate
novel technologies to greatly increase the flux of free neutrons relative to [29], withprojected sensitivities reaching τn→n̄ ≥ 1010 s.

2.5.2

Dinucleon Decay

The same operator 2.4 contributes to dinucleon decay: N N → ππ . The N N →ππ decay rate is calculated with Fermi’s Golden Rule:1
Γ (N N → ππ) =(2π)3 ρNZ

d3 k1 d3 k2 ρ (k1 ) ρ (k2 ) vrel (1 − v~1 · v~2 ) σtot (N N → ππ)(2.18)

where ρN ≈ 0.25 fm−3 is the average nuclear density, ρ (k1 ) and ρ (k2 ) are the densities
of nucleon 1 and 2 in momentum space, v1 and v2 are the (small) velocities of eachnucleon and vrel their small relative velocity, and σ (N N → ππ) is the cross-section

20
ExperimentIMB
KAM Frejus
Material
oxygen oxygen iron
32
3.2
3.0
5.0
Exposure (10 n-yr)
Efficiency
50.0% 33.0% 30.0%
—
0.9
2.5
Scaled background
Data candidates
300
Tn→n̄ (1032 yr)
0.24
0.43
0.65
1.0
1.0
1.4
R (1023 s−1 )8
τn→n̄ (10 s)
0.88
1.2
1.2

SD2
SNO
SK-I
iron deuterium oxygen
21.9
0.55
245
18.0%
—
12.1%
4.5
—
24.1
5
—
24
0.72
0.11
1.9
1.4
0.25
0.52
1.3
1.2
2.7

Table 2.1: Limits on bound neutron oscillation lifetimes from Irvine-Michigan-Brookhaven([31]), Kamiokande([32]), Frejus ([33]), Soudan-II ([34]), Sudbury Neutrino
Observatory([35]), and Super-Kamiokande ([23]). Limits are converted to afree oscillation time via Equation 2.16. See the text for a comparison to thefree neutron experiment from [29]. The values of R for [31–35] are calculated
in [36], while the value of R used for [23] incorporates a more recent calculation from [27]. The uncertainty in R is also incorporated in the conversion
to the free lifetime in [23]. The study in [35] used a different search methodthan the other analyses listed, such that efficiency, background rate, and data
candidates are not illuminating figures.

for the process. A specific calculation within the context of a supersymmetric model
is done in [37], while [38] does the same calculation for a more general ∆B = 2
operator. The result for the lifetime τN N →ππ = 1/Γ (N N → ππ) is

τN N →ππ ≈πMN2 M 108ρN Λ10(2.19)

where MN is the nucleon mass, Λ the same hadronic matrix element from n → n̄ ,and M the suppression scale in Equation 2.4. Recalling Equation 2.6, we can rewrite
this asτN N →ππ ≈πMN2 2 2Λ τnn̄ .
8ρN(2.20)We see, then, that the N N → ππ lifetime scales with the square of the free n → n̄
lifetime, as does the bound n → n̄ lifetime. We can think of the prefactor as a nuclear

21
mode
pp → π + π +pn → π + π 0
nn → π 0 π 0pp → K + K +pp → e+ e+

Material Efficiency Background Candidates
Tlim
Iron
19.0%
2.34
4
7.0 × 1029 yrs
Iron
23.4%
0.31
0
2.0 × 1030 yrs
Iron
36.7%
0.78
0
3.4 × 1030 yrs
Oxygen12.6%
0.28
01.77 × 1032 yrs
Oxygen79.2%≈00
1.0 × 1033 yrs

Table 2.2: Summary of results for N N → ππ from Frejus [39], and pp → K + K + [40] andpp → e+ e+ [41] from Super-Kamiokande. The Frejus searches were performed
with 2.2 × 1031 iron-years of exposure, and the Super-Kamiokande searcheswith 3.06 × 1032 oxygen-years.

suppression factor, although the source of the suppression is different. In n → n̄ ,the suppression arises from the difference in nuclear potentials felt by neutrons andantineutrons, while for N N → ππ it arises from the requirement of overlapping
wavefunctions in two bound nucleons.
Searches for dinucleon decay via any mode are less common than searches for single
nucleon decay and n → n̄ (Table 2.2). A search for the modes discussed in this thesis,
among many other B − L violating modes, was conducted by Frejus [39]. Searches
for pp → K + K + and pp → e+ e+ have been performed at Super-Kamiokande [40,41]. The searches are distinct, both in their experimental signatures and theoreticalframework. For instance, although N N → ππ and pp → K + K + are superficiallysimilar, as they both involve decay to two light mesons, the search in [40] is motivatedby supersymmetric models discussed in [37] in which the BNV operator has two downtype quarks of different generations. For N N → ππ , all quarks are first generation,and thus such supersymmetric models are insensitive to N N → ππ (at least at treelevel). At the experimental level, pions interact heavily with both the residual nucleus
and the surrounding medium, while kaons suffer fewer hadronic interactions, and havedistinctive subsequent decays.

Chapter 3
The Super-Kamiokande Detector
In this chapter, I describe the Super-Kamiokande detector, the experimental apparatus used to perform the searches for ∆B = 2 processes. Full details of the detector
can be found in [42].

3.1

Overview

Super-Kamiokande (hereafter “SK”) is a large cylindrical water Cherenkov detector. It is located underneath Mount Ikenoyama, inside the Kamioka Observatory in
Gifu Prefecture, Japan. Figure 3.1 shows a basic detector schematic. The detector isoperated by the University of Tokyo’s Institute for Cosmic Ray Research. It contains
50 kilotons of ultra-pure water, surrounded by photomultiplier tubes (PMTs) to detect Cherenkov radiation from charged particles. It has been taking data since April
1996.
The main purpose of the SK detector is to search for nucleon decay1 , as well as
study neutrinos from various extraterrestrial sources. Since 2010, it has also been
1

The extra “nde” after “Kamioka” is for “nucleon decay experiment.”

22

3.1

Overview of the Super-K amiokande Detector

T he Super-K amiokande detector [37] is a large water Cherenkov detector located in the
Mozumi zinc mine below the peak of Mt.Ikenoyama, under ∼ 1000m rock overburden as shownin Figure 3.1. T his rock overburden is equivalent to 2700m of water and acts as a shield fromthe cosmic ray muon background. It reduces the cosmic ray f ux to 6 × 10−8 cm−2s−1
23sr−1, a
reduction of 5 orders of magnitude compared to that on the surface of the earth.

Figure 3.1:
T he3.1:
Super-K
amiokande
detector
FigureThe SK
detector [43].
T he Super-K amiokande experiment started data taking in April, 1996 and had continued
usedforas fave
far-detector
thedetector
Tokai-to-Kamioka
long-baseline
neutrino
observation
years untilforthe
maintenancein J uly 2001.
T hisexperiment.
period is referredto Super-K amiokande
I (SK-I).collected
On November
2001, while
f lling
the tank withDatasets havebeen
in four different
detector
configurations.
Thewaterfirst after
period, called “SK-I”, consisted of 1489.2
10livetime days from April 1996 to July 2001.
Following this, there was a planned maintenance for several months. An accident
occurred during water refilling in November 2001, causing over half of the inwardfacing PMTs to implode. The remaining PMTs were quickly re-distributed uniformly
throughout the detector surface, and the detector resumed taking data in October
2002. This second period, called “SK-II”, comprised 798.6 livetime days, endingin October 2005. The original SK-I PMT coverage was restored, and data-taking
resumed in June 2006. This third period, called “SK-III”, comprised 518.1 livetimedays until the summer of 2008. Significant upgrades to the detector electronics were24
made prior to the resumption of data-taking in September 2008. This fourth period,
called “SK-IV”, is the current configuration of the detector. As of March 2015, its
dataset comprises 2166.5 livetime days.
This thesis uses the SK-I-IV dataset from April 1996 to March 2015, comprising4972.4 livetime days, or 307 kiloton-years.

3.2

Structure

The 50 kilotons of water are held in a stainless steel, cylindrical tank. The detector
has a full diameter of 39.3 m and a height of 41.4 m. The tank has two optically separated regions — a smaller concentric cylinder called the inner detector (ID) containing
inward-facing PMTs, and an outer detector (OD) containing outward-facing PMTs.
The two regions are separated by a 55 cm thick stainless steel support structure on
which the PMTs are mounted.

3.2.1

Inner Detector (ID)The ID is 33.8 meters wide by 36.2 meters tall, and contains 32 kilotons of ultrapure water. A large number of 20-inch, inward-facing PMTs are arranged uniformly
on its boundaries. For the SK-I data-taking period, there were 11,146 PMTs, suchthat about 40% of the ID surface was covered by photocathode. For SK-II, there were
5,182 PMTs remaining after the accident, corresponding to about 19% photocathodecoverage. For SK-III and IV, the coverage is nearly the same as SK-I, with 11,129
PMTs.

25

3.2.2

Outer Detector (OD)The OD surrounds the ID on all sides. It has 1,885 outward-facing 8-inch PMTs.
Each PMT is embedded in the center of a 60 cm × 60 cm wavelength-shifting plate
to increase the effective photosensitive area. White Tyvek sheets are placed beneath
the OD PMTs to maximize reflectivity. The main purpose of the OD is to vetocosmic ray muons, as well as to shield the detector from low-energy radioactivity inthe surrounding rock. Though the 1 km rock overburden (2.7 km water equivalent) of
Mount Ikenoyama reduces the flux of cosmic ray muons by five orders of magnitude,
it is still the most persistent source of background at SK, and the OD is indispensable
in rejecting it.

3.3

Cherenkov Radiation

The fundamental phenomenon by which SK reconstructs physics events is Cherenkov
radiation. Cherenkov radiation is light emitted when a relativistic charged particle
passes through a medium at a speed greater than the speed of light within thatmedium. The condition for such radiation is seen in the simple inequality

β>

1
n

(3.1)

where β = v/c, v being the speed of the particle, and n is the index of refraction of
the material. For the ultra-pure water of SK, n = 1.33, so that 1/n ≈ 0.75.
Cherenkov light is emitted at an acute angle with respect to the direction theparticle is traveling. This produces a cone, whose half-angle θC (typically called the26
Cherenkov angle) is simply determined bycos (θC ) =

1
.
nβ

(3.2)

Often we are interested in the ultra-relativistic limit β ≈ 1, in which case θC reaches
a maximum of about 42 degrees. The Cherenkov angle is an important quantity
for determining what kind of particle is traveling through the detector. For instance,lighter particles such as electrons almost always produce a cone at the ultra-relativistic
angle, while heavier particles like muons, pions, and protons are often somewhat
collapsed, due to the higher momentum threshold they must reach.
The Cherenkov radiation spectrum is described by the formula:
2πα
d2 N
= 2
dxdλ
λ



1
1− 2 2
nβ


.

(3.3)

Here N is the number of photons emitted, λ is the wavelength of light, x is the
distance the particle has traveled, and α = 1/137 is the fine structure constant. In
the sensitive wavelength region of the SK PMTs (300-600 nm), about 340 photons
are emitted per centimeter traveled (Figure 3.2).
The Cherenkov light emitted by a relativistic charged particle is collected by the
ID PMTs and converted to an electronic signal of photoelectrons (p.e.’s), or “hit.”
The projection on the ID wall is a ring of hits. The charge and time information ofeach hit is used to determine crucial information such as the event vertex, particle
identification, and momentum. Information from the OD is crucial for determining
whether the event came from outside the tank, and if it had enough energy to escapethe ID. Event reconstruction will be described in detail in Chapter 6.

27

Figure 3.2: Relative spectrum of Cherenkov light in pure water, with quantum efficiency
Figure 3.3: T he spectrum of Cherenkov light through pure water. T he quantum ef ciency ofof the ID PMT overlaid [43].
the 20-inch PMT is also illustrated for comparison.

3.4

3.2

Photomultiplier tubes
Water C herenkov Technique

T he Super-K amiokande detector detects relativistic charged particles by emitted Cherenkov

3.4.1
Inner
Detector
light in water.
When
the speed of particles exceeds the speed of light in the medium, Cherenkov
light is emitted on a cone along the track of the charged particle with a characteristic half:
openingangle
θC 20-inch
The IDuses
diameter Hamamatsu R3600 PMTs (Figure 3.3), developed1(3.1)
cosθC =
nβ with Kamiokande collaborators [44].
by Hamamatsu Photonics K.K. in cooperation
where n is the refractive index of the medium, and β is the speed of charged particles v over the

speedof lightstructure
in vacuumandc. T heCherenkov
radiation
electrons,muons
and response
chargedThe
dynodebleeder
circuitwere threshold
modifiedofto
improvetimingpions are 0.57, 118 and 156 MeV/ c, respectively, because the refractive index of water is about

andcollection
efficiency
1.34.
For particles
with β for' 1,SKθC [45].
is 42◦ .The photocathode of the ID PMT is coated withT he spectrum of Cherenkov light per unit path length dx is given by the following formula.

bi-alkali that has high sensitivity to Cherenkov
lightµ¶ and low thermionic emission.
2πα
1
d2N
= 2 1− 2 2(3.2)
dxdλ is foundλβ 360-400 nm region, as seen in
A peak quantum efficiency of 22%
inn thewhere3.2.
α is Thethe f single
ne structure
constant,
andtimeλ is distributions
the wavelength canof Cherenkov
light.
About
Figurep.e. andtransitbe seen in
Figure3.4.

340 photons/ cm are emitted between the wavelength of 300 nm to 600 nm, which is the mostsensitiveof the PMTspecificationss used in Super-K
amiokandeas shown in Figure 3.3.
Table
3.1 region
lists additional
forthe ID PMTs.
Cherenkov light emitted by charged particles project ring images on the detector wall. Figure
3.4geomagnetic
shows a typicalfield
eventofdisplay
withdetector
Super-K amiokande
detector.
A clearandThe≈450 observedmG at thesite affects
the flowof photosharp Cherenkov ring can be seen. By detecting and analyzing the Cherenkov ring images,
we can reconstruct
events.
For degrades
example, theCherenkov To
ringcompensate,corresponds to26
electronsin the ID the
PMTs,
which
thenumber
timingofresolution.
the number of charged particle. T he Cherenkov ring edge sharpness tells us the particle type,sets of Helmholtz coils are arranged around the tank, reducing the magnetic field to≈50 mG.

12

28
Following the accident in November 2001, ID PMTs have been instrumented withacrylic covers. A clear 12 mm UV-transparent acrylic dome is put over the photocathode area of each PMT, and the side of each PMT is protected with fiber-reinforced
plastic, with holes allowing water to flow freely in and out of the case. The transparency of the acrylic cover for photons at normal incidence is >96% above 350 nm
wavelength.
kuda et al. / Nuclear Instruments and M ethods in Physics Research A 501 ( 2003) 4~ 70000

cable length
φ 116

φ 254 10φ 82 2glass multi-seal

photosensitive area > φ 460< φ 520

cable

water proof structure( 610 20 )

~ 720

(mm)

Fig. 7. Schematic view of a 50cm PM T.

Figure 3.3: A schematic of the 20-inch PMT used in the ID [42].

3.4.2

Outer Detector

The 8-inch OD PMTs are Hamamatsu R1408 PMTs obtained from the decomissioned IMB experiment. The photocathodes of the PMTs are fitted with a 60 cm
× 60 cm wavelength shifter plate [46], which increases the total OD light collection
efficiency by about 50%. The plates degrade the timing resolution of the OD PMTs

29

Figure 3.4: Left: pulse height distribution for a single p.e. in a 20-inch PMT. The peaknear zero is from dark current. Right: relative transit time for a 20-inch PMT
illuminated with light of 410 nm wavelength, for single p.e. intensity. Bothfrom [42].

from 11 ns to 15 ns. However, as the primary function of the OD is to veto cosmic ray muons, light collection efficiency far outranks sharp timing resolution as anexperimental necessity.

3.5

Electronics and Data Acquisition (DAQ)The Super-Kamiokande electronics and data acquisition systems (DAQ) were optimized to handle signals from a large number of channels quickly and efficiently. In
the ID in particular, rapid processing of events occuring closely in time is necessary,such as a stopping muon followed by its decay electron, or a quick burst of neutrinos
from a supernova. Here I describe basic features of the SK electronics and DAQ
system.

30
Shape
Hemispherical
Photocathode area
50 inch diameter
Window material
Bi-alkali (Sb-K-Cs)
Quantum efficiency
22% at 390 nm
Dynodes
11-stage Venetian blindGain
107 at 1500-2000 V
Dark current
200 nA at 107 gain
Dark pulse rate
3 kHz at 107 gain
Cathode non-uniformity <10%
Anode non-uniformity<40%
Transit time
90 ns at 107 gain
Transit time spread
2.2 ns (1 σ) for 1 p.e. equivalent signal
Weight13 kg
Pressure tolerance
6 kg/cm2 water proof
Table 3.1: Specifications for the ID PMTs.

3.5.1

Inner Detector Electronics and DAQ (SK I-III)

For the first three running periods, the ID PMT signals were processed by custom
analog timing modules (or ATMs) of the Tristan KEK Online standard (or TKO). The
ATMs recorded integrated charge and arrival time information of all ID PMTs. EachATM received signals from 12 PMTs. Each ATM was housed in a crate containing
20 ATMs, with a total of 48 crates distributed evenly among four electronics huts
situated above the detector tank.
Each channel of the ATMs has two pairs of Charge to Analog and Time to Analog
Converters (QAC and TAC, respectively). The inclusion of two channels minimizes
dead time, in particular for physics processes in which there may be two stages occurring in rapid succession. Detailed specifications of the ATM are shown in Table 3.2.
Along with the ATMs, each TKO crate contains additional elements of the ID
DAQ system. There is a go/no-go (GONG) trigger control module through which theglobal trigger signal is distributed, and a super control header (SCH). The SCH sends

31
Number of channels
One hit processing time
Charge dynamic range
Timing dynamic range
Charge resolution (LSB)
Charge resolution (RMS)
Timing resolution (LSB)
Timing resolution (RMS)
Temperature dependence (QAC)
Temperature dependence (TAC)
Event number
Data size of one hit
FIFO

12/board∼5.5 µs∼400-600 pC (12-bit)∼1300 ns (12-bit)
0.2 pC
0.2 pC∼0.3-0.4 ns
0.4 ns3 Count/deg. ↔ 0.6 pC/deg.
2 Count/deg. ↔ 0.8 ns/deg.
8-bit
6 bytes
2 kBytes (∼340 hits)

Table 3.2: Specifications for the ATM.

the ATM information to the super memory partner (SMP) when a trigger is received(see Section 3.5.3). It acts as a data buffer to be read out by an online computer
workstation. Each electronics hut contains two such workstations, each handling six
SMP modules. All eight workstations feed into a central online worstation, whichassembles all of the PMT information collected by the ATMs, and sends it away toan offline computer where it can be stored and analyzed. Figure 3.5 shows a schematicof the SK-I-III ID DAQ system.

3.5.2

Outer Detector Electronics and DAQ (SK-I-III)

Signals from OD PMTs are passed through the same coaxial cable that provides
them with high voltage. The high voltage for OD PMTs is distributed through“paddle cards”, and PMT signals are picked off by a capacitor on the same paddle
cards. From the paddle cards, the signal is fed into custom charge-to-time converters(QTCs) containing LeCroy MQT200 chips and comparators. Each QTC contains four
paddle cards, each of which feed out to 12 PMTs. The QTCs measure the charge and32
Inner DAQ
×6

20-inch PMT

×8• PMT × 11146

SCH
ATM

• ATM × 946

× 20

× 240

ATM
GONG

TKO

interface
SMP
×6

trigger signal(to ATMs)HITSUM(from ATMs)• online computer
server × 9
host × 1

onlinecomputer(server)

Ultra-Sparc

VME

trigger logic• SMP × 48

Ultra-Sparc

onlinecomputer(host)

VME
interface
TRG

Ultra-Sparc

FDDI

onlinecomputer(server)

NIM

Fig. 15. The data acquisition system for the inner detector. Arrows show data ﬂow.

Figure 3.5: A schematic of the ID DAQ for SK-I-III [42].

timing information from the OD PMTs, and then send them to a LeCroy 1877 timeto-digital converter (TDC). The TDC modules can record up to eight QTC output
pulses with a resolution of 0.5 ns. The dynamic range of the module is set to 16 µs(10 µs before and 6 µs after the trigger time). The data are then stored in a large
VME dual-port memory buffer (DPM), and sent to an event builder process that runs
on an online host CPU that merges OD and ID data to form a full event. Figure 3.6
shows a schematic of the SK-I-III OD DAQ system.

3.5.3

Trigger (SK-I-III)The ID trigger system is determined by the number of hit PMTs in the ID within a
200 ns time window (HITSUM in Figure 3.5). A “hit” corresponds to about 0.25 p.e.
Three types of triggers are generated, based on the HITSUM signal. The high energy(HE) trigger is signaled if HITSUM exceeds 31 hits (SK-I). For the low energy (LE)33
Local Hitsum

QTC
Modulex40/quadx10/quad

TDC
x5
4 LSBsHigh Voltage~470/quadrant

16 bit event #TDC

FASTBUSPMT

Paddle Card

StruckSupplyFSCC

AUX

84 pin data ribbon cables

Ethernet FSCC Control

Figure 3.6: A schematic of the OD DAQ for SK-I-III [43].

Figure 3.11: The schematic view of the outer detector data acquisition system.
trigger, the threshold is 29 hits (SK-I)2 . Finally, a super low energy (SLE) trigger wase height of the summed signal is proportional to the number of hit PMTs within 200
window.
There
three
types
global period
triggers depending
energythe analysis
implemented
about
a are
year
into
theofSK-I
to lower onthesolarregions:
neutrino
gy trigger (HE), thelow energy trigger (LE), and thesuper low energy trigger (SLE). In
threshold.period,
SK-II
and SK-III
implement
similarschemes,withthetrigger
thresholdsof theHE andLE triggers
correspondto 31detector-dependent
hits and 29
trigger
for the HEvalues.
and LE triggers
are∼thesis,
5 Hz and∼ 11
Hz,trigger
respectively.
The starting
variationstheon
theratesthresholdFor thistheHEis theger threshold is equivalent to a 5.7 MeV electron assuming 50% ef ciency. The SLE tri
point for mplemented
relevant data.
in May 1997 in order to lower theenergy threshold for solar neutrino anal
achieved
theenergythreshold
4.6 MeV equivalent.
the SK-II
period, the
The OD trigger system
worksofsimilarly,
with a In
triggerthresholdofthresho
19 hits within
number of hit PMTs werechanged dueto a half density of PMTs. Thethreshold energy
a 200 nsLEtime
window.
four trigger signals (HE, LE, SLE, OD) are fed into a
triggerin SK-II isAll
8 MeV.
TheOD trigger generation schemeis similar to that of theID. HITSUM signals from alhardware trigger module (TRG) that generates a global trigger signal when any of
C modules are summed up to count the number of hits in the OD within a 200 nsec
The thresholdfor thetriggerthe
is setdifferent
to 19 hits.levels of the global trigger system.
them aredow.
activated.
Figure
3.7OD
shows

3.5.4

SK-IV Electronics and DAQ Upgrade

For the SK-IV data-taking period, the SK front-end electronics and DAQ systemwere upgraded [47, 48]. The main motivations for this upgrade were to increase thenumber of events that could be detected in relevant time windows for neutrinos from2

The LE threshold corresponds to the number of hits generated by a 5.7 MeV electron, assuming
50% trigger efficiency. As there are roughly 5 hits per MeV, the corresponding energy for the HE
trigger is similar. The HE trigger value is very conservative, given that the HE analyses typicallyrequire 30 MeV of visible energy.

34

Analog signal
from PMT
ATM threshold

ADC gate
TDC startand stop
HITSUM

400 nsec width

stop

start

200 nsec width

Global triggerATM
Sum of HITSUM
Master threshold
Global trigger
CentralHut
Figure 3.12: T he overview of the global trigger generation.

Figure 3.7: A schematic of the SK-I-III trigger system [43].

supernova bursts, obtain more precise measurements of solar neutrino parameters,
21

and improve searches for supernova relic neutrinos. Here I describe the main aspectsof the upgrade.
The new front-end electronics modules are QTC-based electronics with Ethernet(or QBEE). Each QBEE contains a custom application-specific integrated circuit(ASIC) with a QTC. The QTC has three measurement ranges (small, medium, large)
that covers a wider dynamic range than the ATMs. The ATMs were charge-saturated
at 600 pC, while the new QTC can reach over 2000 pC. The timing and charge
resolution are similar to that of the ATMs. The QTC outputs a pulse whose leadingedge contains timing information and width is proportional to the total charge. The
ethernet cables improve data flow and reduce congestion relative to the old TKO
readout.
The hardware trigger system has been replaced by a software trigger that contains
all PMT hit information. The software trigger makes selections based on the same

35
hit threshold mechanism described above for the SK-I-III trigger system.

3.6

Water Purification System

The 50 kilotons of water in the SK tank must be as transparent as possible tomaximize the light seen by interesting physics events. The transparency should also
be stable, thereby allowing accurate modeling of light attenuation effects. To achievethis, the water is continuously circulated to remove various sources of impurity at a
rate of about 30 tons/hour, which encompasses the entire tank volume in about 70
days.
The water is first passed through 1 µm mesh filters to reduce dust and otherparticles. It is then passed through a heat exchanger that cools the water from 14.2◦ C
to 12.9◦ C, and supresses the growth of bacteria. Following this, a cartridge polisher
removes heavy ions, and a UV sterilizer kills remaining bacteria. It then goes througha reverse osmosis and vacuum degassifier system, which removes radon and otherdissolved gases (keeping radon levels low is especially important, as radon constitutes
one of the main backgrounds to the low-energy neutrinos from supernovas). A second
filtering stage removes particles larger than ∼10 nm, and a membrane degassifier
further lowers radon levels. The water is then returned to the tank. Figure 3.8 showseach stage of the water purification system.

3.7

Calibration

The detector is calibrated via a number of sources, all of which are essential for its
stable operation. Here I describe the main sources. Calibration procedures for SK-I
are described in [42]. A recent, detailed description of detector calibration for SK-IV

36

S. Fukuda et al. / Nuclear Instruments and M ethods in Physics Research A 501 (2003) 418–462

Fig. 22. A schematic view of the water puriﬁcation system.

efﬁciency
in thepossible radonsourcethewater removal
Figure3.8: inside
The SK
purification
system[42].vacuum destage which follows.
ande detector.
A VD removes dissolved gases inhanger is used to cool the water toconsists of a cylindrical stainless steedark noise levels, and suppress
can be found[49]. also reduce water
vacuum pump (16Torr, 410m3 h 1,
cteria,inwhich
Surviving bacteria are killed by thevessel is 1.4m in diameter and 4.5
stage. Typical water temperatures
Radon gas dissolved in water is a s
exchanger
and after the second
ground event source for solar neu
3.7.1 heat
Water
Calibration
are14.2 C and 12.9 C, respectively.
M eV energy range, while dissolved
courages growth of bacteria in thepolisher (CP) eliminates heavy ions
Laser absorption and scattering

The intensity of light as it travels through the ulta-pure water of SK changes with
distance. It is described simply by the function

I (L) = I0 e−L/Latten(3.4)

where I is the intensity, I0 the intensity at the point the light is emitted, L the
distance traveled, and Latten the attenuation length of the water. Latten is wavelength-

37dependent, and must account for both scattering and absorption. In SK, the relevant
scattering processes are Rayleigh and Mie scattering. Rayleigh scattering is relevantin the case where the scattering object is much smaller than the wavelength of light,
while Mie scattering is relevant when the length scales are comparable (such as with alarge, ∼100 nm impurity in the water). Thus the attenuation length can be describedby three parameters:

Latten (λ) =1αRay (λ) + αM ie (λ) + αabs (λ)

.

(3.5)

To determine these parameters, laser light was injected from an optical fiber fromeight different positions. In SK-I-III, a 337 nm nitrogen laser was used in conjunction
with wavelength-shifting laser dyes, such that data was taken for wavelengths 337,
365, 400, and 420 nm. For SK-IV, a laser diode system was used, with slightly
different wavelengths. The hit and timing information from the laser light was used
to tune the water parameters in Monte Carlo simulations (MC) until good agreementwith the data was found. Figure 3.9 shows measured values of these parameters.

Cosmic ray muons
Downward-going cosmic ray muons occur at a rate of a few Hz at SK, making
up a majority of triggered high energy events. Below energies of ∼100 GeV, they
predictably lose energy via ionization at a rate of about 2 MeV/cm. Thus, they area valuable natural calibration source, and cover a much broader spectrum than thelaser sources described above.
The number of photoelectrons detected by a cosmic ray muon is given by

Q (`) = Q0

f (θ) −`/Latten
e
`

(3.6)

ne the parameters. The parameters were tuned to make the chargeg distributions of the simulated events agree well with that of the lTheparameters obtained by this calibration areshown in Figure5

38

attenuation coefficient(1/m)

Light scattering measurement

10

10

-1

-2
absorption

Rayliegh scattering

10-3
Mie scattering

300

400

500

600

700

wavelength(nm)e 5.14: Absorption and scattering parameters obtained by laser cn. Stars indicate
the results obtainedby thismeasurement.
Figure 3.9: Attenuation
measurements
and fitted
curvesusing the Lines
nitrogen laser [50].
polatedfromthat
result.
Measured values for SK-IV are similar.

where Q0 is a constant, f (θ) is the PMT angular acceptance with θ being the angle

2 Water transparency measurement using cosm
ray muons

with respect to the normal, ` is the photon path length, and Latten is the attenuation
length. From cosmic ray data, the attenuation length is measured to be about 95
r transparency is also measured using Cherenkov light from cosmic

m. Figure 3.10 shows measurements of Equation 3.6, as well as time variation of theattenuation length.

3.7.2

Charge calibration

Various measures must be undertaken to ensure that the PMT response to Cherenkov
radiation is well-understood, for both measurement and simulation. I briefly describe
these measures here. More details on charge calibration can be found in [51].

500

log(Ql/f( θ))log(Ql/f( θ))

39

400

RUN 3106500RUN 3106

P1
P2P1

5.987
-0.9487E-04

0.4609E-03

5.987
0.4444E-06-0.9487E-04

P2

0.4609E-03
0.4444E-06

400
300
200

300

0

1000

2000

3000

4000

200

5000
l (cm)

Figure 5.16: The value log(Q·l/ f (θ)) as a function of l in a typical run

0

1000

2000

3000

4000

5000
l (cm)

150

attenuation length [ m ]Figure 5.16:140Theusingvaluef (θ)) as a function of l in a typical run
bottom log(Q·l/
PMT
130
120
110
100
90
80
70
60
50
01/01/1996 01/01/1998 01/01/2000 01/01/2002 01/01/2004 01/01/2006 01/01/2008 01/01/2010
date

Figure 5.17: Time variation of water attenuation length from SK -I to III.

attenuation length [ m ]150Q(`)·`
Figure
3.10:
Plot
of
log
vs. ` (in cm) for a typical run (top); variation of the
f (θ)
using bottom PMT
140
attenuation length in the SK-I-III periods (bottom). Both from [50]. The
130
120
time variation of the length is sensitive to water quality, and time-dependent
108
110
attenuation lengths are usedin data processing.
100
90
High80voltage determination
70
60
The
50 charge response for a given PMT can be quantified in terms of its gain and01/01/1996 01/01/1998 01/01/2000 01/01/2002 01/01/2004 01/01/2006 01/01/2008 01/01/2010
quantum efficiency. The gain of a given PMT is controlled by the high voltage (HV)
datethat is supplied to its dynodes, as seen in the simple relationship

Figure 5.17: Time variation of water attenuation length from SK -I to III.

108

40

Gain ∝ αVβ .

(3.7)

Here V is the voltage, and α and β are parameters specific to each PMT. The observed
charge, in turn, is proportional to the gain, as well as the number of incident photons
and the quantum efficiency (QE):

Qobs ∝ Nphoton × QE × Gain.

(3.8)

The charge response was most recently calibrated as follows. In 2005, before the startof the SK-III data-taking, a group of so-called “standard PMTs” was calibrated prior
to installation in the ID. They were placed in a dark box with a xenon lamp source
passed through a UV filter requiring λ < 400 nm, and a scintillator ball. A total of
420 PMTs underwent this process. Their HV was adjusted so that the charge output
matched that of a reference avalanche photodiode (APD).
Upon installation, the ID PMTs were placed into 25 different groups, so thateach group had 9-12 of these standard PMTs. The same xenon lamp and scintillatorball was placed in the center of the tank and used for calibration. The PMTs were
adjusted such that their charge response matched the average charge response of thestandard PMTs in a given group.

Gain calibration
Two kinds of gain calibration are performed at SK: absolute and relative. I describe each briefly here.
For the absolute calibration, the single p.e. distribution must be measured, andso a low-energy source is used. Specifically, the calibration used a Californium 252 Cf
source, surrounded by a spherical mixture of nickel oxide and polyethylene. The signal

41from this source is produced as follows: spontaneous fission of the 252 Cf produces
neutrons that are brought to thermal equilibrium by the polyethylene, and thencaptured by the nickel. Upon capture, the nickel then emits a gamma ray of about 9
MeV. The emission yield is about 100 p.e., low enough that most hit PMTs will have
only a single p.e. With this measurement, the gain, or the ratio of the final PMT
charge output to the p.e. input, can be determined.
The relative gain calibration describes the response of each PMT relative to theaverage. Measurements are made with a dye/nitrogen laser, for both high and low
intensity light. The relative gain can be determined for the ith PMT via

Gain(i) ∝Qobs (i),Nhit (i)(3.9)

where Qobs is the observed charge and Nhit is the number of photons reaching thePMT cathode. To get the relative gain, this value is normalized by the mean for all
PMTs.

QBEE Charge linearity
For the QBEE modules in SK-IV, it was necessary to measure the charge linearity,
defined as the ratio of observed to expected charge. The response in the full ∼1-1000
pe range of the QBEEs was measured. The nitrogen laser mentioned previously wasused.
Monitor PMTs with well-known responses were used for reference. Their linearity
was measured by placing the light source at various distances, and also scanning overgain. This monitor PMT charge was used to predict the expected charge for other
ID PMTs via

42

(i,n)

m
Q(i,m)
exp = Qmon ×

Qmean
,
Qnmon

(3.10)

where i indexes the PMT, m and n index the monitor and normalizing runs (runs inwhich the monitor and remaining PMT responses were measured, respectively), Qmon
is the monitor PMT charge, and Qmean is the mean output charge per event for the
ith PMT.
Non-linearity is most noticeable at high charges, and comes from two sources. One
is the fact that the input signal is read out by more than one of the three available
QBEE channels. The second source is the PMTs themselves, whose response falls off
by 2% at larger currents. The QBEE multi-hit effect can be corrected for, such thatthe overall charge non-linearity is ∼1% up to about 100 p.e. It becomes worse as the
charge increases up to the QBEE saturation point around 1200 p.e. Non-linearity
effects are included in Monte-Carlo simulations.

3.7.3

Timing Calibration

The hit times of PMTs are crucial for accurate vertex reconstruction. Several
time scales must be taken into account for timing calibration measurements:• The photon time-of-flight from its emission point along the particle track to the
PMT surface;• The time for the photoelectron cascade to pass through the PMT;• The time to travel through the cable from the PMT to the ATM or QBEE;• The readout time.
The latter three intervals are unique to each PMT. In addition, they exhibit charge
dependence, since events with higher charge will pass the discriminator threshold43
sooner than events with less charge. The purpose of the timing calibration is to measure these time intervals for each PMT as a function of charge, model the dependence,and use this to apply a correction factor such that the hit timing is independent ofcharge. The same nitrogen/dye laser described previously is used for a calibration
light source. A typical plot of the timing charge dependence is shown in Figure 3.11.

Figure 3.11: A plot of the time vs. charge for a typical readout channel [49]. The time ofeach hit is corrected for time of flight.

3.7.4

Energy Scale

Precise knowledge of the detector energy scale is extremely important, as virtually
all SK physics analyses depend on energy reconstruction. Here I describe sources ofenergy scale calibration for the high energy dataset; for low energy, see [52].

44
Stopping Muons
Stopping muons are useful energy calibration sources by way of two quantities:
their Cherenkov angle, and their range. The momentum of any charged particle ofmass m above Cherenkov threshold can be written as a function of the Cherenkov
angle θC (in natural units):
m
p= q
.
2
(n cos θC ) − 1(3.11)The Cherenkov angle reaches a limiting value for higher muon momenta, so only
muons with momenta ∼400 MeV/c or less are useful when considering this relationship. It is a good check on momentum reconstruction, as it depends only on θC andnot any more complicated charge pattern.
At higher energy, the momentum loss of stopping muons can be reliably determined by measuring their range, as muons below ∼100 GeV/c are minimum ionizing
particles that lose energy at a rate of about 2 MeV/cm in water. The range is measured from the entrance point in the detector, to the vertex of the decay electron.
Figure 3.12 shows comparisons of data and MC for both low- and high-momentum
muons.

Decay electrons
As with stopping muons, their resultant decay electrons are also an abundantcalibration source at SK. The decay electron spectrum is useful for calibrating thetens of MeV energy region (Figure 3.13). Not all stopping muons produce decay
electrons, as µ− are susceptible to muon capture by protons. Specific selection criteria
for decay electrons will be described in detail in Chapter 6.

45

Figure 3.12: Left: double ratio RM C /Rdata , where R is the ratio of measured momentum to momentum inferred from the Cherenkov angle, for low momentum
stopping muons. Right: ratio of momentum to range for high momentum
stopping muons. Both from [42]Figure 3.13: Momentum of decay electrons for data (crosses) and MC (solid histogram) [42].

46
Neutral pions
Neutral current interactions in the detector often produce single π 0 ’s, which thendecay into two photons. The photons typically create two showering rings (rings thatexhibit the effects of electromagnetic showers, which will be described in further detailin Chapter 6). The combination of two showering rings with some other requirements,
such as no detected decay electron, creates a mostly pure single π 0 sample. The
invariant mass of the two showering rings can be reconstructed and compared between
data and MC (Figure3.14).
s in Physics
Research A 501 ( 2003) 418–462

Fig. 50. The invariant mass distribution of events with 2e-likerings.
0

Figure 3.14: Invariant mass of the π sample (two showering rings), between data (crosses)
and MC (boxes) [42].
8.11. Absolute energy calibration
As described in Sections 8.5–8.9, Super-K amio-

Summary
Figure 3.15 shows the energy scale uncertainty as a function of momentum for thecalibration samples described above. The agreement between data and MC is generally quite good, such that the uncertainty is less than 3% for all samples considered.

47

Figure 3.15: Energy scale uncertainty vs. momentum range used for calibration measurements in SK-I [42], in order of increasing energy: decay electrons (solid circles), single π 0 ’s (solid triangles), low-energy stopping muons (open squares),and high-energy stopping muons (solid squares).

Chapter 4
Atmospheric Neutrino Simulation
The only background to the ∆B = 2 searches in this thesis comes from neutrino
interactions in the water. The features of Super-Kamiokande that make it an ideal detector for BNV searches — large exposure, natural shielding from most noise sources,
4π coverage — also make it ideal for the study of neutrinos. Neutrino Monte Carlosimulations for each SK-I-IV detector phase are necessary to estimate the backgroundrate and its associated systematic uncertainties for these searches.
The particular type of neutrinos that concern this thesis come from the earth’s
atmosphere. Most of these neutrinos pass through the earth without interacting,but a small fraction do. At Super-Kamiokande, the rate of neutrino events after
data reduction is about 8/day. When summed over nearly two decades of data, thisamounts to a large background, particularly given that ∆B = 2 processes are alreadyconstrained to be quite rare if they occur at all. Thus, it is extremely important tounderstand atmospheric neutrinos as accurately as possible, both their interactionsin the detector and their origins outside of it.

48

49

4.1

Neutrino Flux

The atmospheric neutrino flux starts from the flux of cosmic rays. Primary cosmic
rays — mainly energetic H nuclei — bombard the earth’s atmosphere from all directions. They interact hadronically with air molecules, producing additional hadrons,
mainly pions and kaons. The hadrons decay in stages to produce a large flux ofneutrinos before reaching sea level.
The primary cosmic ray flux and resulting neutrino flux is fairly well-known below
momenta of about 100 GeV/c, which encompasses the relevant scale of this thesis of
2Mnucleon ≈ 2 GeV. For SK analyses, the most recent flux calculated by M. Honda
and collaborators is used [53], which compared to previous works [54, 55] contains
improvements at low energy. The effects of the earth’s geomagnetic field, as well as
the magnetic field from the solar wind, which are especially prominent at low cosmic
ray momenta (∼1 GeV/c), are taken into account in these calculations. Other flux
calculations, namely FLUKA [56] and BARTOL [57], are used to estimate systematic
uncertainties. Figure 4.1 shows the primary cosmic ray flux, and Figure 4.2 shows
the neutrino flux and flavor ratios up to 50 GeV.
Unless otherwise noted, the atmospheric neutrino rates in this thesis are weighted
to include effects of two-flavor neutrino oscillation, namely the µ − τ oscillation probability:22

P (νµ → ντ ) = sin (2θ23 ) sin



1.27 × ∆m2 × L
E


.

(4.1)

Here θ23 is the mixing angle between the second and third neutrino mass eigenstates,
assumed to be maximal (θ23 = 45◦ ), ∆m2 = 2.5 × 10−3 eV2 is the fitted value ofthe mass splitting between these eigenstates, L is the length (in km) between theneutrino’s creation and interaction points, and E (in GeV) is the true neutrino en-

50ergy at the interaction point. Smaller weight factors from the flux in [53] are also
applied, related to effects of the solar wind. In principle a full three-flavor oscillation
probability should be used, although the µ − e oscillation probability is quite small

10

4

Proton Flux x E

2.5

−2

−1

1.5

(m sr GeV )in the atmospheric neutrino energy regime.

10

3
0

10

10

1

10

2

10
Ek (GeV)

3

10

4

10

5

Figure 4.1: The primary cosmic ray flux for protons (at solar minimum), as calculated
mary cosmic ray observation and our model curves for protons at soin [54]. Overlaid data points (see [54] for references) come Webber (crosses),
MASS (open circles), LEAP (closed upward triangles), IMAX (closed downte datafrom Ref. [26], open circles indicate data from MASS [27], cward triangles), CAPRICE-94 (closed vertical diamonds), CAPRICE-98 (openverticaldiamonds),(closedcircles),(closedhorizontal
diamonds),
P [28],
closeddownwardBESS
triangles
IMAX[29],AMS
closedverticaldiamond
BESS-TeV (open upward triangles), Ryan et al (closed horizontal diamonds),
JACEE(large open
downward [31],triangles),(large[15],openclosdiamonds),
verticaldiamonds
CAPRICE-98
closedRUNJOB
circles BESS
Ivanenko et al. (large open upward triangles), and Kawamura et al. (large
open
squares).
lines
indicate [32],calculations
from a fitting
procedure,
S [14],
open
upwardDashedtriangle
BESS-TeV
closed horizontal
diam
and the solid line includes additional corrections explained in [54].

51

10

2
1.5xν µνe
0.75xν e101νµThis WorkHKKMS06
Bartol
Fluka

0

10

−1

10

0

10
10
Eν (GeV)1

Neutrino Flux Ratios

φ νx E ν2 (m−2sec −1sr−1GeV)

5This WorkHKKMS06
BartolFluka

νµ + νµνe + νeν µ ν µ( x1.5)

2

νe νe
1

−1

10

0

10
10
Eν (GeV)1

Figure 4.2: The atmospheric neutrino flux (left) and neutrino flavor ratios (right) up to. Comparison
atmospheric
neutrino
f uxesKamiokaaveragedline
overan
50 of
GeV.
The solid line
indicates
the calculated
calculation forof [53],the dot-dashedearlier calculation from Honda et al. [55], the long dashed line the BARTOL
calculationand theshortpanel),dashedwith
line otherthe FLUKA
calculation[56].
ons (left panel),and the f[57],ux ratios(rightcalculations.
The dashedlincalculationby the Bartol interactionsgroup [20][21], dotted lines for the FLUKA group [22], and da
4.2
Neutrino
our previous calculation (HKKM06).
Four main types of interactions are simulated:• (quasi-)elastic
ν +statistics
N → ` +in
Nthe observation at balloon altitude.
ν̄e below
1 GeV,
duescattering:to the small
00• singlemesonν +altitudeN →`++me muons
at sealevelproduction:or mountain
areN notuseful to examine the atmosphe

no of• theseenergies,
sincethemuonsresult
fromhigher
energy pionsat higher altitud
deep inelastic scattering:ν+N →`+
N 0 + hadrons
Fi • 9coherent
h pionth production:th iν +16 0t →i `+f 16 O + π f

ti

f th

ith

Here ν is a neutrino or anti-neutrino of any flavor, N and N 0 are the incoming
and outgoing nucleons, respectively, and ` is the outgoing lepton. All four of these52interactions can occur in charged current (CC) and neutral current (NC) modes. The
outgoing lepton may be either charged or neutral.
Neutrino interactions are simulated with the NEUT [58] neutrino interaction andnuclear effects simulator. The interactions are described below.

4.2.1(Quasi)-elastic scattering

Charged current quasi-elastic scattering (CCQE) is simulated for both free protons
and bound nucleons, following the models of Llewellyn Smith [59] and Smith andMoniz [60], respectively. The bound nucleon calculation uses a relativistic Fermi gas
model. In both cases, the cross section uses vector and axial vector form factors. The
axial vector mass is set to 1.2 GeV/c2 to align it with neutrino experiments. Neutral
current elastic cross sections are determined by constant coefficient scaling with theircharged current quasi-elastic counterparts.
Charged-current quasielastic scattering is the single most common interaction seenin SK, and is of fundamental importance for both atmospheric and accelerator neutrino analyses. There is typically only one visible Cherenkov ring formed from thecharged lepton. Since the ∆B = 2 processes in this thesis involve two or more energetic final state pions forming more than one ring, CCQE is not a major source of
background 1 . More details on this mode can be found in [61].

4.2.2Single meson production

Single meson production is the dominant hadron production mechanism in theregion W < 2.0 GeV/c2 , where W is the hadronic invariant mass. We follow themodel of Rein and Sehgal [62]. Pions, kaons, and η mesons are produced via
1

It can still constitute background if, for instance, the outgoing nucleon has enough energy tointeract with another nucleon and produce a secondary hadron.

53

ν + N → ` + N ∗ =⇒ N ∗ → N 0 + meson(4.2)

where, as before, ν is the incoming neutrino, ` the outgoing lepton, and N and N 0
are the incoming and outgoing nucleon. Here N ∗ is a short-lived baryon resonance.
In total, 18 resonances under 2.0 GeV/c2 are considered. The pion angular distribution was calculated from the Rein and Sehgal model for the ∆(1232) resonance,
and was assumed to be isotropic for other resonances. Charged current single pion
interactions (CC1π) are a common source of background for N N → ππ , as a pionthat is produced with a sufficiently large angle with respect to the outgoing lepton
can simulate the “back-to-back” signal of two pions decaying from a low momentum
initial dinucleon state. The Pauli blocking effect is considered in the resonance decay,
by requiring that the outgoing nucleon has momentum greater than the Fermi surface
momentum. Finally, absorption of pions is also considered, and about 20% of single
pion production interactions have only an outgoing lepton and nucleon.
Coherent pion production constitutes a small subset of single meson production.
In this interaction, the nucleus is unchanged, but a final state lepton and pion are
still present.

4.2.3

Deep inelastic scattering

Deep inelastic scattering interactions are relevant in the region W > 1.3 GeV/c2 ,and they are the dominant interaction for W > 2.0 GeV/c2 . The charged current
cross section is given by

54

Y . Hayat o
cm2)

νµ p → µ− p π+σ(ν µ)(10

−38

1.2

σ(ν µ)(10

−38

cm2)

2480

1
0.8

1
0.9

νµ n → µ− p π0

0.8
0.7
0.6
0.5

0.6

0.4

0.4

0.3
0.2

0.2

0.1

σ(ν µ)(10

−38

cm2)

0
1
0.9

1

10

0

Eν (GeV)νµ n → µ n π+

1

10Eν (GeV)−ANL Radecky, Phys.Rev.D 25, 1161 (1982)

0.8

ANL Campbell, Phys.Rev.Lett. 30, 225 (1973)

0.7

ANL Barish, Phys.Rev.D 19, 2521 (1979)

0.6

BEBC Allen Nucl.Phys.B 264, 221 (1986)

0.5

BEBC Allen Nucl.Phys.B 176, 269 (1980)

0.4
BEBC Allasia Nucl.Phys.B 343, 285 (1990)

0.3
BNL Kitagaki Phys.Rev.D 34, 2554 (1986)

0.2

FNAL Bell Phys.Rev.Lett. 41, 1008 (1978)

0.1
0

1

10

Eν (GeV)Fig. 2. Cross-sections of (a) νµ p → µ− pπ+ , (b) νµ n → µ− pπ0, (c) νµ n → µ− nπ+ .
Solid lines
the calculated
cross-sections.
Experimental
datalines
pointsare sumFigure
4.3: show
Cross-sections
for various
CC1π interactions,with solid
showing
the fitmarized in theusedtable.
in NEUT [58]. See [58] for details on the data points.

In thedetermination of theangular distribution of pion in thef nal state,
weuseRein and Sehgal’s method for theP 33(1232) resonance. For theother distribution of generated
 is set
2
 pionresonances,ind2 σGdirectional
y to beisotropic
F MN Eν
=
1 − y + y 2 + C1 F2 x, q 2 ± y 1 − + C2 xF3 x, q 2 ,dxdyAdler Frame
π (resonance rest frame). With the Rein 2and Sehgal’s model,
the
xyMN
M2
M`2 each individual resonanceif weyM`2
it ispossibletocalculatethecross-sectionsfor
C1 =−− `2 −,
2EUsing
4E
2MN Eν x
N Eν xνν characteristics,
ignore the4M
interferences.
this
we made the probability2M`functions
theintermediateresonanceisP 33 or not with
C2 = to− determinewhether
,
(4.3)
4M
E
x
N
ν
thesingle-resonancecross-sections. Theangular distribution of π+ has beenmeasured for νp → µ− pπ+ mode[14] and theprediction from thesimulation
whereM` theare data.
the nucleon
andconsider
outgoingthe
leptonmasses,Eν is
N andagreesMwellwith
We alsoPauliblocking
ef theect neutrino
in the2 resonances by requiring that the momentum
decay qofof nucleon
should beenergy,is the squared momentum transfer, and x = −q 2 / (2MN (Eν − E` )) and y =55(Eν − E` ) /Eν are the Bjorken scaling parameters. The Fi (x, q 2 ) are nucleon structure
functions, taken from the GRV98 parton distribution functions [63], with corrections
from Bodek and Yang [64]. For the charged current cross-section, NEUT integrates
Equation 4.3 directly. The neutral current cross section is obtained by multiplying
the CC cross section by linear functions of the neutrino energy, for different energy
ranges. See [58] for details.
In the region 1.3 GeV/c2 < W < 2.0 GeV/c2 , only pions are considered as outgoingmesons, with nπ ≥ 2. Average pion multiplicities are estimated from the Fermilab
15-foot hydrogen bubble chamber experiment [65]:
hnπ i = 0.09 + 1.83ln W 2 .

(4.4)

In addition, the ratio of forward/backward pion production in the hadronic centerof mass is taken from the BEBC experiment [66], using
0.35 + 0.41ln (W 2 )
nFπ
=
.
nB
0.05 + 0.09ln (W 2 )
π

(4.5)

The low W region is particularly important for n → n̄ , since an ideal n → n̄
event has an isotropic spread of pions, e.g. a similar number of pions in the forward
and backward direction. For W > 2.0 GeV/c2 , hadron kinematics are determinedby PYTHIA/JETSET [67], and additional hadrons such as kaons and η mesons are
considered.

4.2.4

Nuclear effects

The interactions of hadrons in the nucleus are an extremely important componentof MC simulation, both for n → n̄ and N N → ππ signals and atmospheric neutrino

µ(3 GeV < E ν < 6 GeV) ,σ(ν̄N → ν̄X )= 0.37 (E ν ≥ 6 GeV) .
σ(ν̄N → µ+ X )56

σ/Eν (10-38cm2 GeV-1)

These values are estimated from the experimental results [29,30].
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

GRV94LOGRV94LO (w/ B-Y corr.)
GRV98LO
GRV98LO (w/ B-Y corr.)

CCFR 90
CDHSW 87
GGM-PS 79

0

10

CHARM 88
BNL 80
CRS 8020

BEBC-WBB 79
IHEP-JINR 96
IHEP-ITEP 79

30 50

CCFRR 84
SKAT

100 150 200 250

Eν (GeV)Fig. 3. Neutrino and anti-neutrino charged current total cross-section. The upper
4 lines
to neutrino
and lower
4 lines(uppercorresponds
the anFigure4.4: corresponds
Total charged-current
cross-section
for neutrinos
lines) andto
antineutrinosmeanings(lower lines).
show
a comparisonof different
neutrino. The
of Theeachlines
lines
andcorresponding
dataparton
pointsdistribution
are showfunctions, with and without corrections from Bodek and Young [64]. See [58]n the f gure.for(CCFR90[18], CDHSW87 [19], GGMPS79 [20], CHARM88 [21details on the data points.
BNL80 [22], CRS80 [23], BEBCWBB79 [24], IHEPJ INR96 [25], IHEPITEP79 [26
CCFRR84[27],Thus,
SK AT[28]). consideration of hadron Final State Interactions (FSI)
backgrounds.
a carefulis necessary following the calculation of outgoing particle kinematics.

In NEUT, both PY THIA/ J etSet and our own codes areused to generat
16
in particular,
the events
interaction
probabilities
insideO are
The
he Fordeeppionsinelastic
scattering
becausePY THIA/J etSetwaslarge.
develope
o simulatehigher
energyareinteractions
followingpion interactions
considered: and does not f t for thelower energies
Therefore, we use the experimental results to generate an event whose W
2. ForQuasi-elastic
scattering,in whichof thechargein theinitialt
those small
W pionevents,wesameassumeK isNO
scalingess•than
2 GeV/ c(QE)determine
thestate,
valuewith
of W
[31].
The
multiplicity of pions is determined b
and finalsomeenergyloss;
he function described before, which was also obtained experimentally. Th• absorption, e.g.asymmetry
π + + n → p;of pion multiplicity(nFπ / nB
orward–backwardπ ) in the hadroni• charge exchange, e.g. π + + n → p + π 0 ;• pion production, e.g. π + + n → π + + π 0 + n.

57

450

π+ 12
6C

400
350

Reactive (TunedFSI)
Reactive (OldFSI)
Quasi-elastic
Absorption
Single CX
π Production

σ (mb)σ (mb)

Figure 4.5 shows these interactions as a function of pion momentum.

400

300

250

250

200

200

150

150

100

100

50

50
200

400

600

800 1000 1200 1400 1600
π+ Initial Momentum (MeV/c)

π- 12
6C

350

300

0
0

Reactive (TunedFSI)
Reactive (OldFSI)
Quasi-elastic
Absorption
Single CX
π Production

450

0
0

200

400

600

800

1000 1200 1400 1600
π- Initial Momentum (MeV/c)+
Figure 4.5:
Pion1.interaction
cross sections
as a function of lab momentum,
FIGURE
NEUT simulationof
π12C scatteringcomparedtodatafrom[e.g.
5, 6, 7, 8,for
9] π (left)−and π (right) [68]. The “TunedFSI” plots are used in the most recent MC
simulation. See [68] for details on the data points.

TUNING AND VALIDATION OF THE PION CASCADE MODEL

To simulate pion interactions in the nucleus, each pion is propagated in a stepiwise TheNEUT
fashion fromits model
starting
point, with cascadethat
interactionspropagates
determined
individual
π FSI
is amicroscopic
theπbyin thef nitesteps

throughprobabilities
the nucleus. The(MFP)
of various
mechanisms
interaction
at meaneach free
step.pathThisis doneuntil intranuclear
the pion escapes
the nuare calculated froma Delta-hole model [4] at low energy (LE), pπ ≤ 500 MeV/c, andcleus,fromor isfreeabsorbed.
The starting
pointatofhighpions
in thenucleusdetermined
MeV/c. Theusing
πp scatteringcross sections
energy(HE),pπ > is500
MFPsaredensity(position)
dependentandaWoods-Saxondistributionisassumedwith
a Woods-Saxon
type distribution:

parameters measured fromelectron scattering data. The microscopic method provides
aconnection between π scattering measurements,
Zρ0 wheretheπ originates fromoutside,ρ(r) = wheretheπ isr−c(4.6)
thenucleus, to neutrino experiments
A 1 + exp aproduced within thenucleus. With
π scatteringdata, wecantuneandconstraintheFSI model, withoutthecomplicationof
whereanyr additional
is the distancenuclear center, Z and A are the proton and mass
primaryfrom
vertexthe
physics.
At LE,
microscopic
MFPs fordensity,
absorption
scattering,tuned fromthe panumbers,
ρ0 isthetheaverage nuclear
andanda andc are wereadditional
densityrameters. For 16 O, Z = 8, A = 16, ρ0 = 0.48m3π , a = 0.41 fm, and c = 2.69 fm.
For DIS interactions, this starting position is shifted slightly by the formation length,
accounting for the fact that hadronization is not instantaneous. The formation length

58
L for a given hadron is defined as L = p/µ2 , where p is the momentum of the outgoinghadron and µ2 =0.08 GeV2 is an experimentally determined parameter.
In considering pion interactions, two momentum regions are used — low momentum (pπ < 500 MeV/c) and high momentum (pπ > 500 MeV/c). For low momenta,
the interaction probabilities per step are calculated according to the model of Salcedo and Oset [69]. The Fermi surface momenta of the nucleons is determined with
−1/3
. Pauli blocking is taken into account by
Equation 4.6 via pF (r) = 23 π 2 ρ(r)requiring that, following an interaction, the nucleon momentum exceeds the Fermi
surface momentum. Thus, the interaction probabilities are density dependent. The
scattering kinematics are drawn from a phase-shift analysis of pion interactions on
free nucleons. Recently, the low momentum interaction probabilities in NEUT weretuned to achieve better agreement with π + scattering on 12 C [68].
For pions above 500 MeV/c, nucleons are treated as quasi-free particles. QE
scattering, which is dominant up to very high pion momenta (Figure 4.5) is definedas [68]1σQE = σπd
2

f ree
σQE

!

f ree
σtot

,

(4.7)

f ree
where σπd is the total cross section for pion scattering on deuterons, and σQE
and
f ree
σtot
are cross sections for scattering on free protons. The hadron production crosssection, which is only considered in the high momentum region, is defined similarly:
f ree
σQE
1
σhad = σπd 1 − f ree
2
σtot

!
.

The same phase-shift kinematics as the low momentum pions is used.

(4.8)

59

4.3

Detector Simulation

After the neutrino interactions and nuclear effects of outgoing particles are modeled by NEUT, final-state particles are put through a custom detector simulation
program. The simulation tracks particles (both from the primary interaction andsecondary particles from interactions in the water), models Cherenkov radiation and
PMT response, and accounts for other detector-specific phenomena, such as reflection
from Tyvek in the detector scaffolding. The simulation is based on CERN’s GEANT3
package [70].
Cherenkov radiation is modeled using inputs from the water calibration procedures
described in the previous chapter. Reflection and absorption off of different detector
surfaces is also modeled from calibration measurements. PMT response is modeledby using the measured quantum efficiency of each PMT. The charge of each PMT istaken to be the sum of all p.e.’s weighted by the single p.e. distribution, and the hit
timing is smeared by the PMT timing resolution.
As with the treatment of hadronic interactions in the nucleus, hadronic interactions in the water are treated differently in the high and low momentum regions.
For pπ < 500 MeV/c, custom code [71] is used. The low momentum simulations
were recently updated and tuned to recent 16 O data, and use the same NEUT interaction model described for intranuclear effects. For high momenta, the GCALOR
package [72] is used.

60

e± pair production
Compton scattering
photoelectric effecte±multiple scattering
ionization and δ-ray production
Bremsstrahlung
positron annihilationµ±decay in flightmultiple scattering
ionization and δ-ray production
Bremsstrahlung
direct e± pair production
nuclear interactions
Hadrons
decay in flightmultiple scattering
ionization and δ-ray production
hadronic interactionsγ

Table 4.1: Interactions modeled for each particle type. Cherenkov radiation is modeled
for e± , µ± , and charged hadrons.

Chapter 5
Signal Simulation
Four signal processes are studied in this thesis. Three fall under the category of
dinucleon decay, or N N → ππ , encompassing the modes 16 O(pp) → 14 C(π + π + ),16O(pn) → 14 N(π + π 0 ), and 16 O(nn) → 14 O(π 0 π 0 ) (truncated throughout to pp →π + π + , nn → π 0 π 0 , and pn → π + π 0 ). The fourth process is neutron-antineutron
oscillation, or n → n̄ . These processes are united by similar underlying physics(∆B = 2, all first-generation quarks), but entail different simulation and signal extraction methods.
The intranuclear simulation for each process follows a similar pattern. Two boundnucleons release ≈2 GeV of energy in the form of pions, which then propagate throughthe residual nucleus as described in Section 4.2.4. Outgoing particles are then passed
through the detector simulation described in Section 4.3.

5.1

Dinucleon Decay

The starting point for the N N → ππ simulation is two bound nucleons. The
nucleons are chosen randomly from the 16 available nucleons in 16 O (for nucleons of61

62
appropriate charge for the outgoing pions), which consists of two s1/2 , two p1/2 , andfour p3/2 protons and neutrons. The nucleon momenta are determined from electron12

C scattering data [73] (Figure 5.1). Nuclear binding energy is also taken into accountby subtracting it from the nucleon masses — 39.0 MeV and 15.5 MeV for s-state andp-state nucleons, respectively. The two nucleons are approximated as starting in thesame nuclear position with the Woods-Saxon distribution (Equation 4.6).

Figure 5.1: Fermi momentum predictions (solid line) and data (crosses) from [73]. The
left shows a p-state nucleon, and the right an s-state in 12 C.

The momenta of the outgoing pions are mostly determined with normal twobody decay kinematics. There is an additional effect called “correlated decay”, inwhich the wavefunction of an additional nucleon overlaps with that of a dinucleon
decay pair. Reference [74] estimates that 10% of single-nucleon decays will have acorrelated nucleon, and we assume the same percentage of correlation in dinucleon
decays. Events with a correlated decay are essentially three-body decays, with themomenta of the outgoing pions affected by the momentum absorbed by the third
nucleon (which escapes notice, as it is below Cherenkov threshold). Figure 5.2 showsthe effect of correlated decay on the invariant mass of the N N → ππ system.

Events

63

104

Invariant mass of pions

3

10

102
10
1
10-1
10-2
500

1000

1500

2000

Invariant mass (MeV/c2)

Figure 5.2: Invariant mass for pp → π + π + . The modes pn → π + π 0 and nn → π 0 π 0are essentially identical. The dotted histogram shows events with a correlated
decay.

The nuclear effects described in Section 4.2.4 affect the final state pions significantly. About 40% of events in each mode remain in their initial pion configurations
upon leaving the nucleus. Table 5.1 shows the number of pions after intranuclear
simulation.

5.2

Neutron-antineutron oscillation

For n → n̄ , the signal process is not the oscillation itself, but the subsequentannihilation. A neutron in 16 O that has oscillated to an antineutron will quickly
annihilate with one of the other nucleons, which results in multiple pions.
The oscillation position in the nucleus is estimated from Equation 2.14 (Figure 5.3), which gives approximately the same result as the Woods-Saxon distribution.
The same Fermi momentum simulation and binding energy modifications described64
pp → π + π +
Nπ
0
1
2
3
0
— 11.8% 45.1% 2.4%
1 2.7% 21.8% 5.2%—
2 2.8% 3.0%——
pn → π + π 0
Nπ
0
1
2
3
0
— 6.8% 12.3% 1.4%
1 7.1% 46.8% 3.7%—
2 10.5% 4.1%——
nn → π 0 π 0
Nπ
0
1
2
0
— 2.6% 3.5%
1 11.5% 25.1% 2.8%
2 43.0% 4.2%—
Table 5.1: Number of pions for pp → π + π + (top), pn → π + π 0 (middle), and nn → π 0 π 0(bottom) after nuclear effects. Percentages indicate the percentage of eventswith the specified number of π + ’s (columns) and number of π 0 ’s (rows). Entries
with a “—” indicate a small contribution (<0.1%). π + ’s are required to be
above Cherenkov threshold. The number of pions prior to nuclear effects isboldened.

for N N → ππ are used for n → n̄ .
Following the oscillation, the antineutron is assumed to have an equal probabilityof annihilating with any of the remaining nucleons. Unlike the N N → ππ modes,
the final state of n → n̄ has many non-negligible contributions for both n̄n and n̄pannihilation.
The final states are determined from antinucleon scattering data. Antineutron
data is sparse, and so the annihilation branching fractions for n̄n and n̄p are mainly
drawn from p̄p and p̄n data, which should be equivalent from isospin considerations(and appropriate adjustments to outgoing charged particles in the case of p̄n). The
analysis in [23] used cross sections from old bubble chamber data [75–77]. More recent

65

Figure 5.3: The integrand in the numerator of Equation 2.14 for an s-state and p-stateneutron in 16 O. The solid and dotted lines show calculations for two different
approximations.

p̄p data from the Crystal Barrel experiment [78, 79] is available, and is especiallyabundant for branching fractions with more than one neutral particle in the final state.
The OBELIX experiment [80] has also collected data for low-energy antineutrons in
n̄p annihilation. Details on the branching fractions can be found in Appendix A.
Regardless of the specific branching fractions used, the annihilation products follow a similar pattern — there are typically ≥3 pions that must then be propagatedthrough the residual nucleus. In addition to the branching fractions, the pion-nucleon
model described in Section 4.2.4 is a different model than that used in [23]. The outgoing pion characteristics are similar. Figure 5.4 shows the pion multiplicities andaverage charged pion momentum for pions emerging from the nucleus, and Table 5.2
shows the percentage of pions that undergo intranuclear interactions. Overall, theupdated pion model is more interactive.

66

charged pions

neutral pions
10000

7000

6000

8000

5000
6000
4000

Old analysis vectors
4000
3000

Updated pion model
2000

2000

Updated pion model + BR
1000
0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0

0

0.5

total pions

1

1.5

2

2.5

3

3.5

4

4.5

5

average charged pion momentum

7000

600

6000

500

5000
400
4000
300
3000
200
2000
100

1000
0

0

1

2

3

4

5

6

7

0

0

100

200

300

400

500 600 700 800
Momentum (MeV/c)

Figure 5.4: Charged, neutral, and total pion multiplicity (top left, top right, and bottom
left), and average charged pion momentum (bottom right) for n → n̄ MC.
The blue histogram shows the simulation of [23], the red histogram shows thesimulation using the same branching fractions as [23] but the updated pionnucleon model described in Section 4.2.4, and the green histogram shows thesimulation using new branching fractions and the updated pion model.

Abs. QE
Ref. [23] (SK-I) 24% 24%∗
This analysis
21% 20%QX

π prod.
3%
23%
3%

No int.
49%
30%

Table 5.2: Percentage of pions undergoing different interactions for the analysis in [23],and this analysis. From left to right, the interactions are absorption, quasielastic scattering, charge exchange, π production, and no interaction. The ∗indicates that QE scattering and charge exchange are combined.

Chapter 6
Reduction and Reconstruction
Dedicated reduction and reconstruction algorithms are required to retain andanalyze interesting physics events at Super-Kamiokande. Prior to any reduction, ∼106
events are recorded. Most of these are from low-energy radioactivity surrounding thedetector, at a rate of about 11 Hz. The high-energy region is dominated by cosmic
ray muons, at a rate of about 3 Hz.
The Super-Kamiokande dataset is split into three categories:• Fully contained events, in which all of the energy is desposited in the ID;• Partially contained events, which originate in the ID but have enough energy
to escape the detector, depositing significant energy in the OD;• Upward-going muons, high-energy muons produced by subterrestrial neutrino
interactions that enter the detector from below the horizon.
While each of these event categories are crucial for a variety of Super-Kamiokande
physics analyses, the ∆B = 2 processes in this thesis must originate inside the watertank, and are not sufficiently energetic to escape. Thus, I restrict attention to the67

68
fully contained (FC) reduction and reconstruction algorithms. A recent descriptionof the reduction and reconstruction of all event categories can be found in [81].

6.1

Fully Contained Reduction Steps

There are five FC reduction stages, each aimed at retaining true atmospheric
neutrinos and rejecting both high-energy cosmic rays and low-energy radioactivity.
Each selection is briefly described below.

6.1.1

First Reduction

The majority of background events are removed by the first reduction, whichmakes the following requirements:
1a) PE300 > 200 (100 for SK-II1 ), where PE300 is the total number of ID photoelectrons seen in a 300 ns time window;
1b) NHITA800 ≤ 50 (55 for SK-IV), where NHITA800 is the total number of OD
tubes hit within a time window (-500 ns, +300 ns) relative to the trigger time.
The first criterion corresponds to a 22 MeV/c electron, and removes most of the
background from low-energy radioactivity. The second criterion removes most cosmic
ray muons. Following the first reduction, about 3000 events/day remain from theinitial ∼106 /day.

6.1.2

Second Reduction

The second reduction also contains a set of simple cuts, and removes most of the
background remaining after the first reduction:
1

SK-II has modified criteria relative to other SK periods, due to the decreased photocoverage.

692a) NHITA800 ≤ 25 for events with PEtotal < 100,000 (50,000 for SK-II);2b) PEmax /PEtotal < 0.5, where PEmax is the largest number of p.e.’s observed in asingle PMT.
Criterion 2a) further removes cosmic ray muons, while still allowing very highenergy events originating in the ID. Criterion 2b) removes events caused by electronicnoise, which tend to have a single PMT discharge containing much more charge thanan average PMT. After the second reduction, a few hundred events per day remain.

Figure 6.1: The second reduction cut 2a) for data (top), MC (middle), and MC normalized
to data events after the selection (bottom) in SK-IV. The small number of
events for which NHITA800 > 25 have PEtotal > 100,000. Taken from [61].

70

6.1.3

Third Reduction

The third FC reduction more specifically targets remaining cosmic ray muon
events, as well as low-energy radioactivity. In particular, cosmic ray muons without much OD activity are targeted. After these more specific selections, about 45
events/day remain.

Through-going muons
Muons that enter and exit the tank may not leave much of a trace in the OD,
depositing most of their energy in the ID. To be considered for this category, theID must have over 1000 hit PMTs, with PEmax more than 230 p.e.’s for at least one
PMT. If these criteria are met, a through-going muon fitter is applied that assumes
an entry point, taken to be the first hit PMT in the ID entrance cluster, and an exit
point, taken to be the centroid of the ID exit cluster2 . A set of expected hit timing
values Ti is determined using MC simulations, and a goodness-of-fit parameter iscalculated:
X 1(ti − Ti )2
1
exp−
goodness = P 1 ×
σi2
3σi2
i σ2
i

!
,

(6.1)

i

where ti and σi are the hit timing and timing resolution of the ith PMT. The thrugoing muon cut removes events that satisfy the following criteria:3a) goodness > 0.75;3b) NHITAin > 10 and NHITAout > 10, where NHITAin,out is the number of hit OD
PMTs within 8 meters of the fitted entry and exit point.
2
Clusters are just sets of contiguous hit ID PMTs, with the largest two corresponding to entrance
and exit.

71
An additional cut is made for high-energy throughgoing muons. Muons have a
critical energy of about 1 TeV, above which their predictable ionization energy lossof 2 MeV/cm becomes secondary to more random processes, such as bremsstrahlung
and pair production. These processes can produce a large number of hits in a shorttime interval. Events satisfying the following cut are removed:3c) NHITA500 > 40,
where NHITA500 is the number of OD hits within a 500 ns sliding time window.
Stopping muons
The stopping muon cut follows the same logic as the through-going muon cut,
except there is no exit point. The direction of the muon can still be reconstructed by
maximizing the charge seen by the PMTs within a 42-degree opening angle from theentry point. A goodness parameter similar to Equation 6.1 is calculated, and events
are rejected as stopping muons if the following criteria are met:
4a) The goodness ≥ 0 and NHITAin ≥ 10;4b) (SK-I only) the goodness > 0.5 and NHITAin ≥ 5;
Figure 6.2 shows the goodness parameter for both thrugoing and stopping muons.
Cable hole muons
One source of missing OD activity comes from slight asymmetries in the OD PMT
distributions. In particular, there are 12 holes in SK, starting in the ID and goingto the electronics huts, through which cables are threaded. These holes interrupt the
distribution of OD PMTs (Figure 6.3), so that muons passing through these holes
may escape notice by the OD.

72

Figure 6.2: The goodness parameter (Equation 6.1) for thrugoing (top) and stoppingmuons (bottom) in SK-IV, normalized to data taken over two days (redcrosses). The blue histogram shows data over the previous two months.

To partially account for muons that pass through these holes, four of the twelve
have plastic scintillator veto counters installed. An event is removed if:5a) the cable hole veto counter is hit;5b) the entry or exit point is within 4 m of the veto counter.
In SK-IV only, a tighter selection is made to reject stopping cable hole muons. Events
that satisfy the following criteria are rejected:

736a) The stopping muon goodness is greater than 0.4;6b) PEtotal > 4000;6c) The muon’s zenith angle is more vertical than horizontal: cos θ < -0.6;6d) The entry point is within 2.5 meters of the nearest cable hole.

Figure 6.3: A schematic of one of the cable holes with a veto counter.

Flasher PMTs
A “flasher” refers to when there is a repetitive discharge in the dynode of a PMT,
which may also affect nearby PMTs. Such events are characterized by abnormallybroad hit timing distributions. They can be rejected if they exhibit too much ID
activity well after the trigger time. Events that satisfy the following detector-specificcuts are removed:

747a) (SK-I) NMIN100 ≥ 15 if Nhit > 800, and NMIN100 ≥ 10 otherwise. Here NMIN100
is the number of hits in the ID in a 100 ns time window from 200 ns to 700 ns
after the trigger time, and Nhit is just the total number of ID hits.
7b) (SK-II to IV) NMIN100 ≥ 20.
Figure 6.4 shows the hit time distributions for a flasher, compared to an atmospheric
neutrino.

Figure 6.4: Typical distributions of NMIN100 for a flasher (top) and an atmospheric neutrino MC event (bottom). The arrows indicate the region that is removed
with the third reduction flasher cut. Taken from [61].

Coincidence muons
Sometimes there is a coincidence between a high-energy cosmic ray muon anda low-energy event. If the low energy event occurs slightly earlier and starts thetrigger, it can avoid the previous cuts designed to remove low-energy events, due tothe energy deposited by the muon, and conversely avoid the high-energy cuts, since

75
the OD activity of the muon is recorded too late. Events passing the following criteria
are rejected:8a) NHITAoff ≥ 20, where NHITAoff is the number of OD hits between 300 ns and
800 ns after the trigger time;8b) PEoff > 5000 (2500 for SK-II), where PEoff is the total number of p.e.’s in the
ID for the same time window.

Low energy events
Finally, a more specific cut aimed at low energy events is applied. Events are
rejected if NHIT50 < 50 (25 for SK-II), where NHIT50 is the number of hit ID PMTs
within a sliding 50 ns time window. The time window uses the residual time. Thistakes into account the time-of-flight of the photons, assumed to come from a single
vertex.

6.1.4

Fourth Reduction

The fourth FC reduction more specifically seeks to eliminate flasher events. Flashers typically have a highly-correlated charge pattern that extends across several event
timing windows, and thus a pattern-matching algorithm is applied to reject them:
1) The ID tank wall is divided into 1450 roughly equal partitions.
2) For each patch, a correlation is calculated:



A
B
B
1 X QAi − hQ i × Qi − hQ i,r=
N i
σA × σB(6.2)

76
where N is the total number of patches, A and B are the two events being comparedin different timing windows, Qi and hQi are the charge in patch i and the averagecharge in the event, and σ is the charge resolution.
3) The distance dmax between the PMTs with the maximum charge in each event iscalculated. For dmax < 0.75 m, the correlation in Equation 6.2 is incremented: r= r + 0.15.
4) Two events are defined as a match if their correlation exceeds a threshold rthresh :rthresh = 0.168 × log10

AB
P Etot+ P Etot2

+ 0.130,(6.3)

where PEtot is the total number of p.e.’s observed in the respective events.
5) Repeat the correlation calculation for 10,000 events near the target event, andreject events with a large number of matches.
Following the fourth reduction, about 18 events/day remain.

6.1.5

Fifth reduction

The fifth reduction is aimed at removing three specific categories of remaining
background: invisible muons, coincidence muons, and long-tail flashers. After thefifth reduction, about 16 events/day remain.

Invisible muons
Muons below Cherenkov threshold will escape notice from any of the criteria
described above, but they can still activate the trigger via their decay electron. They

77
can be rejected as cosmic-ray muons if some OD activity is found. Events satisfying
the following criteria are rejected:9a) PEtotal < 1000 (500 in SK-II), where as before PEtotal is just the total charge in
the ID;9b) NHITACearly + NHITAC500 ≥ 10 if Distclust < 500 cm; NHITACearly > 9 otherwise.
Here NHITAC is the number of hit OD PMTs in the largest cluster. NHITACearly isdefined in a 200 ns time window from -9000 to -200 ns prior to the primary trigger
time, NHITAC500 is the hits corresponding to -200 to 300 ns, and Distclust is the
distance between the centroids of these clusters. Selection 9a) corresponds to about
the maximum p.e. yield for a Michel electron, while 9b) indicates significant early
OD activity.

Coincidence muons
Coincidence muons remaining after the cuts in the third reduction are subjected toanother set of cuts, scanning for low-energy events in the main time window and scanning for off-trigger OD activity in a longer time window than in the third reduction.
Events meeting the following criteria are rejected:10a) PEtotal < 300 (150 for SK-II);10b) NHITAlate ≥ 20, where NHITAlate is the number of hit OD PMTs in the OD in
a 200 ns sliding time window from 300 to 2100 ns.

78Long tail flashers
Flasher events remaining after the cuts in the third reduction are removed byrequiring the Cherenkov ring to be reasonably well-fit, which is typically not the casefor the charge pattern created by flashers. These cuts use a goodness-of-fit parameterthat will be described in detail later in this chapter:11a) NMIN100 ≥ 6 if the goodness-of-fit is less than 0.4, where NMIN100 is as definedin the third reduction flasher reduction;11b) (SK-II to SK-IV only) NMIN100 ≤ 5 if the goodness-of-fit is less than 0.3.

6.1.6

Final FC sample

Of the remaining events after the five reduction stages, a final selection is madeusing more specific reconstruction information. The final sample is defined by thefollowing selections:• dwall > 200 cm, where dwall is the distance from the reconstructed vertex tothe nearest ID wall. This further removes background coming from outside thedetector, and also ensures that the event has a reasonably well-reconstructedvertex;• NHITAC < 16 (10 for SK-I), where NHITAC is the number of hit OD PMTs
in the largest cluster. This further removes high-energy events coming fromoutside the detector;• Evis > 30 MeV, where Evis is the visible energy, defined as the energy of a single
electron that would produce the equivalent amount of charge seen in the ID.
This further removes low-energy background.

79
These cuts are defined as the Fully-Contained Fiducial Volume cuts (FCFV),
which is the starting point for all analyses in this thesis. After these cuts, about 8
events/day remain. The efficiency of each cut is estimated from MC samples, anddefined as the ratio of true FC events that survive each cut. Following the FCFV
selection, the efficiency is estimated to be >99%, with a systematic uncertainty of<1% for each reduction step, and background contamination <1%.

Figure 6.5: A plot of the event rate across each reduction stage in SK-IV, with first (black),
second (red), third (cyan), fourth (dark blue), fifth (yellow), and FCFV (magenta). There is a slight gain increase with time most apparent in the firsttwo reductions. This is due to degrading PMT quality, and is accounted forin event reconstruction.

6.2

Event Reconstruction

Following the FC reduction, remaining events are put through a set of reconstruction algorithms that convert the charge and timing information into kinematicalvariables suitable for physics analyses. The same reconstruction algorithms are applied to data and MC. First, an overall event vertex is determined using PMT timing

80
information3 . Following this, Cherenkov rings are sought using an iterative Houghtransform method. More detailed information on each ring, such as direction, particle identification, and momentum, is then calculated. I describe this process here,
along with some other essential reconstruction algorithms independent of the main
ring-counting program.

6.2.1

Vertex fitting

The event vertex is calculated in three steps. The first is simple, and uses ID
PMT hit residual times (with time-of-flight subtracted) to maximize a goodness offit parameter:(ti − t0 )2
1 X
exp −
GP =
N i
2 (1.5 × σ)2

!
,

(6.4)

where N is the total number of hit PMTs, ti is the residual time of the ith PMT, t0 is afree parameter chosen to maximize the goodness GP , and σ is the timing resolution,chosen to be constant at 2.5 ns. Upon taking the vertex that maximizes GP , theoverall direction is estimated by taking a charge-weighted sum of each ID PMT.
From these initial vertex and direction calculations, the brightest Cherenkov ringis sought. The edge of the ring is determined by

R θedge

P E (θ)Q (θedge ) = 0
×sin θedgedP E(θ)dθ

2(θedge − θexp )2
× exp −
2σθ2

!
.

(6.5)

Here P E (θ) is the angular distribution of the observed charge with respect to opening
3
For most SK analyses, including this one, a single event vertex is assumed, though this issometimes not the case. For instance, a search for pp → K + K + used an algorithm that sought
multiple vertices, corresponding to the decay positions of the charged kaons. See [82] for moredetails.

81
angle θ, taken to be the angle between the charge-weighted direction and the ringedge being tested θedge ; θexp is the expected angle based on the charge in the cone;and σθ is the angular resolution. The charge in P E (θ) is corrected for attenuation
and PMT angular acceptance. The overall ring direction, initially calculated only
from PMT charge and opening angle θedge , are varied to maximize Q (θedge ).
A final, more precise vertex fit is performed that takes into account scattering of
Cherenkov light and the track length of the charged particle. Instead of calculating
the residuals from their location relative to an assumed vertex as in the initial fit ofEquation 6.4, the residuals are calculated as follows:~i − O~ i | − n |P~i − X~ i | for PMTs within the Cherenkov ring edge;ti = t0i − 1c |X
c~ i | for PMTs outside the Cherenkov ring edge.
ti = t0i − nc |P~i − X~ is the vertex position, X~ i is the emission point along the particle track that
Here O
hits PMT i, P~i is the position of the ith PMT, and n is the index of refraction. Thus
PMTs within the ring edge take into account both light emitted along the particle
track, and scattered light (the 1/c and n/c terms, respectively), and PMTs outside
the ring consider only scattered light.
From these residuals, an improved goodness of fit parameter is calculated. For
PMTs within the ring edge, the calculation is similar to Equation 6.4. For thoseoutside, scattering effects are also included. More details can be found in [83].
The quality of vertex reconstruction is assessed using MC. The resolution of thetrue MC vertex with the reconstructed vertex is measured for different samples. The
resolution for multi-ring samples, of interest for this thesis, range from about 50 to 100
cm, although the analyses do not depend very much on precise vertex reconstruction,
so that it is not a significant source of systematic uncertainty.

82

6.2.2

Ring counting

The vertex reconstruction algorithms locate the main Cherenkov ring. Other ringcandidates are sought using a Hough transform method. Figure 6.6 shows the basic
logic of the algorithm. Each hit PMT has a ring with a 42-degree angle drawn aroundit, with the center of the ring taken to be the point at which many of these drawn
rings intersects. The diagram in Figure 6.6 is a simplification, in that the actual
expected charge pattern is more complicated than a circle, and depends on the angle
with respect to the normal line for each PMT.

Figure 6.6: The Hough transform method. The grey area corresponds to hit PMTs, andthe dashed circles are drawn around each hit PMT with a 42-degree opening
angle. The ring center is determined by the point at which the most circles
overlap.

Rings are counted iteratively via a likelihood method. The hypothesis of (N+1)
rings is tested against N rings, up to a maximum of five rings. The likelihood isdefined as:

83

LN =

X

log P

qiobs ,

N
X

!!
exp
αn qi,n

,

(6.6)

n=1

i

where αn is a weight for ring n, allowed to vary to maximize the likelihood, and P isa Gaussian probability:exp

P qiobs , qi

qiobs − qiexp
=√
exp −
2σ 2
2πσ 2
1

2 !
.

(6.7)

Figure 6.7 shows the ring-counting likelihood for data and MC. The starting point
for analyses in this thesis is the multi-ring sample, for which L2 > L1 .

Figure 6.7: Ring-counting likelihod in SK-IV, for the sub-GeV (Evis < 1.3 GeV) sample.
The MC is normalized to data events (crosses) and separated by neutrino
interaction type.

84

6.2.3

Particle identification

Classifying the relativistic charged particles passing through the detector is important for most physics analyses. In SK, Cherenkov rings are classified in two broad
categories, corresponding to muons and electrons, the two leptons produced in CC
neutrino interactions4 The terms e-like and µ-like will be used hereafter.
High-energy electrons produce electromagnetic showers that create diffuse ringpatterns. Specifically, they interact via the chain of bremsstrahlung (e± → e± γ)
and photon pair production (γ → e+ e− ). Though neutral, photons also produce
electromagnetic showers via pair production. The more massive muons (and charged
mesons and protons) interact negligibly via Bremsstrahlung, as the cross section is
proportional to 1/m2 . Muons interact primarily via ionization, and thus produce
rings with sharp, thick edges. Charged mesons, particularly pions, are subject tohadronic interactions that can suddenly alter their path. As a result, the rings they
form are usually thinner than those of muons, though the absence of electromagnetic
showers still gives them sharp edges. Figure 6.8 shows examples of µ-like and e-likerings.
The particle identifcation algorithm (or PID) mainly works by comparing theobserved charge with an expected charge, for the e-like or µ-like hypothesis. The
expected charge distributions for the ith PMT are calculated as follows:4τ leptons are also produced in CC ντ interactions, although they decay rapidly and require a
specialized analysis to properly identify. See [84] for details.

85Super-Kamiokande IV
Run 999999 Sub 0 Event 2
15-09-04:04:33:36
Inner: 1634 hits, 5608 pe
Outer: 4 hits, 3 pe
Trigger: 0x07
D_wall: 1690.0 cm
Evis:

0.0 MeV

Charge(pe)>26.7
23.3-26.7
20.2-23.3
17.3-20.2
14.7-17.3
12.2-14.7
10.0-12.2
8.0-10.0
6.2- 8.0
4.7- 6.2
3.3- 4.7
2.2- 3.3
1.3- 2.2
0.7- 1.3
0.2- 0.7< 0.2
800
1 mu-e
decay

640
480
320
160
0
0

500

1000

1500

2000

Times (ns)Super-Kamiokande IV
Run 999999 Sub 0 Event 7
15-09-04:04:36:08
Inner: 3075 hits, 7369 pe
Outer: 0 hits, 0 pe
Trigger: 0x07
D_wall: 1690.0 cm
Evis:

0.0 MeV

Charge(pe)>26.7
23.3-26.7
20.2-23.3
17.3-20.2
14.7-17.3
12.2-14.7
10.0-12.2
8.0-10.0
6.2- 8.0
4.7- 6.2
3.3- 4.7
2.2- 3.3
1.3- 2.2
0.7- 1.3
0.2- 0.7< 0.2
1500
0 mu-edecays

1200
900
600
300
0
0

500

1000

1500

2000

Times (ns)Figure 6.8: A simulated µ+ (top) and e+ (bottom), both with momentum 800 MeV/c.
The charge pattern is notably sharper for the µ+ and more diffuse for the e+ .
The intensity scale indicates the amount of charge deposited on the PMTs.
The main display shows the ID, while the small display on the top right shows
the OD, in which there is negligible activity. Both displays show the detector“unrolled” from its cylindrical shape. The white +’s indicate the horizontal
and vertical location of the reconstructed vertex, projected onto the detector
wall.

86

qiexp (e) = αe × Qexp (pe , θi ) ×sin2 θxi

qiexp (µ) = αµ × 1.5R
1× ri /L × f (Θ) + qiscat ,
ri
e


ri




! + qiknock dθ
sin θxi + ri dx


x=xi

1
× ri /L × f (Θ) + qiscat .
e

(6.8)

In these equations, αe,µ are normalization factors, θi the angle between the ith PMT
and the ring direction, f (Θ) the PMT acceptance as a function of incident angle Θi ,ri the distance from the vertex to the ith PMT, L the attenuation length, and qiscat
the expected charge for the ith PMT due to scattering.
For the e-like hypothesis, MC simulations are used to estimate Qexp (pe , θi ), theexpected charge as a function of momentum pe and θi . The simulations use a test
vertex located R=16.9 m from the ID wall. The factor of (R/ri )1.5 accounts forthe difference in diffusion between the test radius track length and the actual track
length. The expected charge accounts for scattering, light attenuation, and geometricfactors. For the µ-like hypothesis, the muon track length x is taken into account, withxi referring to the estimated point along the muon track from which photons at the
ith PMT were received. The µ-like hypothesis also includes contributions from q knock ,so-called “knock-on” electrons that are ionized by the passing muons. This effect isestimated with MC simulations.
The PID likelihood function is calculated from the observed and expected charge
for the nth ring as87!
Ln (e, µ) =Yθi <1.5×θCP

exp
qiobs , qi,n(e, µ) +

X

exp
qi,n
0

,

(6.9)

n0 6=nwhere the product is over all PMTs within the Cherenkov cone θC × 1.5, P is theGaussian probability defined in Equation 6.7 and the sum over n0 accounts for lightfrom overlapping rings. The expected e-like and µ-like charges are optimized byaltering the direction and opening angle of each ring to maximize the likelihood.
The charge pattern likelihood 6.9 is converted to a χ2 parameter,χ2n (e, µ) = −2logLn (e, µ) + const,(6.10)

which is then used to construct a probability based on the charge pattern:2(χ2 (e, µ) − min (χ2n (e), χ2n (µ)))
Pnpattern (e, µ) = exp − n2σχ2 2nwhere σχ2n =√!
,(6.11)

2N for N PMTs in the likelihood calculation.

For single-ring events, a second probability is used in the PID determination,
based on the Cherenkov angle:
θobs − θexp (e, µ)
angleP(e, µ) = exp −
2δθ2

2 !
,

(6.12)

where θobs is the reconstructed Cherenkov angle, δθ the angle fitting error, andθexp (e, µ) the expected angle based on the reconstructed e/µ-like momentum.
The final PID probability is the product of these two estimators:

P (e, µ) = P pattern (e, µ) × P angle (e, µ).

(6.13)

Note that for multi-ring events, only the light pattern is used, so that P angle = 1.

88
The final PID likelihood is defined as

LP ID =

pp−logP (µ) − −logP (e).

(6.14)

For positive values, the ring is fit as µ-like, and negative as e-like.

6.2.4

Momentum reconstruction

The momentum of a ring is determined by the charge observed around the ring,and the energy scale calibration described in Chapter 3. Specifically, the correctedcharge in a 70-degree half-angle cone of a ring’s direction, called “RTOT”, is used todetermine its momentum. The charge is restricted to a (-50, 250) ns time window
relative to the time-of-flight subtracted timing peak, to avoid charge due to decay
electrons. It is corrected for angular acceptance, water attenuation, contributions
from scattered light, and PMT gain as follows:
GM C
RTOT =
GDAT A X
X
cos Θiobs
ri /Lα×
qi × e
×
−
Si
f
(Θ
)
i
i
i

!
.

(6.15)

Here GMC/DATA is the relative PMT gain for MC and data, α is a normalization factor,
ri is the distance from the ith PMT to the event vertex, L is the water attenuation
length, f (Θ) is the acceptance of the PMT for incident angle Θ, Si is the expectedcharge for the ith PMT from scattered photons, and the sums are over all PMTs
meeting the timing and angular requirements described above. Figure 6.9 shows thelinear relationship of RTOT and momentum. The conversion between RTOT andmomentum is determined by MC simulation, by simply calculating the RTOT values
for particles of known momenta.

89

RTOT (corrected p.e.)

RTOT vs. Momentum
12000

e-like10000

mu-like

8000

6000

4000

2000

0
0

200

400

600

800

1000 1200 1400 1600 1800 2000 2200

Momentum (MeV/c)

Figure 6.9: Distributons of RTOT vs. momentum up to 2000 MeV/c for both e-like andµ-like rings.

6.2.5

Decay electrons

Decay electrons are a valuable indicator of the presence of a muon — and, in thecontext of this thesis, a charged pion. They provide a relatively independent sourceof event information from the Cherenkov rings produced by primary particles.
The decay electron search window extends to 20 µs after the event trigger. Decay electron candidates are classified as primary or “in-gate” events, or sub-events,
depending on their timing. In SK-I-III, the primary event requirements are as follows:• 0.1 µs < t < 0.8 µs;• NHIT30 ≥ 60 (30 in SK-II).
Here t is the time after the event trigger, and NHIT30 is the number of hits in a 30
ns time window. Requirements for sub-events are defined similarly:• 1.2 µs < t < 20 µs;

90• NHIT30 ≥ 40 (20 in SK-II).
In addition, both primary and sub-event types are required to have at least 50 hit
ID PMTs (25 for SK-II), and the vertex goodness (Equation 6.4) is required to begreater than 0.5. The ID PMT cut is designed to remove low-energy photons radiatedfrom µ− capture in oxygen.
The decay electron detection efficiency varied significantly between SK-I-III and
SK-IV. In SK-I-III, an impedance mismatch in the DAQ electronics required that thetime interval (800 ns < ∆ t < 1200 ns) be excluded from the search, where ∆t is
the time difference from the primary event peak. The upgraded electronics of SK-IV
removed this mismatch, so that this interval is now included. Also, the overall hit
ID PMT cut is removed. Both of these changes increase the decay electron tagging
efficiency. In SK-I-III, the efficiency is 80% for µ+ and 63% for µ− ; in SK-IV, it is
96% for µ+ and 80% for µ− . In all cases, the efficiency is estimated with MC.

6.2.6

π 0 reconstruction

Ideally, a π 0 produces two showering rings, one from each photon from the decay
π 0 → γγ. Sometimes, only one ring will be detected, if one of the two photons
receives very little of the decay energy, or there is a close overlap between the two
photons, such that they do not create distinct rings. In such cases, a π 0 reconstruction
algorithm described in detail in [85] is useful. It does not seek individual Cherenkov
rings, but rather assumes two photons, and constructs a likelihood based purely on
the observed light pattern. This algorithm has been used in atmospheric [86] andlong-baseline [87] neutrino experiments to identify and reject neutral-current single
π 0 events. In the pn → π + π 0 search, it is used to reconstruct the π 0 candidate mass,
which is used as a variable in the analysis.

Chapter 7
Dinucleon Decay Search
The experimental signature of a dinucleon decay to pions event, for all threemodes considered, is two back-to-back pions, each forming at least one Cherenkov
ring of the correct type. Cleanly identifying such events is complicated largely due
to the interactions of charged pions in both the residual 16 O nucleus and the water.
Interactions in the water can suddenly interrupt the pion’s path, which typicallylead to poor momentum reconstruction. Thus, for modes involving at least one π +(pp → π + π + and pn → π + π 0 ), simple cuts on kinematic variables such as the total
momentum and invariant mass that are used in other nucleon decay analyses do notgive optimal sensitivity. For these modes, a multivariate method is applied to better
exploit event variables. Specifically, a boosted decision tree (BDT) method is used.
The method is implemented in the ROOT-based TMVA [88] analysis library. This
was also used in the dinucleon decay search for pp → K + K + in [40].
For nn → π 0 π 0 , the situation is more straightforward, as the π 0 ’s that escape
the nucleus decay to energetic photons before interaction in the water, and their
momentum is well-reconstructed. For this mode, simple kinematic cuts are sufficientfor the analysis.
91

92

7.1

Boosted Decision Trees

Here I describe some relevant details of the BDT, applicable for both pp → π + π +and pn → π + π 0 searches (and the n → n̄ search in the next chapter). The BDT
method was chosen among other multivariate analysis options, such as the artificial
neural network and likelihood methods. It was found that, for the same set of input
variables, BDTs generally performed better in basic signal and background separation.
BDTs have found diverse applications in particle physics. In addition to the pp →K + K + analysis in [40], they have been used for identifying electron neutrinos in
MiniBooNE [89] and single top quark production at D0 [90], among others.
Similar to other multivariate methods, a BDT takes a set of input variables – inthis case, reconstructed information for both dinucleon decay signal and atmospheric
neutrino background – with the goal of producing a single discriminating output
variable that separates signal and background. To get to this single variable, threestages are required. They are referred to as training, testing, and analysis, and each
uses separate sets of signal and background MC. I describe the process briefly.
Boosted decision trees are sets of individual decision trees, simple binary trees
formed by cuts on single variables. Trees are built using the training sample asfollows. Starting with the full training sample, a cut on a single variable is made thatbest separates signal and background. For the analyses in this paper, this optimal
separation is determined by the Gini index g = p(1 − p), where p is the purityof the sample after the cut. Other separation types exist, though they don’t lead tosignificantly different performance [88]. The sample then splits off into two samples, or
nodes, based on this cut. Each of the daughter nodes does the same thing, separately
finding the variable that best separates signal and background, and so forth until
some stopping criterion is reached (such as a minimum number of events remaining

93
on a node). Terminal nodes are declared signal-like or background-like based on thepurity of signal events in the node. Figure 7.1 shows an example tree used in the
training process for the pn → π + π 0 search.

Figure 7.1: An example tree for the pn → π + π 0 analysis. The tree has a depth of four.
The variables used are the reconstructed π 0 momentum (“pi0ptot”), the visible energy (“evis”), the π 0 − π + opening angle (“opang”), and the ratio of
charge carried in the highest-energy ring (“qratio”), all of which are describedin the text. Nodes that split to the left are more signal-like, and nodes thatsplit to the right more background-like.

On their own, decision trees are weak classifiers, providing no more discriminatory
power than a simple set of cuts. They are also unstable, prone to potentially largechanges from statistical fluctuations. The idea behind a BDT is that many suchtrees can combine to form a single, strong classifier. The training process continues
over many such trees (typicially 1000 or so) using the same training sample, withthe prescription that events in subsequent trees are re-weighted, or “boosted.” Upon
re-weighting, the best set of discriminating variables may change. Variables can also

94
be used in more than one cut on a given tree.
Once all the trees are done, additional datasets may pass through the trained setof trees. On an event-by-event basis, the final BDT output is given by

YBDT =

1

N
tree
X

Ntree i=1

wi hi (x).

(7.1)

Here Ntree is the size of the ensemble of trees, wi is the weight of each tree in theensemble, determined by the boosting method, hi is the “performance” of each eventfor the ith tree (whether it ended on a signal or background-like node) and x is the setof input variables. Variables are given a ranking, or “importance,” a numerical valuethat roughly corresponds to the number of times the variable is used in the training
process relative to the other variables in the set.
Once the BDT is trained, two additional, statistically independent sets of signal
and background MC are necessary. The test samples are used to determine optimal
parameters for the BDT. These parameters include:• The boost method — this determines how events are re-weighted, or “boosted”,
in the training process. Several boosting algorithms exist, and are described indetail in [88].1 Two types of boosts are considered in this thesis — the adaptive
boost, and the gradient boost.
The adaptive boost uses a simple formula for determining the weight wi in
Equation 7.1. It takes the misclassification error of the previous tree (the ratioof signal events that ended up in background nodes, and vice-versa) and defineswi (adaptive) = ln
1

See especially Section 7, “Boosting and Bagging.”

1 − err
err


.

(7.2)

95
The gradient boost method is more complicated, and doesn’t have a set formulafor its re-weighting procedure. It attempts to minimize a loss function thatmeasures the difference between the value from a given tree, and the true value
known from the training process2 .
In general, the adaptive boost method works well in situations with a small
misclassification error, thereby maximizing the boosted weight in Equation 7.2.
In noisier settings prone to a higher error, the gradient boost method is often
superior.
• The number of trees — how many trees are used in the training process (Ntree
in Equation 7.1). More trees generally lead to a more stable performance.
• The maximum depth — how many decisions are allowed to be made from thefirst node in each tree. Figure 7.1 shows a tree with depth 4. Setting the
maximum depth too high can lead to overtraining (described below).
• The size of the cut grid — how many cuts are considered for each variable inoptimizing the separation at each node.
• The minimum number of events in a node — the minimum number of events
such that no further decisions will be made.
The boost method is the most consequential paramater for the BDT. Adjusting theother parameters was not found to change signal and background separation verymuch.
2Actually, both adaptive and gradient boosts operate on the principle of minimizing a loss function. However, the adaptive boost loss function leads to the simple formula for re-weighting inEquation 7.2, while the gradient boost formula must be approximated via steepest-descent approach— hence its name.

96
The analysis samples are used to obtain the final sensitivity. Unlike in the training
and testing process, the BDT does not know the composition of the analysis samples.
Both the test and analysis samples are also important in detecting training bias in the
BDT. This bias is a situation in which the BDT has been trained to model statisticalfluctuations in the training sample, as opposed to modeling from general trends inthe input variables. Such overtraining can lead to different performance between MC
samples, even if they are statistically similar.
Table 7.1 shows a set of common parameters used in the N N → ππ analyses, andTable 7.2 breaks down the signal and background MC samples used for each stage.
Separate trees are trained, tested, and analyzed for each SK period. Real data areprocessed through the trained BDTs as a final step.
Value
Parameter
Number of trees
1000
Maximum depth
420Size of cut grid
Minimum events on a node 100
Table 7.1: A set of common parameters used for the BDTs in the pp → π + π + and pn →π + π 0 searches.

training
testinganalysis
N N → ππ25000 ev.
25000 ev.
25000 ev.
atm-ν∼750,000 ev. (150 yr.) ∼750,000 ev. (150 yr.) ∼1,000,000 ev. (200 yr.)
Table 7.2: Independent MC sample sizes used for each stage of the BDT for both the
pp → π + π + and pn → π + π 0 searches, prior to any cuts. Dinucleon decay
sample sizes show only the number of events, while the atmospheric neutrino
sample sizes show the approximate number of events and the equivalent numberof years. The same statistics are used for SKI-IV.

97

7.2

pp → π +π +7.2.1

Pre-selection for pp → π + π +The interactions of π + ’s in the nucleus and the water complicate the event signature for pp → π + π + . A set of precuts is applied to keep pp → π + π + events, andreduce the atmospheric neutrino background. The key characteristics are that theevent has two “back-to-back” rings, as the ring-finding algorithm is still good at reconstructing the ring direction even with the poor momentum resolution, and thatthe rings are identified as µ-like. Hadronic interactions in the water can sometimes
appear like electromagnetic showers, thereby causing π + ’s to be mis-identified as elike. However, the converse – in which rings produced by electrons or photons are
misidentified as µ-like – is rare. The following precuts are applied:(A1) There is more than one Cherenkov ring.
(A2) The two most energetic rings are µ-like. These are assumed to correspond tothe π + candidates.
(A3) The angle between the two most energetic rings is greater than 120 degrees, aminimal requirement for “back-to-back” pions.
(A4) The total visible energy is less than 1600 MeV, approximately the maximum
visible energy expected from a dinucleon decay event (2Mp − 2Mπ+ ), where Mp
is the proton and Mπ+ the pion mass.
Table 7.3 shows the remaining efficiencies, background rates, and data events after
each precut. Figure 7.2 shows the selections graphically.

98
Number of Cherenkov rings

Angle between candidate rings
25

10000
20

6000

EventsEvents

8000

15

4000

10

2000

5

0
1

1.5

2

2.5

3
3.5
4
number of rings

4.5

5

5.5

6

0
0

20

40

60
80
100
120
140
Angle between π+ candidates

160

180

Visible energy
14
12

Events

10
8
6
4
2
0
0

200

400

600

800 1000 1200 1400 1600 1800 2000
visible energy (MeV)

Figure 7.2: Selection criteria (A1) (top left), (A3) (top right), and (A4) (bottom left) for
pp → π + π + MC (blue histogram), atm.-ν background (red histogram), anddata (crosses) for SK-IV. All plots are area-normalized to the data. The green
line indicates the placement of the cuts described in the text.

7.2.2

Multivariate analysis for pp → π + π +After applying these precuts, remaining events are processed by the BDT. The following set of input variables is used, ordered by their relative importance, as describedabove in Section 7.1:(a1) The angle between the two most energetic µ-like rings. It is expected to peakaround a perfect “back-to-back” angle of 180 degrees for the signal, and have afalling spectrum for increasing angle in the background.

99
SK-I SK-II SK-III SK-IV
FCFVEff. (%) 95.1
Bkg. 11889
Data 12299

97.1
6415
6610

95.7
95.7
4206 14266
4355 14444(A1)
Eff. (%)
Bkg.
Data

84.2
3595
3691

85.1
1944
2056

85.3
1266
1292

85.4
4330
4392

(A2)
Eff. (%)
Bkg.
Data

19.7
245
222

18.8
117
106

21.4
96
78

21.2
327
273

(A3)
Eff. (%)
Bkg.
Data

11.3
36
28

10.5
18
14

12.0
14
8

12.1
49
45

(A4)
Eff. (%)
Bkg.
Data

11.2
33
27

10.5
17
14

12.0
13
8

12.1
45
43

Table 7.3: Efficiency, expected background, and data events after each precut for the
pp → π + π + search. The FCFV cut is described in Section 6.1.6, and cuts(A1)-(A4) are described in the text. Expected background is weighted foroscillation effects, and scaled to the appropriate SK detector livetime (from the
200 year analysis sample).

(a2) The ratio of charge carried by the highest-energy ring to the total charge of allrings. This is expected to be smaller for signal, since the pions are expected tohave roughly equal energy. By contrast, we expect most of the energy from acharged-current (CC) atmospheric neutrino event to be in the charged lepton
ring.

100
(a3) The total visible energy. This has a broad distribution up to 1600 MeV for thesignal, but peaks at a lower value for the background around 200 MeV.
(a4) The maximum distance between a decay electron vertex and the primary event
vertex. We expect this to be larger for energetic muons than for pions, since
the muons penetrate water without sudden hadronic interactions. Muons above
Cherenkov threshold sometimes appear in the signal (about 9% of the time),
but are much more common in the background, due to charged-current νµ interactions such as single pion production.
(a5) The maximum angle between any µ-like ring direction vector, and any Michelelectron vertex vector (as pointed to from the primary vertex). Similar to (a1),this is designed to look for “back-to-back” events, and is an additional check on
the topology of the µ-like rings.
(a6) The magnitude of the vector sum of corrected charge associated with all rings(or each ring’s RTOT, as described in Section 6.2.4). This is similar to thetotal momentum, since the corrected charge associated with each ring is used
to determine its momentum. Due to typically poor momentum resolution fromhadronic energy losses, it is preferable to use this variable instead of the total
momentum.
(a7) The number of Cherenkov rings. Though we are trying to isolate two pions, they
typically interact in the water, often producing visible secondary tracks, so thatthe number of rings distribution peaks at three. By contrast, the Cherenkov
ring spectrum for the background falls with increasing ring number.
(a8) The number of decay electrons. Naively we would expect two Michel electrons
from two charged pions. However, the pions could undergo charge exchangeor absorption, leading to zero or one Michel electrons. Though the signal andbackground distributions both peak at one decay electron, this variable is rela-101tively independent of most of the other variables, which depend on Cherenkov
ring-counting.
(a9) The number of µ-like rings. Similar to the total number of rings, this variable
specifically seeks scatters that produce a secondary charged pion.
Figures 7.3-7.5 show distributions of each input variable for SK-IV. The corresponding
plots for SK-I-III are similar, and can be found in Appendix B. The number of events
remaining in the data after the pre-cuts is small, but the distributions generally
agree with the atm.-ν MC. The relative importance of all input variables is shown in
Table 7.4.
Variable
ImportanceAngle between µ-like rings
0.16
Ratio of charge carried by most energetic ring
0.15
Visible energy
0.15
Max. distance to decay electron vertex
0.13
Max. angle between µ-like ring and Michel vertex
0.13
Magnitude of vector sum of corrected charge
0.12
Number of rings
0.071
Number of Michel electrons
0.055
Number of µ-like rings
0.045
Table 7.4: Relative importance of each variable in the pp → π + π + search, averaged acrossSK periods.

The final distributions of the BDT output for all SK periods are shown in Figure 7.6. For pp → π + π + , the gradient boost [88] method is used. The final efficiencies,
backgrounds, and candidate data events are shown in Table 7.5.

7.2.3

Final sample for pp → π + π +The final cut on the BDT output was determined using only signal and backgroundMC, not the real data. The cut takes two basic considerations into account: (1) the

102

Events

SK-IV
8
7
6
5
4
3
2
1
0
120

130

140
150
160
Angle (degrees)

170

180

Figure 7.3: The angle between µ-like rings, the strongest discriminatory for the pp →π + π + search, for signal MC (solid histogram), atm.-ν MC (dashed histogram),and data (crosses). Distributions are shown for SK-IV and normalized to thenumber of events in the data.

√cut maximizes / b, where  is signal efficiency and b is the background rate; and (2)the cut is not too close to the signal peak, since variations are propagated through the
BDT to estimate systematics, and the part of the distribution near the peak changes
the most under these variations. For (2), 0.7 was the closest value to the peak that we
considered. As the efficiency (background) distribution increases (decreases) nearlymonotonically for all SK periods, it was found that a cut at 0.7 satisfied (1) while
obeying the constraint of (2).
The largest component (∼40%) of the remaining atm.-ν background is fromcharged current single pion production (CC1π). In this mode, typically a νµ will produce a muon and a charged pion at a large angle, thus passing the π + candidate angle
requirement. Other contributing background interaction modes include charged- andSK-IV

10
9
8
7
6
5
4
3
2
1
0

EventsEvents

103

0

0.2

0.4

0.6

0.8

SK-IV

10
9
8
7
6
5
4
3
2
1
0

1

0

Charge ratio

SK-IV
12

EventsEvents

10
8
6
4
2
0
0

500
1000
1500
Maximum Michel distance (cm)

200 400 600 800 1000 1200 1400 1600
Visible Energy (MeV)

SK-IV

10
9
8
7
6
5
4
3
2
1
0

2000

0

20

40

SK-IV

8
7
6
5
4

35
30

EventsEventsSK-IV

60 80 100 120 140 160 180
Angle (degrees)

3
2
1
0

25
20
15
10
5
0

0

500 1000 1500 2000 2500
sum of charges (corrected p.e.)

3000

2

2.5

3

3.5
4
4.5
Number of rings

5

5.5

6

Figure 7.4: The ratio of charge carried by the leading ring (top left), the visible energy (topright), the maximum distance between the primary vertex and a decay electron
vertex (middle left), the largest angle between a µ-like ring and decay electron
vertex (middle right), the magnitude of the vector sum of RTOT (bottom left),
and the number of Cherenkov rings (bottom right) for the pp → π + π + search,
for signal MC (solid histogram), atm.-ν MC (dashed histogram), and data(crosses). Distributions are for SK-IV, and are normalized to the number of
events in the data. Details of each variable are described in the text.

SK-IV

24
22
20
18
16
14
12
10
8
6
4
2
0

SK-IV

50
40

EventsEvents

104

30
20
10
0

0

0.5

1

1.5

2

2.5

3

3.5

4

2

Number of Michel electrons

2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8
Number of mu-like rings

4

Figure 7.5: The number of decay electrons (left) and the number of µ-like rings (right) forthe pp → π + π + search, for signal MC (solid histogram), atm.-ν MC (dashedhistogram), and data (crosses). Distributions are shown for SK-IV, and arenormalized to the number of events in the data. Details of each variable aredescribed in the text.

SK-II
6

10

5

8

4

EventsEvents

SK-I
12

6

3

4

2

2

1

0

0
-1

-0.5

0

0.5

1

-1

-0.5

BDT Output

SK-III

3.5

0.5

1

0.5

1

SK-IV

2.5

12

Events3

16
14

Events0
BDT Output

2

1.5
1
0.5

10
8
6
4
2
0

0
-1

-0.5

0
BDT Output

0.5

1

-1

-0.5

0
BDT Output

Figure 7.6: BDT output for pp → π + π + signal (solid), atm.-ν background (dashed), anddata (crosses). The vertical green line indicates the BDT cut value, and thearrow indicates that only events to the right of the cut are kept.

105
SK-I SK-II SK-III SK-IV
Eff. (%)
6.1
5.3
6.4
5.8
Bkg. (MT-yr) 17.8 14.3
17.4
14.2
Bkg. (SK live.) 1.6 0.70
0.56
1.6
Candidates01
01
Table 7.5: Efficiency, expected background events, and candidate data events for SKI-IV
pp → π + π + search. Background is quoted both for the appropriate SK livetime,and per megaton-year.

mode SK-I SK-II SK-III SK-IV
CC1π 45% 42%
51%
42%
24%
34%
CCDIS 30% 31%
NCDIS 10%
9%
6%
13%
9%
9%
10%
10%
CCQE
NC1π
5%
8%
8%
0%
Table 7.6: Atmospheric neutrino interaction modes in the remaining background for the
pp → π + π + search by percentage, as estimated by atmospheric neutrino MC.

neutral-current deep inelastic scattering (CCDIS and NCDIS), charged-current quasielastic (CCQE) and neutral-current single pion production (NC1π). See Table 7.6 fora breakdown of the neutrino interaction modes for remaining background.
A pp → π + π + MC event is shown in Figure 7.7. An atmospheric neutrino event
surviving all cuts is shown in Figure 7.8. Two candidate events were found, one in
SK-II (Figure 7.9) and another in SK-IV (Figure 7.10). No events are observed in SKI or SK-III. Table 7.7 shows values of the BDT input variables for each candidate,compared to the means of the signal and background distributions. While bothcandidate events appear signal-like in some variables, particularly the angle betweenµ-like rings (both candidates) and energy variables (SK-IV candidate), the candidates
are consistent with the expected background rate for each SK period (Table 7.5).
Thus, we conclude that there is no evidence for pp → π + π + in the data.

106Super-Kamiokande IV
Super-Kamiokande IV
13-07-12:04:29:54
13-07-12:04:29:54
Inner:
1832 hits, 7363 pe
Inner:
1832 hits,
Outer:
5 hits,3 pe 7363 pe
Outer: 5 hits, 3 pe
Trigger: 0x00
Trigger: 0x00
D_wall: 735.1 cm
D_wall: 735.1 cm
Evis: 829.6 MeV
Evis: 829.6 MeV

Charge(pe)Charge(pe)
>26.7
>26.7
23.3-26.7
23.3-26.7
20.2-23.3
20.2-23.3
17.3-20.2
17.3-20.2
14.7-17.3
14.7-17.3
12.2-14.7
12.2-14.7
10.0-12.2
10.0-12.2
8.0-10.0
8.0-10.0
6.28.08.0
6.24.76.26.2
4.73.34.74.7
3.32.23.33.3
2.21.32.22.2
1.30.71.31.3
0.70.20.70.7
0.2< 0.2
< 0.2

370
370
22 mu-emu-edecaysdecays

296
296
222
222
148
148
74
74

00
00

500
500

1000
1000

1500
1500

2000
2000

Times (ns)
Times
(ns)

Figure 7.7: A pp → π + π + MC event. The two solid orange rings correspond to true π + ’s,
and have an angle of 164 degrees between them. The dashed cyan ring is fit
as e-like, but is actually a hard scatter of one of the π + ’s.

Super-Kamiokande I
Super-Kamiokande I
12-03-07:01:50:29
12-03-07:01:50:29
Inner:
1726 hits, 3784 pe
Inner:
1726 hits,
Outer:
2 hits,4 pe 3784 pe
Outer: 2 hits, 4 pe
Trigger: 0x03
Trigger: 0x03
D_wall: 397.7 cm
D_wall: 397.7 cm
Evis: 409.1 MeV
Evis: 409.1 MeVCharge(pe)Charge(pe)
>26.7
>26.7
23.3-26.7
23.3-26.7
20.2-23.3
20.2-23.3
17.3-20.2
17.3-20.2
14.7-17.3
14.7-17.3
12.2-14.7
12.2-14.7
10.0-12.2
10.0-12.2
8.0-10.0
8.0-10.0
6.28.08.0
6.24.76.26.2
4.73.34.74.7
3.32.23.33.3
2.21.32.22.2
1.30.71.31.3
0.70.20.70.7
0.2< 0.2
< 0.2

270
270
+1
+1 subevt
subevt
216
216
162
162
108
108
54
54

00
00

500
500

1000
1000

1500
1500

2000
2000

Times (ns)
Times
(ns)

Figure 7.8: Surviving atmospheric neutrino MC event for the pp → π + π + search. Theinteraction type is νµ charged-current single π + production. The uppermostµ-like ring corresponds to a µ− , while the smaller µ-like ring is a π + . The
e-like ring is mis-fit, as it comes from a hard scatter of the π + . The angle
between the two µ-like rings is 170 degrees, thereby mimicking the main event
signature. The momentum of the muon µ-like ring is 402 MeV/c, and themomentum of the π + µ-like ring is 301.3 MeV/c

107

Super-Kamiokande II
Super-Kamiokande II
04-11-10:07:48:37
04-11-10:07:48:37
Inner:
885 hits, 1517 pe
Inner:
885 hits,
Outer:
2 hits,9 pe 1517 pe
Outer: 2 hits, 9 pe
Trigger: 0x07
Trigger: 0x07
D_wall: 456.4 cm
D_wall: 456.4 cm
Evis: 387.3 MeV
Evis: 387.3 MeV

Charge(pe)Charge(pe)
>26.7
>26.7
23.3-26.7
23.3-26.7
20.2-23.3
20.2-23.3
17.3-20.2
17.3-20.2
14.7-17.3
14.7-17.3
12.2-14.7
12.2-14.7
10.0-12.2
10.0-12.2
8.0-10.0
8.0-10.0
6.28.08.0
6.24.76.26.2
4.73.34.74.7
3.32.23.33.3
2.21.32.22.2
1.30.71.31.3
0.70.20.70.7
0.2< 0.2
< 0.2

210
210
+1
+1 subevt
subevt
168
168
126
126
84
84
42
42

00
00

500
500

1000
1000

1500
1500

2000
2000

Times (ns)
Times
(ns)

Figure 7.9: pp → π + π + candidate event in SK-II. The two µ-like rings are strikingly backto-back, at almost exactly 180 degrees to each other. The e-like ring could be
due to a hard scatter. There is one Michel electron, not shown. The µ-likerings have momenta of 399.0 MeV/c (solid) and 280.5 MeV/c (dashed), andthe e-like ring has momentum 42.0 MeV/c. The visible energy is 387.0 MeV,
the total momentum 124.2 MeV/c, and the invariant mass 710.9 MeV/c2 .

Super-Kamiokande IV
Super-Kamiokande IV
12-02-11:20:32:17
12-02-11:20:32:17
Inner:
2097 hits, 7408 pe
Inner:
2097 hits,
Outer:
8 hits,5 pe 7408 pe
Outer: 8 hits, 5 pe
Trigger: 0x10000007
Trigger: 0x10000007
D_wall: 393.5 cm
D_wall: 393.5 cm
Evis: 742.8 MeV
Evis: 742.8 MeV

Charge(pe)Charge(pe)
>26.7
>26.7
23.3-26.7
23.3-26.7
20.2-23.3
20.2-23.3
17.3-20.2
17.3-20.2
14.7-17.3
14.7-17.3
12.2-14.7
12.2-14.7
10.0-12.2
10.0-12.2
8.0-10.0
8.0-10.0
6.28.08.0
6.24.76.26.2
4.73.34.74.7
3.32.23.33.3
2.21.32.22.2
1.30.71.31.3
0.70.20.70.7
0.2< 0.2
< 0.2

440
440
22 mu-emu-edecaysdecays

352
352
264
264
176
176
88
88

00
00

500
500

1000
1000

1500
1500

2000
2000

Times (ns)
Times
(ns)

Figure 7.10: pp → π + π + candidate event in SK-IV. The two µ-like rings are at an angleof 151 degrees to each other, with momenta of 541.4 and 389.5 MeV/c.
The dashed e-like ring could be due to a hard scatter. There are 2 Michelelectrons, not shown. The visible energy is 742.8 MeV, the total momentumis 445.8 MeV/c, and the invariant mass is 1022.3 MeV/c2 .

108
Variablea1 (degrees)a2 (dimensionless)a3 (MeV)
a4 (cm)a5 (degrees)a6 (corrected p.e.)a7 (dimensionless)a8 (dimensionless)a9 (dimensionless)

Candidate
SignalBackground
SK-II SK-IV SK-II SK-IV SK-II SK-IV
177.7 151.8 152.3 152.1 138.3 138.1
0.60
0.46
0.55
0.55
0.66
0.66
400.0 745.6 680.5 693.7 440.2 407.3
1245.6 1420.1 440.7 424.8 523.0 459.2
166.3 111.9
94.2 109.0
93.2 110.9
306.9 1799.5 601.3 1310.0 448.7 797.5
3
3
2.89
2.95
2.39
2.40
1
2
0.91
1.09
1.00
1.21
2
2
2.14
2.16
2.05
2.05

Table 7.7: Candidate variable values, and mean variable values for signal and background
distributions, for each of the variables used in the BDT for the pp → π + π +search. The description of each variable is in the text. Values are given for
SK-II and SK-IV, for which single candidate events were found.

pn → π +π 0

7.3

There are many similarities between the pn → π + π 0 search and the pp → π + π +search. The main difference is that the π 0 provides a new set of discriminatory
variables, and some of the variables only apply to the reconstructed π 0 .

7.3.1

Pre-selection for pn → π + π 0

Following largely the same logic of the pp → π + π + search, a set of pre-cuts is
applied:(B1) There is more than one Cherenkov ring.
(B2) There is at least one e-like and one µ-like ring. The π + is assumed to correspondto the most energetic µ-like ring, while the π 0 is assumed to correspond to thee-like ring (if one e-like ring is found) or the two most energetic e-like rings (ifmore than one is found).

109(B3) The number of Michel electrons is no more than one.
(B4) The “reduced” visible energy, defined as the total visible energy minus theenergy of the π 0 as defined in the previous cut, is less than 800 MeV. This issimilar to cut (A4) used in the pp → π + π + search - the maximum visible energy
from the π + is Mproton − Mπ+ ≈ 800 MeV.
(B5) The angle between the reconstructed π + and π 0 is greater than 120 degrees.

Number of Cherenkov rings

Number of decay electrons

10000
700
600

8000

EventsEvents

500
6000

400
300

4000

200
2000
100
0
1

1.5

2

2.5

3
3.5
4
number of rings

4.5

5

5.5

0
0

6

0.5

Reduced visible energy

1.5
2
2.5
3
3.5
number of decay electrons

4

4.5

5

160

180

Angle between candidate rings
70

80
70

60

60

50

50

EventsEvents

1

40

40
30

30
20

20

10

10
0
0

200

400

600 800 1000 1200 1400 1600 1800 2000
reduced visible energy (MeV)

0
0

20

40
60
80
100
120
140
Angle between π candidates (degrees)

Figure 7.11: Selection criteria (B1) (top left), (B3) (top right), (B4) (bottom left) and(B5) (bottom right) for pn → π + π 0 MC (blue histogram), atm.-ν background(red histogram), and data (crosses) for SK-IV. All plots are area-normalized
to the data. The green line indicates the placement of the cuts described in
the text.

110
Table 7.8 shows the remaining efficiencies, background rates, and data events after
each precut.
SK-I SK-II SK-III SK-IV
FCFVEff. (%) 94.0
Bkg. 11889
Data 12299

96.8
6415
6610

96.1
94.9
4206 14266
4355 14444(B1)
Eff. (%)
Bkg.
Data

78.6
3595
3691

79.7
1944
2056

79.8
1266
1292

79.6
4330
4392

(B2)
Eff. (%)
Bkg.
Data

34.9
1178
1176

36.0
615
645

36.1
428
414

36.3
1469
1437

(B3)
Eff. (%)
Bkg.
Data

32.9
995
1009

34.1
521
554

34.1
361
360

33.6
1101
1091

(B4)
Eff. (%)
Bkg.
Data

32.3
622
665

33.4
330
349

33.2
224
237

33.7
693
691

(B5)
Eff. (%)
Bkg.
Data

21.0
132
136

21.9
69
66

21.6
48
45

21.1
147
171

Table 7.8: Efficiency, expected background, and data events after each precut for the pn →π + π 0 search. Cuts (B1)-(B5) are described in the text. Expected backgroundis weighted for oscillation effects, and scaled to the appropriate SK detector
livetime (from the 200 year analysis sample).

111

7.3.2

Multivariate analysis for pn → π + π 0

As with pp → π + π + , events surviving the precuts are passed into a BDT. The
following set of input variables is used, ordered according to variable importance:(b1) The momentum of the π 0 candidate. The π 0 momentum peaks near 1 GeV forthe signal, which is typically well-reconstructed. The background peaks at amuch lower value of ∼ 200 MeV.
(b2) The angle between the reconstructed π + and π 0 . This peaks near 180 degrees
for the signal.
(b3) The momentum of the π + candidate. The momentum resolution for charged
pions from dinucleon decay is typically poor. Despite this, it still peaks at ahigher value in the signal than the background (470 MeV/c vs. 390 MeV/c),and is relatively independent of the better-reconstructed π 0 momentum.
(b4) The invariant mass of the π 0 . This can be estimated for both one and two-ring
π 0 hypotheses, by means of the specialized algorithm referred to in Section 6.2.6.
(b5) The ratio of charge carried by the highest-energy ring to the total charge of allrings.
(b6) The total visible energy.
(b7) The number of decay electrons.
Variables (b5), (b6), and (b7) were described for the pp → π + π + search. Figures 7.127.13 show distributions of each input variable for SK-IV; analogous distributions for
SK-I-III can be found in Appendix B. The data distributions generally agree with theatm.-ν MC. The relative importance of all input variables is shown in Table 7.9.

112
Variable
Importanceπ 0 candidate momentum
0.19
0+Angle between π and π candidates
0.17+
π candidate momentum
0.16
π 0 candidate invariant mass
0.15
Ratio of charge carried by most energetic ring
0.14
Visible energy
0.14
Number of decay electrons
0.058
Table 7.9: Relative importance of each variable in the pn → π + π 0 search, averaged acrossSK periods.

7.3.3

Final sample for pn → π + π 0

The final BDT outputs are shown in Figure 7.14. For the pn → π + π 0 search,an adaptive boost method [88] was chosen. The discriminatory power of the π 0momentum (Figure 7.12) in particular helped keep the misclassification rate low,
thus increasing the boosted weights for the adaptive boost (Equation 7.2). The final
cut was designed so that less than one background event was expected across all
SK periods, with contributions from each SK period ordered by their livetime. It isdetermined with MC only. The total expected background is 0.75 events, consistent
with the one candidate event found in SK-I. Unlike the pp → π + π + search, the signal
distributions are not sharply peaked, so that the efficiency does not change drasticallyas the cut value is shifted.
The cut on the final BDT output, efficiencies, background rates, and candidates
are shown in Table 7.10. CCDIS and CC1π are the main components of the remaining
background, detailed in Table 7.11.
An example pn → π + π 0 MC event display is shown in Figure 7.15, and an example background MC event is shown in Figure 7.16. One candidate is found in
SK-I, shown in Figure 7.17. Table 7.12 shows the values of the BDT input variables

113

SK-IV

35
30

Events

25
20
15
10
5
0
0

200

400

600

800 1000 1200 1400

π0 momentum (MeV/c)
Figure 7.12: The total momentum of the π 0 candidate in SK-IV, the best discriminating
variable for the pn → π + π 0 search, for signal MC (solid histogram), atm.-ν
MC (dashed histogram), and data (crosses). Distributions are normalized to
the number of events in the data.

for the candidate event, compared with the means of the signal and background distributions. While the candidate event appears closer to the signal mean values forenergy-related variables, it appears more background-like for the π 0 − π + angle andthe reconstructed π 0 mass. There are no candidate events in SKII-IV data. The
probability of observing one or more background event in SK-I, without incorporating systematic uncertainties, is 22%. This is consistent with a background event.
Further, the low π 0 mass of the candidate event, as reconstructed by the specializedalgorithm discussed in Section 6.2.6, suggests that it is not a pn → π + π 0 signal. The
showering ring is more likely to be an electron from, for instance, a CCνe 1π event.
Thus, I conclude that there is no evidence for pn → π + π 0 in the SK-I-IV dataset.

114

18
16
14
12
10
8
6
4
2
0
120

EventsEvents

SK-IV

130

140

150

160

170

SK-IV

45
40
35
30
25
20
15
10
5
0

180

0

π0-π+ angle (degrees)

EventsEvents

SK-IV
45
40
35
30
25
20
15
10
5
0
0

100

200

300

400

200
400
600
800
π+ momentum (MeV/c)

SK-IV

22
20
18
16
14
12
10
8
6
4
2
0

500

0

0.2

0.4
0.6
charge ratio

π0 mass (MeV/#c^2)

SK-IV
12

100

10

80

8
6
4

0.8

1

SK-IV

120

EventsEvents

14

1000

60
40
20

2
0

0
0

200 400 600 800 1000 1200 1400 1600
Visible Energy (MeV/c)

0

0.5

1

1.5

2

Michel electrons

Figure 7.13: The opening angle between the π 0 and π + candidate rings (top left), themomentum of the π + candidate (top right), the invariant mass of the π 0candidate (middle left), the ratio of charge carried by the most energetic
ring (middle right), the visible energy (bottom left), and the number ofdecay electrons (bottom right) for the pn → π + π 0 search in SK-IV, for signal
MC (solid histogram), atm.-ν MC (dashed histogram), and data (crosses).
Distributions are normalized to the number of events in the data. Details ofeach variable are described in the text.

115

SK-II

45
40
35
30
25
20
15
10
5
-0.6

EventsEvents

SK-I

-0.4

-0.2

0

0.2

0.4

0.6

-0.6

-0.2

0

BDT Output

SK-III

SK-IV

0.2

0.4

0.6

0.2

0.4

0.6

60
50

12

EventsEvents-0.4

BDT Output

16
14
10
8
6
4

40
30
20
10

2
-0.6

22
20
18
16
14
12
10
8
6
4
2

-0.4

-0.2

0

BDT Output

0.2

0.4

0.6

-0.6

-0.4

-0.2

0

BDT Output

Figure 7.14: BDT output for pn → π + π 0 signal (solid), atmospheric neutrino background(dashed), and data (crosses). The green vertical line indicates the BDT cut
value, and the arrow indicates that only events to the right of the cut are
kept. One candidate event can be seen in the SK-I data distribution, just to
the right of the cut at 0.19.

SK-I SK-II SK-III SK-IV
Cut
0.19 0.24
0.20
0.17
Eff. (%)
10.2 10.0
9.4
10.4
Bkg. (MT-yr)
2.7
2.3
2.2
2.9
Bkg. (SK live.) 0.25 0.11
0.07
0.32
Candidates
1000Table 7.10: Final BDT cut value, efficiency, expected background, and candidate data
events for the SKI-IV pn → π + π 0 search. Background rates are quoted forthe appropriate SK livetime, and per megaton-year.

116

Super-Kamiokande IV
Super-Kamiokande IV
13-11-12:14:02:55
13-11-12:14:02:55
Inner:
3999 hits, 11637 pe
Inner:
3999 hits,
Outer:
5 hits,3 pe 11637 pe
Outer: 5 hits, 3 pe
Trigger: 0x00
Trigger: 0x00
D_wall: 1124.2 cm
D_wall: 1124.2 cm

Charge(pe)Charge(pe)
>26.7
>26.7
23.3-26.7
23.3-26.7
20.2-23.3
20.2-23.3
17.3-20.2
17.3-20.2
14.7-17.3
14.7-17.3
12.2-14.7
12.2-14.7
10.0-12.2
10.0-12.2
8.0-10.0
8.0-10.0
6.28.08.0
6.24.76.26.2
4.73.34.74.7
3.32.23.33.3
2.21.32.22.2
1.30.71.31.3
0.70.20.70.7
0.2< 0.2
< 0.2

1200
1200
00 mu-emu-edecaysdecays

960
960
720
720
480
480
240
240

00
00

500
500

1000
1000

1500
1500

2000
2000

Times (ns)
Times
(ns)

Figure 7.15: A pn → π + π 0 MC event. The µ-like ring corresponds to a true π + . The e-likering contains two overlapping γ’s from a π 0 . The angle between the µ- ande-like rings is 165 degrees. The e-like ring has a reconstructed momentum of
893 MeV/c, while the original π 0 had momentum 876 MeV/c.

Super-Kamiokande IV
Super-Kamiokande IV
11-10-30:11:31:48
11-10-30:11:31:48
Inner:
3933 hits, 10587 pe
Inner:
3933 hits,
Outer:
3 hits,2 pe 10587 pe
Outer: 3 hits, 2 pe
Trigger: 0x00
Trigger: 0x00
D_wall: 1421.3 cm
D_wall: 1421.3 cm
Evis:
1.2 GeVEvis:
1.2 GeV

Charge(pe)Charge(pe)
>26.7
>26.7
23.3-26.7
23.3-26.7
20.2-23.3
20.2-23.3
17.3-20.2
17.3-20.2
14.7-17.3
14.7-17.3
12.2-14.7
12.2-14.7
10.0-12.2
10.0-12.2
8.0-10.0
8.0-10.0
6.28.08.0
6.24.76.26.2
4.73.34.74.7
3.32.23.33.3
2.21.32.22.2
1.30.71.31.3
0.70.20.70.7
0.2< 0.2
< 0.2

1400
1400
00 mu-emu-edecaysdecays

1120
1120
840
840
560
560
280
280

00
00

500
500

1000
1000

1500
1500

2000
2000

Times (ns)
Times
(ns)

Figure 7.16: A surviving atmospheric neutrino MC event for the pn → π + π 0 search. Theinteraction type is CC1π 0 , νµ n → µ− π 0 p. The two overlapping e-like rings
are photons from a π 0 , with total momentum 939 MeV/c. The µ-like ringcorresponds to the µ− , with momentum 410 MeV/c. The angle between
them is 151 degrees, thereby appearing back-to-back.

117

Super-Kamiokande I
Super-Kamiokande I
98-09-16:12:28:02
98-09-16:12:28:02
Inner:
4493 hits, 10722 pe
Inner:
4493 hits,
Outer:
5 hits,4 pe 10722 pe
Outer: 5 hits, 4 pe
Trigger: 0x07
Trigger: 0x07
D_wall: 836.1 cm
D_wall: 836.1 cm
Evis:
1.3 GeVEvis:
1.3 GeV

Charge(pe)Charge(pe)
>26.7
>26.7
23.3-26.7
23.3-26.7
20.2-23.3
20.2-23.3
17.3-20.2
17.3-20.2
14.7-17.3
14.7-17.3
12.2-14.7
12.2-14.7
10.0-12.2
10.0-12.2
8.0-10.0
8.0-10.0
6.28.08.0
6.24.76.26.2
4.73.34.74.7
3.32.23.33.3
2.21.32.22.2
1.30.71.31.3
0.70.20.70.7
0.2< 0.2
< 0.2

1300
1300
1040
1040
780
780
520
520
260
260

00
00

500
500

1000
1000

1500
1500

2000
2000

Times (ns)
Times
(ns)

Figure 7.17: The pn → π + π 0 candidate in SK-I. The e-like and µ-like rings are at anangle of 140 degrees. The e-like ring has a high momentum of 987 MeV/c,
but its fit π 0 mass is low, at 10 MeV/c2 . The µ-like ring has a momentumof 460.0 MeV/c. The total momentum and invariant mass are 700.1 MeV/c
and 1266.4 MeV/c2 , respectively.

mode SK-I SK-II SK-III SK-IV
CC1π 33%
29%
43%
45%
CCDIS 38%
46%
31%
40%
8%
20%
21%
8%
NCDIS
CCQE
8%
5%
5%
8%
Table 7.11: Neutrino interaction mode percentages for remaining background in the pn →π + π 0 search for SKI-IV.

Variableb1 (MeV/c)b2 (degrees)b3 (MeV/c)b4 (MeV/c2 )b5 (dimensionless)b6 (MeV)b7 (dimensionless)

Candidate Signal Background
986.5
756.0
233.7
142.8
152.9
140.4
459.5
470.3
394.5
10.0
143.0
102.2
0.77
0.56
0.58
1171.61098.6
495.0
0
0.40
0.51

Table 7.12: Candidate variable values, and mean variable values for signal and background
distributions, for each of the variables used in the BDT for the pn → π + π 0
search. Letters for each variable are assigned in the text. Values are given for
SK-I, for which a single candidate event was found.

118

7.4

nn → π 0π 07.4.1

Selection criteria for nn → π 0 π 0The nn → π 0 π 0 search is distinct from the other two searches, in that no multivariate method is used. Similar to other nucleon decay searches at SK, the total
momentum and invariant mass are highly discriminatory variables, so that the search
proceeds with a simple set of cuts. The cuts are similar to those used for the p → e+ π 0analysis [16].
(C1) 2 ≤ Nring ≤ 4;
(C2) all rings are e-like;(C3) no decay electrons;(C4) Ptot ≤ 600 MeV/c;(C5) 1600 MeV/c2 ≤ Minv ≤ 2000 MeV/c2 .
PNringp~i | is the total momentum
Here Nring is the number of Cherenkov rings, Ptot = | i=1
p 2
PNringover all rings, and Minv =Etot − P2tot , where Etot =pi | (the rings are
i=1 |~
assumed to be massless, coming from photons).
Criterion (C1) allows for both one- and two-ring π 0 fits. Criteria (C2) and (C3)ensure that all rings come from true electromagnetic showers, with (C3) removing
charged pions that may have been misidentified as e-like. Criterion (C4) ensures a
low total momentum, and (C5) covers the invariant mass region ∼2MN as tightly
as possible without getting too close to the signal peak. (C5) eliminates nearly all
background, which peaks sharply near Mπ0 . Figure 7.18 shows one-dimensional distributions for (C4) and (C5), and Figure 7.19 shows two-dimensional plots of Ptot vs.

119
Minv for all SK periods combined. Figures 7.20 and 7.21 show typical nn → π 0 π 0 andbackground MC events surviving all cuts.

Final sample for nn → π 0 π 0

7.4.2

The signal efficiency, expected background, and data events after each cut are
shown in Table 7.13. The remaining background after all cuts is mostly NCDIS(Table 7.14). The expected background rate in SK-IV is lower than the other SK
periods despite its longer livetime, due to the increased efficiency of decay electron
tagging in (C3). No candidates are found in the data, consistent with the near-zero
expected background. Thus, I conclude that there is no evidence for nn → π 0 π 0 .
SK-IV

12

600

10

500

EventsEvents

SK-IV

8
6

400
300

4

200

2

100
0

1000 2000 3000 4000 5000
Total Momentum (MeV/c)

6000

0

500

1000

1500

2000

2500

Invariant Mass (MeV/c2)

Figure 7.18: Total momentum (left) and invariant mass (right) for data (crosses), nn →π 0 π 0 MC (solid histogram), and atmospheric neutrino MC (dashed histogram). Cuts (C1)-(C3) are applied to both plots. The invariant masscut (C5) has been applied to the total momentum, and the total momentum
cut (C4) to the invariant mass. The thin green vertical lines indicate thelocations of the final cuts for each variable.

120

SK-I SK-II SK-III SK-IV
FCFVEff. (%) 94.1
Bkg. 11846
Data 12299

96.6
6389
6610

94.4
94.1
4187 14214
4355 14444(C1)
Eff. (%)
Bkg.
Data

76.7
3434
3558

76.4
1846
1969

76.6
1204
1239

76.8
4135
4215

(C2)
Eff. (%)
Bkg.
Data

61.0
2131
2249

59.1
1175
1268

59.6
728
782

60.1
2502
2623

(C3)
Eff. (%)
Bkg.
Data

54.4
1441
1492

52.6
799
837

52.8
497
529

52.1
1543
1627

(C4)
Eff. (%)
Bkg.
Data

30.1
837
853

27.0
470
503

29.1
290
310

29.3
940
1021

(C5)
Eff. (%)
Bkg.
Data

22.1
0.05
0

18.8
0.04
0

20.9
0.03
0

21.4
0.02
0

Table 7.13: Efficiency, expected background, and data events for the SKI-IV nn → π 0 π 0search, for each analysis cut. The cuts are described in the text.

121

Total Momentum (MeV/c)

nn → π0π0 SK-I-IV MC

atm. ν SK-I-IV MC

SK-I-IV Data

1800

1800

1800

1600

1600

1600

1400

1400

1400

1200

1200

1200

1000

1000

1000

800

800

800

600

600

600

400

400

400

200
0

200
0
2000

200
0
2000

500

1000

1500

500

1000

1500

500

1000

1500

2000

Total Invariant Mass (MeV/c2)

Figure 7.19: The total momentum and invariant mass distributions for nn → π 0 π 0 MC
(left), atmospheric neutrino MC (center), and data (right) combined for SK
I-IV. Events have passed cuts (C1)-(C3). The solid box indicates the total
momentum and invariant mass cuts ((C4) and (C5) in the text).

Super-Kamiokande I
14-03-06:00:48:03

Super-Kamiokande I

Inner: 5798 hits, 16444 pe
Outer: 1 hits, 1 pe

Trigger:
0x03
14-03-06:00:48:03
D_wall:Inner:716.0
5798 cm
hits, 16444 pe
Outer:1.8
1 hits,Evis:
GeV 1 pe
Trigger: 0x03
D_wall: 716.0 cm
Evis:
1.8 GeV

Charge(pe)>26.7
23.3-26.7

Charge(pe)20.2-23.3>26.7
17.3-20.2
23.3-26.7
14.7-17.3
20.2-23.3
12.2-14.7
17.3-20.2
10.0-12.2
14.7-17.3
8.0-10.0
12.2-14.7
6.28.010.0-12.2
4.76.2
8.0-10.0
3.34.7
6.22.23.38.0
4.71.3- 2.26.2
3.30.7- 1.34.7
2.20.2- 0.73.3
1.3< 0.22.2
0.7- 1.3
0.2- 0.7
< 0.2

1400
1120
1400
840
1120
560
840
280
560
0
280

0

500

1500

2000

0

500Times
1000(ns) 1500

2000

0

1000

Times (ns)Figure 7.20: A nn → π 0 π 0 MC event. The two overlapping e-like rings on the top left
correspond to two γ’s from a π 0 decay, and the other ring also corresponds
to a π 0 . The true momentum of the π 0 corresponding to the two-ring fit is872.7 MeV/c, and the true momentum of the π 0 corresponding to the onering fit is 936.7 MeV/c. The total reconstructed momentum of this event is
219 MeV/c, and the reconstructed invariant mass is 1793 MeV/c2 .

122Super-Kamiokande I
12-03-20:00:33:35

Super-Kamiokande I

Inner: 6253 hits, 16571 pe
Outer: 1 hits, 0 pe

Trigger:
0x03
12-03-20:00:33:35
D_wall:cm
Inner:1154.3
6253 hits,
16571 pe
Outer:1.8
1 hits,Evis:
GeV 0 pe
Trigger: 0x03
D_wall: 1154.3 cm
Evis:
1.8 GeV

Charge(pe)>26.7
23.3-26.7

Charge(pe)20.2-23.3>26.7
17.3-20.2
23.3-26.7
14.7-17.3
20.2-23.3
12.2-14.7
17.3-20.2
10.0-12.2
14.7-17.3
8.0-10.0
12.2-14.7
6.28.010.0-12.2
4.76.2
8.0-10.0
3.34.7
6.22.23.38.0
4.71.32.26.2
3.30.7- 1.34.7
2.20.2- 0.73.3
1.3< 0.22.2
0.7- 1.3
0.2- 0.7
< 0.2

1900
1520
1900
1140
1520
760
1140
380
760
0
380

0

500

1500

2000

0

500Times
1000(ns) 1500

2000

0

1000

Times (ns)Figure 7.21: Surviving atmospheric neutrino MC event for the nn → π 0 π 0 search. Theinteraction is νµ neutral current deep-inelastic scattering. In particular, theinteraction produced a π 0 and an η meson, from which four γ’s are produced.
All but the left-most ring correspond to true photons from the π 0 and η
meson. Many additional charged pions are also produced, concentrated near
the center of the display, thereby complicating reconstruction. The left-mostring corresponds to a charged pion. There is a ring near the center frommultiple, closely overlapping charged pion tracks, which is not found by thereconstruction program. The total reconstructed momentum is 553 MeV/c,and the reconstructed invariant mass is 1763 MeV/c2 .

mode SK-I SK-II SK-III
NCDIS 63%30%
67%
CCDIS 16%50%24%CC1π 21%
20%
9%

SK-IV
49%
0.0%
51%

Table 7.14: Neutrino interaction mode percentages for the remaining background in the
nn → π 0 π 0 search for SKI-IV.

7.5

Systematic Uncertainties

Systematic uncertainties in each of the N N → ππ searches are large, and considerably affect interpretation of the results. Sources of uncertainty can be split into
three categories — MC simulation, reconstruction/detection, and BDT-related (forpp → π + π + and pn → π + π 0 ).

123

7.5.1

Simulation Uncertainties

SignalThe dominant uncertainty for each N N → ππ mode comes from the NEUT
pion-nucleon interaction model, described in 4.2.4. To estimate the uncertainty, 1σvariations in the form of constant multiplicative coefficients are applied to each interaction cross-section based on fits to pion-nucleon scattering data. There are 24
variations in total, shown in Table 7.15. The same set of variations were used in theT2K νe appearance analysis [87]. Separate MC sets were generated for each set ofvariations, and were processed through each stage of the analysis (nuclear and detector simulation, reduction, reconstruction, and the appropriate N N → ππ selection
criteria). For pp → π + π + and pn → π + π 0 , the MC sets were processed with theoriginal trained BDT. The systematic uncertainty was conservatively taken to be thelargest percent change in signal efficiency from all variations. Due to the high initial momentum of the pions in N N → ππ , the high-energy variations produced thelargest change in efficiency. Figure 7.22 shows a comparison of a standard analysis
distribution with its FSI-varied counterpart.
Other intranuclear uncertainties are also significant. The uncertainty in Fermi
momentum is considered from a difference in Fermi momentum models, in particular
the spectral functions used in the N N → ππ and n → n̄ simulation (Figure 5.1),
and the Fermi gas model used for the atm.-ν simulation. Another major sourceof uncertainty comes from correlated decay, described in Section 5.1. Due to thepoor understanding of correlated decay, it is assigned a 100% uncertainty. However,it only affects 10% of all simulated events. For both Fermi motion and correlated
decay, a re-weighting method is used to estimate the total uncertainty, as opposed
to propagating the uncertainties through each stage of the analysis as for the pion-

124
nucleon uncertainties.
Variation type
Default
Low Energy

QE
1.0
0.6
0.6
0.7
0.7
1.4
1.3
1.5
1.6
Increased Hadron Production 0.6
0.6
0.7
0.7
1.4
1.3
1.5
1.6
Decreased Hadron Production 0.6
0.6
0.7
0.7
1.4
1.3
1.5
1.6

QEH
1.8
1.8
1.8
1.8
1.8
1.8
1.8
1.8
1.8
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
2.3
2.3
2.3
2.3
2.3
2.3
2.3
2.3

INEL
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.5
1.5
1.5
1.5
1.5
1.5
1.5
1.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5

ABS
1.1
0.7
0.7
1.6
1.6
0.6
0.7
1.5
1.6
0.7
0.7
1.6
1.6
0.6
0.7
1.5
1.6
0.7
0.7
1.6
1.6
0.6
0.7
1.5
1.6

CX
1.0
0.5
1.6
0.4
1.6
0.6
1.6
0.4
1.6
0.5
1.6
0.4
1.6
0.6
1.6
0.4
1.6
0.5
1.6
0.4
1.6
0.6
1.6
0.4
1.6

CXH
1.8
1.8
1.8
1.8
1.8
1.8
1.8
1.8
1.8
2.3
2.3
2.3
2.3
2.3
2.3
2.3
2.3
1.3
1.3
1.3
1.3
1.3
1.3
1.3
1.3

Table 7.15: Variations used to estimate systematic uncertainties from ion interactions for
N N → ππ . The coefficients correspond to quasielastic scattering (QE),
inelastic scattering including hadron production (INEL), absorption (ABS),and charge exchange (CX). Both QE and CX are considered in low- and highenergy contexts, with an “H” added for the latter. Because hadron production
is only considered for high momenta (pπ > 500 MeV/c), variations that change
hadron production are considered high-energy variations, which are of themost relevance for N N → ππ .

125
FSI comparison

100

Events

80
60
40
20
0
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6
BDT output
Figure 7.22: The BDT output for pn → π + π 0 signal, for standard (solid histogram) andFSI-varied MC (dashed histogram). The varied MC output is from the decreased hadron production set of variations. This decrease in hadron production leads to a clearer signal.

Background
Neutrino interaction uncertainties are significant for the interaction modes thatsurvive the N N → ππ analysis selection. In particular, single pion production uncertainties are large. A list of the main interaction uncertainties in this analysis areshown in Table 7.16.
SourceUncertainty0±Single pion π /π ratio
40%
Single meson total cross-section
20%
Axial mass in single pion production, CCQE
10%
Table 7.16: Dominant interaction uncertainties that contribute to the overall atm.-ν uncertainty for the N N → ππ searches.

Other simulation uncertainties come from the overall neutrino flux, and the same

126
pion interaction uncertainties considered for the N N → ππ signal. For the flux, thelargest uncertainty comes from the absolute flux normalization in the energy range 1
GeV < Eν < 10 GeV, which is 7%.

7.5.2

Reconstruction and BDT uncertainties

Reconstruction uncertainties are common to both signal and background. The
main sources in these searches come from ring separation and PID, energy scale, andring angular resolution.
For ring separation and PID systematics, results from a recent SK atmospheric
neutrino oscillation analysis in [61] are used, in which the uncertainties are classified
by event topology (single or multi-ring), leading ring PID, and visible energy. The
final event samples in each N N → ππ search are then re-weighted by the associateduncertainties. For pp → π + π + , the sample is µ-like by definition, mostly in the
sub-GeV (Evis < 1330 MeV) energy range. For nn → π 0 π 0 , it is e-like by definition,in the multi-GeV (Evis > 1330 MeV) range. For pn → π + π 0 , the sample is mostly
e-like multi-GeV. The corresponding uncertainties for these samples are shown in
Table 7.17.
Uncertainty
Ring Separation, sub-GeV µ-like
Ring PID, sub-GeV µ-like
Ring Separation, multi-GeV e-like
Ring PID, multi-GeV e-like

SK-I SK-II SK-III
4.5% 8.2%
2.6%
3.9% 2.2%
3.1%
3.1% 1.9%
1.1%
1.7% 1.8%
2.5%

SK-IV
2.3%
2.2%
0.9%
3.4%

Table 7.17: Ring separation and PID errors, from the analysis in [61], for relevant samples
in the N N → ππ analyses. All samples are multi-ring.

The energy-scale uncertainty is common to all energy-dependent variables. It
is estimated from the calibration procedures described in Section 3.7.4. Table 7.18

127
shows the absolute energy scale uncertainty for each SK period. To estimate theuncertainty, each energy-dependent variable was varied by the appropriate energy
scale error (both increased and decreased), and propagated through the analysis. For
pp → π + π + and pn → π + π 0 , this meant being processed by the original, unvaried
BDT, while for nn → π 0 π 0 it simply meant putting putting events with varied total
momentum and invariant mass through the same set of selection cuts.
SK-I SK-II
1.1% 1.7%

SK-III
2.7%

SK-IV
2.3%

Table 7.18: Values for the absolute energy scale uncertainty in each SK period.

The energy scale uncertainty was estimated differently for signal and background.
For signal, the uncertainty was just taken to be the percentage change in efficiency
between the varied and unvaried MC. For background, and particularly the nn →π 0 π 0 search, this same method led to large fluctuations, due to low statistics after all
selection cuts. To ensure that the uncertainty was not overestimated, a function was
fit in the cut region. The uncertainty was then estimated by finding the differencein the integral of the function by shifting the cut value(s), or integration limit(s), by
the energy scale uncertainty. An example of this procedure is shown in Figure 7.23.
The ring angular resolution was taken into account for the pp → π + π + and pn →π + π 0 searches, as the angle between pion candidates was crucial both in the preselection stage and the BDT stage. However, the angular resolution for rings frompp → π + π + and pn → π + π 0 simulation is quite good, and so varying the angle
between pion candidates produced only a very small uncertainty.
Finally, a systematic uncertainty for the BDT was estimated for the pp → π + π +and pn → π + π 0 searches. This was done by assessing the difference in the BDT
output distributions for the test sample and the final analysis sample (Table 7.2).

128This uncertainty is essentially a measure of the amount of overtraining in the BDTs.
Tables 7.19, 7.20, and 7.21 show the systematic uncertainty contributions fromeach source described above, as well as the total uncertainty.
invariant mass confined to total momentum cut region104χ2 / ndf

47.32 / 47

Prob
p0
p1

0.4596
9.57 ± 0.07
-0.005829 ± 0.000094

103

102

10

1

10-1

10-2
0

200

400

600

800

1000

1200

1400

1600

1800

2000

Figure 7.23: The invariant mass distribution of the nn → π 0 π 0 atm.-ν background in
SK-IV, after selections (C1)-(C4), with a function of the form exp (p0 + p1 x)
fit to the region 600 MeV/c2 < Minv < 2000 MeV/c2 . The uncertainty wasestimated by integrating the function between the invariant mass cut values,
and finding the difference in the integral when the cut values were shifted.
A similar procedure was done for the total momentum, and the larger value
was taken for the uncertainty.

129

SK-I

SK-II

SK-III

SK-IV

Simulation
Pion FSI
Fermi motion
Correlated decay

26.9%
20.4%
9.9%

27.9%
18.8%
10.0%

25.1%
20.2%
9.4%

30.5%
21.5%
9.4%

Reconstruction
Ring separation
Ring PID
Energy scale
Ring angle

4.5%
3.9%
<1%
<1%

8.1%
2.1%
1.0%
2.0%

2.6%
3.1%
1.3%
<1%

2.1%
2.4%
<1%
<1%

BDT

3.6%

2.2%

4.4%

2.0%

Total

35.9%

36.2%

34.1%

38.7%

Simulation
Neutrino interaction
Pion FSI
Neutrino flux

22.9%
16.4%
7.3%

22.9%
16.5%
7.3%

27.1%
22.2%
7.3%

19.7%
16.2%
7.2%

Reconstruction
Ring separation
Ring PID
Energy scale
Ring angle

4.5%
3.9%
1.2%
<1%

7.9%
1.9%
1.3%
<1%

2.6%
3.2%
2.4%
<1%

2.2%
2.3%
2.4%
<1%

BDT

6.8%

1.0%

4.3%

1.4%

Total

30.5%

30.3%

36.4%

26.8%

SignalBackground

Table 7.19: Systematic uncertainties in the pp → π + π + search.

130

SK-I

SK-II

SK-III

SK-IV

Simulation
Pion FSI
Fermi motion
Correlated decay

26.8%
17.2%
9.6%

24.6%
18.4%
9.6%

24.5%
10.9%
9.5%

28.7%
17.2%
10.1%

Reconstruction
Ring separation
Ring PID
Energy scale
Ring angle

1.7%
<1%
2.5%
1.3%

<1%
<1%
1.7%
<1%

<1%
1.6%
4.9%
2.1%

<1%
2.3%
4.7%
1.9%

BDT

<1%

1.6%

<1%

<1%

Total

33.4%

32.3%

28.9%

35.3%

Simulation
Neutrino interaction
Pion FSI
Neutrino flux

18.2%
10.2%
7.3%

14.0%
12.2%
7.2%

20.1%
11.0%
7.2%

20.9%
13.7%
7.2%

Reconstruction
Ring separation
Ring PID
Energy scale
Ring angle

1.8%
<1%
<1%
<1%

<1%
<1%
1.8%
<1%

<1%
<1%
3.3%
<1%

<1%
2.0%
3.2%
<1%

BDT

6.3%

7.4%

10.3%

11.3%

Total

23.1%

21.3%

26.3%

28.6%

SignalBackground

Table 7.20: Systematic uncertainties in the pn → π + π 0 search.

131

SK-I

SK-II

SK-III

SK-IV

22.2%
19.6%
9.6%

26.8%
19.3%
9.5%

30.1%
19.8%
9.8%

24.9%
19.6%
9.6%

Reconstruction
Ring separation <1%
Ring PID <1%
Energy scale 1.5%
Total 31.2%

<1%
<1%
1.7%
34.4%

<1%
<1%
4.0%
37.6%

<1%
<1%
3.6%
33.3%

7.3%
8.9%
7.3%

10.9%
8.2%
7.3%

6.0%
11.0%
7.3%

9.1%
7.7%
7.2%

Reconstruction
Ring separation <1%
Ring PID <1%
Energy scale 10.9%
Total 17.5%

<1%
<1%
18.1%
24.0%

<1%
<1%
28.9%
32.3%

<1%
<1%
24.3%
28.0%

SignalSimulation
Pion FSI
Fermi motion
Correlated decay

Background
Simulation
Neutrino interaction
Pion FSI
Neutrino flux

Table 7.21: Systematic uncertainties in the nn → π 0 π 0 search. Note that the energy scale
uncertainty is dominant for the background, unlike other searches in which
the simulation uncertainties are dominant.

Chapter 8
Neutron-Antineutron Oscillation
SearchThe search for n → n̄ follows much of the same pattern as the N N → ππ searches.
In particular, similar to pp → π + π + and pn → π + π 0 , a multivariate analysis is used,with a boosted decision tree (BDT). Like N N → ππ , the final-state particles ofn → n̄ are mostly pions, and the interactions of charged pions in particular candistort the event signature. Unlike N N → ππ , in which two relatively back-to-backpions were sought, there is no particular desired event topology in n → n̄ , due to themany non-negligible final states.

8.1

Pre-selection

A set of selection criteria is first applied. The requirements are more lax than the
N N → ππ searches, and have no PID or topology requirements.
(D1) There is more than one Cherenkov ring;(D2) The visible energy is between 400 and 1600 MeV;
132

133
(D3) The total momentum is less than 500 MeV/c.
Figure 8.1 shows plots of each event sample across selections (D1)-(D3), and Table 8.1
shows the remaining events in each sample following the selections.
For the n → n̄ search, the atm.-ν background is weighted differently than in the
N N → ππ search. The N N → ππ search used the two-flavor oscillation probability,
plus small correction factors from the Honda flux, as described by Equaton 4.1 andSection 4.1. For n → n̄ , a more accurate background weight estimation is used thattakes into account the full three-flavor oscillation analysis in [86] — namely, it uses
the fitted “pulls” from all the systematic error sources in the oscillation analysis. This
new weighting procedure tends to increase the normalization (Figure 8.2).

8.2

Multivariate analysis

Ten variables are used for the n → n̄ multivariate analysis. Some have been
described in the N N → ππ analyses, while others are new to this analysis. Table 8.2
shows their relative importance in the BDT training. A larger set of variables wasconsidered initially. Variables were removed on the basis of high correlation withother variables, and their relative importance. For instance, the total visible energy
is highly correlated with the invariant mass. Because the invariant mass was a higherranked variable, the visible energy was removed.
Figures 8.3-8.4 show plots of each of the variables in SK-IV; analogous plots for
SK-I/II/III can be found in Appendix B. For the BDT, the same set of parameters
shown in Tables 7.1 and 7.2 are used. The gradient boost method [88] was chosen,
as the n → n̄ signal is much more difficult to separate from the atm.-ν backgroundthan in any of the N N → ππ analyses (detailed further below).

134
number of rings

visible energy
140

12000

120
10000
100
EventsEvents

8000
6000
4000

80
60
40

2000

20

0
1

1.5

2

2.5

3
3.5
4
number of rings

4.5

5

5.5

6

0
0

200

400

600

800 1000 1200 1400 1600 1800 2000
visible energy (MeV)

total momentum

70
60

Events

50
40
30
20
10
0
0

200

400

600
800
1000
total momentum (MeV/c)

1200

1400

Figure 8.1: Selection criteria (D1)-(D3) for n → n̄ MC (blue histogram), atm.-ν background (red histogram), and data (crosses) in SK-IV. All plots are areanormalized to the data. The green line indicates the placement of the cuts
described in the text.

8.2.1

Standard variables

The following variables are derived from the standard reconstruction algorithms
described in Section 6.2:
• The total momentum, already required to be less than 500 MeV/c by selection(D3). This peaks at a higher value for the background than the signal;• The invariant mass, which has a sharp peak at the π 0 mass for background andis more spread out for signal;

135
SK-I SK-II SK-III SK-IV
FCFVEff. (%) 92.0
Bkg. 11889
Data 12299

95.3
6415
6610

91.7
92.2
4206 17411
4355 17664(D1)
Eff. (%)
Bkg.
Data

82.6
3595
3691

84.2
1944
2056

82.2
1266
1292

82.7
5283
5415

(D2)
Eff. (%)
Bkg.
Data

61.6
1456
1444

62.1
784
813

60.6
516
513

61.7
2117
2145

(D3)
Eff. (%)
Bkg.
Data

39.6
446
343

39.7
238
252

39.1
159
139

40.5
660
706

Table 8.1: Efficiency, expected background, and data events after each precut for then → n̄ search. The FCFV cut is described in Section 6.1.6, and cuts (D1)(D3) are described in the text. Expected background is weighted by oscillation
factors determined from the analysis in [86], and scaled to the appropriate SK
detector livetime (from the 200 year analysis sample).

• (Etot − Eprimary ) /Etot , where Etot is the total energy and Eprimary the energyof the primary ring in the event. This peaks in a higher value for the signal
than the background, since the energy is more evenly distributed among rings,
while the background (especially for a CC interaction) will typically have moreenergy contained in a single ring;• The PID likelihood difference of the first ring. The leading ring likelihood isslightly higher in the background (meaning the background is slightly more
likely to be µ-like);

136

Background weight
6000

5000

4000

3000

2000

1000

0
0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Figure 8.2: A comparison of the atm.-ν weight using two-flavor oscillation and small corrections from the flux in [53] (blue curve), and the atm.-ν weight using thefull oscillation analysis in [86] (red curve), for all atm.-ν events passing theselection pre-cuts (D1)-(D3).

• The number of decay electrons. Typically non-zero in the signal due to thepresence of charged pions, with a falling spectrum for the background;• The maximum distance between a decay electron vertex and the primary event
vertex, expected to be longer in the background particularly if it’s a CC interaction with an energetic muon.

8.2.2

Non-standard variables

The standard ring-counting algorithm is limited with respect to the n → n̄ search.
In particular, it has a maximum of five rings. It is often the case that there are more

137
than five visible particles in the final state of a n → n̄ interaction. Instead of trying
to pick out individual particles, it is useful to extract variables from the detectorwide charge pattern, which are relatively independent of variables that rely on thestandard ring-counting algorithm.

Sphericity
As its name implies, the sphericity S is a measure of how spherical an event
is. It is commonly used to characterize hadronic jets in accelerator experiments. A
perfectly isotropic event will have S = 1, while a perfectly one-directional event will
have S = 0. An ideal n → n̄ event is isotropic, and while this is often not the case due
to nuclear effects, it is still more isotropic than a typical atm.-ν event. The sphericity
is defined in terms of the sphericity tensor, defined as:

S

ab

P a b
p p
= Pi i 2 i .
i pi

(8.1)

Here the sum is over bins in Hough space (the same space in which Cherenkov rings
are found), pi is the momentum in bin i, and a, b = 1, 2, 3 are the three cartesian
directions. For our purposes, the “momentum” is actually just the charge, corrected
for light attenuation and PMT angular acceptance. The sphericity is then defined interms of the three eigenvalues of this tensor, λ1 ≥ λ2 ≥ λ3 :S=

3
(λ2 + λ3 ) .
2

(8.2)

The corresponding eigenvector for λ1 spans the event axis, while those of λ1 and λ2
span the event plane. Another variable, aplanarity, defined as A = 23 λ3 , can also be
defined from the sphericity tensor, although this is highly correlated with sphericity
and is not used in the analysis.

138
Potential rings
While the ring-counting algorithm saturates at five rings, bins in Hough spacethat exceed a certain charge threshold are considered as potential rings, or “seeds.”
This variable has no upper limit, and thus is sensitive to charge deposited by particles
not found via normal ring-counting. An n → n̄ event typically has more seeds thanatm.-ν background.

Fox-Wolfram moments
Fox-Wolfram moments are another set of variables used in accelerator physics
that measure correlations between particle momenta. The `th Fox-Wolfram moment
is defined as:H` =X X |~pi ||~pj |
i

j

2
Evis

P` (cosθij ) .

(8.3)

Here p~ are the momenta, Evis is the visible energy, P` is the `th Legendre polynomial, and θij is the angle between particle i and j. The sum is taken over potential
rings, described above1 . As with sphericity, the momentum is the corrected charge ina given bin. The visible energy is the sum of all corrected charges. For n → n̄ , H1
and H2 are used.

8.3

Final selection

The final selection on the BDT output for the n → n̄ search differs from thecriteria used for the N N → ππ searches. For the latter, background was minimized
without regard for systematic uncertainties. For n → n̄ , the background rate is high1

In principle this calculation could be taken over all bins in Hough space, but it is computationallylaborious and provides no additional discriminatory power between signal and background.

139

SK-IV

80

SK-IV

140

70

120

60

EventsEvents

100
50
40
30

60
40

20

20

10
0
0

80

50

100

150

200

250

300

350

400

450

0
0

500

200

400

600

800

1000

1200

1400

1600

300

400

500

1600

1800

2000

Invariant mass (MeV/c2)

Total momentum (MeV/c)

SK-IV

SK-IV

90

120

80
100

EventsEvents

70
60
50
40
30

80
60
40

20
20
10
0
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0
-500

1

-400

-300

(Etot-Eprimary ring)/Etot

-200

-100

0

100

200

Likelihood difference

SK-IV

SK-IV
120

400
350

100

EventsEvents

300
250
200
150

80

60

40

100
20

50
0
0

0.5

1

1.5

2

2.5

3

3.5

Number of decay electrons

4

4.5

5

0
0

200

400

600

800

1000

1200

1400

Maximum decay electron distance (cm)Figure 8.3: The total momentum (top left), invariant mass (top right), (Etot Eprimary )/Etot (middle left), PID likelihood difference of the leading ring (middle right), number of decay electrons (bottom left), and maximum distance
between the primary vertex and a decay electron vertex (bottom right) for then → n̄ search in SK-IV, for signal MC (solid histogram), atm.-ν MC (dashedhistogram), and data (crosses). Distributions are normalized to the numberof events in the data. Details of each variable are described in the text.

140

SK-IV

SK-IV
160

100

140
120

EventsEvents

80

60

40

100
80
60
40

20
20
0
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0
0

1

2

4

sphericity

6

8

10

12

14

16

18

20

number of seeds

SK-IV

SK-IV
220

160

200
140
180
160

EventsEvents

120
100
80
60

140
120
100
80
60

40

40
20
0
-1

20
0

1

2

3

first fox-wolfram moment

4

5

0
-1

0

1

2

3

4

5

second fox-wolfram moment

Figure 8.4: The sphericity (top left), number of potential rings (top right), first FoxWolfram moment (bottom left), and second Fox-Wolfram moment (bottom
right) for the n → n̄ search in SK-IV, for signal MC (solid histogram), atm.-ν
MC (dashed histogram), and data (crosses). Distributions are normalized to
the number of events in the data. Details of each variable are described in the
text.

for virtually all cut values. Figure 8.5 shows the BDT output and the remaining
efficiency and scaled background as a function of BDT cut. Ideally, of course, thebackground rate is as close to zero as possible, such that the observation of even asingle signal candidate event would be interesting. In the case of the n → n̄ search,
reducing the background rate to <1 expected event would require a cut on the BDT
output such that the signal efficiency is <1%. This is undesirable, given that there
are high uncertainties associated with both signal and background processes.

141

SK-I
200
180
160
140
120
100
80
60
40
20
0

EventsSignalBackground

-1

-0.5

0

0.5

1

BDT Output

SK-I

40

500
450

35

Remaining signal efficiency

400

30

Remaining scaled background

350

25

300

20

250
200

15

150

10

100

5

50

0

0

-1

-0.5

0

0.5

1

Figure 8.5: BDT output for n → n̄ signal and atm.-ν background (top), and the remaining
signal efficiency and scaled background in SK-I as a function of BDT cut(bottom). The BDT outputs are area-normalized

142
Variable
ImportanceInvariant mass
0.110
First Fox-Wolfram moment
0.109
Total momentum
0.105
Sphericity
0.104
Maximum distance from primary vertex to decay electron vertex
0.103(Etotal -Eprimary )/Etotal
0.102
PID likelihood difference of leading ring
0.099
Second Fox-Wolfram moment
0.090
Number of decay electrons
0.089
Number of potential rings
0.089
Table 8.2: Relative importance of each variable in the n → n̄ search, averaged across SK
periods.

The high expected background for reasonable values of the signal efficiency, andlarge systematic uncertainties associated with each, suggests that an optimal cut
should take more into account more than just the raw signal and background acceptance. One such value is the sensitivity, defined as

S=

∞
X

τ (, b, λ, ncand ) P (b, ncand ) .

(8.4)

ncand =0

Here τ is the lower lifetime limit in the absence of signal, calculated with a Bayesian
method described in Appendix C. It is a function of efficiency , background b, and exposure λ and their uncertainties, as well as the number of data candidates ncand . The
sum is weighted by the Poisson probability P (b, ncand ) of observing ncand candidates
assuming the background-only hypothesis. Due to the high remaining background of
the n → n̄ search, S is very similar to a less cumbersome calculation, which is simply
τ evaluated under the assumption that ncand = nbkg . This is the variable that we
maximize with the final cut value.

143

8.3.1

Systematic uncertainties

Like N N → ππ , the dominant systematic uncertainties come from MC simulation in both signal and background, with smaller uncertainties associated withreconstruction.

SignalThe signal uncertainty is overwhelmingly due to the pion FSI variations describedin Section 7.5.1. Specifically, the second variation from the top in Table 7.15 leads
to the largest change in signal efficiency — decreased absorption and decreased QE
scattering both contribute to a clearer multi-pion signal. The other main simulation uncertainty, from Fermi motion, is relatively small. Unlike in N N → ππ , theFermi motion uncertainty was estimated by generating a separate set of MC using the
Fermi gas model, as opposed to the spectral functions (Figure 5.1) used in the standard simulation. The same reconstructon and BDT-related uncertainties describedin Section 7.5 are considered.

BackgroundThe background uncertainty is also simulation-dominated. Given that the remaining background is much higher in n → n̄ than N N → ππ , it is important to treat
background uncertainty carefully.
Neutrino interaction uncertainties depend on the value of the final cut. The tailregion is dominated by both CC and NC DIS uncertainties (Figure 8.6). For the
initial neutrino energy region relevant for this analysis (Eν < 10 GeV), there are large
energy-dependent uncertainties associated with the DIS cross-section (Table 8.3).
Remaining sources of systematic uncertainty from pion interaction, oscillation,

144
reconstruction, and BDT performance are estimated in the same way as the N N →ππ search.

Atm-nu MC BDT output
CCDIS
NCDIS
CC1pi
NC1pi
CCQE
Other

2500

2000

1500

1000

500

0
-1

-0.5

0

0.5

1

Figure 8.6: atm.-ν BDT output for SK-I MC, categorized by neutrino interaction mode.

Energy 0-2 GeVν30%
anti-ν
5%

2-3 GeV50%
20%

3-5 GeV10%5%5-10 GeV
40%
10%Table 8.3: Values for the DIS interaction uncertainty for Eν < 10 GeV.

8.3.2

Final cut

Figure 8.7 shows the value of τ (ncand = nbkg ) as a function of BDT cut. Since
there is only a single value of τ (ncand = nbkg ) calculated across all SK periods (see145

Lifetime limit vs. BDT cut

1.4

1

32

Limit (X 10 yrs)

1.2

0.8
0.6

Limit = 1.49 × 1032 years

0.4
0.2
0
-0.8

-0.6

-0.4

-0.2

0
BDT cut

0.2

0.4

0.6

0.8

Figure 8.7: Lifetime limit vs. BDT cut assuming the number of candidates is equal to thenumber of background events. The value is maximized for a cut of 0.35.

Appendix C), the same cut value is used for each. Figure 8.8 shows the BDT output
for all SK periods. Table 8.4 shows the final efficiencies, background rates, and data
candidates. Table 8.5 shows the interaction modes of the remaining background,and Table 8.6 shows the systematic uncertainties for both n → n̄ signal and atm.ν background. There is no excess of events in the data relative to the expectedbackground, and thus there is no evidence for n → n̄ oscillation.

8.3.3

Data discrepancy

While the data is consistent with the atm.-ν background in terms of shape, there
is a significant deficit in the number of data events relative to expected background at

146

EventsEvents

SK-I
40
35
30
25
20
15
10
5
0
-0.5

24
22
20
18
16
14
12
10
8
6
4
2
0

0

0.5

1

-1

-0.5

0

BDT Output

BDT Output

SK-III

SK-IV

0.5

1

0.5

1

100

EventsEvents

-1

SK-II

45
40
35
30
25
20
15
10
5
0

80
60
40
20
0

-1

-0.5

0
BDT Output

0.5

1

-1

-0.5

0
BDT Output

Figure 8.8: BDT output for n → n̄ signal (blue), atm.-ν background (red), and data(crosses). All histograms are normalized to the number of data entries. The
vertical green line indicates the BDT cut value, and the arrow indicates that
only events to the right of the cut are kept.

SK-I SK-II SK-III SK-IV
Eff. (%)
11.0 10.5
9.5
12.0
Bkg. (MT-yr)
223
217
204
254
Bkg. (SK live.) 20.5 10.7
6.5
33.9
Candidates
65326
Table 8.4: Efficiency, expected background events, and candidate data events for the SKIIV n → n̄ search. Background is quoted both for the appropriate SK livetime,and per megaton-year.

the cut value for the first three SK periods, most seriously in SK-I. This can be seen in
Table 8.4 and Figure 8.9. The sources of these discrepancy are detector-dependent.
The better overall performance of SK-IV can be explained by a better data-MC

147
mode SK-I SK-II SK-III SK-IV
NCDIS 37% 39%
35%38%
CCDIS 35% 38%40%38%CC1π 17% 15%
17%
14%
NC1π
3%
3%
3%
3%
CCQE <1% <1%
1% <1%
Table 8.5: Atmospheric neutrino interaction modes in the remaining background for then → n̄ search by percentage, as estimated by atmospheric neutrino MC.

agreement in the multivariate input variables, evident from basic comparisons suchas the mean difference between data and MC distributions (Table 8.7) and a χ2 test
statistic (Table 8.8).
One source of discrepancy comes from the pre-cut on total momentum. Particularly for SK-I, this cut leads to a large discrepancy between data and atm.-ν MC.
This can be seen in Table 8.1, as well as Figure 8.9, in which the data is outside theone-sigma systematic uncertainty for all cut values. In SK-I, shifting the momentumof the background distribution by the difference of the mean with the data prior toapplying selection (D3) is sufficient to explain the discrepancy. Figure 8.10 shows the
SK-I total momentum distribution after selections (D1) and (D2), with and without
this shift applied. Figure 8.11 shows the remaining events in SK-I as a function of
BDT cut, analogous to Figure 8.9, but with the shift applied. The number of events
is consistent between data and atm.-ν MC at the cut value.
An additional uncertainty is added to account for the total momentum shift.
Similar to the energy scale uncertainty described in Section 7.5, both signal andbackground MC have the data—atm.-ν MC shift applied to the total momentum
distributions, and this variation is propagated through the BDT. Table 8.9 shows theshift values, and the resulting uncertainties for signal and background. Predictably,
SK-I has the largest associated uncertainty. Ideally, this uncertainty would have been

148
SK-I

SK-II

SK-III

SK-IV

33.1%
4.9%

41.3%
5.1%

43.9%
5.0%

45.1%
5.0%

1.0%
<1%
1.0%
3.3%

1.3%
<1%
2.1%
<1%

<1%
<1%
1.6%
3.7%

<1%
<1%
2.5%
8.7%

33.6%

41.7%

44.4%

46.2%

Simulation
Neutrino interaction
Pion FSI
Neutrino flux

23.7%
18.0%
7.2%

24.5%
14.5%
7.2%

24.9%
19.4%
7.1%

25.2%
17.9%
7.2%

Reconstruction
Ring separation
Ring PID
Energy scale
BDT

<1%
<1%
1.6%
3.1%

1.3%
<1%
2.9%
2.4%

<1%
<1%
3.6%
3.2%

<1%
<1%
5.6%
3.8%

Total

30.8%

29.7%

32.8%

32.5%

SignalSimulation
Pion FSI
Fermi motion
Reconstruction
Ring separation
Ring PID
Energy scale
BDT
Total
Background

Table 8.6: Systematic uncertainties in the n → n̄ search.

considered in the calculation of the expected lifetime limit (Figure 8.7), but because
this uncertainty is associated with a pre-cut, it is relatively independent of the final
BDT cut.
The discrepancy in SK-I is clear from the pre-selection stage. As there is no
significant difference between data and expected background for SK-II and SK-III
after selections (D1)-(D3), the sources of the discrepancies are not immediately clear.

149

SK-I

SK-II
Remaining events

Remaining events

103

102

102
10
1

10-1

-1 -0.8 -0.6 -0.4 -0.20

0.2 0.4 0.6 0.8

10

1

10-1

-1 -0.8 -0.6 -0.4 -0.2

BDT Cut

SK-III

0.2 0.4 0.6 0.8

SK-IV

103

Remaining events

Remaining events0

BDT Cut

102

102

10

1

10-1

-1 -0.8 -0.6 -0.4 -0.20

0.2 0.4 0.6 0.8

10
1

10-1

-1 -0.8 -0.6 -0.4 -0.2

BDT Cut0

0.2 0.4 0.6 0.8

BDT Cut

Figure 8.9: Remaining events as a function of the BDT cut, for data (crosses) and atm.ν MC (red histogram). The atm.-ν MC is normalized by the appropriate
livetime. The cyan curves show the one-sigma systematic uncertainties (Table 8.6).

Variable
Total momentum
Invariant mass
(Etotal -Eprimary )/Etotal
PID likelihood diff. of leading ringNumber of decay electronsMax. distance from primary vertex to Michel vertex
Sphericity
Number of potential rings
First Fox-Wolfram moment
Second Fox-Wolfram moment

SK-I
0.039
0.054
0.006
0.068
0.048
0.085
0.007
0.018
0.054
0.019

SK-II SK-III SK-IV
0.026 0.005 0.012
0.097 0.001 0.062
0.041 0.038 0.044
0.046
0.11
0.11
0.10
0.17 0.002
0.095
0.20 0.078
0.075 0.016 0.082
0.11 0.086 0.047
0.14
0.13 0.009
0.098 0.078 0.028

Table 8.7: Values of | (Meandata − Meanatm.-ν ) /Meanatm.-ν | for each variable in the n →n̄ analysis, following selections (D1)-(D3).

150
Variable
SK-I SK-II SK-III SK-IV
Total momentum
0.95 0.92
0.97
0.99
0.95 0.75
1.0
0.96
Invariant mass
(Etotal -Eprimary )/Etotal
0.94 0.98
0.99
0.68
PID likelihood diff. of leading ring
0.92 0.93
0.93
0.92
0.16 0.78
0.59
0.95
Number of decay electronsMax. distance from primary vertex to Michel vertex 0.86 0.66
1.0
0.99
0.99 0.96
0.99
0.77
Sphericity
Number of potential rings
0.79 0.76
0.70
0.93
0.96 0.92
0.95
0.99First Fox-Wolfram moment
Second Fox-Wolfram moment
0.86 0.38
0.81
0.92
Table 8.8: Values of the χ2 test statistic between data and atm.-ν MC, whose value in[0, 1] gives the probability that differences in distributions are due to statisticalfluctuations, for each variable in the n → n̄ analysis, following selections (D1)(D3).

The sources could come from individual BDT input variables, or parameters that
dictate the training and testing of the BDT itself (as listed in Section 7.1). To
address the discrepancies, individual SK-II and SK-III variables were removed fromthe training process, starting with variables that have the largest values in Table 8.7.
In addition, the number of trees used in the training process was varied (Ntrees inEquation 7.1). Increasing the number of trees generally improved data and atm.ν MC agreement. The resulting data and expected background was then assesseduntil a sufficient set of adjustments was found, meaning that the data and expectedbackground were consistent at the cut value. Then, as with the total momentum
and energy scale errors, additional uncertainties were estimated in both the signal
and background due to these adjustments. Table 8.10 summarizes the sources of
discrepancy and associated uncertainties in SK-I/II/III.

151

Total momentum
90
80
70
60
50
40
30
20
10
0
0

200

400

600

800

1000

1200

1400

1600

1800

2000

Figure 8.10: Total momentum in SK-I, for data (crosses), atm.-ν MC (red), and atm.-ν
MC shifted to match the data (green).

Remaining events

SK-I, Shifted total momentum
102
10
1

10-1

-1 -0.8 -0.6 -0.4 -0.20

0.2 0.4 0.6 0.8

BDT Cut
Figure 8.11: Remaining events as a function of the BDT cut, for data (crosses) and atm.ν MC (red histogram). The atm.-ν MC is normalized by the appropriate
livetime. The cyan curves show the one-sigma systematic uncertainties (Table 8.6).

152

SK-I
SK-II
SK-III
SK-IV
Shift 48.1 MeV/c -12.3 MeV/c 28.9 MeV/c -22.8 MeV/c
17.4%
3.8%
9.1%
6.8%
σsig
σbkg
20.2%
7.0%
15.6%
9.5%
Table 8.9: Shift value in data and atm.-ν MC total momentum, and associated signal andbackground uncertainties in the n → n̄ search.

SK-I
SK-II
SK-III
20.5 ± 6.3
10.7 ± 3.2
6.5 ± 2.2
6
5
3
Shift total Remove 4 variables, Remove 2 variables,
Adjustment(s)momentumincrease Ntrees
increase Ntrees
12.2 ± 3.5
7.1 ± 2.0
Exp. bkg. (adj.) 23.2 ± 6.7
Data (adj.)
20
9
7
17.4%
22.6%
18.3%
σsig
σbkg
20.2%
13.8%
8.5%

Exp. bkg. (orig.)
Data (orig.)

Table 8.10: Summary of the adjustments made in SK-I/II/III to account for the discrepancy seen between data and atm.-ν MC, with original and adjusted efficiency
and background rates at the final cut value, and corresponding signal andbackground uncertainties. The number of variables removed starts from variables with the largest values of the mean difference between data and atm.-ν
MC (Table 8.7). Background rates show the nominal values ± 1σ systematic
uncertainty, calculated with all the sources in Table 8.6.

Chapter 9
Results and discussion
Searches for dinucleon decay to pions (N N → ππ ) via pp → π + π + , pn → π + π 0 ,and nn → π 0 π 0 , and a search for neutron-antineutron oscillation (n → n̄ ), have been
performed by the Super-Kamiokande detector. A total of 282 (307) kiloton-years offully-contained data for the N N → ππ (n → n̄ ) searches was used. No evidence
for any of the sought-after processes was found, and the data are consistent withexpectations from atmospheric neutrino background. Here I present the main results,
and discuss some of the implications of these searches, both for global BNV searches
and issues more specific to the analyses.

9.1

Lifetime limits

In the absence of evidence for either N N → ππ or n → n̄ , lower limits are
set on the time scales for each process. The limits are calculated using a Bayesian
method [91], described in detail in Appendix C.

153

154

9.1.1

Dinucleon decay

Table 9.1 shows the lower lifetime limits at 90% confidence level for N N → ππfrom this analysis, compared to the previous results obtained in [39]. The resuts inthis analysis are about two orders of magnitude more stringent than those of [39],reflecting the large exposure of SK. Note also that the results of [39] do not incorporate
systematic uncertainties, so that the results of this analysis are more conservative.
Mode
pp → π + π +pn → π + π 0
nn → π 0 π 0Frejus limit (56 Fe)
7.0 × 1029 yrs
2.0 × 1030 yrs
3.4 × 1030 yrs

This analysis (16 O)
7.2 × 1031 yrs
1.7 × 1032 yrs
4.0 × 1032 yrs

Table 9.1: Lower lifetimes on N N → ππ modes from this analysis, compared to the resultsof [39].

9.1.2

Neutron-antineutron oscillation

Table 9.2 shows the 90% C.L. lower (bound) neutron oscillation lifetime τ90 for
this analysis, as well as the previous SK analysis in [23]. In addition, the value ofthe limit assuming ncand = nexp. bkg. is shown. This is similar to the sensitivity for ahigh-background search like n → n̄ .
SK-I analysis [23]τ90τ90 (ncand = nexp. bkg. )1.9 × 1032 yrs
1.9 × 1032 yrs

This analysis
τ90
τ90 (ncand = nexp. bkg. )2.4 × 1032 yrs
1.5 × 1032 yrs

Table 9.2: Bound n → n̄ limit τ90 and τ90 (ncand = nexp. bkg. ) for the SK-I analysis in [23],and this analysis. Note that τ90 (ncand = nexp. bkg. ) for this analysis does notinclude the additional systematic uncertainties described in Table 8.10, while
τ90 with the actual number of candidates does.

155

9.2

Discussion

9.2.1

Global BNV searches

The searches described in this thesis comprise a subset of searches for baryon
number violating processes. As with all BNV searches, the results of this thesis arenull, but they still represent important advances and updates relative to previous
searches.
This thesis described the first search for the processes pp → π + π + , pn → π + π 0 ,and nn → π 0 π 0 in a water Cherenkov detector, and only the second search for theseprocesses ever made. Due mainly to Super-Kamiokande’s large nucleon exposure, thenull results of this thesis are about two orders of magnitude more stringent than thoseof [39]. As the world’s current largest water Cherenkov detector, Super-Kamiokande
is uniquely well-situated to study these decay modes. The mode corresponding to
the simplest analysis, nn → π 0 π 0 , also has the most stringent limit, and its potential for discovery or further constraint is the most promising of the three N N → ππ
modes studied. Further exposure both at SK and the similar but much larger proposed Hyper-Kamiokande detector [92] could provide valuable continuations of thesesearches.
This thesis described the most recent of several searches for n → n̄ with boundneutrons (Table 2.1). While this search contains important updates relative to previous searches, such as more data, an improved pion interaction model, and a novel
search method, it still illustrates the severe limitations of bound n → n̄ searches.
Even though this thesis contains a larger exposure by more than a factor of three
than the next largest search, it does not place a more stringent limit on n → n̄ .
Indeed, its limit under the assumption ncand = nbkg is slightly lower than the previous
SK-I result. The reasons for this will be discussed in detail shortly.

156
Unlike N N → ππ , n → n̄ can be searched for with free neutrons. To comparewith free neutron results from [29], the nuclear suppression factor R (Equation 2.16)
is used to simply relate bound and free lifetimes. Table 9.3 shows the free n → n̄ limit
from [29], as well as the calculated limits from [23] and this thesis. The calculations
use the value R = 0.517 × 1023 s−1 from [27], and also take into account theoreticaluncertainty, which [27] estimates to be 15%. The tradeoff between exposure andbackground as described in Section 2.5.1 for essentially zero background free neutron
experiments, and large but background-limited bound neutron experiments, has so
far led to more stringent limits from bound neutrons, by a factor of 3-4. However,the very low background of free neutron experiments means that they will likely bemore competitive in the future, provided they can increase their exposure by morethan this factor, which planned experiments like NNbarX [30] are poised to do.
Free neutron [29]0.86 × 108 s

SK-I analysis [23]2.7 × 108 s

This analysis
3.8 × 108 s

Table 9.3: Free n → n̄ lifetime limits, as measured in [29], and as calculated in [23] andthis thesis. The calculations incorporate theoretical uncertainty on the nuclear
suppression R as calculated in [27].

9.2.2

Comparison with previous SK n → n̄ analysis

Given that they were performed with the same detector and a subset of the samedata, it is instructive to compare the n → n̄ search of this thesis to the previous
search done in SK-I [23], not only in terms of the final result, but also the analysis
method.
There are several differences in the simulation stage, described in Section 5.2 andAppendix A. Most significantly, the n̄-nucleon annihilation branching fractions andthe pion-nucleon interaction model have both been updated. While the outgoing pion

157
multiplicities are similar, the increased interactivity of the updated pion model leads
to a smaller averaged charged pion momentum (Figure 5.4). This generally leads toless visible energy and fewer Cherenkov rings.
The search method following reconstruction is quite different. The original SK-I
analysis used a set of simple cuts, following the FCFV cuts (Section 6.1.6):
A) Nring > 1;B) 700 MeV < Evis < 1300 MeV;
C) Ptot < 450 MeV/c;
D) 750 MeV/c2 < Minv < 1800 MeV/c2 .
Here Ptot and Minv are defined as in Section 7.4.1. Selections C) and D) can be seenin Figure 9.1.
It is straightforward to apply the cut-based method of the SK-I analysis to the
new MC, both for n → n̄ signal and atm.-ν background, to compare the cut-based
method and the multivariate method described in Section 8.2. Table 9.4 shows theefficiencies, background rates, systematic uncertainties, and data candidates for the
SK-I analysis, the cut-based method applied to the new analysis MC, and the fullmultivariate analysis described in Chapter 8.
The first two rows of Table 9.4 are useful for comparing changes in the MC simulation between the SK-I analysis in [23] and this analysis, as they both correspond to
the SK-I period and use the same analysis method. The lower signal efficiency andhigher signal systematic uncertainty in the new MC simulation can both be understood as consequences of the more interactive pion model. The increased interactions
lead to fewer particles above Cherenkov threshold and lower visible energy (selectionsA) and B)), and also further distorts the isotropic spread of outgoing pions (selection

158

Figure 9.1: Total momentum vs. invariant mass following selections A) and B) for n → n̄
signal (top), atm.-ν background (middle), and SK-I data (bottom) for the
analysis in [23], with boxes indicating selections C) and D).

C)). The slightly higher background rate can be understood from the new weighting
procedure, which takes the full three-flavor oscillation analysis into account as opposed to the standard oscillation weight (Equation 4.1), and tends to increase thenormalization (Figure 8.2). The updated pion model also leads to fewer particles
above Cherenkov threshold and lower visible energy in the background interactions
involving pions. However, it also leads to more background events passing selection
C), as pions produced in neutrino interactions are generally not isotropic.

159
Eff.

σeff

Bkg. (SK live)

Bkg. (MT-yr)

σbkg

ncand

12.1%

22.9%

24.1 ± 5.7

263 ± 62

23.7%

24

New MC,cut method
SK-I
SK-II
SK-III
SK-IV

8.35%
8.49%
8.04%
8.05%

38.1%
43.3%
43.5%
45.4%

27.9 ± 7.6
14.9 ± 4.2
10.7 ± 2.8
43.5 ± 11.7

304 ± 83
303 ± 85335 ± 88327 ± 88

27.2%
28.0%
26.2%
26.9%

20
14
4
44

New MC,new method
SK-I
SK-II
SK-III
SK-IV

11.0%
10.5%
9.5%
12.0%

37.9%
47.6%
48.9%
46.9%

20.5 ± 7.5
10.7 ± 3.6
6.5 ± 2.4
33.9 ± 11.0

223 ± 82
217 ± 73204 ± 76254 ± 83

36.8%
33.5%
37.3%
32.5%

6
5
3
26

SK-I analysis

Table 9.4: Comparison of efficiencies, background rates, systematic uncertainties, anddata candidates from the SK-I analysis in [23], the new analysis MC using
the same cut method, and the new analysis as described in Chapter 8. The
background rates are shown with ± 1-σ systematic uncertainties to better compare to the number of data events.

It is clear from Table 9.4 that there are several advantages in the multivariate
approach used in this analysis relative to the cut-based method of the SK-I analysis.
Comparing the results using the new MC, the efficiencies are higher by 15-50% for
the multivariate method relative to the cut method, with about the same systematic
uncertainties. The expected backgrounds are also lower by 20-40% for the multivariate
method relative to the cut method, although the systematic uncertainties are higher.
A crucial aspect in which the multivariate approach performs worse than the cutbased approach is only apparent upon looking at the real data. Data and expectedbackground are consistent within about 1σ for the cut-based method, but they are
not for the multivariate method in SK-I-III, reasons for which were discussed in

160
Section 8.3.3. While there is no ambiguity in interpreting the boosted decision tree
output of the data as being consistent with the atm.-ν background in terms of shape(Figure 8.8), the number of events in the data is well below the expected background.
Overall, the increased systematic uncertainties especially in the signal relative to
the SK-I search lead to a lower sensitivity than that analysis. Given the overwhelmingrole of pions in both n → n̄ and N N → ππ — and to a lesser extent, the atm.-ν
background that usually contains at least one produced pion — it would of course begood to reduce pion-nucleon interaction uncertainties for future analyses. As a moregeneral prescription, multivariate analyses should be used with caution, particularly
for searches with a high expected background. They are useful for increasing efficiency
and reducing background relative to cut-based methods for analyses in which thesignal process is difficult to extract. However, some more quantitative studies between
input variables for MC and real data should be done to ensure the background is wellpredicted, while also keeping the analyses sufficiently blind and not choosing any final
selection based on the performance of the data.

Appendices161

Appendix An → n̄ Branching Fractions
The final state of an antineutron annihilating with a proton or a neutron at low
momentum in the residual 16 O nucleus is important for a good n → n̄ simulation.
There are many non-negligible branching fractions in the final state, mostly involving
multiple pions.
Previous n → n̄ searches have taken branching fractions from experimental data,
and this thesis does the same. Data for antineutron annihilation is scarce, but thereis plentiful data for low-energy antiproton annihilation. From isospin symmetry, theannihilations p̄n → (anything) and p̄p → (anything) are assumed to correspond ton̄p → (anything) and n̄n → (anything), respectively. In the former case, the sign ofcharged pions is reversed.
The n → n̄ search in [23] used p̄p and p̄d data from bubble chamber experiments(where d is deuterium). Branching fractions used in [23] are listed in Table A.1. This
analysis used more recent data from the Crystal Barrel experiment [78, 79], which collected data on a large number of exclusive final states. The OBELIX experiment [80]has also collected data for low-energy antineutrons in n̄p annihilation, though for
fewer branching fractions.
162

163
For n̄n annihilation, over half of the branching fractions are purely pionic (Table A.2). About 25% of branching fractions involve an intermediate resonance including ω, ρ, and η mesons (Table A.3), which mostly decay to pions as well. A
smaller percentage of branching fractions involve strange particles (Table A.4). Nuclear effects are simulated for ω and η mesons, while ρ mesons decay rapidly before
any interactions. More details on the treatment of resonances can be found in [16].
For n̄p annihilation, the branching fractions are largely the same as the old SK
analysis. Some updates are made for particular branching fractions measured in the
OBELIX experiment, particularly for the final state 2π + π − mπ 0 , for 0≤m≤2 [80].
Tables A.2 and A.3 show the branching fractions used for n̄p annihilation.
The updated branching fractions have less of an effect on the analysis than the
updated pion model. Table A.5 shows the efficiency after each of the nn → π 0 π 0 selection pre-cuts for MC using the old SK analysis simulation, MC using the updated
pion model but the same branching fractions as the old SK analysis, and the MC
used in this analysis (both pion model and branching fractions updated). Though afull analysis was not performed with the two former MC sets, the efficiency following each selection pre-cut is more different between the old SK analysis simulationand simulations that use the updated pion model than it is between the latter two
simulations.

164

n̄p
+ 0

π π
π + 2π 0
π + 3π 0
2π + π − π 0
2π + π − 2π 0
2π + π − ω
3π + 2π − π 0

n̄n
1%
8%
10%
22%
36%
16%
7%

+ −

π π
2π 0
π+π−π0π + π − 2π 0
π + π − 3π 0
2π + 2π −2π + 2π − π 0π+π−ω2π + 2π − 2π 0

2%
1.5%
6.5%
11%
28%
7%
24%
10%
10%

Table A.1: Branching fractions used in the SK-I n → n̄ analysis in [23]. The values aretaken from bubble chamber experiments [75–77]

n̄p
0

2π
3π 0
4π 0
5π 0
6π 0
7π 0
π+π−
π+π−π0π + π − 2π 0
π + π − 3π 0π + π − 4π 0
π + π − 5π 0
2π + 2π −2π + 2π − π 0
2π + 2π − 2π 0
2π + 2π − 3π 0
3π + 3π −3π + 3π − π 0

n̄n
0.07%
0.8%
0.3%
1.0%
0.01%
0.1%
0.3%
1.6%
13.0%
11.2%
3.3%
1.4%
6.0%
13.5%
16.6%
0.6%
2.2%
2.0%

π+π0
π + 2π 0
π + 3π 0
π + 4π 0

0.1%
0.7%
14.8%
1.4%

2π + π −2π + π − π 0
2π + π − 2π 0
2π + π − 3π 0
3π + 2π −3π + 2π − π 0

2.0%
17.0%
10.8%
30.1%
5.5%
3.2%

Table A.2: Purely pionic branching fractions used in this analysis. The data for n̄n isfrom data in [78]. Some of the data for n̄p has been updated from results in
the OBELIX experiment, though most is replicated from [23].

165

n̄p

n̄n

ρπ
ρ+/− π −/+
ωω
ρ0 ω
π0π0ω
π+π−ω
ηω
π+π−η

1.8%
3.7%
3.5%
2.4%
2.8% π + π 0 ω
7.1% 2π + π − ω
1.6%
1.7%

0 0

2.0%
12.4%

Table A.3: Branching fractions with one or more intermediate resonances this analysis.
Data for n̄n annihiliaton is taken from [78] and [79].

K +K −
ωK + K −KS KS
π +/− K −/+ KL
π +/− K −/+ KS
K 0 K +/− π −/+ π 0
KS KL π + π −

0.1%
0.2%
0.1%
0.3%
0.3%
1.0%
0.3%

Table A.4: Branching fractions with one or more strange particles in this analysis for n̄nannihilation, from data in [78].

Pre-cut
Nring > 1
400 MeV < Evis < 1600 MeV
Ptot < 500 MeV/c

Old SK Updated pion model This analysis
87.7%
83.1%
82.7%
70.5%
62.6%
61.8%
45.1%
39.0%
40.5%

Table A.5: Efficiency after each of the n → n̄ selection pre-cuts, for three different MC
sets: MC from the old SK simulation, MC with the new pion model but samebranching fractions as the old SK simulation, and MC with both updated pionmodel and branching fractions (this analysis).

Appendix B
Plots of Multivariate Input
Variables for SK-I-III
The main text contains plots of each multivariate input variable for both n → n̄and N N → ππ for the current data-taking period, SK-IV. This appendix contains
plots of the same variables for the SK-I-III data-taking period for reference. Though
the plots are similar across detectors, they contain different data sets and Monte
Carlo simulations.

166

167

SK-I
7

5

6

4

5

EventsEvents

SK-I
6

3
2

4
3
2

1

1

0
120

0
130

140
150
160
Angle (degrees)

170

180

0

0.2

SK-II

0.6

0.8

1

0.8

1

0.8

1

SK-II

3

4

EventsEvents0.4

Charge ratio

2

3
2

1
1
0
120

0
130

140
150
160
Angle (degrees)

170

180

0

0.2

0.4

0.6

Charge ratio

SK-III

SK-III

EventsEvents

2

1

0
120

2

0
130

140
150
160
Angle (degrees)

170

180

0

0.2

0.4

0.6

Charge ratio

Figure B.1: The angle between µ-like rings (left) and the ratio of charge carried by themost-energetic ring (right) for the pp → π + π + search, for signal MC (solid
histogram), atm.-ν MC (dashed histogram), and data (crosses). Distributions
are normalized to the number of events in the data. Details of each variable
are described in the text, in Section 7.2.2.

168

SK-I

SK-I

6
12
10

4

EventsEvents

5

3

8
6

2

4

1

2

0

0
0

200 400 600 800 1000 1200 1400 1600
Visible Energy (MeV)

0

SK-II
4

EventsEvents

2000

SK-II

4
3
2
1

3
2
1

0

0
0

200 400 600 800 1000 1200 1400 1600
Visible Energy (MeV)

0

2

0

500
1000
1500
Maximum Michel distance (cm)2000

SK-III

Events

SK-III

Events

500
1000
1500
Maximum Michel distance (cm)500200 400 600 800 1000 1200 1400 1600
Visible Energy (MeV)

0

500
1000
1500
Maximum Michel distance (cm)2000

Figure B.2: The visible energy (left) and the maximum distance between the primaryvertex and a decay electron vertex (right) for the pp → π + π + search, for signal
MC (solid histogram), atm.-ν MC (dashed histogram), and data (crosses).
Distributions are normalized to the number of events in the data. Details ofeach variable are described in the text, in Section 7.2.2.

169

SK-I

SK-I
7

10

6

EventsEvents

8
6
4

5
4
3
2

2

1

0

0
0

20

40

60 80 100 120 140 160 180
Angle (degrees)

0

SK-II

500 1000 1500 2000 2500
sum of charges (corrected p.e.)

3000

SK-II
6

EventsEvents

6

4

4

2

2

0

0
0

20

40

60 80 100 120 140 160 180
Angle (degrees)0

SK-III

500 1000 1500 2000 2500
sum of charges (corrected p.e.)

3000

SK-III

EventsEvents

5

0

2

0
0

20

40

60 80 100 120 140 160 180
Angle (degrees)0

500 1000 1500 2000 2500
sum of charges (corrected p.e.)

3000

Figure B.3: The largest angle between a µ-like ring direction and a decay electron vertex (left) and the magnitude of the vector sum of RTOT (right) for the
pp → π + π + search, for signal MC (solid histogram), atm.-ν MC (dashedhistogram), and data (crosses). Distributions are normalized to the numberof events in the data. Details of each variable are described in the text, inSection 7.2.2.

170

SK-I

SK-I

20

EventsEvents

25

15
10
5
0
2

2.5

3

3.5
4
4.5
Number of rings

5

5.5

18
16
14
12
10
8
6
4
2
0

6

0

0.5

1

SK-II

2.5

3

3.5

4

3.5

4

3.5

4

12

12

10

10

EventsEvents

2

SK-II

14

8
6

8
6

4

4

2

2

0

0
2

2.5

3

3.5
4
4.5
Number of rings

5

5.5

6

0

0.5

1

1.5

2

2.5

3

Number of Michel electrons

SK-III10

SK-III
6
5

Events8

Events1.5

Number of Michel electrons

6
4

4
3
2

2

1

0

0
2

2.5

3

3.5
4
4.5
Number of rings

5

5.5

6

0

0.5

1

1.5

2

2.5

3

Number of Michel electrons

Figure B.4: The number of Cherenkov rings (left) and the number of decay electrons(right) for the pp → π + π + search, for signal MC (solid histogram), atm.-ν
MC (dashed histogram), and data (crosses). Distributions are normalized to
the number of events in the data. Details of each variable are described in
the text, in Section 7.2.2.

171

SK-I

SK-II
16
14

30

EventsEvents

25
20
15
10
5

12
10
8
6
4
2
0

0
2

2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8
Number of mu-like rings

42

2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8
Number of mu-like rings

4

SK-III
10

Events

8
6
4
2
0
2

2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8
Number of mu-like rings

4

Figure B.5: The number of µ-like rings for the pp → π + π + search, for signal MC (solid
histogram), atm.-ν MC (dashed histogram), and data (crosses). Distributions
are normalized to the number of events in the data. Details are described in
the text, in Section 7.2.2.

172

SK-I
25

EventsEvents

20
15
10
5
0
0

200

400

600

800 1000 1200 1400

18
16
14
12
10
8
6
4
2
0
120

SK-I

130

π0 momentum (MeV/c)

140

0

200

400

600

800 1000 1200 1400

3
2
1
0
120

130

140

SK-III

150

160

170

180

170

180

SK-III
7

10

6

8

5

EventsEvents180π0-π+ angle (degrees)

12

4

170

8
7
6
5
4

π0 momentum (MeV/c)

6

160

SK-II

EventsEvents

SK-II
18
16
14
12
10
8
6
4
2
0

150

π0-π+ angle (degrees)

4
3
2

2

1

0
0

200

400

600

800 1000 1200 1400

π0 momentum (MeV/c)

0
120

130

140

150

160

π0-π+ angle (degrees)

Figure B.6: The π 0 candidate momentum (left) and the angle between the π 0 and π +candidate rings (right) for the pn → π + π 0 search, for signal MC (solid histogram), atm.-ν MC (dashed histogram), and data (crosses). Distributions
are normalized to the number of events in the data. Details of each variable
are described in the text, in Section 7.3.2.

SK-I

24
22
20
18
16
14
12
10
8
6
4
2
0

SK-I
35
30

EventsEvents

173

25
20
15
10
5
0

0

200
400
600
800
π+ momentum (MeV/c)

1000

0

100

SK-II

8

EventsEvents

10

6
4
2
0
200
400
600
800
π+ momentum (MeV/c)

1000

0

EventsEvents

100

200

300

400

500

400

500

SK-III

6
4
2
0
200
400
600
800
π+ momentum (MeV/c)500

π0 mass (MeV/#c^2)

8

0

400

18
16
14
12
10
8
6
4
2
0

SK-III

10

300

SK-II

12

0

200

π0 mass (MeV/#c^2)

1000

20
18
16
14
12
10
8
6
4
2
0
0

100

200

300

π0 mass (MeV/#c^2)

Figure B.7: The π + candidate momentum (left) and the invariant mass of the π 0 candidate (right) for the pn → π + π 0 search, for signal MC (solid histogram),atm.-ν MC (dashed histogram), and data (crosses). Distributions are normalized to the number of events in the data. Details of each variable aredescribed in the text, in Section 7.3.2.

174

SK-I
12

12

10

EventsEvents

SK-I
16
14
10
8
6
4

8
6
4
2

2
0

0
0

0.2

0.4
0.6
charge ratio

0.8

1

0

SK-II

200 400 600 800 1000 1200 1400 1600
Visible Energy (MeV/c)

SK-II
7

10

6

EventsEvents

8
6
4

5
4
3
2

2

1

0

0
0

0.2

0.4
0.6
charge ratio

0.8

1

0

SK-III

200 400 600 800 1000 1200 1400 1600
Visible Energy (MeV/c)

SK-III6

10
5

EventsEvents

8
6
4

4
3
2

2

1

0

0
0

0.2

0.4
0.6
charge ratio

0.8

1

0

200 400 600 800 1000 1200 1400 1600
Visible Energy (MeV/c)

Figure B.8: The ratio of charge carried by the most energetic ring (left) and the visibleenergy (right) for the pn → π + π 0 search, for signal MC (solid histogram),atm.-ν MC (dashed histogram), and data (crosses). Distributions are normalized to the number of events in the data. Details of each variable aredescribed in the text, in Section 7.3.2.

175

SK-II
40
35
30

EventsEvents

SK-I
90
80
70
60
50
40
30
20
10
0

25
20
15
10
5
0

0

0.5

1

1.5

2

0

0.5

Michel electrons

1

1.5

2

Michel electrons

SK-III30

Events

25
20
15
10
5
0
0

0.5

1

1.5

2

Michel electrons

Figure B.9: The number of decay electrons for the pn → π + π 0 search, for signal MC (solid
histogram), atm.-ν MC (dashed histogram), and data (crosses). Distributions
are normalized to the number of events in the data. Details of each variable
are described in the text, in Section 7.3.2.

176

SK-I

12

500

10

400

EventsEvents

SK-I

8
6
4

0

1000 2000 3000 4000 5000
Total Momentum (MeV/c)

6000

0

500

1000

1500

2000

2500

2000

2500

2000

2500

2

Invariant Mass (MeV/c )

SK-II

SK-II

300

7
6

250

5

200

EventsEvents

200
100

2

4
3

150
100

2

50

1
0

1000 2000 3000 4000 5000
Total Momentum (MeV/c)

6000

0

SK-III
3

Events

2.5
2
1.5
1
0.5
0

1000 2000 3000 4000 5000
Total Momentum (MeV/c)

500

1000

1500

Invariant Mass (MeV/c2)

3.5

Events300

6000

SK-III

200
180
160
140
120
100
80
60
40
20
0

500

1000

1500

Invariant Mass (MeV/c2)

Figure B.10: Total momentum (left) and invariant mass (right) for data (crosses), nn →π 0 π 0 MC (solid histogram), and atmospheric neutrino MC (dashed histogram). Cuts (C1)-(C3) are applied to both plots. The invariant masscut (C5) has been applied to the total momentum, and the total momentum cut (C4) to the invariant mass. The thin green vertical lines indicate
the locations of the final cuts for each variable. Details of the selection
criteria can be found in Section 7.4.1.

177

SK-I

SK-I

50
45
40

40

EventsEvents

35
30

20

30
25
20
15
10

10

5
0
0

50

100

150

200

250

300

350

400

450

0
0

500

200

400

800

1000

1200

1400

1600

1400

1600

1400

1600

Invariant mass (MeV/c2)

Total momentum (MeV/c)

SK-II

SK-II

35

45

30

40
35

Events25

Events

600

20
15

30
25
20
15

10

10
5
0
0

5
50

100

150

200

250

300

350

400

450

0
0

500

200

400

600

800

1000

1200

Invariant mass (MeV/c2)

Total momentum (MeV/c)

SK-III

SK-III

25

20
18

20

14

EventsEvents

16

12
10
8

15

10

6
5

4
2
0
0

50

100

150

200

250

300

350

Total momentum (MeV/c)

400

450

500

0
0

200

400

600

800

1000

1200

Invariant mass (MeV/c2)

Figure B.11: The total momentum (left) and invariant mass (right) for the n → n̄ search,
for signal MC (solid histogram), atm.-ν MC (dashed histogram), and data(crosses). Distributions are normalized to the number of events in the data.
Details of each variable are described in the text, in Section 8.2.

178

SK-I

50

SK-I

70
60

40

EventsEvents

50
30

20

40
30
20

10
10
0
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0
-500

1

-400

(Etot-Eprimary ring)/Etot

-300

-200

0

100

200

300

400

500

300

400

500

300

400

500

Likelihood difference

SK-II

35

-100

SK-II
100

30
80

EventsEvents

25
20
15

60

40

10
20
5
0
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0
-500

1

-400

(Etot-Eprimary ring)/Etot

-300

-200

-100

0

100

200

Likelihood difference

SK-III

SK-III
30

18
16

25

EventsEvents

14
12
10
8

20

15

10

6
4

5
2
0
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

(Etot-Eprimary ring)/Etot

0.8

0.9

1

0
-500

-400

-300

-200

-100

0

100

200

Likelihood difference

Figure B.12: (Etotal -Eprimary )/Etotal (left) and the PID likelihood difference of the leading
ring (right) for the n → n̄ search, for signal MC (solid histogram), atm.-ν
MC (dashed histogram), and data (crosses). Distributions are normalized
to the number of events in the data. Details of each variable are describedin the text, in Section 8.2.

179

SK-I

220

SK-I

200

50

180
40

EventsEvents

160
140
120
100
80

30

20

60
40

10

20
0
0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0
0

5

Number of decay electrons

200

400

600

800

1000

1200

1400

1600

1800

2000

Maximum decay electron distance (cm)

SK-II

SK-II

180

25

160
20

120

EventsEvents

140

100
80

15

10

60
40

5

20
0
0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0
0

5

Number of decay electrons

200

400

600

800

1000

1200

1400

1600

1800

2000

Maximum decay electron distance (cm)

SK-III

SK-III

100

16
14

80

EventsEvents

12
60

40

10
8
6
4

20

2
0
0

0.5

1

1.5

2

2.5

3

3.5

Number of decay electrons

4

4.5

5

0
0

200

400

600

800

1000

1200

1400

1600

1800

2000

Maximum decay electron distance (cm)Figure B.13: The number of decay electrons (left) and the maximum distance from theprimary vertex to a decay electron vertex (right) for the n → n̄ search,
for signal MC (solid histogram), atm.-ν MC (dashed histogram), and data(crosses). Distributions are normalized to the number of events in the data.
Details of each variable are described in the text, in Section 8.2. The distanceis only shown for events with one or more decay electron.

180

SK-I

220

SK-I

200

50

180
40

EventsEvents

160
140
120
100
80

30

20

60
40

10

20
0
0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0
0

5

Number of decay electrons

200

400

600

800

1000

1200

1400

1600

1800

2000

Maximum decay electron distance (cm)

SK-II

SK-II

180

25

160
20

120

EventsEvents

140

100
80

15

10

60
40

5

20
0
0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0
0

5

200

400

600

800

1000

1200

1400

1600

1800

Number of decay electrons

Maximum decay electron distance (cm)

SK-III

SK-III

100

2000

16
14

80

EventsEvents

12
60

40

10
8
6
4

20

2
0
0

0.5

1

1.5

2

2.5

3

3.5

Number of decay electrons

4

4.5

5

0
0

200

400

600

800

1000

1200

1400

1600

1800

2000

Maximum decay electron distance (cm)Figure B.14: The sphericity (left) and the number of potential rings (right) for the n → n̄
search, for signal MC (solid histogram), atm.-ν MC (dashed histogram), anddata (crosses). Distributions are normalized to the number of events in thedata. Details of each variable are described in the text, in Section 8.2.

181

SK-I

SK-I

100

80
70

80

EventsEvents

60
50
40
30
20

60

40

20

10
0
-1

0

1

2

3

4

0
-1

5

first fox-wolfram moment

1

2

3

4

5

4

5

4

5

second fox-wolfram moment

SK-II

SK-II

90

70

80

60

70

50

60

EventsEvents

0

40
30

50
40
30

20
20
10
0
-1

10
0

1

2

3

4

0
-1

5

first fox-wolfram moment

0

1

2

3

second fox-wolfram moment

SK-III

SK-III
45

35

40

30

35

EventsEvents

25
20
15

30
25
20
15

10
10
5
0
-1

5
0

1

2

3

first fox-wolfram moment

4

5

0
-1

0

1

2

3

second fox-wolfram moment

Figure B.15: The first and second Fox-Wolfram moments (left and right, respectively)
for the n → n̄ search, for signal MC (solid histogram), atm.-ν MC (dashedhistogram), and data (crosses). Distributions are normalized to the numberof events in the data. Details of each variable are described in the text, inSection 8.2.

Appendix C
Limit Calculations
In each of the searches in this analysis, we seek to measure the lifetime (or inversely,
the rate) of the process. The results of the searches of course depend on whether any
signal events are observed. If signal events are observed in significant excess of the
expected background, then the rate Γ is calculated simply as

Γ=

nsig
,
NT 

(C.1)

where nsig is the number of observed signal events, N the number of targets (for thesesearches, either 16 O nuclei or individual neutrons within the nuclei), T the running
time of the experiment, and  the signal detection efficiency.
In the absence of evidence for any signal process, lower limits are set on the timescale of the process. There are two ways to do this, one using purely frequentist
statistics and one using Bayesian inference. The Bayesian method is used for theseanalyses, but the frequentist approach is simple and instructive, and so I briefly
discuss it first.
In setting the limit, a confidence level is also set. For a counting experiment,
182

183
this entails finding a parameter µC.L. associated with the signal process that is themean of a Poisson distribution, such that any value taken from this distribution has
a probability of C.L. to be larger than µC.L. . The confidence level in these searches is
set to 90%. For a signal process with some expected background b and a number ofobserved candidates n, µ90 can be found by solving
Pn
0.9 = 1 −m−(µ90 +b) (µ90 +b)
m=0 e
m!
Pn
bm
−b
m=0 e
m!

.

(C.2)

The value of µ90 from the solution to Equation C.2 can then be substituted for
nsig in Equation C.1.
The main limitation of the frequentist approach is that it has no mechanism
for incorporating systematic uncertainties. Intuitively, our notion of “confidence”
should include uncertainties on the signal and background processes, which for the
searches in this thesis are quite substantial. By assuming prior distributions for theseparameters, the Bayesian approach naturally incorporates systematic uncertainties,
although it makes the calculation much more complicated.
The limits are calculated using a Bayesian method [91] as follows. The probabilityof observing n events, given a true process rate of Γ, a true exposure λ (the productof N and T in Equation C.1), a true efficiency , and a true background rate b, iscalculated with the Poisson formula:e−(Γλ+b) (Γλ + b)n.
P (n|Γλb) =
n!

(C.3)

As the parameter of interest is the true rate Γ, we apply Bayes’ theorem to get

P (Γλb| n) =1
P (n|Γλb) P (Γλb) ,
A

(C.4)

184
where A is a normalization constant. The process rate, efficiency, exposure, andbackground are all independent parameters, so that P (Γλb) can be written as a
product P (Γ) P (λ) P () P (b). The probability density function for Γ is obtained byintegrating over the exposure, efficiency, and background rate:

1
P (Γ|n) =Ae−(Γλ+b) (Γλ + b)n
P (Γ) P (λ) P () P (b) dλddb.
n!

Z Z Z

(C.5)

A confidence level can be obtained by integrating P (Γ|n) up to an upper limit on therate, Γlimit :Z Γlimit
P (Γ|n) dΓ,

C.L. =(C.6)

0

and the lower lifetime limit τlimit is just the inverse of Γlimit .
The benefit of this approach is that it allows incorporation of systematic uncertainties. The probabilities for λ, , and b are taken to be Gaussian, with symmetric σ
values for  and b taken from the systematic uncertainties described in Sections 7.5;
the uncertainty in λ is assumed to be negligible in each search. The probability forΓ is assumed to be uniform for Γ > 0.
Since the searches were performed for four different detectors, Equation C.5 mustbe generalized for detector-specific n, λ, , and b:

4

1Y
P (Γ|n1 , n2 , n3 , n4 ) =
A i=1

Z Z Z

e−(Γλi i +bi ) (Γλi i + bi )ni
P (Γ) P (λi ) P (i ) P (bi ) dλi di dbi .
ni !
(C.7)

In this calculation, systematic uncertainties are conservatively assumed to be fully
correlated across SK detectors. This is a reasonable assumption, since the dominant

185
systematic uncertainties in these searches are related to MC simulation, and thus areroughly detector-independent.
Finally, for very low background searches, the statistical uncertainty is taken into
account by convoluting the Gaussian prior with a Poisson distribution:Z
P (b) =!
e−bM C (bM C )nb(bC − bM C )2
dbM C ,
exp −
nb !
2σb2

(C.8)

where nb is the number of background in the un-normalized MC, bM C is the numberof true un-normalized background, and C is the MC oversampling factor.

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Curriculum Vitae
Jeffrey D. Gustafson
590 Commonwealth Avenue
Department of Physics, Boston University
Boston, MA 02215, U.S.A.

Mobile: (651)357-6771
Office: (617)353-6040
E-mail: jdgustaf@bu.edu

EDUCATION• Boston University, Boston, Massachusetts– PhD in Experimental Particle Physics, expected in May 2016
– MA in Physics, January 2012
• Macalester College, St. Paul, Minnesota– Bachelor of Arts in Physics, May 2009
– Additional Major in Mathematics, May 2009
RESEARCH EXPERIENCE• Super-Kamiokande– Simulation software:∗ Update of Monte Carlo generator for dinucleon decay∗ Design of custom Monte Carlo generator for neutron-antineutron oscillation– Reconstruction and Analysis software:∗ Update and maintenance of upward-going muon reduction and analysis software∗ Custom analysis modules for dinucleon decay and neutron-antineutron
oscillation analyses
TEACHING EXPERIENCE• Boston University Academy– Discussion and lab teaching assistant for high-school physics, Spring 2015
• Boston University
193

194
– Discussion session teaching assistant for physics, Fall 2010 - Fall 2011
• Macalester College– Tutoring center, Fall 2007 - Spring 2009
PRESENTATIONS• Searches for ∆ B = 2 processes at Super-Kamiokande
Jeff Gustafson
International Conference on Baryon and Lepton Number Violation Conference,
Parallel Talk
Amherst, MA, April 2015
SELECTED PUBLICATIONS• Search for dinucleon decay into pions at Super-Kamiokande
J. Gustafson et al. [Super-Kamiokande Collaboration]
Phys. Rev. D 91:072009 (2015)
OTHER PUBLICATIONS• Limits on sterile neutrino mixing using atmospheric neutrinos inSuper-Kamiokande
K. Abe et al. [Super-Kamiokande Collaboration]
Phys. Rev. D 91:052019 (2015)• Test of Lorentz invariance with atmospheric neutrinos
K. Abe et al. [Super-Kamiokande Collaboration]
Phys. Rev. D 91:052003 (2015)• Search for trilepton decay via p → e+ νν and p → µ+ νν in the SuperKamiokande experiment
V. Takhistov et al. [Super-Kamiokande Collaboration]
Phys. Rev. Lett. 113:101801 (2014)
REFERENCES• Prof. Edward Kearns
Boston University
Department of Physics
590 Commonwealth Avenue
Boston, MA 02215
Office Phone: (617)353-3425
E-mail: kearns@bu.edu