The Intermediate Detector
and Neutrino Phenomenologyof the Hyper-Kamiokande
Experiment

Nicholas William Prouse
29th September 2018

Submitted in partial fulfilment of the requirements of the Degree of Doctor ofPhilosophy

Queen Mary University of London

Particle Physics Research Centre, School of Physics and Astronomy
Faculty of Science and Engineering
andUniversity of Southampton

School of Physics and Astronomy
Faculty of Physical Sciences and Engineering

I, Nicholas William Prouse, confirm that the research included within this thesis is myown work or that where it has been carried out in collaboration with or supported byothers, that this is duly acknowledged below and my contribution indicated. Previously
published material is also acknowledged below.
I attest that I have exercised reasonable care to ensure that the work is original, anddoes not to the best of my knowledge break any UK law, infringe any third party’s
copyright or other Intellectual Property Right, or contain any confidential material.
I accept that the College has the right to use plagiarism detection software to check theelectronic version of the thesis.
I confirm that this thesis has not been previously submitted for the award of a degreeby this or any other university.
The copyright of this thesis rests with the author and no quotation from it or information
derived from it may be published without the prior written consent of the author.
Signature:
Date:
Details of collaboration and publications:Due to the highly collaborative nature of all parts of the work of this thesis, a fullaccount of the individual contributions is given here:
Chapters 1 and 2 provide an overview of background work, which does not constituteoriginal research of this thesis.
The work of Chapters 3 and 4 was completed as part of the Hyper-Kamiokande, TITUS
and E61 collaborations.
In Chapter 3, the high-energy reconstruction software was developed by the author, withsome input also into the low-energy reconstruction that was mainly developed by TITUS
collaborator W. Ma. The selections described were also developed by the author, whichwere then provided to TITUS collaborator R. Shah to perform the analysis on theireffect on oscillation sensitivities. The remaining software described in the chapter wasdeveloped by other members of the TITUS, Hyper-Kamiokande and T2K collaborations.
The simulation and reconstruction software used in Chapter 4 was mainly developed byother members of the E61, Hyper-Kamiokande, Super-Kamiokande and T2K collaborations, except for the addition of Gadolinium support to the simulation software, which
was by the author. The primary event selection was developed by other members of theE61 and νPRISM collaborations. The remaining steps of the analysis: the neutron selections, background separation, efficiency and background correction and the productionof the final results were performed by the author.
The work of Chapters 5 and 6 were performed in collaboration with S.F. King of the
University of Southampton and P. Ballett, S. Pascoli and T. Wang of the University ofDurham, as published in [1, 2]. The simulations of Hyper-Kamiokande in both of thesechapters were developed by the author, while the simulations of DUNE were developed
by T. Wang.

2

The simulations of Chapter 5 use the existing GLoBES library, with the additional
code to perform the simulations produced by the author, except for the simulations ofSection 5.4 which were produced by T. Wang.
The results for new sum-rules presented in Chapter 6 and Appendix D were found by
the author, with the rest of the theoretical and phenomenological development of themodel by S.F. King. The fits to existing data in this chapter were performed by theauthor. The simulations of future experiments were performed using GLoBES, withmodifications by the author to allow fitting to the oscillation models being tested, andwith the additional code to perform the simulations also developed by the author.

3

Abstract
In this thesis, a number of aspects of neutrino oscillation physics are investigated, focusing on the design and phenomenology of next generation neutrino beam experiments.
In working to optimise the sensitivity of the Hyper-Kamiokande experiment, new reconstruction software has been developed to meet the goal of fast, flexible reconstructionof particle positions, directions, energies and species. This is demonstrated for the
TITUS intermediate detector, producing samples of neutrino interactions to constrainsystematic uncertainties of oscillation analyses. Inclusion of these samples achieves areduction of over 50 % of systematic errors in measurements of θ23 and δ and increases
the parameter space for CP violation discovery after 10 years at 5 σ from 51 % to 74 %.
A full analysis is presented of a potential neutron measurement at the E61 detector,
including full simulations of all major backgrounds, a likelihood method of backgroundremoval, and a procedure for correcting for efficiency and backgrounds. Large, puresamples are produced, with the ability to accurately reproduce true distributions ofneutron capture multiplicity, distance and angle relative to neutrino interactions, withtrue and reconstructed distributions agreeing within ∼ 2 %.
The sensitivities of Hyper-Kamiokande, including with a possible second tank in Korea,
in combination with the DUNE experiment, have been thoroughly investigated. Several areas of strong synergy are identified, with the optimal combination of experiments
possessing the ability to definitively resolve all remaining unknowns of 3-neutrino oscillations: determining both the octant of θ23 and the mass-ordering in under 2 years anddiscovering CP violation at 5 σ for 50 % of parameter space after 5 years.
The highly predictive Littlest Seesaw flavour models of neutrino masses and mixing aretested against current oscillation data, finding no tension even at 1 σ. The ability of thenext-generation experiments’ oscillation measurements to probe these models is investigated, with all strands of the programme, including long baseline beam experiments
and short and medium baseline reactor experiments, found to show high potential to
exclude the models both individually and in combination.

4

Acknowledgements
Firstly, I would like to give enormous thanks to both my supervisors, Jeanne Wilson
and Steve King. Your knowledge, guidance, support and encouragement throughout
this PhD have been invaluable and I feel immensely fortunate to have had not one but
two supervisors who always found the time for discussions and advice whenever I needed
it. I can’t imagine any better people to have led me through this journey.
I am also indebted to the many friends and collaborators who I have worked with on thevarious projects: In particular to Ryan, Matthew and Wing for guiding me through myfirst PhD project in Chapter 3, to Peter, Silvia and TseChun, for their insights into theworld of phenomenology through our collaboration in my second year in Chapters 5 and
6, to Mark S, Mark H and Ben, for their help throughout my final projects in Chapter
4, to Susana for her help discussing the theory of Chapter 1, to the staff at Whipps
Cross for their help with my appendix, to Alex for his computing help and getting thethesis to actually print, and to Teppei and Francesca who have given help and guidanceat various times throughout. Special thanks should go to Soph and Pierre, who were
there from beginning to end, and Paul and Stephane, who were there at the beginningand at the end, all always there to help with the physics, computing, making the mostof our trips to Japan and academic life in general.
Thank you to the many others who have helped me along the way, joining me forcountless lectures, conferences, workshops, meetings, lunches, films, football matches,pizza parties, drinks in the pub, and travels around London, the UK, Europe, Asia andMesoamerica. Starting from my undergrad, I would not be where I am today without
the friends who joined though the years, especially to Jon, continuing to wander round
London with me throughout both our PhDs and beyond, and to Guy, Richard, and Tom.
To Tom, and the rest of Team Penetrate, thank you for helping me stay sane while we’veadventured through insanity. In my final undergrad year I was sent down this path byworking with the Imperial T2K group; thanks especially to Yoshi and Morgan who
made me into a neutrino physicist for life, despite my experiment at not being one.
That was at Betgenius, where I must thank Pob, Kieron, Alex and all the others there;
my experience working with you – and meeting up at the pub, cinema and the DCPs
since – certainly helped make this PhD possible. You also helped see me through mypostgrad course, where I met incredible physicists in Alejandro, Pete, Zoe, Giulia, Tom,Cian, Simon, Marcos, Andrew and all the rest, who are still great friends even if not all
still physicists. Our shared QFFF experience has led me directly to this PhD. Thankyou in particular to Alejandro who has never failed to be there whether I’ve been in agood place, bad place or the normal place. This PhD itself has also allowed me to meetmany great people; I already mentioned some, but thank you as well to Tom S, Andres,
Billy, Shivesh, Rodrigo, Eddie, Tom C, Jesal and everyone else who made so much timespent in the SCR so enjoyable. And thank you to all those others around the world
who have made collaboration meetings and conferences something to look forward to
regardless of the inevitable last minute late nights preparing talks.
Finally, I would like to give the biggest thanks to my family, who have been supportingme my entire life, in every way. Thank you to all my (nth ) cousins, uncles, aunts,
brothers, sister and brother-in-law, step-parents and, most of all, my parents, without
whom I would not have been able to exist, let alone complete this PhD.
5

6

Contents
List of Figures

11

List of Tables

15

Introduction

17

1 Neutrino Physics

19

1.1

1.2

Neutrinos beyond the Standard Model . . . . . . . . . . . . . . . . . . .

21

1.1.1

Neutrino mixing and flavour oscillations . . . . . . . . . . . . . .

21

1.1.2

Neutrino oscillations in matter . . . . . . . . . . . . . . . . . . .

25

1.1.3

The origin of neutrino mass . . . . . . . . . . . . . . . . . . . . .

27

1.1.4

Family symmetry models . . . . . . . . . . . . . . . . . . . . . .

30

Detection of neutrino interactions . . . . . . . . . . . . . . . . . . . . . .

32

1.2.1

Neutrino interactions with matter . . . . . . . . . . . . . . . . .

33

1.2.2

Water Cherenkov detectors . . . . . . . . . . . . . . . . . . . . .

35

1.2.3

Neutron capture and Gadolinium doping . . . . . . . . . . . . . .

36

2 The Hyper-Kamiokande Experiment

39

2.1

The J-PARC neutrino beam . . . . . . . . . . . . . . . . . . . . . . . . .

40

2.2

The Hyper-Kamiokande far detectors . . . . . . . . . . . . . . . . . . . .

44

2.3

Near detectors for Hyper-K . . . . . . . . . . . . . . . . . . . . . . . . .

44

2.4

Intermediate Water Cherenkov Detector . . . . . . . . . . . . . . . . . .

45

2.4.1

The TITUS detector . . . . . . . . . . . . . . . . . . . . . . . . .

46

2.4.2

The E61 detector . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

Physics potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

2.5.1

50

2.5

Neutrino oscillation physics . . . . . . . . . . . . . . . . . . . . .

7

2.5.2

Nucleon decay . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 Intermediate detector simulation, reconstruction & selection

57
61

3.1

Monte-Carlo event generation and detector simulation . . . . . . . . . .

62

3.2

Low-energy event reconstruction . . . . . . . . . . . . . . . . . . . . . .

63

3.3

High-energy event reconstruction . . . . . . . . . . . . . . . . . . . . . .

64

3.3.1

Ring searching . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

3.3.2

Track fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

3.3.3

Energy reconstruction . . . . . . . . . . . . . . . . . . . . . . . .

68

3.3.4

Likelihood fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

3.3.5

Performance. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

Event Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

3.4.1

Charged lepton selection . . . . . . . . . . . . . . . . . . . . . . .

73

3.4.2

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

83

Effect on sensitivities of Hyper-K . . . . . . . . . . . . . . . . . . . . . .

83

3.4

3.5

4 Measuring neutrons at E61
4.1

89

Event simulation and reconstruction . . . . . . . . . . . . . . . . . . . .

91

4.1.1

WCSim detector simulation . . . . . . . . . . . . . . . . . . . . .

91

4.1.2

Low-energy background and background neutron simulations . .

94

4.1.3

Event reconstruction . . . . . . . . . . . . . . . . . . . . . . . . .

96

4.2

Neutron selection and background rejection . . . . . . . . . . . . . . . .

97

4.3

Background and efficiency correction . . . . . . . . . . . . . . . . . . . .

107

4.4

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

109

5 Sensitivities and synergies of T2HK and DUNE
5.1

5.2

117

Oscillation phenomenology at DUNE and T2HK . . . . . . . . . . . . .

120

5.1.1

Mass ordering, CPV and the octant of θ23 . . . . . . . . . . . . .

121

5.1.2

Precision on δ . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

122

Simulation details. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

125

5.2.1

DUNE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

125

5.2.2

T2HK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

128

5.2.3

Experimental run times and ν : ν ratios . . . . . . . . . . . . . .

131

5.2.4

Statistical method . . . . . . . . . . . . . . . . . . . . . . . . . .

132

8

5.3

5.4

5.5

Sensitivity to mass ordering, CPV, non-maximal CPV, and octant . . .

134

5.3.1

Mass ordering sensitivity. . . . . . . . . . . . . . . . . . . . . .

134

5.3.2

CP violation sensitivity . . . . . . . . . . . . . . . . . . . . . . .

138

5.3.3

Sensitivity to maximal CP violation . . . . . . . . . . . . . . . .

141

5.3.4

Octant degeneracy and the precision on θ23 . . . . . . . . . . . .

141

Complementarity of precision measurements of δ . . . . . . . . . . . . .

147

5.4.1

Normalising by number of events . . . . . . . . . . . . . . . . . .

151

5.4.2

Normalising by run time . . . . . . . . . . . . . . . . . . . . . . .

153

5.4.3

Impact of systematic errors . . . . . . . . . . . . . . . . . . . . .

157

Impact of potential alternative designs . . . . . . . . . . . . . . . . . . .

157

5.5.1

Experimental run times and ν : ν ratios . . . . . . . . . . . . . .

160

5.5.2

Mass ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

160

5.5.3

CPV and MCP sensitivity . . . . . . . . . . . . . . . . . . . . . .

162

5.5.4

Precision on δ . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

165

5.5.5

Optimal configuration . . . . . . . . . . . . . . . . . . . . . . . .

167

6 Testing Littlest Seesaw models of lepton flavour
6.1
6.2

6.3

173

Littlest Seesaw models of neutrinos . . . . . . . . . . . . . . . . . . . . .

176

6.1.1

Sum rules of LS . . . . . . . . . . . . . . . . . . . . . . . . . . . .

178

Probing LS with existing data . . . . . . . . . . . . . . . . . . . . . . . .

179

6.2.1

Predictions of oscillation parameters with fixed η = ±2π/3 . . .

179

6.2.2

Predictions of oscillation parameters with η free

. . . . . . . . .

180

6.2.3

Fitting LS models to global fit data . . . . . . . . . . . . . . . .

182

Sensitivity of future experiments . . . . . . . . . . . . . . . . . . . . . .

186

6.3.1

Future neutrino oscillation experiments . . . . . . . . . . . . . .

187

6.3.2

Statistical method . . . . . . . . . . . . . . . . . . . . . . . . . .

188

6.3.3

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

189

Conclusions

199

Appendices

205

A Further details of likelihood distribution fits

205

9

B Obtaining total multiplicity distributions

209

C Further details of T2HK GLoBES simulation

211

C.1 Energy bins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

211

C.2 Channel systematic uncertainties . . . . . . . . . . . . . . . . . . . . . .

211

C.3 Comparison with published event rates . . . . . . . . . . . . . . . . . . .

212

D Exact expressions for LS sum rules

213

Bibliography

215

10

List of Figures
1.1

The Standard Model extended to include family symmetry. . . . . . .

31

1.2

Feynman diagrams for neutrino-electron scattering . . . . . . . . . . . .

33

1.3

Feynman diagrams for CCQE neutrino-nucleus scattering . . . . . . . .

34

1.4

Neutron capture time and gamma cascade energy for Gd and H . . . . .

37

2.1

Target station at J-PARC . . . . . . . . . . . . . . . . . . . . . . . . . .

41

2.2

Effect of off-axis angle on neutrino beam energy spectrum . . . . . . . .

42

2.3

Neutrino beam flux energy spectra at Hyper-K . . . . . . . . . . . . . .

43

2.4

The T2K near detectors . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

2.5

Unoscillated flux ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

2.6

The TITUS detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

2.7

MRD muon charge reconstruction efficiency . . . . . . . . . . . . . . . .

48

2.8

The positions of the E61 detector . . . . . . . . . . . . . . . . . . . . . .

50

2.9

Reconstructed appearance neutrino energy spectra at Hyper-K . . . . .

54

2.10 Reconstructed disappearance neutrino energy spectra at Hyper-K . . . .

55

2.11 Appearance neutrino energy spectra for various δ . . . . . . . . . . . . .

56

2.12 Octant and mass ordering sensitivity at Hyper-K . . . . . . . . . . . . .

58

3.1

Hough transform used for Cherenkov ring searching. . . . . . . . . . .

65

3.2

Spherical Hough transform bin partitioning schemes . . . . . . . . . . .

66

3.3

Flight of charged particle and emitted Cherenkov photon to PMT . . .

67

3.4

Cherenkov photon emission probability density functions . . . . . . . . .

70

3.5

Particle position, direction and energy resolutions in TITUS . . . . . . .

72

3.6

Particle identification for muon and electrons in TITUS . . . . . . . . .

73

3.7

TITUS reconstruction performance with towall and dwall cuts . . . . .

75

11

3.8

TITUS particle identification dependence on distance to tank wall . . .

76

3.9

TITUS νµ reconstruction performance with dwall and PID cuts . . . . .

77

3.10 TITUS νe reconstruction performance with dwall and PID cuts . . . . .

78

3.11 Vertex resolution against reconstructed lepton and neutrino energy . . .

79

3.12 Reconstructed energy distributions of various interaction types . . . . .

80

3.13 Effect of TITUS on the sensitivities of Hyper-K . . . . . . . . . . . . . .

84

4.1

Comparison of models of Gd neutron capture . . . . . . . . . . . . . . .

93

4.2

Effect of Gd neutron capture models on reconstruction . . . . . . . . . .

94

4.3

Time and PMT charge distributions for neutron captures in E61 . . . .

99

4.4

Distance and time relative of signal and background neutron captures .

100

4.5

Distributions used to construct signal neutron likelihood . . . . . . . . .

102

4.6

Distributions used to construct sand neutron likelihood . . . . . . . . .

103

4.7

Distributions used to construct NC event neutron likelihood . . . . . . .

104

4.8

Distributions used to construct cosmic neutron likelihood . . . . . . . .

104

4.9

Log-likelihood cut for distinguishing signal and sand neutrons . . . . . .

105

4.10 Effect of unfolding various bins of multiplicity distribution . . . . . . . .

111

4.11 Distance and angular distributions of neutron captures . . . . . . . . . .

112

4.12 Distance and angular 2D distributions of neutron captures . . . . . . . .

113

4.13 Difference between corrected and true capture distance and angle . . . .

114

5.1

Approximate ∆δ against δ from ν & ν̄ appearance in vacuum . . . . . .

123

5.2

Fluxes & oscillation probabilities at DUNE . . . . . . . . . . . . . . . .

126

5.3

Fluxes & oscillation probabilities at T2HK . . . . . . . . . . . . . . . . .

129

5.4

Fluxes & oscillation probabilities at T2HKK . . . . . . . . . . . . . . . .

130

5.5

DUNE & T2HK mass ordering sensitivity against δ . . . . . . . . . . . .

136

5.6

DUNE & T2HK mass ordering sensitivity against run time . . . . . . .

137

5.7

Effect of local minima on mass ordering sensitivities . . . . . . . . . . .

139

5.8

DUNE & T2HK sensitivities to discover CP violation. . . . . . . . . .

142

5.9

DUNE & T2HK sensitivities to exclude maximal CP violation . . . . . .

143

5.10 DUNE & T2HK sensitivities to the octant of θ23 . . . . . . . . . . . . .

145

5.11 DUNE & T2HK sensitivities to exclude maximal θ23 . . . . . . . . . . .

146

5.12 DUNE & T2HK precision on sin2 θ23 . . . . . . . . . . . . . . . . . . . .

148

12

5.13 DUNE & T2HK combined precision on sin2 θ23 & δ . . . . . . . . . . . .

149

5.14 DUNE & T2HK precision on δ . . . . . . . . . . . . . . . . . . . . . . .

150

5.15 DUNE & T2HK precision on δ for fixed S . . . . . . . . . . . . . . . . .

152

5.16 DUNE & T2HK precision on δ for fixed N . . . . . . . . . . . . . . . . .

154

5.17 DUNE & T2HK precision on δ with fixed run time . . . . . . . . . . . .

155

5.18 DUNE & T2HK precision on δ against run time . . . . . . . . . . . . . .

156

5.19 Effect of size of sysetematics on DUNE & T2HK precision on δ . . . . .

158

5.20 Effect of size of T2HK sysetematics on precision on δ . . . . . . . . . . .

159

5.21 DUNE & T2HK redesigns’ sensitivity to mass ordering . . . . . . . . . .

161

5.22 DUNE & T2HK redesigns’ CPV & MCP sensitivity against δ . . . . . .

163

5.23 DUNE & T2HK redesigns’ CPV & MCP sensitivity against time . . . .

164

5.24 DUNE & T2HK redesigns’ precision on δ . . . . . . . . . . . . . . . . .

166

5.25 Optimal combination’s mass ordering sensitivity . . . . . . . . . . . . .

168

5.26 Optimal combination’s CPV & MCP sensitivity . . . . . . . . . . . . . .

169

5.27 Optimal combination’s precision on δ . . . . . . . . . . . . . . . . . . . .

170

6.1

Predictions on θ13 , m2 and m3 of the LS models . . . . . . . . . . . . .

179

6.2

Allowed regions of ma and mb due to θ13 , ∆m221 and ∆m231 . . . . . . .

180

6.3

LS predictions of oscillation parameters with η fixed . . . . . . . . . . .

181

6.4

Allowed regions of ma and mb with η fixed. . . . . . . . . . . . . . . .

181

6.5

Allowed regions of ma /mb and η . . . . . . . . . . . . . . . . . . . . . .

182

6.6

Results of the fits to LS of the NuFIT neutrino oscillation data . . . . .

184

6.7

LS allowed values for oscillation parameters with η fixed . . . . . . . . .

185

6.8

LS allowed regions for oscillation parameters with η fixed . . . . . . . .

186

6.9

Sensitivity to exclude LS with fixed η from individual parameters . . . .

191

6.10 Sensitivity to exclude LSA from parameter combinations . . . . . . . . .

192

6.11 Sensitivity to exclude LSB from parameter combinations . . . . . . . . .

193

6.12 Sensitivity to exclude LSA from parameter combinations . . . . . . . . .

194

6.13 Sensitivity to exclude LSB from parameter combinations . . . . . . . . .

195

C.1 T2HK neutrino appearance reconstructed energy spectrum . . . . . . .

212

13

14

List of Tables
1.1

Three neutrino flavour oscillation parameter values . . . . . . . . . . . .

24

2.1

Expected appearance event rates at Hyper-K . . . . . . . . . . . . . . .

53

2.2

Expected disappearance event rates at Hyper-K . . . . . . . . . . . . . .

53

2.3

Systematic errors at Hyper-K . . . . . . . . . . . . . . . . . . . . . . . .

55

2.4

Sensitivity to the octant and mass ordering at Hyper-K . . . . . . . . .

57

2.5

Sensitivities to nucleon decay at Hyper-K . . . . . . . . . . . . . . . . .

59

3.1

TITUS reconstruction resolutions . . . . . . . . . . . . . . . . . . . . . .

81

3.2

TITUS single ring electron and muon selections . . . . . . . . . . . . . .

82

3.3

Effect of TITUS on the sensitivities of Hyper-K . . . . . . . . . . . . . .

85

3.4

TITUS neutron tagged selections . . . . . . . . . . . . . . . . . . . . . .

86

4.1

Activity levels of radioactive impurities simulated in E61 . . . . . . . . .

96

4.2

Definitions of parameters used to construct likelihoods . . . . . . . . . .

101

4.3

Construction of the signal and background likelihoods . . . . . . . . . .

103

4.4

Signal and background neutron capture selection at E61 . . . . . . . . .

106

4.5

Bins of distance and angle used in E61 neutron analysis . . . . . . . . .

110

5.1

Run times and ν : ν ratios for T2HK & DUNE . . . . . . . . . . . . . .

132

5.2

True oscillation values used in the simulations . . . . . . . . . . . . . . .

134

5.3

Effect of systematics on DUNE & T2HK precision on δ

. . . . . . . . .

156

6.1

Best fit values for the LS model parameters and predictions . . . . . . .

184

6.2

Precision on oscillation parameters of future reactor experiments . . . .

188

A.1 Fitted distributions neutron selection likelihoods . . . . . . . . . . . . .

207

A.2 Probability density functions used in fits . . . . . . . . . . . . . . . . . .

208

15

C.1 Systematic errors used for T2HK simulation . . . . . . . . . . . . . . . .

16

211

Introduction
In the two decades since the observation of neutrino oscillations gave the first measurements directly contradicting the Standard Model of particle physics [3] – less timethan it took to first observe the neutrino [4] after its proposal [5] – experiments haveprovided information on all six independent neutrino oscillation parameters [6]: Precision measurements of the size of every mixing angle and mass-squared difference havebeen made, with some angles of the CP -violating Dirac phase also now starting to be
excluded, leaving only the sign of the larger mass splitting, the octant of the atmosphericmixing angle and the precise value of the Dirac phase left undetermined. This has beenachieved through a diverse array of experiments observing solar neutrinos from nuclear
fusion reactions in the Sun [7–9], atmospheric neutrinos from interactions of cosmic rays
with nuclei in the upper atmosphere [3], reactor neutrinos from fission reactions at nuclear power stations [10] and accelerator neutrino beams from decaying muons producedin collisions of protons accelerated to high energies [11, 12].
Oscillations involving all three neutrino flavours have now been observed: solar andreactor electron neutrino disappearance, atmospheric and accelerator muon neutrino
disappearance and appearance into electron neutrinos [13, 14] and tau neutrinos [15],for neutrinos having travelled distances spanning eight orders of magnitude: reactor
neutrinos oscillating in a couple of kilometres [13] to solar neutrinos travelling 150 million
kilometers to Earth, with energies spanning four orders of magnitude: MeV scale solar
neutrinos up to several GeV accelerator neutrinos (or ten orders of magnitude when
including PeV scale cosmic neutrinos [16]).
In the three neutrino oscillation picture that has emerged, the few remaining missing
pieces are expected to be resolved with the next-generation accelerator neutrino projects [17,18]. In particular, for a precision measurement of the Dirac phase they provide
the only opportunity to improve on the hints to its value coming from the current neutrino beam experiments. In order to achieve this, an unprecedented quantity of neutrino
oscillation data will be needed to reduce statistical errors, through new detectors withlarger mass and higher power beams than are available today, as well as a corresponding reduction in systematic errors from improved detector capabilities and additional
measurements to constrain model uncertainties.
This thesis will investigate several aspects of the goals of the next-generation programmeof neutrino oscillation experiments, starting from the design of the Hyper-Kamiokande
experiment and how its performance can be optimised through an intermediate water
Cherenkov detector, then studying its performance in the context of running alongside
the DUNE experiment, and investigating the ability of these and other future exper-

17

iments to probe theoretical models of flavour introduced to explain the addition ofneutrino mass and mixing to the Standard Model.
The first two chapters provide a review of background material relevant to the remaining chapters: Chapter 1 gives an overview of neutrino physics covering the theoreticalgrounding, current state and future prospects of neutrino masses and mixing, as wellas the physics of the detection of neutrinos. Chapter 2 then details the design, physics
goals and sensitivities of the Hyper-Kamiokande project. The following two chaptersthen focus on two intermediate detector designs; Chapter 3 describes the developmentof the reconstruction software for the TITUS detector and its use in selecting event
samples to constrain systematic errors and Chapter 4 demonstrates a potential neutron
measurement at the E61 detector. Chapter 5 then provides a thorough analysis of thesensitivities of Hyper-Kamiokande and DUNE and the complementarity of their measurements. The ability of these and other experiments to potentially exclude the Littlest
Seesaw class of neutrino mass models is then investigated in Chapter 6. The thesis thencloses with some final concluding remarks and comments on future work and prospects.

18

Chapter 1

Neutrino Physics

19

Introduction
The history of the neutrino begins with Pauli’s proposal [5] of an uncharged spin- 12
particle to allow the continuous β-decay spectrum to be explained in a way which conserves energy and spin. While Pauli originally conceived of this particle as being a
massive particle contained within the nucleus, Chadwick’s discovery [19] of the neutron
two years later suggested a mass much heavier than Pauli’s requirement of “not larger
than 0.01 proton masses”. However, Fermi included Pauli’s much lighter neutral particle
as an essential component of his theory of the weak interaction [20, 21], predecessor to
our current understanding of the weak interaction, also originally developed to describe
β-decay; he named this particle the neutrino.
The first direct detection of neutrinos came more than two decades after Fermi’s theory, with the Cowan-Reines experiment [4] detecting the inverse β-decay interactions
predicted to occur when a neutrino interacts with a proton, producing a neutron andpositron. The particles detected in this experiment (and produced in beta decay) turned
out not to be simply neutrinos, but to be electron anti-neutrinos: With further understanding of the weak force, the development of electroweak unification as part of theStandard Model, the neutrino lost its mass but gained flavour and chirality [22, 23]; themodel now contained three distinct neutrino fields (νe , νµ , ντ ) all with left-handed (LH)chirality, along with three corresponding right-handed (RH) anti-neutrino fields (ν e , ν µ ,ν τ ), associated with the three flavours of charged leptons, electron (e− ), muon (µ− ) andtau(τ − ), and their antiparticles (e+ , µ+ , τ + ).
In the Standard Model, the weak force interacts only with the LH chiral componentof particles, and the RH component of anti-particles. The (anti)neutrinos are masslessparticles with LH (RH) chirality and no electric or colour charge, interacting only withthe weak force and gravity. They are partners of the massive, charged (anti)leptons,
which together form three LH electroweak doublets. The RH components of the charged
leptons (and LH components of the charged antileptons) do not interact with the weakforce and form electroweak singlets, with no RH components of the neutrinos or LH
components of the antineutrinos.
The Standard Model, with massless neutrinos, has no need to include RH neutrinos or
LH antineutrinos; all massive Dirac fermions must have both RH and LH components,
but this is not true of massless fermions. If RH neutrinos were to be included, theywould be required to form electroweak singlets (like the RH charged leptons) so would
not interact with the weak force, and being neutral would be unable to interact withthe other fundamental forces (except gravity), thus making it effectively impossible forthem to be directly detected.
The next section describes how this picture fails to account for observations of the nonconservation of neutrino flavour, and details some attempts to extend the Standard
Model to explain this phenomenon. Section 1.2 then describes the physics of the detection of neutrinos as applied to the detector designs described in the next chapter andused in the work of this thesis.

20

1.1

Neutrinos beyond the Standard Model

The Homestake experiment provided the first indication [24] that the Standard Model
neutrino did not have the properties of the neutrinos we observe; the experiment measured a flux of electron neutrinos, originating from the sun, that was significantly lower
than expected. The possibility of electron neutrinos changing into other flavours, in an
analogous way to the oscillations of neutral mesons, had already been suggested [25, 26]
and could explain the reduction of the observed flux due to the experiments inability todetect neutrino flavours other than νe .
However, this solution was not confirmed until further observations – the directional
measurement of the atmospheric νµ flux at Super-Kamiokande (Super-K) [3], and measurements of νe -only and total (all flavours) 8 B solar neutrino fluxes at Super-K [7] andthe Sudbury Neutrino Observatory (SNO) [8, 9] – conclusively confirmed that neutrinos
were changing flavour: Super-K found that the atmospheric νµ flux coming from above
was about twice that of the flux coming from below (where the neutrinos have had totravel a much greater distance). SNO compared their observed solar neutrino flux seenin charged-current νe interactions to the total solar neutrino flux observed through thedetection of interactions of all three neutrino flavours; a comparison was first made toelastic scattering interactions measured at Super-K (and later also at SNO), sensitive
mostly to νe but also with some sensitivity to other flavours, and the conclusive comparison was then made to neutral-current interactions on deuteron in SNO itself, sensitive
equally to all neutrino flavours. They found that the total flux matched the predictedsolar neutrino flux, without the deficit observed in measurements of the νe flux alone.
These observations suggested that both the atmospheric νµ and solar νe were changing
in flavour as they propagated from their source to the detector. However, the mechanism
for this change in flavour, through flavour oscillations, requires that the neutrino has
more than one mass state. Since the Standard Model neutrino does not – and cannot– have any non-zero mass (the reasons for which will be detailed in Section 1.1.3) thismeant that the observation of neutrino oscillations provided the first (and currently
only) clear and direct measurement of particle physics beyond the Standard Model.

1.1.1

Neutrino mixing and flavour oscillations

For three massive neutrino flavours, the flavour eigenstates need not coincide with themass eigenstates of the neutrino fields. In general, the three neutrino fields in theflavour basis να , α ∈ {e, µ, τ }, form linear combinations of the three fields in the mass
basis, νi with masses mi , i ∈ {1, 2, 3}, related by the 3 × 3 unitary matrix Uαi , known
as the PMNS matrix after Pontecorvo introduced neutrino oscillations [25] and Maki,
Nakagawa and Sakata introduced the 3 × 3 mixing matrix [26]:
να =

3
X

Uαi νi(1.1.1)i=1Equivalently, this relationship can be applied to the flavour and mass eigenstates asvectors of the fields:|να i =

3
X
i=1

21

∗
Uαi
|νi i

(1.1.2)

This matrix was originally introduced as a 2 × 2 matrix with two states; however wehave now observed three states, requiring a 3 × 3 matrix. Of the nine independent realparameters describing an arbitrary 3 × 3 unitary matrix, for the PMNS matrix threeof those can be absorbed as phases into the definition of the fermion fields, leaving six
physical parameters.
The matrix can then be parametrised in the standard factorisation [27] asc12 c13s12 c13α21iδc12 c23 − s12 s23 s13 eiδ ei 2
U = −s12 c23 − c12 s23 s13 e
s12 s23 − c12 c23 s13 eiδ
−c12 s23 − s12 c23 s13 eiδ

s13 e−iδ
s23 c13 
α31
c23 c13 ei 2


1
0
c13
0 s13 e−iδ
c12 s12 0
1
0
0
α




i 221
0
1
0  −s12 c12 0 0 e
= 0 c23 s23  
00 1
0 −s23 c23−s13 e−iδ 0
c13
0
0








0

0 ,


α31

ei 2
(1.1.3)where sij ≡ sin θij and cij ≡ cos θij , θij are the three mixing angles, δ is the Dirac CP
phase [28] and α21 and α31 are the Majorana phases, which are physical only if theneutrinos have a Majorana mass [29].
The evolution in time of a neutrino state vector |ν(t)i from time t = 0 to time t is (innatural units)|ν(t)i = e−iHt |ν(0)i .
(1.1.4)
The threemass eigenstates are eigenstates of the Hamiltonian, H, with eigenvaluesq2
Ei = p + m2i , where p is the neutrino momentum.
Then taking the plane wave solution and assuming the neutrino is ultrarelativistic som2
m2
that1 Ei ≈ p + 2pi ≈ p + 2Eiν , where Eν is the total energy of the neutrino, gives|νi (t)i =e−i(Ei t−p·x) |νi (0)i ,
m2i L
−i 2E

|νi (L)i =eν|νi (0)i ,(1.1.5)(1.1.6)

where L ≈ ct is the distance travelled by the neutrino. Then, for a neutrino of a specificflavour eigenstate |να (t)i, Equations (1.1.1) and (1.1.6) combine to give|να (L)i =

3
X

Lm2i
∗ −i 2Eν
Uαi e|νi (0)i(1.1.7)i=1

It should be noted here that while this standard derivation provides a demonstration of how the oscillation formula can be obtained, the approximations made are not entirely mathematically or physicallyconsistent; the correct formula is eventually reached only through some rather fortuitous cancellationof effects that this derivation overlooks but are seen in more rigorous treatments. More details on thisissue and its resolution using more complete derivations with wavepackets and/or a full field theoretical
approach can be found in [30, 31].
1

22

The probability of observing a neutrino of a given flavour β at a distance L from where
it was created as flavour α is thenPα→β (L) =|hνβ (L)|να (0)i|2=

3
X(1.1.8)Lm2 3−i 2E i Xhνi (L)| Uβi e

i=1

=

3
X

ν

0m2j
∗ −i 2Eν
|νj (0)iUαj e

2(1.1.9)j=1
Lm2 2−i 2E i∗νUαi Uβi e(1.1.10)i=1=δαβ − 4X∗∗
Re(UαiUβi Uαj Uβj) sin2

i>j

+2

X

∗
∗
Im(UαiUβi Uαj Uβj
) sin

i>j

L∆m2ij4Eν

L∆m2ij,2Eν

(1.1.11)

using hνi |νj i = δij and where ∆m2ij ≡ m2i − m2j and mi , i ∈ {1, 2, 3} are the three massstates. For neutrino oscillations to occur, it is therefore required that there is at least
one non-zero mass squared difference ∆m2ij in addition to non-zero mixing angles.
The Majorana phases appear only on the diagonal elements of the mixing matrix, whichin Equation (1.1.11) are always multiplied by their complex conjugate, and so they
cancel and are not observable in oscillation experiments. The Dirac phase δ, however,remains when α 6= β. The probability for antineutrinos is identical to that of neutrinos
given in Equation (1.1.11), except that the replacement U → U ∗ must me made, whichis equivalent to replacing δ → −δ as can be seen from Equation (1.1.3). This makes clear
how a non-zero Dirac phase violates CP symmetry by altering the oscillation probability
for RH antineutrinos compared to LH neutrinos, which must be identical if CP is to beconserved.
This gives a total of six independent free parameters describing the oscillations between
three neutrino (and anti-neutrino) flavours; three mixing angles θ12 , θ23 and θ13 , two
independent mass squared differences, usually ∆m221 , and either ∆m231 or ∆m232 , and theCP violating Dirac phase δ. Table 1.1 gives a summary of the experimentally determinedvalues of these parameters from a recent fit to global data [6].
The mixing angles θ12 , θ13 and θ23 are often referred to as the solar, reactor and atmospheric mixing angles respectively; all of these angles are now known to be non-zero [6].
The parameter which has been constrained the least is δ, which has only recently had
any angles excluded by combined fits of experimental data at the 3 σ level [32]. The
possibility of discovering CP violation in the lepton sector through a non-zero value ofsin δ was opened up after the discovery of a large θ13 , the last of the three angles to bemeasured, first hinted at by the T2K experiment [33] and then confirmed by the Daya
Bay experiment [34].
The prospect of CP violations in neutrinos is of particular interest due to the possibilityof it having a role in leptogenesis and the origin of the matter-antimatter asymmetryof the universe [35, 36]. Leptonic CP violation can arise through the three neutrino
23

Normal Ordering

Inverted Ordering

Parameter

best fit ±1 σ

3 σ range

best fit ±1 σ

3 σ rangesin2 θ12

0.307+0.013−0.0120.272 → 0.346

0.307+0.013−0.0120.272 → 0.346

sin2 θ23

0.538+0.033−0.0690.418 → 0.613

0.554+0.023−0.0330.435 → 0.616

1.981 → 2.436

2.227+0.074−0.0747.40+0.21−0.20+0.032-2.465−0.031
278+26−29

2.006 → 2.452

sin2 θ

13[10−2 ]

∆m221 [10−5 eV2 ]∆m23l [10−3 eV2 ]δ [°]2.206+0.075−0.0757.40+0.21−0.20+0.0332.494−0.031
234+43−31

6.80 → 8.02
2.399 → 2.593
144 → 374

6.80 → 8.02
-2.562 → -2.639
192 → 354

Table 1.1: Three neutrino flavour oscillation parameter values determined from a recent
fit to global data [6]. The large mass-squared difference, ∆m23l , refers to ∆m231 fornormal ordering or ∆m232 for inverted ordering.
mixing phases δ, α21 and α31 ; CP violating effects in neutrino oscillations, however, areonly dependant on δ due to the Majorana phases appearing only on the diagonal of themixing matrix. A large value of θ13 is also required for leptonic CP violation to be seenin oscillations as δ always appears in U with sin θ13 . In fact all three oscillation angles
are required to be non-zero to allow CP violating effects of δ.
The CP violating term, which appears in the probability of oscillation from να to νβ
for α 6= β, comes from the third term in Equation (1.1.11). Its effect on the difference
between oscillation probabilities for neutrinos and anti-neutrinos can be written in full
as
∗
∗
Uβ1
Uβ2 ) sinP (να → νβ ) − P (ν α → ν β ) =16 Im(Uα1 Uα2= ± 16J sin∆m221 L∆m232 L∆m231 L
sin
sin
2Eν
2Eν
2Eν
(1.1.12)∆m221 L∆m232 L∆m231 L
sin
sin
,
2Eν
2Eν
2Eν

(1.1.13)

where the positive (negative) sign is taken for (anti-)cyclic permutations of e, µ and τin α and β. J is the Jarlskog invariant [37], the invariant measure of CP violation givenby∗∗±J = Im(Uα1 Uα2
Uβ1
Uβ2 )

(1.1.14)= sin θ12 cos θ12 sin θ23 cos θ23 sin θ13 cos2 θ13 sin δ.

(1.1.15)This invariant is proportional to the sine of each of the three mixing angles, clearly
demonstrating that they must all be non-zero for CP violation to be possible throughneutrino oscillations. With the current best-fit values, J ' 0.035 sin δ, allowing forsignificant CP violation effects to be seen in oscillations; the size of these effects is givenby|P (να → νβ ) − P (ν α → ν β )| ' 0.56 sin δ sin
24∆m221 L∆m232 L∆m231 L
sin
sin
.
2Eν
2Eν
2Eν

(1.1.16)

The ordering of the sizes of the mass eigenstates is another aspect of neutrinos which
is not yet fully determined. Although the oscillation probability does not depend on
the sign of the mass squared differences, experiments can still probe the ordering of themasses through the identity ∆m221 + ∆m232 = ∆m231 . The effects of neutrino oscillations
in matter detailed below also give oscillation experiments sensitivity to the sign of thesemass splittings, potentially providing a more experimentally accessible way to determine
the signs of the mass-squared differences. While the sign of ∆m221 is known, the signof ∆m231 is still currently unknown; experimental data is consistent with both normal
ordering where m1 < m2 < m3 and inverted ordering where m3 < m1 < m2 . Determining which of these is the case is another major goal of current and future neutrino
oscillation experiments.
Whether the value of the mixing angle θ23 is greater or less than 45°, known as theoctant of θ23 , is also considered a important question for experimental neutrino physics
to answer. The reason for this is that the octant, along with the value of δ and the massordering, has implications for investigations into understanding the origin of neutrino
masses [38], which may in turn guide experimental and theoretical advances in particle
physics beyond the Standard Model.
Further details on the phenomenological aspects of neutrino oscillations will be coveredin Chapters 5 and 6.

1.1.2

Neutrino oscillations in matter

The discussion above assumed the neutrinos propagate through a vacuum, however thesituation is more complex for neutrinos passing through matter due to matter effects
introduced by Mikheyev, Smirnov and Wolfenstein [39, 40], also known as the MSW
effect. In fact, this effect is required to correctly explain the flavour changes seen insolar neutrinos at different energies [41].
The effect arises as electron neutrinos have different interaction cross-sections to muonor tau neutrinos when travelling through matter containing electrons. The dominant
contribution to the effect comes from elastic forward scattering νe + e → νe + e which
creates a difference in potentials for νe compared to νµ or ντ , given by [39]√Ve = ± 2GF ne ,(1.1.17)
where GF is the Fermi constant, ne is the electron density in the matter, and the positive
sign is taken for νe or negative sign for ν e . This potential modifies the Hamiltonian
from that of the vacuum, and therefore modifies how the neutrinos propagate in time,leading to a modification of the oscillation formula of Equation (1.1.11). In the vacuum,
Equation (1.1.4) can be rewritten asd|νi i =Hm |νi i = Ei |νi i ,dt(1.1.18)d|να i =Hf |να i = U Hm U † |να i ,dt(1.1.19)iiwhere Hm and Hf are the Hamiltonian in the mass and flavour bases respectively.
For particles propagating through matter, the Hamiltonian is modified as H0 → H 0 =25

H0 + H int , where H int accounts for neutrino-matter interactions and H0 is the vacuum
Hamiltonian. This leads to the new time-evolution of neutrino flavour states in matterdi |να i = Hf0 |να i = (U Hm U † + Hfint ) |να i ,(1.1.20)
dt
where Hfint is the modification to the Hamiltonian in the flavour basis and can simply
be given by√
2GF ne , 0, 0 ,(1.1.21)
Hfint = diag (Ve , Vµ , Vτ ) = diag
after removing the effect of neutral current interactions that all neutrino flavours undergo
and so do not affect oscillations.
This new form of the Hamiltonian will have different eigenvalues and eigenstates to thevacuum and so the propagating neutrino states no longer match either the flavour statesor the mass states, so determining the new oscillation probabilities requires solving theflavour evolution equation given in Equation (1.1.20). The full three-flavour solution iscomplicated and so approximations are usually used in practice, but in the two neutrinocase the oscillation formula in matter matches the two neutrino vacuum oscillation
formula with modified parameters [40]:∆m2m L
,4Eν

(1.1.22)sin2 2θ
,(cos 2θ − A)2 + sin2 2θ

(1.1.23)!

P (νe → ντ ) = sin 2θm sin
2

2

where
sin2 2θm =q(cos 2θ − A)2 + sin2 2θ,√
2 2Eν GF ne
A=,∆m2
and in the case of antineutrinos in matter A → −A.
∆m2m = ∆m2(1.1.24)(1.1.25)

It can now be seen that a positive value of A, which is proportional to the electron
density of the medium through which the neutrinos propagate, has two effects; boththe oscillation amplitude and oscillation length are increased, by increasing sin2 2θ anddecreasing ∆m2 . The oscillation amplitude is also seen to become equal to 1, even for
arbitrarily small vacuum oscillation angles, when the resonance condition A = cos 2θ ismet. For antineutrinos (or for a negative value of A), the effect is reversed, decreasing
the oscillation amplitude and oscillation length.
The only difference between the oscillation probabilities of neutrinos and antineutrinos,
other than from this matter effect, comes from CP violating terms involving the Diracphase δ. This means that when attempting to determine δ through measurements ofoscillations of neutrinos and antineutrinos, the matter effect can make this more difficult
by mimicking the CP -violating difference in oscillation probabilities. Note that thematter effect does not violate CP symmetry; the difference comes from the fact thateveryday matter contains electrons and not positrons2 .
However, the fact that everyday matter contains electrons and not positrons is itself also evidenceof CP -violation: that of the baryon asymmetry of the universe, which to be explained requires CP violation, which may well itself be largely due to the CP -violating effects of δ!
226

Since the sign of A depends on the sign of ∆m2 , the matter effect also provides sensitivity
to the ordering of the neutrino masses. This is particularly apparent in the case ofresonant matter effects; the resonance condition can only be met for neutrinos when ∆m2
is positive or for anti-neutrinos when ∆m2 is negative. This fact, with the observationof solar neutrino oscillations, requires that the sign of ∆m221 is positive [39, 40], andwill contribute to the sensitivity of future experiments to determine the mass ordering
through measuring the sign of ∆m232 , which will be covered in Chapter 5.

1.1.3

The origin of neutrino mass

While the phenomenological model of neutrino oscillations presented above has beenhighly successful in describing the observed flavour changes of neutrinos over a wide
variety of sources, energies and distances, it leaves the open question of where theneutrino mass originates. As mentioned previously, the Standard Model forbids neutrino
masses. Understanding the reasons for this will help identify the possible extensions of
the Standard Model to allow for neutrino masses.
The simplest attempt to include a neutrino mass could be to add a Dirac mass term to
the Standard Model Lagrangian,L = · · · − mD νL νR + h.c.,(1.1.26)

where mD gives the neutrino mass and h.c. refers to the Hermitian Conjugate term. As
with the other fermions, adding such terms breaks electroweak gauge invariance unlessgenerated through Yukawa interactions with the Higgs,
L = · · · + yD LHνR + h.c.,(1.1.27)

where yD is the Yukawa coupling constant, L the LH lepton doublet and H the Standard
Model Higgs doublet. The presence of the RH neutrino singlet νR in Equations (1.1.26)
and (1.1.27) clearly demonstrates that this mass term requires the RH neutrino’s existence, which is explicitly not included in the Standard Model.
As mentioned previously, these right-handed or sterile neutrinos, do not interact with the
weak force (since they are right-handed) or the strong or electromagnetic forces (since
they are uncharged). While there is no a priori reason why such particles cannot simply
be introduced to the Standard Model to allow Dirac mass terms for the neutrinos, thisoption is not usually considered satisfactory. The first reason for this is that it provides
no insight into why the neutrino masses (or, more fundamentally, their unitless Yukawa
coupling constants to the Higgs, yD ) must be so drastically small; it might be expectedthat they be of similar orders of magnitude to the other fermions whose masses are
generated through the same mechanism. A second reason comes from the fact that
there is an alternative type of mass available only to neutral particles: the Majorana
mass term.
A Majorana fermion is an alternative type of massive particle to the Dirac fermions of
the standard model, constructed such that they are their own antiparticle. This canonly be the case for a neutral particle, since particles and antiparticles have charges of
opposite signs; for a particle and its antiparticle to be the same, those charges must
therefore be zero. Since neutrinos are the only neutral fermions of the Standard Model,
they provide the only observed candidate for Majorana particles.
27

A mass term of a LH Majorana neutrino can be generated using only the LH neutrino
field, through a mass term of the form3
1
L = · · · − mL νL νLC + h.c.,2(1.1.28)

where mL now gives the neutrino mass and νLC = Cν TL is the charge conjugated LH
neutrino. Again this mass term, like the Dirac mass terms, breaks electroweak gauge
symmetry and so will need to be generated after symmetry breaking. However, thistime it is not possible to generate such a term using the Standard Model Higgs doublet:
Attempting to add a term like yM νL HνL fails to conserve weak isospin and hypercharge,
while using a product of Higgs doublets as in yW LHHL (the 5-dimensional Weinberg
operator [42]) introduces a non-renormalisable coupling yW . Only by introducing a new
Higgs triplet could a renormalisable Higgs interaction term be added to generate the
LH neutrino’s Majorana mass term.
On the other hand, returning to the above approach of introducing a RH neutrino fieldto allow a Dirac mass, its second potential issue can now be seen more clearly; the RH
neutrinos, being electroweak singlets, could also have a Majorana mass term,
1
C ν + h.c.,
L = · · · − MR νR
R
2(1.1.29)

where MR is the Majorana mass of the RH neutrino. With no symmetry or invariancepreventing such a term, it could be considered an extreme case of fine-tuning to set it
to zero.
So far, to generate the neutrino masses, the options available include introducing RH
neutrinos to generate a Dirac mass through Yukawa interaction with the Higgs, or generating a Majorana mass either by using the non-renormalisable Weinberg operator, byextending the Higgs sector of the Standard Model, or by finding some other mechanism.
The seesaw mechanism combines the first and last of these options to generate neutrino
masses in a way which resolves both of the issues of the Dirac mass approach describedabove. While there are many other approaches to generating neutrino masses and several versions of the seesaw mechanism itself (for a recent review, see [43]), here only thetype I seesaw mechanism will be considered.
If a RH neutrino is introduced, as in the Dirac neutrino case, but both the RH and
LH neutrinos are considered Majorana particles, then there are three potential massterms which could be generated: the two Majorana mass terms in Equations (1.1.28)
and (1.1.29) and the Dirac mass term in Equation (1.1.26) (plus its Hermitian conjugate). As explained previously, while there is nothing preventing the addition of the RH
neutrino’s Majorana mass term and the Dirac mass term can be generated after electroweak symmetry breaking through the Higgs Yukawa interactions, the LH neutrino’s
mass term is forbidden by electroweak gauge invariance and cannot be generated using
renormalisable interactions with the Standard Model Higgs. Instead, an effective Majorana mass of the LH neutrino can be generated in a different way, combining the twoother mass terms, while also providing a natural explanation for the small size of the
LH neutrino mass.
the factor of 12 here is required for the Majorana field ν = νL + νLC to satisfy the Dirac equation
when using the conventional Majorana neutrino kinetic term iνL γ µ ∂µ νL = 21 iνγ µ ∂µ ν.
328The neutrino mass sector of the Lagrangian, including Hermitian conjugates, then looks
likeC ν − m ν ν − m ν C ν C + h.c.
L = · · · − MR νR
R
D L R
D R L



= − νL

C
νR



0 mD
mD MR

!

νLC
νR

!

+ h.c.,(1.1.30)

where on the second line the terms have been collected into matrix form for convenienceand the zero in the upper left element of the matrix corresponds to the lack of Majorana
mass term for the LH neutrino.
The Dirac mass, mD , being generated after symmetry breaking from interactions with
the Higgs, is expected to be of similar size to the other fermions of the Standard Model.
The Majorana mass, MR , however, is not restricted to being generated only at theelectroweak symmetry breaking scale and so could be reasonably expected to be muchlarger. For example, it could generated through some mechanism that occurs at a higher
GUT scale, suggesting a higher value then the Dirac masses generated at the electroweak
unification scale.
Since Equation (1.1.30) contains terms in the off-diagonal elements of the mass matrix,
this implies that the basis of the chiral fields used up to now do not have a definite
mass, but are superpositions of mass states. To obtain these mass states – the physicalparticles we can observe – the chiral basis is changed to the mass basis by diagonalising
the mass matrix. The physical particles will then be given by superpositions of thechiral fields, with masses given by the eigenvalues of the mass matrix,
λ± =

MR ±

q

MR2 + 4m2D2

(1.1.31).

With the assumption that MR  mD , based on the arguments given above, the massstates found correspond to a heavy mostly right-handed particle with effective Majorana
mass λ+ ≈ MR , and a much lighter mostly left-handed particle with effective Majorana
mass λ− ≈ −m2D /MR . The lighter mass state can of course be identified with the massesof the nearly massless LH neutrinos that have been observed, while the heavy mass state
would be some new as yet unobserved particle.
The above derivation of the type I seesaw mechanism was completed for just a single
LH neutrino and single RH neutrino, however this could be applied using three RH
neutrinos to provide a mass to each of the LH neutrinos4 . In this case, the neutrinomass terms of the Lagrangian becomeC
L = · · · − νL νR

0 mD
mTD MR

!

νLC
νR

!

+ h.c.,(1.1.32)

which differs from Equation (1.1.30) as each νL and νR (or νL and νR ) term now corresponds to a column (or row) vector of chiral neutrino fields, and each element of theUsing just two RH neutrinos results in one massless and two massive neutrinos, which is possible
given that the absolute neutrino mass scale has not been measured, and using more than three RH
neutrinos is also possible.
429

matrix corresponds to a matrix itself with the element in the lower left now the transposeof the element in the upper right.
In the approximation used above, MR  mD (which now applies to the relative sizesof the elements of MR compared to those of mD ), the 3 × 3 mass matrix of the three
LH light effective Majorana neutrinos, mν , after applying the seesaw mechanism, isgiven by mν = −mD MR−1 mTD . Furthermore, since the diagonalisation of the full mass
matrix in Equation (1.1.32) links the chiral states to the mass states, the matrix used
to diagonalise mν can be identified as (approximately) the PMNS matrix introduced in
Section 1.1.1.
By introducing RH Majorana neutrinos to the existing LH neutrinos (now also specified
to be Majorana particles) the type I seesaw mechanism has provided a possible originof the neutrino masses, with a natural explanation of their much smaller size comparedother fermion masses, as well as a link to the neutrino oscillations and the misalignmentof the (chiral) flavour and mass eigenstates.

1.1.4

Family symmetry models

All fermions of the Standard Model have been found to come in families each containing
three flavours – three copies of varying mass but otherwise identical properties – but
the theoretical framework provides no explanation as to the hierarchical nature of theirmasses, either between the different families of fermions or between the different flavours
within each family, nor an explanation of the mixing patterns that occur between theflavours, nor even a reason for the number of flavours of each fermion always being three.
Elsewhere throughout the Standard Model, its structure – the eight gluons and three
colours of QCD, the pattern of mesons and baryons formed of quarks, the three W bosons
and the B boson of the electroweak interaction and their relation to the photon, W ±and Z 0 bosons, and the pattern of weak isospins, weak hypercharges and electric chargesof the fermions – is governed by the local gauge symmetries and the representations of
their corresponding symmetry groups.
The introduction of a flavour symmetry (or family symmetry) aims to extend this framework, building upon the existing symmetries of the Standard Model in an attempt tosolve the mystery of its flavour structure. This is depicted in Figure 1.1, showing aminimal extension of the Standard Model to include three RH neutrinos and a flavour
symmetry. While the formal details of the theoretical framework are beyond the scopeof this thesis5 , a basic overview of the technique, as applied to the models investigatedin Chapter 6, is presented here. This overview is based largely on the more detailed
presentation given in [45].
The family symmetry group GF added to the Standard Model acts on the generations offermions, with each family transforming as a triplet, restricting the choice of group GF
to one which admits triplet representations. Having the fermions transform as a tripletof the family symmetry by construction enforces the number of generations to be three.
The more ambitious goal than simply determining the number of generations, however,is to explain the pattern of masses and mixing of these generations.
5

See e.g. [38, 44] for reviews.

30

brR
srR

trL
tgL

crL

cbL

L

ug
L

dr

ub

L

dg

(3)

tL
νµL

b
cL
b

L

uL

db
L

b
dL

νeL

C

νe sL
L

ντ L
ντ L

b

bL

b

L

SU

b

ubL

U (1)YSU (2)L

tbL

ugL

ur

νµL
τL

νeL
µL

eL

sgR
bbR
g
r
b
bR
dR
sR
ur
b
R
tR ν3R
r
b
s
bgR b
d
R
ug R
cR bb ν2R
drR R
sg
ν3R
R
b R b
u
ν
g R
uR sb 1R
R
d
2
ν
R
R
dr
b
τR
R
ν1
bR
dbR
R b ν1R
dg
µ
R
R
sR
τR
b
db
e
R
R
dR
SU
µR
drR

cgL

urL

bgR

(3)

eL

C

Gf

eR

eR
Gf

Figure 1.1: An example of how a family symmetry can extend the symmetry structureof the LH and RH fermions of the Standard Model, here including three RH neutrinos.
Particles are represented by blocks, which are connected along a direction when thoseparticles form multiplets of the group indicated to represent that axis: Each family ofquarks and leptons forms a triplet of the family symmetry group Gf , the left-handed
quarks and leptons all form doublets of the gauge symmetry group SU (2)L , and boththe LH and RH quarks all form triplets of the gauge symmetry group SU (3)C . For theU (1)Y gauge symmetry, each set of connected blocks has a different weak hypercharge(and particles within these sets have the same weak hypercharge).
Since a family symmetry is not actually observed – we observe generations with different
masses and mixing and so they are not invariant under the family symmetry – we knowthat the symmetry must be broken. This can be done in an analogous way to thespontaneous symmetry breaking of the Higgs mechanism, using new Higgs-like particles
known as flavons, whose vacuum expectation values will end up determining the fermion
mass matrices after family symmetry breaking, and hence their masses and mixing
parameters.
The gauge fields do not interact with the flavons, ensuring that the part of the family
symmetry that we do observe is retained: the gauge (strong, weak, electromagnetic)
interactions remain identical for each generation.
Consider now the case relevant to this thesis – lepton mixing: The various flavon fields
can be arranged, through careful selection of the charges associated with the symmetriesof the model assigned to each particle, such that one set of flavon fields appear only in
Lagrangian terms responsible for the charged lepton masses, with others appearing in
the only terms for the neutrino masses. The vacuum expectation values (VEVs) of theseflavon fields are generally 3-dimensional vectors, since the flavons also form triplets of
Gf , and are therefore known as vacuum alignments.
In the direct approach, the group Gf splits into subgroups corresponding to the symmetry of the charged lepton and neutrino sectors: The VEVs of the flavons associatedwith generating the charged lepton masses are those that preserve the generators of
the subgroup whose symmetry determines the form of the charged lepton mass mat-31

rix. Similarly the VEVs of the flavons associated with the neutrino masses preserve thegenerators of the subgroup determining the form of the neutrino mass matrix.
For example, the discrete symmetry group S4 has generators S, U and T ; if the flavons
with VEVs that preserve T appear with the charged lepton mass terms in the Lagrangian,while those of S and U appear with the neutrino mass terms, then this results in a
diagonal charged lepton mass matrix, enforced by the symmetry group generated by
T , and tri-bimaximal mixing [46] is found for the neutrinos, enforced by the symmetrygroup generated by S and U . The patterns of mixing predicted by this approach, while
often fairly similar to the observed mixing pattern, have now mostly been experimentally
excluded by precision measurements such as the value of θ13 [38].
In the semi-direct approach, the residual symmetry of Gf includes some, but not all, of
its generators. This allows for models which lead only partially to the patterns foundin the direct approach.
In the indirect approach, none of the generators of Gf remain unbroken in the residual
symmetry. In this case, the VEVs of the flavons still determine the form of the resulting
charged lepton and neutrino mass matrices, but must themselves be determined throughsome alternative mechanism.
Both the indirect and semi-direct approaches have been used in separate models to
produce versions of the Littlest Seesaw forms of the neutrino mass matrix [47, 48], thephenomenology of which will be investigated in Chapter 6.

1.2

Detection of neutrino interactions

In order to better understand the properties of the neutrinos, to try to guide the buildingof models extending the Standard Model to account for their mass and explain theirflavour structure, a number of experimental measurements can be made. While attempts
are already underway to measure the neutrino masses directly [49] and to search forneutrinoless double β-decay [50], aiming to provide information directly on the nature ofneutrino masses, our current knowledge of neutrino properties comes almost exclusively
from measurements of the oscillation parameters [51]. This thesis will focus on thesemeasurements and in particular on the future measurements of the Hyper-Kamiokande(Hyper-K) experiment, whose main goals are to perform the first precision measurementof the Dirac phase, δ, and provide improved measurements of the mixing angles andmass-squared differences.
To measure the neutrino oscillation parameters, experiments aim to detect neutrinos
coming from a known source, where they are created with a known energy distribution
in a specific flavour state; most commonly these are electron neutrinos produced in thesun, electron anti-neutrinos produced in nuclear reactors, muon neutrinos produced in
the atmosphere, and muon neutrinos or anti-neutrinos produced by accelerators. By
measuring the rates of neutrino interactions of each flavour at some distance from thesource, the oscillation probabilities can be determined and the values of the oscillation
parameters can be constrained.
The neutrinos themselves are detected via the products of their interactions with matterin the detectors. The systematic errors on the neutrino fluxes, determined either from32

theoretical calculation or measured interaction rates, will depend heavily on our understanding of these interactions and their cross sections. To make precision measurementsof oscillation parameters it is therefore important to understand these interactions as
thoroughly as possible. As seen in Section 1.1.2, when neutrinos pass through matter
additional terms to the oscillation probability are introduced by the MSW mechanism,
so understanding these interactions is also important to understand this effect. Since
the sign of the additional terms is different for neutrinos and for anti-neutrinos, theeffect could mimic the CP -violating consequence of a non-zero sin δ. The sign of theadditional terms also depends on the mass ordering, and so sensitivity to δ will also be
improved if the mass ordering is known.

1.2.1

Neutrino interactions with matter

The weak interactions between neutrinos and matter have very small cross-sections
and so occur very rarely, as compared to those of the strong nuclear force and theelectromagnetic force. Neutrino interactions seen in detectors generally occur between
a neutrino and either an electron or a nucleus of the target material. Lower energy solar
neutrinos are often seen through the former, through elastic scattering (ES) of neutrinos
on electrons via exchange of either a Z 0 boson in neutral current (NC) interactions ora W ± Boson in charged current (CC) interactions. NC ES interactions occur for all
neutrino flavours, however the flavours of the interacting neutrinos are experimentallyindistinguishable as only the outgoing electron is visible. These interactions are shownin Figure 1.2. CC scattering off an electron can also occur for all flavours, so long as theenergy is high enough to produce the charged appropriate charged lepton. At the higher
energies, of atmospheric or beam neutrinos, interactions on nuclei dominate, separated
again into CC interactions via exchange of a W ± boson, with an outgoing charge lepton,or NC interactions via a Z 0 boson, which have no outgoing charged lepton, making themmore challenging to detect.
νl

νl

νe

νµ

W±

Z0
e−

e−

e−

W±
νe

e−

µ−

e−

νe

Figure 1.2: Feynman diagrams for neutrino-electron scattering. Neutral current elastic
scattering (left) can occur for all neutrino flavours, with the flavour of the incoming neutrino experimentally indistinguishable. Charged current elastic scattering for νe (middle)
and inverse muon decay for νµ (right) produce electrons and muons respectively, which
can be distinguished in experiment. Similar interactions occur for antineutrinos, with
the negatively charged leptons in the CC case being replaced with positively charged
antileptons. S-channel diagrams with W ± propagator are also allowed for electron scattering of ν e , however at low neutrino energies this is suppressed by the mass of theW ±.
Accelerator neutrino beam experiments most commonly use the CC quasielastic (CCQE)
channel, as shown in Figure 1.3, νl + n → l− + p for neutrinos and ν l + p → l+ + n
33

for antineutrinos, which is the interaction mode with the highest cross section below
1 GeV. These CCQE interactions are the dominant neutrino interactions of interestfor the long-baseline accelerator neutrino experiments covered throughout this thesis.
This interaction mode is useful as the neutrino energy can be reconstructed from thekinematics of the observed outgoing lepton:
Erec =mN El − m2l /2mN − El + pl cos θ(1.2.1)

where mN is the mass of the hit nucleon, El , ml and pl are the energy, the mass andthe magnitude of the momentum of the charged lepton, and θ is the angle between theincoming neutrino beam and the outgoing charged lepton. An important considerationof these interactions is that due to the chiral nature of the weak force, when calculating
the cross-sections for ν + n → l− + p and for ν + p → l+ + n, the cross-section forneutrinos is three times larger than that of antineutrinos, at first order [52].
νl

l−

νl

W±
d
n d
u

l+
W±

u
d p
u

u
p d
u

d
d n
u

Figure 1.3: Feynman diagrams for CCQE neutrino-nucleus and antineutrino-nucleus
scattering. Neutrinos interact with a neutron in the nucleus converting it into a proton(left), while antineutrinos convert a proton in the nucleus into a neutron (right).
Close to the peak energy for CCQE interactions, inelastic resonance production starts
to become significant. These interactions are where the hit nucleon is excited into a
resonance which then decays, for example a ∆ resonance may be produced which thendecays into a neutron or proton and a pion. These may be difficult to distinguishfrom CCQE interactions if, for example, the outgoing pion is not observed. At higher
energies deep inelastic scattering (DIS) dominates, where the neutrino scatters with aquark within a nucleon, fragmenting the nucleon into any of a wide range of possible
final states.
While the neutrino energy reconstruction method of Equation (1.2.1) gives a reasonableapproximation in the region dominated by CCQE interactions, there is the implicit
assumption that the hit nucleon is initially at rest, which is generally not the case.
Furthermore, non-CCQE interactions, such as resonance production described above, orinteractions involving a pair of correlated nucleons rather than a single nucleon within
the target nucleus, can appear as CCQE interactions, especially when only the outgoing
lepton is detected. Since the oscillation probability is dependent on the neutrino energy,
accurately reconstructing this quantity is essential and so these nuclear effects need to
be understood. If a detector can identify outgoing nucleons in addition to the charged
lepton then this could provide useful information to help understand the hadronic systemand the cross-sections of the different interaction modes.
341.2.2

Water Cherenkov detectors

One of the most successful methods of detecting neutrino interactions is from the Cherenkov light emitted by the charged lepton produced in CC interactions (or scattered in
ES interactions). By using target material such as water, which is cheap, transparent,and has a high enough refractive index for the Cherenkov effect (i.e. charged particles
at the energy produced by the neutrino interactions of interest meet the Cherenkov
threshold energy determined by this refractive index), the water Cherenkov techniqueis well suited to the large detector sizes required to see a significant number of neutrino
events.
The Cherenkov effect [53] occurs when a charged particle travelling through a dielectricmaterial is moving faster than the speed of light in that material [54]. This light isemitted at the Cherenkov angle, θC , given bycos θC =1βn()(1.2.2)where βc is the particle’s velocity and n is the material’s refractive index which is afunction of the emitted photon’s energy .
In a dispersive material such as water, the refractive index increases with photon energy,
and beyond a certain energy these photons will be reabsorbed into the material, giving amaximum angle at which the light is emitted which depends on the particle velocity [55].
This results in a cone of light that propagates through the water, emanating from theparticle track, with the opening angle of the cone decreasing as the particle loses energy.
Below the threshold of β = 1/n the particle will not emit Cherenkov light. A particleof charge ±1e will produce N photons along its track x at a rate given by [51]
d2 N
α
α
1
=
sin2 θC =1− 2 2,ddx~c~c
β n ()

(1.2.3)where α is the fine structure constant.
A water Cherenkov detector consists of a large tank of water, surrounded by photomultiplier tubes (PMTs) which detect the Cherenkov radiation. A ring of hits on the PMTs
indicates a cone of Cherenkov light, where the pattern, timing and total number of photoelectrons observed in the PMTs, as a result of photons emitted as Cherenkov radiation,
is used to determine the particle’s position, direction and energy.
The type of particle can also be determined, often with very high accuracy dependent
on the particle’s energy, by the type of ring produced: A muon will produce a sharpcone of Cherenkov light, with an opening angle which closes as the muon travels throughthe detector and loses energy. The muon will also usually decay producing an electronwith a second ring at a later time. An electron, however, will continually interact in thewater, producing an electromagnetic shower containing electrons with slightly differing
directions. These electrons will each produce cones, but due to their directional variation
they will not entirely overlap, and so the ring seen by the detector will be distinctly lesssharp than that produced by a muon.

35

1.2.3

Neutron capture and Gadolinium doping

While water Cherenkov detectors’ performance in identifying charged leptons is excellent, they are unable to distinguish the charge and often do not have the ability todetect the nucleons in the final state. This results in a complete inability to distinguishneutrino and antineutrino events, as well as limited capability in separating out some
non-CCQE interactions. The detection of neutrons through the few-MeV γs released
after the neutron captures on a nucleus provides an additional signal for water Cherenkov detectors beyond just the outgoing lepton. Such measurements could potentiallyprovide a number of benefits:• The statistical separation of neutrino and anti-neutrino events which are otherwise
indistinguishable in a water Cherenkov detector.
• A significantly enhanced ability to detect inverse β decay interactions from low
energy ν interactions such as those from the diffuse supernova neutrino background(supernova relic neutrinos).
• Additional information to distinguish the interaction modes of higher energy neutrino interactions.
• Additional information to help improve the neutrino energy reconstruction.
• Observations of the number, energies and directions of neutrons produced in theseinteractions to help improve models for the neutrino interaction cross-sections andsubsequent final state interactions within the nucleus.
The use of the Gadolinium added to the water target through a Gadolinium salt allows
high efficiency tagging of neutrons produced in neutrino interaction events. Neutrons
produced from neutrino interactions have in the past been detected from their capture on
Hydrogen, however a neutron tagging efficiency of only 18 % has been achieved [56] due tothe relatively small amount of visible energy produced, 2.2 MeV, and long capture time ofaround 200 µs. Neutron capture on Gd has much higher cross-section of around 48 000 b(compared to 0.33 b on H) [57], leading to more observable neutron captures within
the detector, with a shorter capture time of around 20 µs, with a cascade of photons
produced with total energy of about 8 MeV, providing 4 MeV to 5 MeV of visible energythat can be detected with much greater efficiency [58]. Figure 1.4 compares the capture
time distributions on Hydrogen and on Gadolinium, as well as the energy released after
a capture on Gadolinium.
For an oscillation analysis this can provide purer samples of CCQE interaction events
for when a neutrino is interacting (initially producing no neutrons) or when an antineutrino is interacting (initially producing 1 neutron). The number of neutrons thatwould actually be observed then depends on interactions of these initial neutrons (andalso protons) within the nucleus, as well as the possibility of nucleons scattering withother nuclei after exiting the initial nucleus, potentially causing additional neutrons toescape. This neutron tagging also enhances the purity of samples with respect to theinteraction mode, due to the different neutron multiplicities for non-CCQE interactions,allowing for measurements of cross-sections for interactions as a function of the numbers
neutrons in the final state.

36

Number of events / 10 µs

900
800
700
600
500
400
300
200
100

10

Number of Events

Number of Delayed Signals

100 200 300 400 500 600 700 800∆ T (µs)Data2

τ = 21.2 ± 6.1 µs

35

E = 4.3 ±0.1 MeV30

Bar: Data
Hatched: MC

25
20
15

10

10
5
1

0

100

200

300

0

400
500
∆T [µs]

0

1

2

3

4

5

6

7 8 9 10
Energy [MeV]

Figure 1.4: Time for delayed thermal neutron capture signal on Hydrogen (top, datain black with expected background rate in red) is around 200 µs while on Gadolinium(bottom left) is around 20 µs. The capture signal on Gadolinium consists of an 8 MeV
gamma cascade, with 4 MeV to 5 MeV visible energy (bottom right), compared to thebarely detectable single 2.2 MeV gamma for capture on Hydrogen.

37

R&D to monitor the feasibility and response of the detector when the water is dopedwith Gadolinium has been ongoing through the EGADS project [59, 60], with Super-K
now undergoing refurbishment work in preparation for being filled with Gd-doped water
for the SK-Gd project [61, 62].

38

Chapter 2

The Hyper-Kamiokande
Experiment39

Introduction
The Hyper-Kamiokande (Hyper-K) project [17] is a proposed next generation long
baseline neutrino oscillation experiment consisting of the Hyper-Kamiokande detector– a new megatonne-scale water Cherenkov detector to be constructed near the existing
Super-K detector in Kamioka – together with the megawatt neutrino beam, produced at
the J-PARC synchrotron 295 km away in Tokai and directed 2.5° off-axis to the Hyper-K
detector. In addition to the far detector, a suite of near detectors (ND), located 280 m
from the beam source, and an intermediate water Cherenkov detector (IWCD), 1 km to
2 km from the beam source, will be used to constrain the neutrino beam and interaction
cross-section systematics.
The primary aim of the experiment is to study CP violation in the lepton sector bymeasuring the oscillation probabilities for neutrinos and antineutrinos to determine theoscillation phase δ. Additional physics at Hyper-K will include increasing the precision
on measurements of other neutrino oscillation parameters, probing the proton life-timeat an order of magnitude beyond the current limit [63, 64], and studying atmospheric
neutrinos and neutrinos from astrophysical sources in addition to those produced in the
J-PARC beam.
This chapter provides an overview of these physics goals, as well as the detector design ofHyper-K. Sections 2.1 to 2.4 detail the experiment’s technical design, with the neutrino
beam covered in Section 2.1, the far detector in Section 2.2, near detectors in Section 2.3,
and intermediate detector in Section 2.4. Section 2.5 then discusses in more detailthe physics goals and projected sensitivities of the experiment and how these will beachieved.

2.1

The J-PARC neutrino beam

The neutrino beam produced at J-PARC (Japan Proton Accelerator Research Complex) has been in use for the current T2K neutrino oscillation experiment [65] and is
expected to continue operation for use with Hyper-K. The facility consists of a linear
accelerator which accelerates protons to 400 MeV before they are injected into a rapid
cycling synchrotron. Here the protons are accelerated to 3 GeV, after which they enter
the main ring synchrotron where they are further accelerated up to 30 GeV. They arethen fed through to the target station, shown in Figure 2.1 where they collide with agraphite target producing pions, amongst other hadrons. The charged pions are thenfocused using three magnetic horns and enter a decay volume where the neutrino beam
is formed from the neutrinos produced as the pions decay.
The magnetic horns focusing the pions are capable of running with a forward horncurrent to focus positively charged pions that decay producing neutrinos, or reverse
horn current to focus negatively charged pions that decay producing antineutrinos. This
allows the production of a beam consisting mostly of either neutrinos or antineutrinos,
so that the oscillation probabilities for neutrinos and antineutrinos can be separately
measured for the determination of δ. A small contamination (less than 1 % of theneutrino flux) of νe or ν e in the beam and ν µ (νµ ) in the νµ (ν µ ) beam result from thedecays of kaons produced and focused alongside the numerous pions; these more massive
40

Beam dump
Target station
Muon monitor

(2)

(4)

(5)

(6)

(1) (3)(1) Beam window(2) Baffle(3) OTR(4) Target andfirst horn(5) Second horn(6) Third horn

Figure 2.1: Target station at J-PARC [65].
kaons have larger branching ratios to decay modes that include undesired neutrino
flavours. These neutrino flavours can also enter the beam from the decay of the µ+ (µ− )produced in the pion decay, however the majority of these µ± are stopped after reaching
the end of the decay volume before decaying, minimising this background component of
the beam. The antineutrino beam in particular has a larger background contamination(and smaller signal flux) than the neutrino beam, due to the (positively charged) proton
beam naturally producing more positively charged pions than negatively charged pions.
The beam power has recently been upgraded from 240 kW to 480 kW, in the process ofreaching the design power of 750 kW, with the potential for further upgrades to 1.3 MW
planned [66, 67].
The neutrino energy spectrum of the beam is dependant on the angle from the centreof the beam axis to the point where the neutrino is observed; at larger angles, theenergy distribution has a narrower peak, with the peak located at lower energies. For
water Cherenkov detectors, it is desirable to have fewer higher energy neutrinos where
deep inelastic scattering becomes more common, as opposed to the lower energy CCQE
interactions which give a cleaner signal in the detector, making it easier to reconstruct
the incoming neutrino energy. At 2.5°, the distribution is peaked around 600 MeV,
close to where CCQE interactions are most dominant. This is leveraged by the T2K
experiment, directing the beam such that Super-Kamiokande, the far detector, is located
2.5° off-axis at the distance that coincides with the first oscillation maximum at thisenergy, and the same technique will be used for Hyper-K. Figure 2.2 demonstrates thistechnique as used in the T2K experiment. The final energy spectra of the fluxes of eachneutrino flavour in both the neutrino and antineutrino enhanced beams at Hyper-K areshown in Figure 2.3.

41

P(νµ→ νe)

P(νµ→ νµ)1
sin22θ23 = 1.0
sin22θ13 = 0.12∆m32 = 2.4 × 10-3 eV2

0.5

0.10

1

2

1

2

NH, δCP = 0
NH, δCP = π/2

IH, δCP = 0
IH, δCP = π/2

3

0.05

Φ295km
(A.U.)
νµ0
1

3

OA 0.0°OA 2.0°OA 2.5°

0.5

0
0

1

Eν (GeV)2

3

Figure 2.2: The muon neutrino (top) and electron neutrino (middle) survival probabilities with the unoscillated neutrino flux at different off-axis angles (bottom) for the T2K
experiment [68].

42Flux [/(50 MeV⋅ cm2⋅ 1e21 POT)]

Hyper-K Flux for Neutrino Mode

νµ
νµ
νe
νe

106
105
104
103
102
0

1

2

3

4

5

6

7

8

9 10
Eν [GeV]

Flux [/(50 MeV⋅ cm2⋅ 1e21 POT)]

Hyper-K Flux for Antineutrino Mode

νµ
νµ
νe
νe

106
105
104
103
102
0

1

2

3

4

5

6

7

8

9 10
Eν [GeV]Figure 2.3: The (anti)neutrino flux energy spectra for the neutrino enhanced beam (top)and antineutrino enhanced beam (bottom) at Hyper-K [17].

43

2.2

The Hyper-Kamiokande far detectors

The Hyper-K detector itself is a water Cherenkov detector planned to be constructed in
the Mozumi mine at a depth of 650 m, near to the existing Super-Kamiokande detector
in Kamioka, 295 km west of the J-PARC accelerator complex. The baseline design forthe detector consists of two separate cylindrical tanks 60 m tall and 74 m diameter, with
one tank planned for the initial phase and the second tank planned to start taking data
six years after the first.
Each tank will be filled with pure water, circulated through a purification system keeping
Radon concentration below 1 mBq m−3 in order to minimise low-energy background fromradioactive decays. The tanks will be divided into two optically separated regions; theinner detector (ID) and outer detector (OD). The ID of each tank will comprise all
but the outermost 1 m of the tank around the barrel and 2 m at the top and bottom,
giving a total volume of 258 kt of water, and will be surrounded by 40 000 PMTs eachof 50 cm diameter, resulting in a photocoverage (proportion of the surface covered by
PMTs) of 40 %. The fiducial volume of each tank – the volume in which signal events
are selected (a cylinder 2 m inside of the PMTs on all sides of the ID) – contains 187 kt
of water. While the development of the PMTs to be used is still under development,
the design goal is for each PMT to have a single photon detection efficiency of 24 % andtiming resolution of 1 ns. The OD region contains 6700 20 cm outward facing PMTs andits purpose is mainly to reject the incoming background from cosmic-ray muons. The
possibility of adding Gadolinium to the detector for the neutron capture signal discussed
in Section 1.2.3 is also being considered.
Another possibility being investigated is placing the second tank in Korea [69], with just
one tank built in Mozumi. At the baselines of 1000 km to 1300 km being considered, thiswould place the tank close to the second oscillation maximum, making it the first longbaseline experiment to have a detector at this physically interesting point. Doing so is
expected to increase sensitivity in a number of areas; in particular, the increased baseline
would result in an increased matter effect, which would allow for improved sensitivity
to the mass ordering as well as to the measurement of δ. The phenomenology of thisand a detailed analysis of the effect on various sensitivities is presented in Chapter 5.

2.3

Near detectors for Hyper-K

A suite of detectors, shown in Figure 2.4 located 280 m downstream of the J-PARC
neutrino beam target have been and continue to be used by the T2K experiment [65]and it is expected that Hyper-K will continue to use these in an upgraded form. The
detectors consist of an on-axis detector, INGRID, and 2.5° off-axis detector, ND280.
INGRID consists of 16 modules, arranged in a cross pattern centred on the beam axis,
to measure the neutrino interaction rate at each module’s location with the primary
purpose of precisely constraining the beam direction. ND280 is a magnetised detector
consisting of a π 0 detector, three time-projection chambers and two fine grain scintillator
detectors, surrounded by electromagnetic calorimeters and a side muon range detector
built within the coils of the magnet. Together, these are capable of measuring the charge
and momentum of the leptons produced in CC interactions and the protons and pions

44

produced in both CC and NC interactions, with the primary purpose of constraining theneutrino flux, neutrino interaction cross-sections, and beam contamination of neutrinosof the wrong sign and wrong flavour. The ND280 detector is expected to be upgradedfor future runs of the T2K experiment [70] and this upgraded version will be used with
Hyper-K, with possible further upgrades made if required [17].

Figure 2.4: The ND280 off-axis near detectors (left) and INGRID on-axis near detector(right) of the T2K experiment [65].

2.4Intermediate Water Cherenkov Detector

In order to reach the low level of systematic uncertainties required for Hyper-K’s measurements, the addition of an intermediate water Cherenkov detector is planned [17].
While the near detectors of ND280 are able to accurately distinguish neutrino andantineutrinos through the outgoing lepton’s charge, as well as detect particles below
Cherenkov threshold, the intermediate detector’s 4π acceptance, water target, and water Cherenkov detection method better match the far detector. This makes the IWCD’s
measurements directly applicable in constraining uncertainties at Hyper-K in a way thatwould be far more challenging if only using the existing near detectors.
The key measurements for the IWCD are the (mostly unoscillated) flux × cross-section
for both νe and νµ (as well as ν e and ν µ ), since these dominate the uncertainties on
oscillation measurements at Hyper-K. In order to achieve this, the detector’s size needs
to be large enough to contain the majority of muons so that their momentum can be
determined, with its location then needing to be far enough from the beam source thatthe flux is low enough that pile-up is kept minimal.
An additional feature of the IWCD design is to dope the tank’s water with Gadolinium,
to allow high-efficiency detection of neutron capture as detailed in Section 1.2.3. This
would allow possible outgoing neutrons in both CC and NC interactions to be detected
when captured on the Gadolinium, allowing a large, high-efficiency and high-purity
sample of neutron captures to be observed. Such a measurement can then be used tobetter understand the neutron capture signal in the far detector, providing the benefits
described in Section 1.2.3, both for its uses in beam neutrinos as well as atmospheric
neutrinos. The potential for such a measurement will be analysed in detail in Chapter 4.
45

The IWCD can also be used for other physics purposes including detection of supernovae
neutrinos, sterile neutrino studies and understanding the background for proton decay
searches.
The design of the IWCD has taken several forms over the past few years; two of these– TITUS and E61 – have been studied in the work of this thesis.

2.4.1

The TITUS detector

TITUS (Tokai Intermediate Tank for the Unoscillated Spectrum) [71] was a proposed
design for the intermediate water Cherenkov detector, which has now merged with thealternative νPRISM [72] design to form the E61 detector design discussed in the next
subsection. The detector’s location is approximately 2 km from the J-PARC neutrino
beam, at the same 2.5° angle off axis from the beam as the far detector. TITUS is acylindrical water Cherenkov detector, containing approximately 2 kt of water doped witha 0.2 % concentration of Gadolinium salt, leading to a 0.1 % concentration of Gadolinium.
The detector is also partially surrounded with a magnetised iron muon range detector(MRD) located at the downstream part of the tank, to detect muons exiting that regionof the tank.
Figure 2.5 shows the ratio of the flux at different baselines for TITUS and the flux
at Hyper-K, where it can be seen that the flux at ∼ 2 km is very similar to that at
Hyper-K. Due to being located much closer to the beam source, where the beam is notwell approximated as a point source like it is at the distance of Hyper-K and TITUS,
the ND280 detectors do not measure a flux as similar to that expected at Hyper-K.
Additionally, the water target of TITUS will help reduce systematic errors that are
difficult to constrain using ND280 due to the differing nuclei of neutrino interaction
targets at ND280 compared to Hyper-K.
The design of TITUS has been under continual development, with the baseline design,
shown in Figure 2.6 used for the work of this thesis: The 23 m long, 13 m diameter
cylindrical tank, aligned with the axis of the beam, consists of a 22 m by 11 m innerdetector surrounded by a 1 m outer detector on all but the downstream end of the tank.
The downstream end is instead adjacent to the 2 m thick MRD, which covers this entire
end of the tank with the additional possibility of a 7 m by 4 m side-MRD covering a
portion of the barrel of the tank. The ID is surrounded by a 30 % photocoverage of 1200
PMTs. An alternative possibility is the use of multi-PMT (mPMT) modules, similar
to those used by KM3NeT [73], allowing for finer granularity and additional directional
information to aid in reconstruction.
Due to the size of TITUS, around 18 % of the muons coming from beam neutrino interactions escape the tank. These muons come from neutrinos in the higher end of thespectrum. It is therefore important to quantify their energy after they ranged out ofthe detector to help in understanding the high energy background. A Magnetised Muon
Range Detector (MRD) with magnetic field of 1.5 T can provide energy and charge reconstruction. Figure 2.7 shows the charge reconstruction efficiency dependent on neutrinoenergy. Combined with the neutron tagging this could give very high purity samples aswell as providing a method for validating and calibrating the neutron tagging.

46

Φ ND RND / Φ HK R2HK

2

1.35
1.3

1.25
1.2
1.15
1.1
1.05

+320 kA νµ
1000m
1838m
2036m

1
0.95

Φ ND RND / Φ HK R2HK

0.9
10− 1

1

10

Eν (GeV)

1.35

2

1.3

1.25
1.2
1.15
1.1
1.05

-320 kA antiνµ
1000m
1838m
2036m

1
0.95
0.9
10− 1

1

10

Eν (GeV)Figure 2.5: Unoscillated flux ratios (Near Detector / Nominal Hyper-K) at baselinesof 1000 m, 1828 m, and 2036 m, for νµ with neutrino enhanced beam (top) and ν̄µ withantineutrino enhanced beam (bottom) [71].

47Figure 2.6: The TITUS detector [71].

Figure 2.7: MRD charge reconstruction efficiency for muons coming from the interactionof the neutrinos within the tank [71].

48Chapter 3 covers the development of the reconstruction software used for this detector
and the results of its use in producing neutron-tagged measurements of the unoscillated
flux for Hyper-K.

2.4.2

The E61 detector

The E61 experiment [17] is the result of the merging of the TITUS and νPRISM [72] collaborations, combining features of both previous detector designs into a unified approachto the IWCD of Hyper-K.
Like the TITUS detector design, the E61 design is planned to have Gadolinium doping
for the detection of neutrons through neutron capture. However, sensitivity studies with
the IWCD designs found that the νe cross section was the key measurement, being morecrucial to measurements at Hyper-K than the νµ cross-sections. For this reason, theE61 detector does not include the muon range detector, instead focusing on features ofthe νPRISM detector designed to optimise the νe cross-section measurements.
The novel feature of the νPRISM detector, included also in the E61 detector, is to span arange of off-axis angles, allowing measurements of beam neutrinos over a range of energy
spectra. A linear combination of measurements at different off-axis angles can then be
combined to effectively produce measurements for a desired energy spectrum, such as
the expected flux observed at the far detector, or a gaussian shaped flux centred aroundsome energy of interest. The ability to detect interactions with a tuneable flux also
provides additional information to use in developing cross-section models. Additionally,at higher off-axis angles, the νe proportion of the flux relative to νµ increases, allowing
for a more precise measurement of the ratio of νe to νµ cross-sections – a significant
source of uncertainty for νµ → νe appearance measurements.
The current design for phase-1 of the detector, used in the work in Chapter 4, is for acylindrical tank, vertically oriented, with the ability to be raised and lowered using anelevator in order to cover a range of off-axis angles spanning 1° to 4°, located 1 km fromthe beam source. The detector itself consists of an inner detector of height 10.42 m andradius 3.71 m, surrounded by 832 multi-PMT modules, each containing 19 inward facing
300 PMTs, and an outer detector surrounding the ID by 1 m on all sides. This tank would
be situated at various depths in a pit, with the volume beneath the OD filled with purewater and the volume between the top of the OD up to the surface also filled with up
to 6 m of water (and any remaining volume above this water filled with air). The total
depth of the pit below the surface is 57 m, with the seven vertical positions specified as
being centered at the off-axis angles of 1.24°, 1.66°, 2.08°, 2.5°, 2.92°, 3.34° and 3.76°.
These positions are shown in Figure 2.8.

2.5

Physics potential

In this section the physics programme of Hyper-K is detailed along with the projected
sensitivities of the planned measurements. While the focus is on the neutrino beamprogramme of Hyper-K, a brief overview is also given for two of the other significant
physics goals; the atmospheric neutrino oscillation analysis and nucleon decay search.
The vastly increased volume of the Hyper-K detector also allows for a wide range of49

Figure 2.8: The seven off-axis positions of the E61 detector, centered about 1.24°, 1.66°,
2.08°, 2.5°, 2.92°, 3.34° and 3.76°, shown from left to right. Volumes filled with water(ID, OD and any water above or below the OD) are shown in blue.
new or improved measurements in other areas of neutrino physics not covered here [17],including observations of solar, supernova and other astrophysical neutrinos, indirect
dark matter searches, and neutrino geophysics.

2.5.1

Neutrino oscillation physics

In the past two decades, all three neutrino mixing angles and the magnitude of both
mass-squared differences (and the sign of one of them) have been successfully measured [51]. The last remaining parameter is the CP violating phase δ, which up until
recently was entirely unknown and has still had no value excluded above around the 4 σ
level, even in global fits combining data from all neutrino oscillation experiments [74].
The key goal of the neutrino beam programme of Hyper-K is to perform the first precision measurement of this parameter, allowing the potential for discovery of CP -violationin the lepton sector. The combination with observations of atmospheric neutrino oscillations will make possible measurements of the atmospheric mixing parameters – θ23 ,including its octant, and ∆m232 , including its sign – and improved measurements of thesolar mixing parameters will be possible through observations of the 8 B solar neutrinos.
2.5.1.1

Beam neutrinos

The measurement of δ will be achieved through studying the oscillation of muon neutrinos, produced using the J-PARC accelerator, into electron neutrinos detected at HyperK. This is known as the νµ → νe appearance channel. After including the matter effect,50

the oscillation probability is given by [75]P (νµ → νe ) =4c213 s213 s223 1 +2
2a2
2 ∆m31 L
(1
−
2s
)
sin
13
4Eν
∆m231



+8c213 s12 s13 s23 (c12 c23 cos δ − s12 s13 s23 ) cos−8c213 s213 s223

∆m232 L∆m231 L
∆m221 L
sin
sin
4Eν
4Eν
4Eν

aL
∆m231 L∆m232 Lsin
(1 − 2s213 ) cos
4E
4Eν
4Eν

+4s212 c213 (c212 c223 + s212 s223 s213 − 2c12 c23 s12 s23 s13 cos δ) sin2
−8c213 c12 c23 s12 s13 s23 sin δ sin∆m221 L4Eν∆m232 L∆m231 L
∆m221 L
sin
sin
,
4Eν
4Eν
4Eν

(2.5.1)

with the replacements δ → −δ and a → −a for the probability for antineutrinos.
The location of the far detector and peak of the neutrino energy spectrum are chosensuch that the value of L/Eν corresponds to the first maximum of this oscillation probability. Maximising the oscillation probability in this way optimises the sensitivity tothe parameters in Equation (2.5.1).
The final term in Equation (2.5.1) contains sin δ and so is sensitive to the sign of δ,which modifies the probability for neutrinos and antineutrinos. This will allow Hyper-K
to determine the value of δ by combining measurements of the electron neutrino appearance rates when the unoscillated beam is formed mostly of muon neutrinos and when
it is mostly muon antineutrinos. As in the two √neutrino mixing case described in Section 1.1.2, the sign of the terms containing a = 2 2GF ne Eν will also differ for neutrinos
and antineutrinos, creating an asymmetry between neutrinos and antineutrinos that is
not directly CP violating and not related to δ. The sign of these terms is also changed
depending on the mass ordering, which is the effect that enables the mass ordering to
be determined through observations of atmospheric neutrino oscillations.
By measuring accurately the unoscillated neutrino flux at near detectors and the oscillated flux at the far detector, then the value of δ is determined by fitting to Equation (2.5.1).
Additional information is available from the νµ → νµ disappearance channel, governedby the survival probability
P (νµ → νµ ) ' 1 − 4c213 s223 [1 − c213 s223 ] sin2∆m232 L.
4Eν(2.5.2)

With the approximation of c213 = 1, this becomes
P (νµ → νµ ) ' 1 − 4 cos2 2θ23 sin2∆m232 L,4Eν(2.5.3)

which has a degeneracy in θ23 (the octant degeneracy) either side of the maximal mixing
value of θ23 = 45°. This means that while the measurement of νµ → νµ disappearance
provides information on the value of θ23 , using this information alone it is difficult to51

determine the octant. However, the appearance measurement above is also sensitive
to sin2 2θ13 s223 and short-baseline reactor experiments have provided very precise measurements of sin2 2θ13 . The combination of these measurement can therefore provide
sensitivity on the octant of θ23 .
Sensitivity to the mass ordering in long-baseline neutrino beam experiments can comefrom two complementary places. The first is due to the matter effect, as described above
and in Section 1.1.2, however at Hyper-K this effect is expected to be relatively small.
The second is through comparison between reactor neutrino experiments’ measurementsof∆m2ee = c212 ∆m231 + s212 ∆m232(2.5.4)
with a νµ disappearance measurement of [76, 77]∆m2µµ = s212 ∆m231 + c212 ∆m232 + cos δs13 sin 2θ12 tan θ23 ∆m221 .

(2.5.5)Determining the mass ordering through this comparison is challenging, requiring very
low uncertainties on both measurements.
In order to perform these measurements, samples are produced of CCQE-like eventswith either e-like or µ-like rings for the appearance and disappearance samples observedin the far detector. The reconstructed neutrino energy can be calculated assuming
the events are CCQE interactions, using a version of Equation (1.2.1) which includes
corrections for the average nucleon binding energy V , given for neutrino interactions on
neutrons by2(mn − V )El + m2p − (mn − V )2 − m2l
Eνrec =.
(2.5.6)2(mn − V − El + pl cos θ)
These samples, together with measurements at the near detector to determine the unoscillated flux and interaction cross-sections, can then be fitted to the oscillation formulae
to determine the oscillation parameters.
The expected signal and background rates and reconstructed neutrino energy spectra
for the appearance channel are given in Table 2.1 and Figure 2.9, with those for thedisappearance channel given in Table 2.2 and Figure 2.10. Figure 2.11 shows the different
appearance rates as the value of δ changes, showing how this measurement will allow
the determination of the value of δ.
The main systematic errors for these measurements come from uncertainties on theincoming neutrino flux estimate, neutrino interaction cross-sections, and far detector
efficiency and reconstruction. The first two of these are constrained directly throughmeasurements at the near and intermediate detectors. The far detector uncertainties
are constrained mainly through the use of atmospheric neutrino control samples andthe calibration systems of the detector. Table 2.3 summarises the expected systematic
uncertainties in observed event rate.

52

Signal

Backgroundνµ → νeνµ → νe

νµ CC

ν µ CC

νe CC

ν e CC

NC

Total

Total

ν modeEventsEff. (%)

1643
63.6

15
47.3

7
0.1

0
0.0

248
24.5

11
12.6

134
2.4

400
1.6

2058
—

ν modeEventsEff. (%)

206
45.0

1183
70.8

2
0.03

2
0.02

101
13.5

216
30.8

196
1.6

517
1.6

1906
—

Table 2.1: Expected number of event candidates selected in the νµ → νe and ν µ → ν e appearance channels at Hyper-K [17]. Rates
are for 13 × 107 MW s, corresponding to 2.7 × 1022 POT from the 30 GeV proton beam, assuming normal mass hierarchy, sin2 2θ13 = 0.1,
sin2 θ23 = 0.5 and δ = 0.
53νµ CCQE

νµ CC non-QE

ν µ CCQE

ν µ CC non-QE

νe + ν e CCNCνµ → νeTotal

ν modeEventsEff. (%)

6043
91.0

2981
20.7

348
95.6

194
53.5

6
0.5

480
8.8

29
1.1

10080
—

ν modeEventsEff. (%)

2699
88.0

2354
20.1

6099
95.4

1961
54.8

7
0.4

603
8.8

4
0.7

13726
—

Table 2.2: Expected number of event candidates selected in the νµ → νµ and ν µ → ν µ disappearance channels at Hyper-K [17], under thesame assumptions as for Table 2.1.

Number of events/50 MeV

Appearance ν mode
350

Total
Signal ν µ → ν e
Signal ν µ → ν e
BG ν e + ν e
BGνµ + νµ

300
250
200
150
100
50
00

0.2

0.4

0.6

0.8

1

1.2

Reconstructed Energy Erecν

Number of events/50 MeV

Appearance ν mode
300

Total
Signal ν µ → ν e
Signal ν µ → ν e
BG ν e + ν e
BGνµ + νµ

250
200
150
100
50
00

0.2

0.4

0.6

0.8

1

1.2

Reconstructed Energy Erecν

Figure 2.9: Reconstructed neutrino energy spectra for the νµ → νe (top) and ν µ → ν e(bottom) appearance channels at Hyper-K [17].
For Hyper-K running for a total of 1 × 108 s with a beam power of 1.3 MW, the valueof δ is expected to be determined with a precision of 7.2° at δ = 0° or 180° and a
precision of 23° at δ = ±90°, under the assumption that the mass ordering has been
determined. This will allow the observation of CP violation for 76 % of possible valuesof δ at 3 σ and 57 % of values at 5 σ. Measurements of the other oscillation parameters
will include determining sin2 θ23 with a precision of 0.017 at sin2 θ23 = 0.5 or of 0.006
at sin2 θ23 = 0.45, and determining ∆m232 with an uncertainty of less than 1 %. Thesesensitivities of the beam programme of Hyper-K will be explored in detail in Chapter 5.
2.5.1.2

Atmospheric neutrinos

Atmospheric neutrinos provide a flux of νµ (before oscillations) covering a wide range
of energies from 100 MeV to 10 GeV, having travelled a distance varying with zenith
angle. Unlike the νµ → νe appearance from beam neutrinos, the longer distance for54

Number of events/50 MeV

Disappearance ν mode
900

Total
νµ
νµ
νe + νe

800
700
600
500
400
300
200
100
00

1

2

3

4

5

Reconstructed Energy Erecν

Number of events/50 MeV

Disappearance ν mode
Total
νµ
νµ
νe + νe

1000
800
600
400
200
00

1

2

3

4

5

Reconstructed Energy Erecν

Figure 2.10: Reconstructed neutrino energy spectra for the νµ → νµ (top) and ν µ → ν µ(bottom) disappearance channels at Hyper-K [17].

ν modeν modeAppearance

Disappearance

Appearance

Disappearance

Flux & ND-constrained cross-section
Flux ND independent cross-section
Far detector

3.0 %
0.5 %
0.7 %

3.3 %
0.9 %
1.0 %

3.2 %
1.5 %
1.5 %

3.3 %
0.9 %
1.1 %

Total

3.2 %

3.6 %

3.9 %

3.6 %

Table 2.3: Systematic errors on the expected number of events at Hyper-K [17].

55Number of events/50 MeV500
δ=0δ = 90δ = -90
δ = 180

400
300
200
100
0
0

Difference of events/50 MeV

Antineutrino mode: appearance

0.2

0.4

0.6

0.8

1

1.2

Reconstructed Energy Erec
(GeV)
ν

150
100

δ = 90 / δ = 050

δ = 180/ δ = 0

Difference of events/50 MeV

Number of events/50 MeV

Neutrino mode: appearance

δ = -90/ δ = 0

0
-50
-100
-150
0

0.2

0.4

0.6

0.8

1

1.2

Reconstructed Energy Erec
(GeV)
ν

400
350

δ=0
δ = 90δ = -90
δ = 180

300
250
200
150
100

50
0
0

0.2

0.4

0.6

0.8

1

1.2

Erec
ν (GeV)150
100

δ = 90 / δ = 050

δ = 180/ δ = 0

δ = -90/ δ = 0

0

-50
-100
-150
0

0.2

0.4

0.6

0.8

1

1.2

Reconstructed Energy Erec
(GeV)ν

Figure 2.11: Reconstructed neutrino energy spectra for the νµ → νµ (top) and ν µ → ν µ(bottom) appearance channels at Hyper-K, with differing values of δ [17]. The top plotsshow the rates for the four values δ ∈ {0°, ±90°, 180°} while the bottom two plots show
the difference from the δ = 0° case with error bars representing the statistical errors.

56the atmospheric neutrinos which travel through the earth is significantly affected by
the matter effect. In addition to characterising the atmospheric neutrino background ofother measurements, the key goals of the atmospheric neutrino programme at Hyper-K
are measurements of the octant of the atmospheric mixing angle θ23 and determining
the mass ordering. These measurements are particularly powerful when combined intoa joint analysis including the observations of beam neutrinos described above.
While both the atmospheric and beam neutrino observations are of the same types ofoscillations, νµ → νe appearance, these two measurements are highly complementary.
The beam neutrinos have the benefit of having a relatively narrow energy spectrum
and a well known baseline; with a well understood L/Eν this results in the ability tomeasure θ23 with high precision. However, the approximate degeneracy in the octant of
θ23 described above, and the approximate degeneracy between the octant and the massordering, mean that these two properties of the oscillations are difficult to determine
through beam neutrino observations alone. Atmospheric neutrinos, on the other hand,
have a wide range of energies and direction dependent baseline, as well as significant
contributions from the matter effect. This range of energies and baselines, providing
additional data points in measurements of the oscillations, and the matter effect giving
sensitivity to the mass ordering, allow the degeneracy to be broken. By performing a
combined fit to the atmospheric neutrino data with the beam neutrino data describedabove, sensitivity to the mass ordering and octant can be dramatically improved overthat of atmospheric or beam neutrinos alone. Figure 2.12 shows the projected sensitivities of the combined analysis to reject the wrong θ23 octant and to reject the wrong
mass ordering, as a function of experimental run time, and Table 2.4 gives the expected sensitivities comparing the atmospheric only analysis to the combined beam andatmospheric analysis.
OctantMass ordering

True value of sin2 θ23

0.45

0.55

0.40

0.60

Atmospheric analysis
Combined analysis

2.2 σ
6.2 σ

1.6 σ
3.6 σ

2.2 σ
3.8 σ

4.9 σ
6.2 σ

Table 2.4: Expected sensitivities of the atmospheric neutrino only analysis and of a
combined beam and atmospheric neutrino analysis for rejecting the wrong octant of θ23
and wrong mass-ordering after 10 years [17].

2.5.2

Nucleon decay

Another of the main physics goals of Hyper-K is the search for currently unobserved
nucleon decays. Proton decay is predicted by many theories beyond the Standard
Model [78], but has so far remained undiscovered with the best limits in many channels
coming from Super-K [51]. With the number of protons within the Hyper-K tank anorder of magnitude more than Super-K, Hyper-K is expected to be able to probe thelifetime of the proton to an order of magnitude larger than Super-K has.
The main channels for the proton decay search at Hyper-K are p → e+ π 0 and p → νK + ,although the experiment will also be sensitive to a large number of other modes. For
57

∆χ2 Wrong Octant Rejection

7
6
5
4
3
Atm + Beam (True Normal) sin θ23 =0.45
22

Atm + Beam (True Normal) sin2 θ23 =0.55

1

Atm + Beam (True Inverted) sin θ23 =0.45
2

Atm + Beam (True Inverted) sin θ23 =0.55
2∆χ2 Wrong Hierarchy Rejection

0

1

7

2

3

4

5

6

7

8
9
10
Running Time (Years)

5

6

7

8
9
10
Running Time (Years)

2

Atm + Beam (True Normal) sin θ 23 =0.4
2

Atm + Beam (True Normal) sin θ 23 =0.5
2

Atm + Beam (True Normal) sin θ 23 =0.6

62

Atm + Beam (True Inverted) sin θ 23 =0.4
2

Atm + Beam (True Inverted) sin θ 23 =0.5
2

Atm + Beam (True Inverted) sin θ 23 =0.6

5

4

3

2
1

2

3

4

Figure 2.12: Projected sensitivities of a combined beam and atmospheric neutrino analysis for rejecting the wrong octant of θ23 (top) and wrong mass-ordering (bottom), asa function of experimental run-time, for different true values of sin2 θ23 [17]. Blue lines
correspond to the true normal ordering case and red lines to true inverted ordering.

58

both of these decay modes, as well as many others, the main background is dominatedby atmospheric neutrino interactions. For example, for p → e+ π 0 , neutrino interactions
can produce the exact same final state or other final states with the same event topology:
one Cherenkov ring for the e+ and two (which may overlap into one) for the π 0 . The
main method for reducing this background is to reconstruct the proton’s invariant massfrom the kinematics of the observed outgoing particles, however further reduction canbe achieved through neutron tagging; neutrino interactions are significantly more likely
than proton decays to include a neutron in the final state.
The sensitivity for various nucleon decay modes are given in Table 2.5, along with thecurrent limits from Super-K.
Partial lifetime [1034 years]
Modep → e+ π 0
p → νK +p → µ+ π 0
p → e+ η 0
p → µ+ η 0
p → e+ ρ 0
p → µ+ ρ0
p → e+ ω 0
p → µ+ ω 0
n → e+ π −n → µ+ π −Sensitivity (90 % CL)

Current limit

7.8
3.2
7.7
4.3
4.9
0.63
0.22
0.86
1.3
2.0
1.8

1.6
0.7
0.77
1.0
0.47
0.07
0.06
0.16
0.28
0.53
0.35

Table 2.5: Sensitivity of Hyper-K to various nucleon decay modes for 1.9 Mt · year exposure [17] and the current limits from Super-K [79–81].

59

60

Chapter 3

Intermediate detector event
simulation, reconstruction andselection

61

Introduction
The neutrino interaction event generation, detector simulation and event reconstruction
for the detectors of Hyper-K are an essential part of the software used for any physics
analysis. For a neutrino oscillation analysis, the energy and type of the incoming neutrino must be determined in order to measure the values of the oscillation parameters,
due to the oscillation probabilities’ dependence on all of these inputs. The interaction
position, timing and direction are also required to identify neutrinos originating fromthe beam and separate these from interactions from other sources. For experiments inthe design and development stage, such as Hyper-K, the simulation and reconstruction
software is required to determine its potential physics sensitivity and guide decisions on
the experiment’s design.
This chapter details the development of simulation and reconstruction software for the
TITUS intermediate detector, and its use in determining the detector’s capabilities.
Existing event generators can be used for events in both the near and far detectors ofHyper-K, however at the time of this analysis the existing detector simulation and reconstruction software used for the Hyper-K detector itself was not suitable for TITUS. The
simulation package, WCSim, did not have a working implementation of the Gadolinium
doped water for neutron capture, and the reconstruction package, fiTQun, requires asignificant amount of hand-tuning to a particular detector configuration which has notbeen completed for the TITUS detector. For the development of the detector design, thesoftware described and used in this chapter had the additional requirements of being
fast and simple to run for different tank designs, photodetector types and coverages,
while these properties had not been fully determined.
The detector simulation software WChSandBox, developed for the ANNIE Gadoliniumdoped water Cherenkov detector [60], was used for the simulation of events in TITUS.
Since no existing suitable reconstruction software was available, a new reconstruction
package was been developed as part of this work, for use with the WChSandBox simulation package. The simulation and reconstruction for the MRD were both unavailable,
and so this part of the detector has not been included in this analysis.
The neutrino event generation and simulation software used for this analysis are described in Section 3.1, followed by the low-energy and high-energy event reconstruction
software described in Section 3.2 and Section 3.3, respectively. Section 3.4 then presents
the results of event selections developed to optimise the performance of the reconstruction and sample efficiency and purity.

3.1

Monte-Carlo event generation and detector simulation

The neutrino flux used for event generation was produced by the FLUKA target simulation and JNUBEAM beamline simulation used for the T2K neutrino flux [82]. Thisis then used for both the GENIE [83] and NEUT [84] Monte-Carlo event generators,
however all results shown here have used events generated using NEUT version 5.3.3.
The interaction models used by NEUT are the CCQE model of Benhar [85], the modelof Rein and Seghal for coherent pion production [86, 87] and resonance production [88],62with form factors of Graczyk and Sobczyk for resonance decaying to a nucleon andpion [89], and the GRV98 parton distribution function [90] with corrections by Bodek
and Yang [91] for DIS. The nuclear model used is the relativistic Fermi gas model.
The detector simulation package WChSandBox uses the GEANT4 framework [92], version 4.9.6, to simulate all particles and processes within the water tank. The simulation
starts with the outgoing particles of the initial neutrino interaction, after final stateinteractions in the nucleus, provided by NEUT. These are then tracked through thedetector as they interact, where elastic and inelastic electromagnetic and hadronic scattering and decays, as well as neutron captures the subsequent gamma cascades, are all
simulated according to the models of GEANT4. Any charged particles will produce
Cherenkov radiation, as discussed in Section 1.2.2, when their speed is above the Cherenkov threshold. The optical photons produced are themselves tracked through thedetector, allowing for scattering and reabsorption, until they reach the detector wall.
Once the event has been fully simulated, the photons arriving at the detector wall are
discarded unless stopped at a position covered by a PMT. The remaining photons havethe possibility (according to the PMT’s wavelength-dependant quantum efficiency) of
being recorded as producing a photoelectron at the PMT’s position at a time which issmeared by the PMT’s timing resolution. Randomised dark noise can then be added
to each PMT, although in these studies dark noise was not included. These final steps,
which are dependant on the PMTs’ properties and positioning, are completed separatelyfrom the rest of the simulation to allow for a single set of simulation results to be usedwith a number of different configurations of PMT types and coverages.

3.2

Low-energy event reconstruction

All events are initially reconstructed using reconstruction software [93] that has beendeveloped for low-energy events simulated in WChSandBox.
For these events, a basic estimate of the energy of the event is calculated as a simple
function of the number of hit PMTs registering at least one photoelectron, where only
the photoelectrons occurring within a timing window of 50 ns are included, to ensurethat only the hits from the particle of interest are counted.
In order to find this time window, a hit clustering algorithm is first used. The algorithm
is the same as that of MiniBooNE [94]: Initially any cluster of at least 10 PMT hits
is found, with consecutive hits separated by no more than 10 ns. This cluster is thenextended to include as many later hits as possible while requiring that no consecutive
two hits are separated by more than 20 ns and that there are not more than two pairsof consecutive hits separated by 10 ns to 20 ns.
The interaction vertex (position and time) of the event is determined using an algorithm
similar to that used in SNO [95], where all photons hitting a PMT are assumed to havecome from a point source. This assumption combined with the speed of light in water
allows the four unknowns of the vertex to be constrained with the time and positionof each hit, based on the time of flight from the vertex to the PMT. Four PMT hits
are chosen at random and using their positions and times the system of four equations
can be solved to find a single candidate vertex. This is repeated for a large number ofquadruplets of hits, with the best candidate vertex chosen by taking the solution with63the highest goodness,(t0 − t0 )2
G=
exp − i 22σ
i=1,2,3,4!

X,(3.2.1)

where σ is the detector timing resolution, t0 is the reconstructed interaction time andt0i is the hit timing residual, given by
t0i = ti −xi − xreco
c/nref(3.2.2)where ti is the hit time, xi −xreco is the distance from the reconstructed vertex position to
the hit PMT, and c/nref is the speed of light in water. Note that the form of this goodness
expression looks similar to a likelihood, but the sum rather than product helps to reducethe impact of outliers, which could occur due to dark noise, for example. Alternative
methods of choosing the best vertex, using a likelihood function or the geometric centreof the ensemble of candidate vertices, were also investigated, but found to give poorer
performance.
To determine the particle’s direction, the unit vectors pointing from the reconstructedvertex to each hit PMT are summed, and the direction of the result is taken as the
direction of the particle. Since Cherenkov light is emitted as a cone around the particle’s
direction, this average direction from the vertex to the point where the light is observed
should point to the centre of the ring and give a good approximation of the particle’s
direction.
The Cherenkov ring’s opening angle, determined by Equation (1.2.2), is also reconstructed as this can give information on particle type; the angle of electrons travelling close
to c in water will be around 42°, whereas muons will have a smaller angle due to theirhigher mass. In a similar process to the vertex reconstruction, the angle is reconstructedby choosing three hits at random and solving the system of three equations which describe the opening angle. Repeating this for many triplets of hits gives a distribution of
opening angles, which is fitted with a Gaussian whose mean is taken as the reconstructed
angle.

3.3

High-energy event reconstruction

A new set of tools has been developed to provide the reconstruction of events where thevisible energy is above a threshold of 60 MeV, chosen to exclude the majority of secondary
sub-events such as Michel electrons and neutron capture signals. After an event has beenreconstructed with the low-energy reconstruction tools described in Section 3.2, if theenergy estimate is above 60 MeV, then it is passed to these high-energy reconstruction
tools with the results of the low-energy reconstruction used as the seed for the interaction vertex position, direction and energy. In practice this energy threshold simply
corresponds to a given number of photoelectrons in the event, since the reconstructedenergy from the previous step is a simple function of the number of photoelectrons seen
in a window of 50 ns. However, the number of photoelectrons corresponding to the reconstructed energy threshold will vary according to the configuration of PMTs in thedetector.

64

While low energy particles can be approximated as a point source, for high energy
particles the non-zero track length must be taken into account, along with the possibilityof multiple particles producing Cherenkov rings, to give an accurate reconstruction. The
high energy reconstruction consists of first searching for rings in the pattern of PMT
hits and identifying which observed photoelectrons (PE) belong to each found ring.
Then the timing information of each photoelectron hitting a PMT is used to improve
the reconstructed vertex position and direction. The final part of the reconstruction iscompleted for both a muon and electron hypothesis, using a lookup-based method todetermine the energy and a likelihood-based method of particle identification (PID) to
determine whether the ring is electron-like or muon-like.

3.3.1

Ring searching

A Hough transform, similar to that used in Super-K [96], is used to search for rings. The
algorithm, depicted in Figure 3.1 converts rings imaged on the detector wall into peaksin Hough space by first mapping the PMTs onto a spherical surface and then producing
a circle centred around each hit PMT. The center of the ring will be the point wherethese circles coincide. By filling bins that fall on circles around the hit PMTs, peaks
will form corresponding to the centres of any rings.
e/µ

γ

PMT

Figure 3.1: Left: Hough transform finds the centre of a ring-shaped pattern of PMT hitsby identifying the intersection (red point) of rings drawn around each hit PMT (bluepoints). Right: Multiple possible track directions can produce a photon arriving at a
PMT at a given time and location; the circle drawn around each hit PMT corresponds
to the cone formed by these possible tracks.
To implement the Hough transform, the sphere of possible directions of the track mustbe partitioned into bins. Using regular bins as a histogram in the zenith angle θ andazimuthal angle φ results in bins which do not cover an equal solid angle. Correcting
for this either by weighting bin contents by the solid angle covered by each bin, or bybinning in cos θ instead of θ, still leaves the bins unevenly distributed leading to poor
performance of the Hough transform. Instead, the sphere is partitioned into bins basedon the method of evenly distributing points on a sphere along a spiral [97]. Figure 3.2
shows how this binning provides bins of equal area that are distributed more evenly
than other methods. Each bin centre is spaced along the spiral with a fixed change inthe z-coordinate between consecutive bins and the rate that the spiral descends along
the sphere is chosen so that the number of complete revolutions is equal to the number65

of bins around the sphere’s equator, to ensure they are evenly distributed and all coveran equal area. The bin in which any given direction (as a pair of θ and φ coordinates)
is contained is then determined simply by calculating how far along the spiral strip thepoint falls. This gives this binning scheme the additional advantage of having a single
array of bins with a fast and simple filling mechanism, which is not the case for someother regular binning schemes.

Figure 3.2: Possible ways of partitioning the unit sphere into bins for the spherical
Hough transform. Left: Regular binning in the zenith angle θ and azimuthal angle φsuffers from unequal sized bins, with a smaller solid angle covered by bins closer to thepoles of the sphere. Centre: Correcting for the unequal area by using regular binningin cos θ rather than θ ensures all bins cover the same solid angle, but are still unevenly
distributed with long and narrow bins at the poles. Right: Using a spiral to distributepoints on the sphere and constructing the bins along this spiral gives evenly equal-areabins that are evenly distributed.
Given this binning of the possible directions, the Hough transform proceeds by creatinga circle, for each photoelectron, of the possible particle directions that could result in aphoton arriving at the PMT at the observed time. The circles are then mapped onto thesphere of possible directions, and the corresponding bins are filled. The radius of thesecircles is dependant on the angle between the direction of the particle’s track and the
direction from the interaction vertex and the hit PMT, shown in Figure 3.3. This angle,
α, is determined as the solution for the expression for the expected time-of-flight (TOF)t, using the distance d between reconstructed interaction vertex and hit PMT, and theCherenkov angle θC (which is approximated by that of a charged particle travellingthrough water at c):t=s1 nref s2
d sin(θC − α) nref d sin α
+
,
+
=
c
c
c sin θCc sin θC

where nref is the refractive index. The solution for the angle α is√A − B A2 − B 2 + 1
tan α =,
B 2 − A2
nref − cos θCA=,sin θCct
B= .
d(3.3.1)(3.3.2)

By performing this for a large number of photoelectrons, the bins at the centre of theCherenkov rings, where the circles of the Hough transform coincide, will have the highest
value. The direction of the first Cherenkov ring is therefore taken to be the centre of
the highest value bin.
66

e/µ
θC
γ
s1

PM
T

s2

α

d

Figure 3.3: The Hough transform and track fit use time-of-flight based on the lepton
travelling a distance s1 before emitting a photon at angle θC , which travels distance s2
before arriving at a PMT located distance d from the interaction vertex at an angle α
to the lepton track.
To search for additional rings, photoelectrons are first removed if they are observed ata time consistent with having hit a PMT after being emitted by the particle travellingin the direction of any previously identified ring centres (within twice the PMT’s timing
resolution). The direction for the next ring is then taken as the bin with highest value
after the removal of the photoelectrons from the previously found rings. If the ringis found with at least 9 % as many observed photoelectrons as the first ring then it isconsidered genuine, if there are fewer than 9 % as many photoelectrons then the ring is
discarded and the ring searching process ends. While this simple method of determining
which secondary rings to keep and which to discard, using the threshold of 9 %, wasfound to give acceptable results in these early studies, a more refined approach could
be developed by examining the improvement in quality of a fit of all hits to the foundrings as each additional ring is added. The remainder of the reconstruction is applied
to individual rings, using only the photoelectrons that were identified as belonging tothat ring.

3.3.2

Track fit

To improve the reconstructed vertex, the PMT hit timing information is used in a fit
taking into account the track length of the particle. The algorithm used is based onthat used by Super-K [96] using the goodness test value given by(t0 − t0 )2
G=
exp − i 2
2σ
i∈PE
X

!

.

(3.3.3)This is the same as Equation (3.2.1) used for the low-energy reconstruction, exceptthat the sum is now over all observed photoelectrons, and t0i = ti − t̃i is the timing
residual determined by the expected total TOF (t̃i ) and PMT hit time (ti ) of the ith
observed photoelectron, and t0 is determined by taking the mean value of the t0i . Givena candidate vertex, direction and Cherenkov emission angle, the expected TOF for each
hit is calculated according to Equation (3.3.1), assuming the lepton travels a distances1 at speed c along its track before emitting a photon at the Cherenkov angle θC thattravels distance s2 to the centre of the PMT, as shown in Figure 3.3. Maximising G
has the effect of making the distribution of timing residuals more sharply peaked when
the fitted vertex position and direction are close to the true values. The maximisation67

is performed using the MIGRAD algorithm of MINUIT2 [98]. Taking the seed vertex
position from the low energy reconstruction and the seed direction from the centre of
the bin at the peak of the Hough transform, first the direction is allowed to vary withthe position fixed, followed by varying the position with the direction fixed. These
steps are iterated twice, with the interaction time and Cherenkov angle allowed to varythroughout.

3.3.3

Energy reconstruction

The energy estimate from the low energy reconstruction becomes less accurate at higher
energies and so an improved estimate is required. For this, a simple lookup table is usedbased on the total number of photoelectrons observed in the PMTs and the reconstructeddistance from the interaction vertex to the wall of the tank. For a higher energy particle,
the number of photons emitted will increase as per Equation (1.2.3), and the distance tothe wall will affect how many of these photons are detected. The energies for a similar
number of photoelectrons and distance to the wall will also be different depending on theparticle type due to the higher mass of a muon giving a higher Cherenkov threshold thanthat of an electron. The lookup tables are generated from a Monte-Carlo simulationof CCQE beam events in TITUS. These events are binned according to the numberof observed photoelectrons and distance to the wall, taking the mean true charged
lepton energy of these events for the given bin, separately for muons and electrons. The
reconstructed energy is taken from these tables, linearly interpolated, using the observed
number of photoelectrons and reconstructed vertex. An improved energy reconstruction
should be possible by including the energy as a free parameter in the likelihood fit
described in the next section. An alternative improved energy reconstruction has also
been developed, after this study had been completed, using a machine-learning based
multivariate analysis [99].

3.3.4

Likelihood fit

To perform the particle identification and to improve on the vertex position and particle
direction from the track fit, a likelihood method is used. The method used is similar
to those used in SNO [100] and MiniBoone [101]. The likelihoods are constructed fromtwo separate probability distributions; the probability of observing the number of photoelectrons at each PMT and the probability of the time that each photoelectron isobserved.
The likelihood values for the number of photoelectrons at each PMT are determinedusing probability density functions, generated by Monte-Carlo for a given particle typeand energy, of the photon emission angle relative to the particle track (Cherenkov angle)
and the distance along the track from the interaction vertex to the point of emission.
These are then used to calculate the expected number of photons received at any given
point, given its distance from the interaction vertex and angle between the particle’s
track and the line from the interaction vertex to this point. For each PMT together witha candidate vertex position, particle direction, particle type and energy, these geometricquantities are known, which allows the expected number of photoelectrons at the PMT
to be looked up directly from these pre-calculated values. The difference for electrons68and muons is clearly seen in these distributions as shown in Figure 3.4. This arises due
to a number of physical differences in the Cherenkov process for these particles; muons
will travel a reasonable distance emitting photons close to the Cherenkov angle, whichcloses as the muon loses energy close to the threshold, while an electron will produce
an electromagnetic shower, with each electron within the shower travelling in slightly
different directions, giving a broader range of effective photon emission angles. Due tothe non-Poissonian nature of the number of photoelectrons received at a PMT, it was
found that the reconstruction performed better when only using the relative numbers ofphotoelectrons at each PMT, i.e. the expected number of photoelectrons at each PMT
is normalised such that the total summed over all PMTs is equal to the observed total.
The likelihood is then based on a multinomial distribution, where each photoelectron
has a probability of appearing at each PMT, giving the log-Likelihood function for thepattern of PMT hits,log Lpattern =Xni log pi (x, E, l),(3.3.4)i∈PMTñi (x, E, l)pi (x, E, l) = P,
j ñj (x, E, l)(3.3.5)where ni is the observed number of photoelectrons at the ith PMT and pi is the probability of an observed photoelectron at the ith PMT. The value ñi (x, E, l) is the expected
number of photoelectrons observed at a PMT coming from a lepton of type l with track
starting at point x with energy E, taken from the pre-calculated lookup table produced
by Monte-Carlo simulation. Due to the total number of photoelectrons not contributing to the likelihood fit in this form, the fit is not able to improve on the estimatedparticle energy since the most significant effect of the energy is on the total number ofphotoelectrons observed.
The likelihood for the time that a photoelectron is observed at a PMT is calculatedusing similar look-up tables to those used for the expected total number of photoelectronsobserved at a PMT. Again using Monte-Carlo simulation a table is produced of expectedarrival times of a photon at a point, given the photon’s distance from the interaction
vertex and the angle between the particle’s track and the line from the interaction vertex
to this point. This provides the mean for the probability distribution of the time thata photoelectron is observed in a PMT, which is taken as a Gaussian distribution with
σ = 4 ns + σPMT , where σPMT is the PMT timing resolution and the value of 4 ns wasdetermined from the standard deviation of the distribution of timing residuals whenusing the true photon arrival time in simulated events. This gives the log-Likelihood
function for the timing of PMT hits given bylog Ltime = −X (ti − t0 − t̃i )2
i∈PE

2σ 2

(3.3.6)

where ti is the arrival time of the ith photoelectron, t0 is the reconstructed interaction
time and t̃i is the expected arrival time.
The total log-Likelihood log L = log Lpattern + log Ltime is maximised, while allowing thereconstructed interaction vertex position, time and direction to vary, for both electron
and muon hypotheses. This maximisation is completed using the same MIGRAD algorithm of MINUIT2 as for the track fit, and is completed in two minimisation passes;
69

160
140
-;

10 5

120
100

Angle of emission [°]Angle of emission [°]

180

80

180
160
140
-;

10 5

120
100
80

-;

10 6

60
40

40

20

20
10-;7

50

100

150

200

250

300

350

400

Distance along track [cm]

90

1000

80

900
800

70

700

60

600

50

500

40

400

30

300

20

200

10
0
0

0
0

450

Angle from track to PMT [°]Angle from track to PMT [°]

0
0

-;

10 6

60

100

Distance to PMT [cm]

100

150

200

250

300

350

400

10-;7

Distance along track [cm]

1000

80

900
800

70

700

60

600

50

500

40

400

30

300

20

200

10

100

200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400

Distance to PMT [cm]Figure 3.4: Probability density functions for photon emission at a given distance along
particle track and angle of emission (top) and expected number of photons received at
a PMT given its distance from and angle from the particle track (bottom) for electrons(left) and muons (right) with 600 MeV of kinetic energy.

70

450

90

0
0

200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400

50

first the direction is varied together with the interaction time and vertex position only
along the track direction, to correct for biases along the track direction, before then also
allowing the other components of the vertex position to also be varied in the second
pass. The difference between the log-Likelihood value for the electron hypothesis andthat of the muon hypothesis is used to discriminate between particle types and the final
reconstructed vertex position and direction are taken to be those which maximise thelog-Likelihood for the determined particle type.

3.3.5

Performance

The resolutions obtained for the vertex, direction and energy reconstruction are shownin Figure 3.5 with the particle identification performance shown in Figure 3.6, for muonsand electrons generated with uniformly distributed random positions at least 1 m within
the TITUS tank, isotropically distributed random directions, and kinetic energies of
50 MeV to 2 GeV. At low energies the reconstruction performs poorly, as seen in thethree plots of Figure 3.5; this is mainly due to the small number of Cherenkov photons
produced. For muons, the vertex reconstruction becomes poor below around 200 MeV,
seen in the upper plot of Figure 3.5, and the direction and energy reconstruction failat around 100 MeV, seen in the middle and lower plot of Figure 3.5. Above 200 MeV,
the direction resolution is reasonable to high energies above 2 GeV, however the vertex
resolution starts to become worse and energy reconstruction fails above around 1 GeV,
resulting in the step-like behaviour of the lower plot of Figure 3.5. This is partly due tothe long tracks of higher energy muons resulting in them exiting the tank before depositing the majority of their energy, and partly due to the energy reconstruction lookup
tables focusing on the lower energy ranges resulting in inaccuracies from interpolationand extrapolation at higher energies. For those muons which exit the tank into the
MRD it should be possible to use the additional information from the MRD to improve
the reconstruction. Future improvements to the reconstruction by including the energy
as a parameter in the likelihood fit should also improve the reconstruction, particularly
at higher energies. For electrons, the resolutions are generally worse, but are stable fora wider range of energies; at low energies, the direction and energy resolutions start tobecome significantly poorer below around 100 MeV, while all parts of the reconstruction
remain reasonable beyond 2 GeV. This is due to the electromagnetic showers depositing
all the electron’s energy within the tank, even for initially high energy electrons.

3.4

Event Selection

For these studies event selections have been developed for single ring muon and electron
events, with binary neutron tagging (no tagged neutrons or at least one tagged neutron). Separate selections are produced for the beam in neutrino-enhanced mode and in
antineutrino-enhanced mode. Since the number of interactions seen in TITUS will bedominated by νµ (ν̄µ ) events with the beam in neutrino (antineutrino) mode, generating
events according to this distribution would result in a much larger uncertainty on results for the non-dominant flavours. Events are instead generated with an equal numberof each of the four neutrino flavours (νµ , νe , ν̄µ , ν̄e ); 100,000 events of each flavour, foreach beam mode, using the expected energy distribution of each respective flavour. The
71

Vertex resolution [cm]

45
40
35
30
25
20

0

500

1000

1500

2000

Direction resolution [°]Kinetic energy [MeV]

10
9
8
7
6
5
4
3
2
0

500

1000

1500

2000

Energy resolution [%]Kinetic energy [MeV]

60
50
40
30
20
10

0

500

1000

1500

2000

Kinetic energy [MeV]

Figure 3.5: Resolutions of the reconstructed vertex position (top), lepton track direction(middle) and kinetic energy (bottom) for muons (red) and electrons (blue) generated at
kinetic energies of 50 MeV to 2 GeV for electrons and 150 MeV to 2 GeV for muons.

72

Events

6000

Muons
Electrons

5000

4000

3000

2000

1000

0

− 3000− 2000− 1000

0

1000

2000

3000

ln Lµ - ln Le

Figure 3.6: Particle identification for muon and electrons simulated in TITUS.
selected events are then normalised according to the correct flavour distribution to givethe final selections.
The selection criteria are split into the charged-lepton selection, for selecting events thatappear to contain a single muon-like or electron-like ring, and the neutron selection, fortagging those events that appear to contain at least one neutron in the final state.

3.4.1

Charged lepton selection

The primary goal of TITUS is to constrain the flux and interaction cross-sections ofneutrinos coming from the J-PARC beam. Understanding both the νµ and νe flux andcross-section is important for determining their expected rates at Hyper-K, in particular for the νµ → νe appearance measurement used for measuring δ. To achieve this
it is important to obtain high-purity samples of CCQE events producing electrons andproducing muons, where CCQE events are desired due to the ability to reconstruct theneutrino energy according to Equation (1.2.1). The charged lepton selection criteria
have therefore been optimised according to the performance of the reconstructed lepton
properties – most importantly the direction and energy, for neutrino energy reconstruction, and the particle identification to ensure high purity – while also considering thedifferent expected fluxes for each flavour to optimize the cuts for the selection efficiency
and purity of the flavour that is desired.
For both the muon and electron selections, the event must be identified as having asingle Cherenkov ring, according to the ring searching described in Section 3.3.1. This
increases the purity of the sample with respect to CCQE events, as other interaction
modes may have several outgoing particles above Cherenkov threshold, or hadrons which
decay into multiple Cherenkov-producing particles such as a neutral pion decaying to73

two gammas, each resulting in an electromagnetic shower containing electrons that willappear as electron-like Cherenkov rings.
After selecting single ring events it is useful to define a fiducial volume within which theselected events must occur, so that events are discarded if they are reconstructed close
to the wall of the tank where reconstruction performs poorly. In addition to using the
distance between the interaction vertex position and the tank wall (dwall), it can also
be useful to select events based on the distance the particle would need to travel along
its track to reach the wall (towall). The effect of these cuts on the reconstruction isshown in Figure 3.7. The vertex reconstruction actually appears slightly worse further
from the wall. This is understood to be due to the shorter time of flight from theinteraction to the PMTs providing less possibility to vary the vertex position away from
this, however the variation is small and the resolutions are reasonable throughout. For
all other aspects of the reconstruction, there is significant improvement when excluding
events close to the wall, with a dwall cut of 1 m sufficient for good reconstruction. Usinga towall cut, unless very large, does not appear to allow a reduced dwall cut, and so
only a dwall cut has been used for these selections. Figure 3.8 shows how the particle
identification also becomes more difficult close to the wall of the tank.
Allowing for the potential need for different cuts for the µ-like and e-like samples, thesample purities and efficiencies are considered when determining the position of the cut
for PID, together with the dwall cut. In Figures 3.9 and 3.10 it can be seen that when
optimising the purity × efficiency, the loss of the efficiency with any dwall cut dominates
over any gains in purity. However, from the reduction in reconstruction performanceclose to the wall noted earlier and the need to reduce backgrounds entering the tankwhich have not been simulated here, a cut of 1 m has been chosen. For the PID, for both
samples the optimal point for purity × efficiency is to cut the log-likelihood differenceat around -200. Since there will be such a small νe component of the beam, analyses
requiring a higher purity in the e-like sample could use a tighter cut requiring a differencein log-likelihoods that more favours electrons, without too great a loss in efficiency.
In Section 3.3.5, it was seen that the performance of the reconstruction also depends
on the lepton energy. Figure 3.11 shows how the vertex resolution varies with the reconstructed lepton and neutrino energy. The other reconstructed variables show similar
trends but with a slightly smaller effect. At low energies, if a muon is reconstructed withenergy below around 200 MeV or an electron reconstructed with energy below 100 MeV,
then these events are cut due to the poor reconstruction. The reconstruction for electrons
with energies above around 2 GeV also starts to deteriorate. For muons this is expected
to also be the case but, due to these muons often ranging out of the detector before
depositing the majority of their energy, their reconstructed energy is lower than thetrue value and so cutting the high energy muons does not give significant improvement.
The reconstructed energy of the neutrino itself, Eνrec , can also be used, where the reconstruction is poor for νµ events with Eνrec & 2 GeV or for νe events with Eνrec & 2.5 GeV.
Requiring a lower reconstructed neutrino energy of Eνrec < 1.25 GeV for the µ-like selection also significantly reduces the number of non-CCQE and wrong flavour events inthe νµ (ν̄µ ) samples when the beam is in neutrino (anti-neutrino) enhanced mode, due
to their differing neutrino energy spectra, as seen in Figure 3.12.
The final selection criteria are summarised in Table 3.2, with the resulting resolutions for reconstructed quantities given in Table 3.1. Note that the signal rates for74ν evertex resolution
24.5
24

500

23.5

400

23

towall cut [cm]towall cut [cm]

ν µ vertex resolution
600

600
29

500

28

400

27

22.5

300

25

21.5

200

26

300

22

200

24

21

100
0

100

20.5
20

0

50

100

150

200

250

0
0

300

dwall cut [cm]

23
22

50

150

200

250

300

dwall cut [cm]ν e direction resolution

600
2.15

500

2.1

400

2.05

300

2

towall cut [cm]towall cut [cm]ν µ direction resolution

100

600
2.8

500

2.7
2.6

400

2.5

300

1.95

200

2.4
2.3

200

2.2

1.9

100

100

2.1

1.85

0

0

50

100

150

200

250

0
0

300

dwall cut [cm]

2

50

150

200

250

300

dwall cut [cm]

ν eelectron energy resolution [%]

600

0.28
0.26

500

0.24

400

0.22

towall cut [cm]towall cut [cm]ν µ muon energy resolution [%]

100

0.2

300

600
0.28

500

0.26

400

0.24

300

0.22

200

0.2

100

0.18

0.18

200

0.16
0.14

100

0.12

0

0

50

100

150

200

250

0
0

300

dwall cut [cm]

0.16

50

0.26

500

0.24

400

0.22

300

150

200

250

300

dwall cut [cm]

ν eneutrino energy resolution [%]600

towall cut [cm]towall cut [cm]

ν µ neutrino energy resolution [%]

100

600
0.38

500
0.36

400

0.34

0.2

300

0.32

200

0.18

200

0.3

100

0.16

100

0.28

0

0

50

100

150

200

250

300

0.14

dwall cut [cm]

0
0

0.26

50

100

150

200

250

300

dwall cut [cm]Figure 3.7: Reconstruction performance with varying towall and dwall cuts for simulated
CCQE νµ (left) and νe (right) beam events in TITUS. In all cases except for the vertex
resolution in νµ events, the reconstruction performs better when cutting events further
from the wall.

75

Reconstructed distance from wall [cm]500

ν µ CCQEν e CCQE

450
400
350
300
250
200
150
100
50
0
−1500

−1000

−500

0

500

1000

1500

Reconstructed distance to wall [cm]LnLµ-LnLeν µ CCQE

1400

ν e CCQE

1200
1000
800
600
400
200
0
−1500

−1000

−500

0

500

1000

1500

LnLµ-LnLe

Figure 3.8: The likelihood difference used for particle identification of CCQE νµ events(red) and CCQE νe events (blue), against the distance from the interaction vertex to thenearest tank wall (top) or distance from the interaction vertex to the tank wall in the
track direction (bottom). Close to the wall the two lepton flavours become less clearly
separated.

76ν µ sample efficiency
0.98

350

0.96

300

0.94

250

0.92

200

0.9
0.88

150

0.86

100

Distance from wall cut [cm]Distance from wall cut [cm]ν µ sample purity
400

400
0.9

350

0.8

300

0.7

250

0.6

200

0.5

150

0.4
0.3

100

0.84

50
0

0.2

50

0.82

-;800

-;600

-;400

-;200

0

200

0

400

PID cut position

0.1

-;800

400
350
10-;1

300
250
10-;2

200
150
100

10-;3

0

400

PID cut position

200

400

PID cut position

0.8
0.7
0.6

250

0.5

200
0.4

150

0.3

100

0

-;200

200

300

0

-;400

0

350

50
-;600

-;200

400

50
-;800

-;400

ν µ sample purity ×efficiency
Distance from wall cut [cm]Distance from wall cut [cm]ν µ sample ν emis-ID rate

-;600

0.2
0.1

-;800

-;600

-;400

-;200

0

200

400

PID cut position

Figure 3.9: For CCQE νµ events, the sample purity (top left), efficiency (top right), νe
mis-ID rate (bottom left) and purity × efficiency (bottom right), from equal numbersof simulated νµ and νe beam events, dependant on the position of the dwall and PID
selection cuts.

77ν esample efficiency
0.9

350

0.8

300

0.7

250

0.6

200

0.5
0.4

150

0.3

100

-;600

-;400

-;200

0

200

400

0.9

350

0.8

300

0.7

250

0.6

200

0.5
0.4

150

0.3
0.2

50

0.1

-;800

400

100

0.2

50
0

Distance from wall cut [cm]Distance from wall cut [cm]

ν esample purity
400

0

0

PID cut position

0.1

-;800

-;600

350
10-;1

300
250

10-;2

200
150

10-;3

100
50
0

-;800

-;600

-;400

-;200

0

-;200

0

200

400

0

PID cut position

ν esample purity ×efficiency

400

Distance from wall cut [cm]Distance from wall cut [cm]

ν e sample ν µ mis-ID rate

-;400

200

400

10-;4

PID cut position

400

0.8

350

0.7

300

0.6

250

0.5

200

0.4

150

0.3

100

0.2

50

0.1

0

-;800

-;600

-;400

-;200

0

200

400

PID cut position

Figure 3.10: For CCQE νe events, the sample purity (top left), efficiency (top right), νµmis-ID rate (bottom left) and purity × efficiency (bottom right), from equal numbersof simulated νµ and νe beam events, dependant on the position of the dwall and PID
selection cuts.

78

Vertex resolution [cm]Vertex resolution [cm]

45
40
35
30
25
20

50

40

30

20

15
10

10
5
0

200

400

600

800

1000

1200

1400

1600

1800

2000

0

Reconstructed muon energy [MeV]

50
40
30

1000

1500

2000

2500

3000

Reconstructed electron energy [MeV]Vertex resolution [cm]Vertex resolution [cm]

60

500

20

45
40
35
30
25
20
15
10

10

5

0

500

1000

1500
2000
2500
3000
Reconstructed neutrino energy [MeV]

0

1000

2000

3000

4000

5000

Reconstructed neutrino energy [MeV]

Figure 3.11: Vertex resolution for CCQE νµ events (left) and νe events (right), dependingon the reconstructed electron/muon energy (top) and neutrino energy (bottom). At
higher energies the reconstruction performs poorly for both flavours, as well as at low
energies particularly for νµ events where the muon is close to the Cherenkov threshold.

79

Reconstructed energy for neutrino beam
Events1200
ν µ CCQE
ν µ Other1000νe / νe / νµ

800

600

400

200

0
0

500

1000

1500

2000

2500

3000

EventsReconstructed neutrino energy [MeV]Reconstructed energy for antineutrino beam
1200

ν µ CCQE
ν µ Other1000νe / νe / νµ

800
600
400
200
0
0

500

1000

1500

2000

2500

3000

Reconstructed neutrino energy [MeV]

Figure 3.12: Reconstructed energy distributions for events identified with a single µ likering from a neutrino-enhanced beam (top) and antineutrino-enhanced beam (bottom),with the events shown here normalised to give an equal number of each flavour prior to
selection criteria.

80the antineutrino-enhanced beam are significantly lower than the signal rates in theneutrino-enhanced beam due to a combination of two effects; the cross section is larger for neutrino interactions compared to antineutrinos in the detector and the flux ofantineutrinos produced in the antineutrino-enhanced beam is smaller than the flux ofneutrinos in the neutrino-enhanced beam, as discussed in Section 2.1.
Reconstruction resolution
Selection
Single muon-like ringSingle electron-like ring

Vertex [cm]Direction [°]Energy [%]

18.4
22.1

2.1
2.8

11.8
18.4

Table 3.1: Reconstruction resolutions for CCQE interaction beam events passing thesingle muon- and electron-like ring selections.

81Single muon-like ring selection
Neutrino-enhanced beam mode
SignalAntineutrino-enhanced beam mode

Background

SignalBackground

Selection

νµ
CCQE

νµ
CC otherν̄µCCνe + ν̄e
CC

NC

ν̄µ
CCQE

ν̄µ
CC otherνµCCνe + ν̄e
CC

NC

All
All in FV

1 718 260
1 090 130

1 784 790
1 133 550

97 08661 62276 01848 1491 364 740
867 219486 873308 239322 821206 129455 639
288 673

37 17323 641

553 909353 630

dwall < 1 m+ 1 ring+ µ-like+ Eµrec > 200 MeV+ Eνrec < 1.25 GeV

955 919731 938
723 484
692 934
623 093857 306
415 780
303 093
240 457
191 558

55 99536 34931 98423 32211 61142 241
21 481
1743
688
261

230 22185 32333 64627 73221 252287 067
212 924
208 934
196 807167 208173 483
93 42875 79959 77945 475

250 403139 258
109 447
81 21045 13621 387
11 255963
292
102

106 01339 36213 836
11 5708878

Single electron-like ring selection
82

Neutrino-enhanced beam mode
SignalAntineutrino-enhanced beam mode

Background

Selection

νe
CCQE

νe
CC otherν̄eCCνµ + ν̄µCC

All
All in FV

21 81913 92147 45729 921

6742
4307

dwall < 1 m+ 1 ring+ e-like+ Eµrec > 100 MeV+ Eνrec < 2.5 GeV

13 665
8257
7559
6302
5510

28 70211 96011 105
6786
5433

4301
2392
2160
972
766

SignalBackground

NC

ν̄e
CCQE

ν̄e
CC otherνeCCνµ + ν̄µCCNC

3 600 140
2 285 290

1 364 740
867 219

6970
4429

9688
6136

20 51413 076

1 265 330
803 041553 909353 630

1 994 500
1 225 620
139 81187 03572 040271 71592 88658 20050 30047 684

4414
2680
2427
1832
1644

5956
2944
2716
1545
1281

12 819
6074
5579
2913
2232

748 300
460 495
55 59234 34327 477

124 884
43 38428 95125 18223 295

Table 3.2: The number of expected events at TITUS, 2 km from a 1 MW beam running for 107 s, passing each selection criteria forsingle muon- and electron-like ring events. The signal is for CCQE events of the desired flavour; νµ (ν̄µ ) or νe (ν̄e ) for the muon-like orelectron-like selections respectively, with the beam in (anti)neutrino mode. The rates given in the fiducial volume (FV) are based on thetrue vertex position but the rates passing the selection criteria include those occurring outside the FV if reconstructed within the FV.

3.4.2

Neutron Tagging

To improve the purity of samples with respect to the wrong sign neutrino background,i.e. the antineutrino (neutrino) background contamination of the beam in (anti)neutrino
enhanced mode, neutrons produced in the event are identified through the delayed signal
after a neutron captures on Gadolinium. As described in Section 1.2.3, before final stateinteractions, there will be a single neutron produced for CCQE antineutrino interactions
while none are produced for CCQE neutrino interactions. Although the number ofneutrons in an interaction will change depending on final state interactions within thetarget nucleus, the purity of the samples can still be improved by requiring that there
are no (at least one) neutron tagged in (anti)neutrino event samples.
To tag neutrons from their capture, a timing and energy window is chosen based on the
distributions shown in Figure 1.4. The timing window used is 1 µs to 100 µs with a visibleenergy of 2 MeV to 10 MeV. For neutrons simulated with an initial energy of 50 MeV,
randomly placed within the TITUS tank, the tagging efficiency using these criteria is
91 %. The majority of neutrons that were not tagged either captured on Hydrogen,converted into protons through interactions with nuclei, or escaped the water tank. The
effect on the selections is shown in Table 3.4. For the neutrino beam the contaminationof ν̄µ (ν̄e ) events in the single muon-like (electron-like) sample is reduced by a factor of 9,
while retaining 70 % (76 %) of CCQE νµ (νe ) interactions. For the antineutrino beam thecontamination of νµ (νe ) events in the single muon-like (electron-like) sample is reduced
by a factor of 2.5 (2.2), while retaining 85 % (86 %) of CCQE ν̄µ (ν̄e ) interactions.

3.5

Effect on sensitivities of Hyper-K

The selections described in the previous section have been used in determining the effectof TITUS on oscillation sensitivities at Hyper-K; the full details of this analysis can befound in [71].
Using the VALOR framework, which is also used for official T2K analyses, the near and
far detector samples are simultaneously fit to the expected rates, binned in reconstructed
neutrino energy and accounting for both statistical and systematic uncertainties, todetermine the oscillation parameters. A total of 189 systematic error sources are included
coming from both TITUS and Hyper-K detector response as well as the neutrino beam
flux prediction and neutrino interaction cross-section models. The parameters of thesesystematics are allowed to vary in the fit at the same time as the oscillation parameters,with the expectation that these systematic parameters will be constrained in the fit
by the near and intermediate detector samples thereby reducing the overall errors in
the oscillation parameters constrained by observations in the far detector. The fit has
been performed using TITUS with Hyper-K, Hyper-K without TITUS, and Hyper-K
without TITUS and without any systematic uncertainties, allowing the effect of the
TITUS samples in reducing systematic uncertainties to clearly be seen. The resultsof the study are given in Figure 3.13 and Table 3.3. Improvements are seen in all of
the main oscillation measurements; the precision measurement of δ, the search for CP
violation, and the measurements of the atmospheric mixing parameters θ23 and ∆m232 .

83

∆χ2
|∆ m232|

4
3.5
3 90%
2.5
2
1.5
1σ
1
0.5
0
−3
−2
×10

No Systematics
HK + TITUS
HK only

−1

0

1

−3

2.56

No Systematics
TITUS + HK
HK only

2.55

2

3

δcp
90% CL
68% CL

2.54
2.53
2.52
2.51
2.5
2.49
2.48
2.47

0.47 0.48 0.49

0.5

0.51 0.52 0.53 0.54 0.55 0.56

sin2(θ23)

Figure 3.13: Sensitivities at Hyper-K only, Hyper-K with TITUS, and Hyper-K without
systematic errors for excluding CP conserving values of δ (top) and for measurementsof θ23 and ∆m232 (right) [71].

84HK only
TITUS + HK
HK, no Systematics

Error on δ

CP violation

δ=0

δ = π4

δ = − π2Non-maximal θ32

3σ

5σ

3σ

5σ

8.0°
6.6°
4.7°

12.1°
9.2°
6.6°

23.2°
19.0°
16.9°

73 %
78 %
84 %

51 %
61 %
74 %

90 %
92 %
93 %

87 %
90 %
92 %

Table 3.3: Sensitivities at Hyper-K only, Hyper-K with TITUS, and Hyper-K without
systematic errors. The the sensitivities shown are for the precision on the measurementof δ, the proportion of δ space for which the CP conserving values will be excluded, andthe proportion of sin2 θ23 space for which the value sin2 θ23 = 0.5 can be excluded.

85

Single muon-like ring selection
Neutrino-enhanced beam mode
SignalAntineutrino-enhanced beam mode

Background

Selection

νµ
CCQE

νµ
CC otherν̄µCCνe + ν̄eCCAll
All in FV

1 718 260
1 090 130

1 784 790
1 133 550

97 08661 622

1 µ-like ringNo neutron tag
1+ neutron tag

623 093438 500
184 594

191 558
99 38292 177

11 6111322
10 289SignalBackground

NC

ν̄µ
CCQE

ν̄µ
CC otherνµCCνe + ν̄eCCNC

76 01848 1491 364 740
867 219486 873308 239322 821206 129455 639
288 673

37 17323 641

553 909353 630

261
171
91

21 2524396
16 857167 208
24 859142 349

45 4752938
42 53745 13626 75718 379

102
33
69

8878
2056
6822

86

Single electron-like ring selection
Neutrino-enhanced beam mode
SignalAntineutrino-enhanced beam mode

Background

SignalBackground

Selection

νe
CCQE

νe
CC otherν̄eCCνµ + ν̄µ
CC

NC

ν̄e
CCQE

ν̄e
CC otherνeCCνµ + ν̄µ
CC

NC

All
All in FV

21 81913 92147 45729 9216742
4307

3 600 140
2 285 290

1 364 740
867 219

6970
4429

9688
6136

20 51413 076

1 265 330
803 041553 909353 630

1 e-like ringNo neutron tag
1+ neutron tag

5510
4202
1308

5433
2512
2922

766
83
683

72 04037 84434 19647 68423 24624 437

1644
236
1408

1281
97
1184

2232
1239
993

27 4777692
19 78523 29513 361
9934

Table 3.4: The number of expected events at TITUS, 2 km from a 1 MW beam running for 107 s, passing the selection criteria for single
muon- and electron-like ring events, as in Table 3.2, with the addition of neutron tagging.

Conclusion
In this chapter, event selections of νµ , νe , ν µ and ν e interactions have been developed,
which have then been used to demonstrate the important role of an intermediate water
Cherenkov detector for the Hyper-K oscillation analyses.
The selection criteria developed for these selections have demonstrated how the detector
is capable of producing relatively pure, high-statistics data sets. In order to produce
these samples, the requirement of a new high-energy reconstruction software package
was met, providing fast, adaptable reconstruction suitable for the changing design ofthe detector.
While the design of the IWCD has changed since this analysis was produced, with adesign now at a stage of development where the more advanced existing reconstruction
software of Super-K and Hyper-K can be used, the tools and techniques described here
could be applied to other detector design projects where speed and flexibility is a requirement. The reconstruction software developed for TITUS as part of this work was
already been repurposed for use with the ANNIE experiment [60].
A simple neutron tag has also been implemented to demonstrate the possibilities ofusing the neutron capture signal on Gadolinium. The use of neutron tagging in thisanalysis is relatively basic, simply separating events by whether there appears to bea neutron capture anywhere in the tank within a fairly long time window, then using
this to produce the two corresponding samples to include in the oscillation fit. However,there are a number of additional uses of neutron tagging expected to have a significantly
greater impact in reducing errors at Hyper-K. A full neutron multiplicity measurement
can provide new information to use in constraining cross-section models, which had notbeen implemented for the analysis of Section 3.5. Additionally, such a measurement
could help understand the neutron capture signal that is intended to be used in the far
detector (again not implemented for the analysis in Section 3.5) as well as to inform amore complete use of neutron tagging to statistically separate different interaction modes
and separate neutrinos from antineutrinos. The next chapter investigates the possibilityof producing such a neutron multiplicity measurement in the IWCD of Hyper-K.

87

88

Chapter 4

Measuring neutrons at E61

89

Introduction
As described in Section 2.4.2, the E61 experiment is the current design for an intermediate water Cherenkov detector as part of the Hyper-K project. The main motivation
for these detectors is to perform measurements providing the inputs required to reduce
systematic errors in measurements at the far detector. By providing measurements
performed on the same water target, using the same Cherenkov ring imaging detector
technology, the information provided by E61 can be applied to the far detector in a way
which allows a greater reduction in systematic errors than existing near detectors haveprovided in the T2K experiment.
In T2K, the most significant sources of error come from uncertainties in the neutrino
interaction cross-sections, the neutrino beam flux, the reconstructed incoming neutrinoenergy, and the contamination of neutrinos (antineutrinos) in the antineutrino (neutrino)
beam. For the precision measurements of Hyper-K, these sources of error must be dealt
with in order to reduce the total overall level of systematic errors to match the decreasein statistical errors from the increased event rate. The detection of neutron captures, ina water Cherenkov detector, tagged to specific neutrino interactions, has the potential
to provide information helping to reduce all of these sources of errors; the details of thiswere covered in Section 1.2.3. To achieve this in practice, it is essential that the neutron
capture signal is well understood, and it is hoped that this understanding can be gained
through the observation of neutron captures in the E61 detector.
The benefits described previously of the E61 design in providing information on theneutrino beam flux and cross-sections would also apply to a neutron measurement;
namely, the large sample of neutrino interactions from the largely unoscillated neutrino
beam, the ability to make measurements over a range of different neutrino energy spectra
through the varying off-axis angle, and the compatibility with far detector measurements
due to having the same target and detector technology.
In this chapter, simulations of a neutron-capture measurement at E61 are presented,
including event generation, detector simulation, reconstruction, selection and analysis,
providing a proof-of-concept demonstrating the possibility of such a measurement andthe information it could provide. The basic approach is to first identify and selectprimary events – the initial signal seen by the lepton (or other Cherenkov producing
particles) produced by the initial neutrino interaction – then to search for neutron
capture signals in a specified time window following the primary event. The primary
event selection follows the same process as implemented for other E61 analyses, while
neutrons are selected through a newly developed procedure attempting to maximise
the efficiency and purity of the selected neutron capture samples. These results forma neutron multiplicity measurement, as a function of distance from the primary event
and angle between the beam direction and the direction from the primary event vertex
to the capture position. This can then be corrected to account for backgrounds anddetector effects including efficiency and reconstruction resolutions.
Section 4.1 describes the tools used to produce the simulation and perform the reconstruction of both signal and background events. Section 4.2 details the event selections developed to separate the signal from background, with Section 4.3 discussing themethods used to correct for the signal efficiency of these selections and to subtract theremaining backgrounds. The results are then given in Section 4.4.
90

4.1

Event simulation and reconstruction

In order to perform the full analysis of this chapter, simulations are required to produce
the flux of neutrinos entering the E61 detector tank, generate the neutrino interactions
occurring in the detector, simulate the results of these interactions in the detector itself– the tracking of the particles through the detector, the detection of photons in the
PMTs, and the subsequent electronics and data acquisition – and finally reconstruct theinteractions from the detector simulation results.
2.39 × 1020 POT per off-axis position of simulated data has been produced for the signal
neutrino interactions in the tank. The neutrino flux and interaction Monte-Carlo software used for this is the same as the previous chapter, except that the newer version 536
of NEUT [84] is used. The remaining software used for detector simulation and event
reconstruction differ from those of the previous chapter.

4.1.1

WCSim detector simulation

The detector simulations performed for this analysis use the existing WCSim simulation
package [102], which is the standard water cherenkov simulation package used by theHyper-Kamiokande collaboration. The WCSim software itself has been modified toprovide additional capabilities for the addition of Gadolinium to the water, and for thesimulation (and for validating the simulation) of neutron captures. Additionally, theE61 detector geometry used within WCSim has been developed separately by the E61
collaboration. This geometry matches the description given previously in Section 2.4.2,
with the assumption of 0.1 % Gadolinium doping (0.2 % Gadolinium sulphate) of thewater in both the ID and OD. Further modifications to WCSim as well as the use ofother simulation software allowed the simulation of various background events, which
will be discussed in more detail in the next section.
To simulate events in WCSim, the incoming particles – either from the output of theneutrino interaction event Monte-Carlo or from separate background Monte-Carlo – areprovided as the initial input. For each event, these particles are then simulated using
Geant4 [92], version 4.10.1, in the same way as described for the WChSandbox simulation
software in Section 3.1, up until each particle is either destroyed in a decay or interaction
with the detector material, exits the detector volume or, for optical photons, enters a
PMT. The standard Geant4 physics list FTFP_BERT is used by default in WCSim todefine which physical processes are simulated and the models used. However, the highprecision FTFP_BERT_HP physics list is required to accurately model neutrons and so forthe neutron simulations this has been used instead.
A more advanced treatment of the photodetectors themselves is employed in WCSim
than in the WChSandbox software used in the previous chapter; in WCSim the detection,
digitisation, triggering and data acquisition are all simulated. Each photon reaching a
PMT has a chance of being registered as a hit, depending on the quantum efficiency andcollection efficiency of the PMT, which varies with the incoming photon’s wavelength
as well as its incident angle on the PMT. The digitiser is then simulated, providing foreach hit a delayed and smeared hit time and smeared observed charge. At this point,
dark noise of each PMT is also simulated and included as additional hits, before thetrigger and data acquisition simulation determines whether a collection of hits produce a91

trigger event, or fail to trigger with these hits going unobserved. The models used for the
PMT response, digitiser, trigger and data acquisition system are those of the nuPRISM
branch of WCSim developed by the E61 (and previously nuPRISM) collaboration [103].
The only modification of these made for this analysis is the parameters of the trigger; inorder to successfully observe the lower energy produced by neutron captures, a minimumof only 5 digitised hits above the expected dark rate (of 100Hz at each 300 PMT) is
required within a moving window of 50 ns, compared to the default requirement of 25
hits in 200 ns.
4.1.1.1

Neutron capture models in Geant4

An important consideration for this analysis is the modelling of neutron captures, which
in Geant4 has some shortcomings that will need to be addressed for a neutron multiplicity measurement using neutron captures on Gadolinium. There are two existing models
generally in use for simulations of Gd neutron captures, the default model of Geant4
and a custom model originally created to be used with the GLG4sim package [104], as
well as a third model under development based on experimental data from the ANNRI
experiment [105, 106].
In terms of the effect on the simulation and reconstruction of neutron captures, thedifference between the models is in the number, individual energies, and total energyof the γ-rays emitted from the excited nucleus as a result of the capture. In all threemodels, for the case of more than one emitted photon, they are assumed to be emittedwith no angular correlation.
Figure 4.1 shows the differences between the models, with all three models predicting
different distributions of multiplicities and energies. For the default GLG4sim model,
there are always at least two γs produced and the total energy is always one of the twocorrect values for the nuclear de-excitation of the two isotopes of Gadolinium, 8.5 MeV
for 156 Gd and 7.9 MeV for 158 Gd, after neutron capture. The ANNRI model typically
predicts a larger γ multiplicity, mostly with the same total energies, but with a small
number of events having lower energies. The Geant4 model, however, predicts a muchlarger number of captures having lower energies, except when only a single γ is produced
and the correct total energy is given to that γ. The Geant4 model also does not predict
the specific γ lines in the individual γ energy spectrum, seen as spikes in the γ energy
spectra, other than those for a single γ with the full energy. These spikes correspondto de-excitations between energy levels of the nucleus and have been included in the
GLG4sim and ANNRI models based on experimental observation, having been added
into these models explicitly but not included in the Geant4 model.
Since the charge seen in a PMT is directly correlated with the energy of the Cherenkov
producing particles, both the total energy and how that energy is distributed amongst
γs will affect this. An increased total charge provides more information for use inreconstruction and in Figure 4.2 the effect of the different models on the reconstructedcapture position (vertex) can be seen. As expected from the typically lower total energy
released in the default Geant4 model’s captures, the reconstruction of these captures
is significantly worse than the others. While the reconstruction of captures simulatedusing the ANNRI and GLG4sim models perform similarly, the GLG4sim captures do

92

Events

35000

ANNRI
GLG4simGeant4

30000
25000
20000
15000
10000
5000

Events

0
0

1

2

3

4

5

6

7

8

9

10

Capture gamma multiplicity
ANNRI
GLG4simGeant4

50000
40000
30000
20000
10000

Events

0
0

1

2

3

4

5

6

7

8

9

10

Capture total energy [MeV]ANNRI
GLG4simGeant4

50000
40000
30000
20000
10000
0
0

1

2

3

4

5

6

7

8

9

10

Gamma energy [MeV]

Figure 4.1: Comparison of the ANNRI, GLG4sim and Geant4 (default) models of Gadolinium neutron capture for 100 000 simulated events. The multiplicity (top), total energy (middle) and individual energy (bottom) of the γs emitted after simulated neutron
captures are shown for the three models.

93

Reconstructed distance from primary vertex [cm]have slightly improved reconstruction, due to this model predicting fewer low-energy γs(below Cherenkov threshold).
80
70
60

ANNRI
GLG4simGeant4

50
40
30
20
10
0
0

50

100

150

200

250

300

Capture distance from wall [cm]Figure 4.2: The vertex reconstruction as a function of capture distance from the tankwall for Gadolinium neutron captures simulated with the ANNRI, GLG4sim and Geant4
models.
With the current level of uncertainty in the number and energies of γs released after acapture on Gd, as demonstrated by the significant differences between these models, itwill be important to take the effect of these into account as a source of systematic error
on the neutron measurement. While this analysis does not include systematic errors,
the conservative approach is taken of using the default Geant4 model (which produces
captures with the least well-performing reconstruction). For future analyses the ANNRI
model, once its development progresses and a full version of the model is released, would
be preferred due to being based on experimental observation.

4.1.2

Low-energy background and background neutron simulations

The low energy background rates for a neutron capture measurement are expected tobe significant, due to the relatively long time window required for the search for neutron
captures, the relative proximity to the beam source resulting in a high rate of beam
induced backgrounds, and the lack of overburden above the detector to protect fromcosmic rays. While a cosmic ray muon may appear to mimic the muon coming froma neutrino interaction, they are high enough energy that they would be seen entering
the OD region of the detector and can be vetoed. However they can still produce abackground for the neutron measurement by interacting in the material surrounding
the detector, producing neutrons which then travel undetected into the detector before
capturing and appearing like a neutron produced in a neutrino interaction. In addition
to backgrounds caused by incoming neutrons not associated with the signal neutrino
interaction, any low-energy Cherenkov light producing particles can also mimic the signal of a neutron capture. The most significant source of this comes from the decays ofradioactive isotopes from impurities within the detector. For this analysis, all the expected major sources of backgrounds have been fully simulated through custom software or94

extensions of the WCSim simulation software described in the previous section. Background sources which form a significantly smaller contribution to the total rate have notbeen included; this includes skyshine neutrons (high-energy neutrons produced in thebeamline at J-PARC and subsequently escaping into the air and propagating until theyreach the detector), delayed decays of unstable nuclei activated in or near the detector
by cosmic ray muon spallation, neutrons produced by decays of naturally occurring radioactive isotopes in the surrounding material and triggers produced purely throughPMT dark-noise. Basic estimates of the rates of these backgrounds have suggested that
their rates are at least two orders of magnitude smaller than the background sourcesthat have been included. These estimates were based on extrapolations from measurements of the rates seen at other experiments; the decays of activated nuclei estimate isbased on that seen at Super-K [107] and the neutron skyshine is based on that seen at
K2K [108].
4.1.2.1

Neutrino beam induced backgrounds

The most significant source of background for this measurement comes from neutrons
created in or near the detector tank as a result of neutrino beam interactions separate
from the neutrino interaction identified in the primary event selection. This includes
both neutrino interactions in the surrounding detector material, including the outerdetector region of the tank and the sand and air surrounding the tanks, and neutrino
interactions within the inner detector producing no detectable primary event (mostly
NC events). Both of these sources have been simulated separately; for the ‘invisible’interactions in the inner detector, no additional simulation was needed as these neutrons
were included in the signal event simulation and so it is only required to identify theneutrons from those interactions with no visible primary event. For the neutrino interactions in the surrounding material, these were first generated using the same tools as
the signal interactions in the tank, using a neutrino flux plane located 34 m upstream of
the tank, with the simulation geometry extending from this plane to 5 m downstream of
the tank, from 77 m below the surface to 25 m above the surface and a total of 22 m wide,
centred horizontally with the detector. The products of these neutrino interactions arethen simulated through a custom-built simplified Geant4-based software package which
tracks all particles through the sand, air and water volumes, without simulating theentire detector geometry. This allows for a fast simulation to determine the particles
entering the detector volume, which are then themselves entered into a full WCSim
simulation. These simulations assume throughout that the volume below the surface,surrounding the pit, consists only of silica (SiO2 ), with air above the surface. Below the
OD down to the bottom of the 51 m pit is filled with pure water, as is up to 6 m of thepit above the top of the OD but no higher than the surface, with any remaining volumeof the pit above this water filled with air. The water in the ID and OD detector volumes
are Gadolinium doped, but any water above and below the OD is not. This geometry
was shown for each off-axis angle position in Figure 2.8. A total of 1019 POT has been
simulated for these interactions for each position of the detector.

95

4.1.2.2

Cosmic ray induced backgrounds

With the top positions of the detector having no overburden and the lower positions
having only 6 m of water directly above, the rate of cosmic ray backgrounds will belarge. For a neutron measurement, the majority of cosmic muons will not affect themeasurement other than to temporarily blind the detector from observing low-energy
events while muons pass through. However, these muons can also interact outside thedetector, producing neutrons which may enter the detector, capture and produce thesame signal as that of the neutrons from neutrino interactions. In order to simulate thisbackground, the CRY library [109] has been used, within WCSim, to generate cosmic
ray showers through a 50 m × 50 m plane at the surface (or at the top of the tank in theposition where that is above the surface). The resulting particles of these showers arethen simulated through the full WCSim simulation using an extended geometry, similar
to that used by the beam induced background simulation described above, to include
the sand and air surrounding the water tank. This has been performed for a total of 10
million cosmic ray muons and neutrons, corresponding to the amount expected throughthe 50 m × 50 m plane in approximately 3.3 s, then simulated with the detector at eachof the seven off-axis positions.
4.1.2.3

Radioactive decays

The most significant source of non-neutron backgrounds to this measurement comesfrom electrons and gammas produced in decays of radioactive isotopes in the detector.
A large proportion of the visible decays come from impurities in the PMT glass, however
the smaller rate coming from impurities within the water itself is also significant due
to the greater difficulty in separating these from signal events. In these simulations,only the impurities from these two sources have been considered, however additional
impurities in the material surrounding the detector, the detector support structure andthe materials of the multi-PMT modules may also contribute. A modified version of
WCSim which uses the Geant4 radioactive decay models has been used to simulate
the radioactivity background. The activity levels assumed are given in Table 4.1. The
simulation runs for a fixed time window of 100 µs and was performed for 100 000 such
windows.

PMT glass
Water tank

Isotope

Activity level

208 Tl

0.067 Bq/PMT
0.64 Bq/PMT
2.2 Bq/PMT

214 Bi40 K

214 Bi

940 Bq

Table 4.1: Activity levels of radioactive impurities simulated in E61.

4.1.3

Event reconstruction

As with the analysis in the previous chapter, separate reconstruction software is used
for high energy or low energy events. Initially, all events are passed through the fiTQun
96

high-energy event reconstruction software, which was developed for use with the Super-K
detector [110] and, like the high-energy reconstruction software described in the previous chapter, is based on the algorithms originally developed for the MiniBooNE experiment [101]. This is the standard reconstruction software which has been re-tuned by theE61 collaboration for use with the E61 detector. The results of this are used to identify
and select the primary events, but are not used for subsequent triggers when looking forneutron captures.
The neutron capture reconstruction uses the BONSAI low-energy reconstruction software [111], also originally developed for the the Super-K detector; this has also been
re-tuned for the E61 detector. All triggers with a total number of hits between 5 and
500 are passed through the BONSAI reconstruction and the results used in identifying and selecting neutron captures. Currently only the vertex position reconstructionof BONSAI is used. However, further work to tune the energy reconstruction for E61
could also provide more useful information than the total PMT charge which has beenused here instead (the direction reconstruction is unlikely to be of much use due tothe variable number of cherenkov-producing particles emitted after a neutron capture
in different directions). The vertex reconstruction of BONSAI works by maximising alikelihood function based on the hit timing: A likelihood function of the timing residual of each hit, defined in the same way as previously in Equation (3.2.2), is fitted tothe observed distribution for a large number of events simulated in the detector. This
likelihood function differs only according to the detector geometry and so only needs tobe computed once for a given detector design. The likelihood which BONSAI maximises for each event is then produced by computing the product of the likelihoods of thetiming residual of each of the PMTs’ hits, after a hit selection process which attempts
to remove dark noise and other outliers. The maximisation algorithm itself is designed
to deal with the large number of local maxima which frequently occur in low-energy
events with a small number of hits. Initially a grid of candidate vertices is compiledusing quadruplets of hits, in a similar process to the algorithm described in Section 3.2.
Then, a search for progressively better candidates is performed, traversing a tree of thepotentially local maxima and attempting to maximise the likelihood close to each ofthese, until the global maximum is determined.

4.2

Neutron selection and background rejection

For this analysis, there are two components to the event selection; the primary event
selection to select e.g. CCQE νµ events, and the neutron selection to select the subsequent neutron captures associated with a given primary event. For measurements onother primary event types, such as νe events or non-CCQE events, both the primary andneutron selections could change, so long as the efficiency corrections are also changed
accordingly (see next section).
The primary event selection used in this analysis is the standard E61 single ring muonlike selection, which was originally developed for the νµ flux and cross-section measurement at νPRISM. Except where stated otherwise, the results presented in the remainderof this chapter are only for events passing this selection and for the 2.5° off-axis positionof the detector. The selection criteria for the primary event selection use the fiTQun
reconstructed variables, with the following requirements:97

1. A single muon-like Cherenkov ring is observed
2. The reconstructed muon momentum is at least 200 MeV
3. The muon is contained within the tank
4. The reconstructed interaction vertex is at least 1 m away from the tank wall
5. The vertex is at least 2 m away from the wall in the direction of the muon
6. There is no more than one sub-event (i.e. only no additional particles other thanthe decay electron)
This gives in an efficiency of 38 % for selecting fully contained CC νµ events while
rejecting 99 % of other background events, resulting in a 97 % pure sample containing a
total of 107 262 events from the 2.39 × 1020 POT simulated at 2.5° off-axis.
The above event selection has been performed for events which were simulated as occurring individually, with no pile-up of multiple events occurring together. To account for
this, the rate is adjusted by assuming that if multiple visible events occur together in asingle spill of the beam then both those events will be rejected. The rate of events withina given spill is assumed to be Poisson with the rate determined by the total number ofvisible events simulated divided by the number of spills corresponding to the simulated
POT. The expected rate of visible events per spill is 0.94, giving the proportion ofvisible events occurring with no other visible events in the same spill of 39 %.
The neutron selection begins by searching for triggers which occur within a time windowof 200 ns to 100 µs, the region in which the majority of captures occur while rejecting
non-neutron triggers associated with the primary event, as shown in the upper plotof Figure 4.3. In order for the trigger to be passed to the low energy reconstruction,a total charge seen in the PMTs is required to be equivalent to between 10 and 100
photoelectrons; the lower plot of Figure 4.3 shows that this includes the vast majority ofneutron captures while also ensuring reasonable reconstruction performance. This PMT
charge cut will also help reduce the low-energy background from non-neutron sources
such as radioactive decays.
The remaining selection criteria depend only on the time (in addition to the fixed timing
cut above) and the reconstructed position, both relative to the reconstructed primary
event position and relative to the tank’s geometry. Since the majority of the remaining
background are genuine neutron captures, but from neutrons produced from externalsources, this timing and positional information is the only information that can be used
to distinguish the signal from these backgrounds. In order to maximise the effectiveness of a cut using this information, a likelihood is constructed for each of the main
background neutron sources (sand neutrons, NC event neutrons, and cosmic neutrons),
as well as signal neutrons. In Figure 4.4, which shows the distributions of the timeand distance from the primary event for signal and backgrounds, it can be seen how
a likelihood cut that can follow the contours of these distributions has more power toseparate signal from background than individual cuts.
The definitions of the quantities used to construct the likelihoods are given in Table 4.2.
Each likelihood is then constructed as the product of four probability distributions –three for position and one for time – which may be simply flat distributions included toallow direct comparisons between likelihoods, as tabulated in Table 4.3. Additionally,98Events

3500
3000

Neutron captures
Other

2500
2000
1500
1000
500

10−2

10−1

1

102

10

Time after primary event [µs]

200

2000

180

1800

160

1600

140

1400

120

1200

100

1000

80

800

60

600

40

400

20

200

0
0

20

40

60

80

100

EventsReconstruction resolution [cm]

0
10−3

0
120

Total PMT charge [PE]Figure 4.3: Top: Time distribution of neutron capture triggers (blue) and non-neutron
triggers (black) after the primary event trigger. Bottom: Reconstructed distance fromtrue capture position (black) and event rate (blue) against total PMT charge in units
equivalent to single photoelectrons arriving at the PMT.

99

Reco. distance from primary event [cm]

300

180

250

160
140

200

120

150

100
80

100

60
40

50

20

0
0
Reco. distance from primary event [cm]

200

10

20

30
40
50
60
Reco. time after primary event [µs]

1000

0

220
200
180

800

160
140

600
120
100

400

80
60

200

40
20

0
0

20

40

60
80
100
Reco. time after primary event [µs]0Figure 4.4: Distance and time relative to primary vertex for signal (top) and background (bottom) neutrons. The likelihood method used allows a cut which accounts
for the shape of these distributions, providing better separation than individual cuts
corresponding to vertical and horizontal cuts on these plots.

100

to ensure compatibility between likelihoods, although the spatial part of the likelihoods
are each built using probability distributions of variables defined in polar or cylindricalcoordinates, these must be reparametrised through a change of variable into probability
distributions of a single consistent coordinate system. The probability distributions
have been determined by fitting the various distributions using Monte-Carlo simulated
events, as shown in Figures 4.5 to 4.8 with full details of the fitted distributions given in
Appendix A. For the case of sand neutrons and NC event neutrons, the timing part ofthe likelihood is complicated by the possibility that these neutrons’ primary events could
have occurred at a different time in any of the beam spill’s bunches1 . It is constructedfrom the distribution of the time the neutron capture is observed relative to its primary
event, convoluted with a uniform distribution corresponding to the time within thebunch that the primary event could have occurred, giving the time distribution relative
to the start of the bunch, tb . Assuming that the background neutron’s primary event
could then have occurred with equal likelihood in any of the eight bunches, a mixture
distribution is constructed of the eight time-shifted distributions of tb .
Position relative to primary vertexdθφDistance from primary event to neutron capture
Angle between beam direction and dAzimuthal angle about beam direction
Position relative to tank geometryyrϕVertical position in tank
Radial position in tankAzimuthal angle about tank axis
Time

t
tb

Time relative to primary vertex
Time relative to start of each beam bunch

Table 4.2: Definitions of the parameters used to construct the likelihoods given in
Table 4.3. All parameters are based on the reconstructed time and position of theneutron capture candidate and, where appropriate, the reconstructed time and positionof the primary event vertex.
While these likelihoods have been built in this analysis using Monte-Carlo simulations,with separate likelihoods for each background type, the background distributions could
instead be fit using data collected from spills in which no primary event is observed.
This would remove the model dependence as well as provide a single full likelihood that
includes all backgrounds from neutron sources as well as from other sources. With an
expected visible primary event rate of 0.94 per spill, approximately 39 % of spills will
have no visible primary event – slightly more than the number of spills with a single
visible event, many more than the number of selected primary events – which should
provide ample data to fit the distributions of the likelihood.
Calibration with neutron sources could also be used to help fit the likelihoods for thesignal. Neutron sources are under development [17] which could be inserted into the
The bunch structure of the T2K beam has been assumed, with eight bunches, each 58 ns long,
separated by 581 ns gaps.
1101

p.d.f

0.008
0.007
0.006
0.005
0.004
0.003
0.002
0.001

p.d.f.

0
0

100
200
300
400
500
600
Reconstructed neutron distance from reconstructed primary vertex [cm]

0.03

0.025

0.02

0.015

0.01

0.005

p.d.f

0
0

20
40
60
80
100
120
140
Reconstructed neutron time after reconstructed primary vertex [µ s]

1.2

1

0.8

0.6

0.4

0.2

0
−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Reconstructed cosine of angle from beam direction

Figure 4.5: Distributions of the distance d to neutron capture (top), time t of neutron
capture (middle) and angle θ of neutron capture (bottom) relative to primary eventfor signal neutrons, used to construct the signal neutron likelihood function. Fitted
distributions are shown in red with Monte-Carlo simulated data shown in blue.

102

Position from vertex

Position in tank

Time

d

θ

φ

y

r

ϕ

t

tb

SignalFittedFitted

Flat

—

—

—

Fitted—Cosmic
Sand
NC

—
—
—

—
—
—

—
—
—

FittedFittedFittedFittedFittedFittedFlat
FittedFittedFlat———
FittedFitted0.002

p.d.f.

p.d.f

Table 4.3: Construction of the signal likelihood and sand neutron, cosmic neutron andNC event neutron background likelihoods using distributions of the quantities given in
Table 4.2. Each likelihood is based on probability distributions corresponding to theposition and time of the neutron capture candidate. The table shows which quantities
are used and whether the distributions are fitted or flat.

0.0018

16

×10−6

14

0.0016120.00140.0012

10

0.001

8

0.000860.000640.000420.0002
0

−400

−200

0

200

0
0

400

20

p.d.f.

p.d.f.

0.025

0.4

0.02

0.3

0.015

0.2

0.01

0.1

0.005

0.5

1

1.5

2

2.5

80

100

120

×103
2

0.03

0.5

0
0

60

Reconstructed radial position squared [cm ]Reconstructed y-position [cm]
0.6

40

0
0

3

Angle of reconstructed position in tank from beam dir [rad]

20

40

60

80

100

120

140

Capture time [µs]Figure 4.6: Distributions of the vertical position y (top left), radial position r2 (topright), azimuthal angle ϕ (bottom left), and time tb (bottom right) of neutron capturesin the tank for sand neutrons, used to construct the sand neutron likelihood function.
Fitted distributions are shown in red with Monte-Carlo simulated data shown in blue.

103

p.d.f.

p.d.f.

0.0012

10

×10−6

9
8
7

0.001

6
0.0008540.0006320.0004

1
−400

−200

0

200

0
0

400

20

40

80

100

120

×103
2

Reconstructed radial position squared [cm ]p.d.f.

p.d.f.

Reconstructed y-position [cm]
0.4

60

0.35

10−1

0.3
10−2
0.25
0.2

10−3

0.15
10−4

0.1
0.05
0
0

10−5
0.5

1

1.5

2

2.5

3

0

20

40

60

80

100

Angle of reconstructed position in tank from beam dir [rad]

120

140

Capture time [µs]0.0024

p.d.f.

p.d.f.

Figure 4.7: Distributions of the vertical position y (top left), radial position r2 (topright), azimuthal angle ϕ (bottom left), and time tb (bottom right) of neutron capturesin the tank for NC event neutrons, used to construct the NC event neutron likelihood
function. Fitted distributions are shown in red with Monte-Carlo simulated data shownin blue.

0.0022

×10−6
25

0.002
0.0018200.00160.0014150.00120.001
100.00080.000650.00040.0002
0

−400

−200

0

200

0
0

400

Reconstructed y-position [cm]

20

40

60

80

100

120

×103

Reconstructed radial position squared [cm2]

Figure 4.8: Distributions of the vertical position y (left) and radial position r2 (right)of neutron captures in the tank for cosmic neutrons, used to construct the cosmic event
neutron likelihood function. Fitted distributions are shown in red with Monte-Carlosimulated data shown in blue.

104

detector to emit neutrons at known energy, location and time, which could then be used
to measure precisely the capture time and distance distributions. These distributions
would then be used to form the likelihoods that in this analysis have been determined
purely by simulation.
For each neutron capture candidate, a separate cut is then made on the difference
between the signal log-likelihood and each of the three background log-likelihoods. Figure 4.9 demonstrates this, showing the signal neutron log-likelihood log Lsig against thesand neutron log Lsand , with the cut on√log Lsig − log Lsand . The value of each of thesecuts has been optimised to maximize S/ S + B, where S and B are the accepted signal
and background rates respectively.

− 15

log L

sig

Note that the use of the time and, in particular, the position of the neutron capture
relative to the primary event vertex introduces a significant model dependence to themeasurement. This is due to the large theoretical uncertainties in the momenta ofneutrons produced in neutrino-nucleus interactions and will be dealt with as part of theefficiency corrections described in the next section.

− 20

− 25

− 30

− 35
−27

−26

−25

−24

−23

−22

−21

−20

log L

sand

Figure 4.9: Signal neutron and sand neutron log-likelihoods for signal events (blue) andbackground events (red), with cut on the log-likelihood difference log Lsig − log Lsand(black).
The final samples after likelihood cuts contain a total of 105 032 neutron capture candidates scaled to 1020 POT, with a sample purity of 92 % and efficiency2 of 75.6 %.
The effect of each of the cuts on the sample, with rates for signal and each type ofbackground, are given in Table 4.4.
Despite the relatively high rate of incoming background neutrons, the selections used
above demonstrate that it is still possible to make very pure samples, with high signal
efficiency, of neutrons associated with neutrino interactions. The samples here are for νµinteractions, which are expected to generally produce fewer neutrons than ν µ interactions
or other interaction modes, so the neutron sample for ν interactions is expected to have
Efficiency here is defined as the number of selected signal neutron captures divided by the true
number of signal neutrons which capture inside the ID.
2105Signal

Background106

Selected

Efficiency

PuritySand

CosmicNC

Radioactivity

Other

Total

All captures in ID
Captures on Gd
Captures on H

138939121830
17087

—
—
—

—
—
—

80806
67736
13037

761
658
102

9497
8253
1240

—
—
—

—
—
—

91064
76647
14379

All triggers
OD veto10 < PMT charge < 100
Reconstruct in ID
200 ns < t < 100 µs
Likelihood

132478131862125736115225112621105032

95.3 %
94.9 %
90.5 %
82.9 %
81.1 %
75.6 %

17.7 %
23.5 %
32.3 %
40.6 %
48.6 %
92.1 %

340828180017118131
86739
78779
5919

36598
8487
1386
1371
1368
134

15336
15336
13423
10398
8657
1106

159623
159623
86049
30223
30165
1827

64875
64826
44574
39713
0
0

6172594282882635631684451189708986

Table 4.4: Table of signal and background true and selected neutron capture candidate rates for 1021 POT of running E61 at the 2.5° offaxis position. This corresponds to 175 402 selected primary events. Backgrounds include triggers from neutrino interactions in surrounding
material (sand), cosmic rays, neutrino interactions in the tank with no visible primary event (NC), radioactive decay of impurities in PMT
glass and water, and other non-neutron capture triggers from the signal primary event.

an even higher signal purity, allowing for less strict cuts to increase the amount of data
available. However, in order to achieve the high purities and efficiencies seen here, aselection method has had to be used that produces efficiencies (and therefore event rates)that are highly model dependent, as mentioned above. To deal with this, a non-trivial
method of background and efficiency correction needs to be used.

4.3

Background and efficiency correction

In order to understand the origin of the model dependencies introduced through theselection criteria of the previous section, and how they can be addressed, it is useful tolook at the various sources of systematic effects that cause the shifts from the distributions of true variables to the distributions of observed variables. If the observables ofinterest are defined as the number of neutron captures and their positions relative tothe primary event, then there are four main sources of these effects:
1. Increase in neutron multiplicity due to addition of background events.
2. Decrease in neutron multiplicity due to efficiency of signal events:(a) from neutrons failing to trigger or reconstruct(b) from neutrons failing to pass the selection cuts
3. Changes in reconstructed neutron capture position due to detector and reconstruction resolutions
Fundamentally, the size of each of these effects varies only according to the position of
the neutron capture within the detector, e.g. the background rates, triggering efficiency
and reconstruction resolutions all depend on the distance from the tank wall. In terms of
the observables defined above, however, this dependence translates into a dependence on
the capture position relative to the primary event, which is where the model dependence
comes from: The overall effect of the above listed systematics depends on the capture
position relative to the primary event, which depends on the direction neutrons areejected from the nucleus and their momenta (which directly affects the distance they
travel while thermalising before capturing), which both have large model uncertainties.
The fact that these effects depend on the model due to the effective dependence on
the distance d and angle θ of the emitted neutron means that even if only the total
multiplicity is of interest, the multiplicity must be measured as a function of d and θ inorder to correct for the effects in a way that takes this dependence into account.
In effect, the model dependence can be removed by performing the measurement as
a function of d and θ, followed by a correction of the systematic effects which also
depend on d and θ. The model dependence is effectively removed due to the fact that,once the position of the capture relative to the primary event is specified, i.e. theposition of the capture within the tank is fixed, the size of the above effects for agiven event do not depend on the models’ predictions, only on the understanding of
the detector’s capabilities and the background rates at that position. This requires athorough understanding of the how each of the above effects depends on the position
within the tank. While Items 1 and 2b in the above list could be calculated using
the likelihoods described in the previous section, which are required to perform the107selection anyway, a thorough understanding of Items 2a and 3 depend on the calibration
programme of the detector, which is essential to producing a valid final result.
Performing the correction required to remove the model dependency is more complex for
multiplicity measurements; whereas it is usually only the case that detector resolution
effects like Item 3 cause migration between bins, with the background and efficiency
effects of Items 1 and 2 only causing an increase or decrease in the observed event rate,
for a multiplicity measurement all of these effects cause migration between bins, since
the rate (i.e. multiplicity) is itself an observable whose distribution is being measured.
So to correct for these effects, rather than simply subtracting the expected backgroundrates and dividing by the expected efficiency, a method is required to correct for bin
migration. In this analysis, the iterative unfolding method of D’Agostini3 [112] is used,with the implementation provided by RooUnfold [113]. To simplify the analysis, theneutron capture position observables are left as reconstructed quantities, meaning that
only Items 1 and 2 are corrected for and not Item 3, however a full analysis also unfolding
the detector resolutions on capture position should be possible using the same approach.
The basic principle of this unfolding method is as follows: Both the true and observed
distributions being measured are binned into N truth bins Ti , i = 1 . . . N and M observed
bins Oj , j = 1 . . . M (here it is assumed that N = M ). It is then assumed that the effects
in Items 1 to 3 above are known and well understood, which allows the calculation ofa response matrix describing the conditional probability P (Oi |Tj ) of an observation in
bin Oi given that the truth was in bin Ti . The aim of unfolding is to find the inverse
conditional probability P (Ti |Oj ) to use in statistically determining the true distribution
P (Ti ) from the observed distribution P (Oi ). Bayes’ theorem states that
P (Oj |Ti )P (Ti )P (Ti |Oj ) = P.
k P (Oj |Tk )P (Tk )(4.3.1)

The iterative unfolding method initially assumes some prior distribution for P0 (Tk ),which can then be updated through the iterative procedure:
P (Oj |Ti )Pn−1 (Ti )Pn (Ti |Oj ) = P,
k P (Oj |Tk )Pn−1 (Tk )Pn (Ti ) =X

Pn (Ti |Oj )P (Oj )(4.3.2)(4.3.3)jAfter a number of iterations, Pn (Ti ) gives the unfolded distribution. Using too higha number of iterations in this procedure can result in the amplification of statisticalfluctuations, however in the case of the multiplicities being unfolded in this analysis,
where the amount of data is fairly large, the number of multiplicity bins is small andwith steeply falling distributions, this has not been observed to occur.
To implement the unfolding in this analysis, tables of signal efficiency (x, y, z, d, θ) and
expected background rates µ(x, y, z, d, θ) are calculated, as a function of the primary
event position in the tank (x, y, z) and of the bins in d and θ, using the likelihoods ofthe previous section. With the assumption that the number of observed signal events
sig
sig
Nobs
follows a Binomial distribution, Nobs
∼ B(Ntrue , ), where Ntrue is the true numberThis is often referred to as Bayesian unfolding due to its use of Bayes’ theorem in constructing themethod.
3

108

bgof signal neutron captures, and the observed background rate Nobs
follows a Poissonbg
distribution, Nobs
∼ Pois(µ), the response matrix for the observed number of neutron
capture candidates Nobs can be calculated asmin(no ,nt )P (Nobs = no |Ntrue = nt ) =X
ns =0min(no ,nt )

=

X
ns =0

fN sig (ns ; nt , )fN bg (no − ns ; µ)
obs

obs

(4.3.4)!

µno −ns e−µnt ns (1 − )nt −ns,
ns(no − ns )!

where fN sig and fN bg are the binomial and Poisson probability mass functions, for
obs

obs

sig
bg
Nobs
and Nobs, respectively, and the sum over ns is over the possible number of observed neutrons that come from signal (with the remaining observed neutrons comingfrom background). This response matrix can then be used to unfold the multiplicity
distribution in each bin of d and θ.

The approach described here differs from typical uses of unfolding in that the unfolding
matrix is generated from analytic formulae of the distributions described above. Unlike
the more common case where the unfolding matrix is generated by simply filling ahistogram of the relevant quantities calculated by Monte-Carlo simulation, the approach
used here helps avoid the problem of over-fitting of statistical fluctuations from thesimulations. Note that this means that unfolding the same simulated sample of eventsused in generating the unfolding matrix, to reproduce exactly the true distributions, is
not relevant for the unfolding used here.
Note that this unfolding procedure is only required for multiplicity distributions; for the
distributions only of d and/or θ but not the multiplicity (i.e. the distribution of theposition of a neutron capture, regardless of how many neutron captures occur in total),
the correction is much simpler: The number of neutrons observed for each event can becorrected by simply subtracting the expected background and dividing by the efficiency
in each bin of d and α.

4.4

Results

For the selections described in Section 4.2, the selected neutron capture candidates arebinned in the reconstructed distance d from the primary event to the capture and thecosine of the reconstructed angle θ between the direction of the beam and the directionof d, according to the binning scheme given in Table 4.5. Distributions are then found
for the observed multiplicity of neutron capture candidates per event in each bin of dand cos θ, with corrections applied as described in the previous section.
Figure 4.10 shows the unfolding of the multiplicity distributions themselves for several
distance and angle bins. The effect of unfolding varies between bins, but overall it isfound to improve with increasing number of iterations up to five iterations, beyond which
there is no significant change. The remaining results in this section that use unfolding
use a fixed number of five iterations.

109

Bins in cos θBins in d [cm]-1.00 – -0.60-0.60 – -0.25-0.25 – 0.00
0.00 – 0.25
0.25 – 0.40
0.40 – 0.55
0.55 – 0.70
0.70 – 0.80
0.80 – 0.90
0.90 – 1.00

0 – 2525 – 40
40 – 50
50 – 60
60 – 75
57 – 95
75 – 115
115 – 140
140 – 180

Table 4.5: Bins of distance d and angle cos θ used in E61 neutron analysis
Using the multiplicity distributions in each distance and angle bin to produce a total
multiplicity distribution, for the number of neutrons observed across all bins, is a complex challenge. This is because the rates of neutrons observed at each distance and angle
are far from being independent of each-other; Appendix B discusses this in more detail,along with the approaches that could be used to address the problem.
Figure 4.11 shows the neutron multiplicity as a function of distance and as a functionof angle. Comparing the signal-only and signal plus background distributions it can beseen that the background is mostly included at the larger distances, as could be expected
due to the majority of signal neutrons capturing closer, as well as the primary event
selection ensuring that primary events are at least 1 m from the wall, which is close
to where the majority of background neutrons will capture. It can also be seen thatthe efficiency and background correction is successfully transforming the distribution
closer to the true distribution, with some shift of the capture distance towards largerd remaining. This shift is mainly due to the smearing effect of the detector resolution,
which has not been corrected for in this analysis, and is seen to also occur to a lesser
extent in the angular distribution.
The multiplicity as a 2D function of both distance and angle is shown in Figure 4.12.
Since the plots of both Figures 4.11 and 4.12 are not functions of the neutron multiplicity
itself – they show the average multiplicity as a function of distance and angle – the full
unfolding method of efficiency and background correction is not required, as describedin the previous section. However, as an additional check of the full unfolding method(applied for the full 3D distributions of multiplicity, distance and angle), the average
multiplicity in each distance and angle bin can be found by integrating over the multiplicity axis of the unfolded 3D distribution. This is also shown in Figure 4.12, alongside
the equivalent plot using the simple correction, demonstrating that both methods ofcorrecting for efficiency and background appear to work reasonably well.
The difference between the observed and the true distributions, before and after theefficiency and background correction, are shown in Figure 4.13. The correction is seen
to remove the general reduction in rates due to efficiency, while some differences remain.
The main difference is the shift from shorter to longer capture distances, and from more
forward going to less forward going neutrons, due to the smearing effects of detector

110d = 50 cm to 60 cm, cos θ = 0.25 to 0.4
Observed - true events

Observed - true events

d = 25 cm to 40 cm, cos θ = 0.9 to 1
300
200
100
0

100

50

0

−100

−50

−200
−100
−300
0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5

6

Neutron multiplicity

7

8

9

10

Neutron multiplicity

Observed - true events

Observed - true events

d = 75 cm to 115 cm, cos θ = −0.6 to −0.25 d = 140 cm to 180 cm, cos θ = 0.55 to 0.75
100
80
60
40
20
0

Selected sig+bg

150

Unfolded 1 iteration
Unfolded 2 iterations

100

Unfolded 5 iterations

50

0

−20
−40

−50

−60
−100

−80
0

1

2

3

4

5

6

7

8

9

10

Neutron multiplicity

0

1

2

3

4

5

6

7

8

9

10

Neutron multiplicity

Figure 4.10: Effect of unfolding in various d and cos θ bins of the full 3D multiplicity
distribution. The difference between true and observed number of events against number of neutron captures is shown for selected signal plus background neutron capture
candidates (red) and the unfolded rates after 1, 2 and 5 iterations (green solid, dashedand dotted respectively).

111

Neutron captures [event-1m-1]

0.7

True captures
Selected sig

0.6

Selected sig+bg
0.5

Corrected sig+bg

0.4
0.3
0.2
0.1
0
0

20

40

60

80

100

120

140

160

180

Neutron captures [event-1sr-1]

Distance from primary vertex [cm]True captures

0.1

Selected sig
Selected sig+bg

0.08

Corrected sig+bg
0.06

0.04

0.02

0
−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Cosine of angle from z-direction

Figure 4.11: Neutron multiplicities as functions of distance d and of angle θ. Shown inblue is the true distributions, as a function of the true distance and angle from primary
event. The observed signal-only distributions are shown in magenta, the signal withbackground distributions shown in red, and the background and efficiency correcteddistributions shown in green, all as functions of the reconstructed distance and angle
from the primary event. Errors on all distributions are statistical only.

112

0.09

0.6

0.08

0.4

0.07

0.2

0.06

0

0.05

−0.2

0.04

−0.4

0.03

−0.6
−0.8
40

60

80

100

120

140

160

180

0.09

0.6

0.08

0.4

0.07

0.2

0.06

0

0.05

−0.2

0.04

−0.4

0.03

0.02

−0.6

0.02

0.01

−0.8

0.01

−1
0

0

20

40

60

80

Distance from primary vertex [cm]

0.8

0.09

0.6

0.08

0.4

0.07

0.2

0.06

0

0.05

−0.2

0.04

−0.4

0.03

−0.6
−0.8
20

40

60

80

100

120

140

160

120

140

160

180

0

Unfolded signal + background
0.1

180

0.1

0.8

0.09

0.6

0.08

0.4

0.07

0.2

0.06

0

0.05

−0.2

0.04

−0.4

0.03

0.02

−0.6

0.02

0.01

−0.8

Cosine of angle from z-direction

1

Neutron captures [event-1m-1sr-1]

Cosine of angle from z-direction

Corrected signal + background
1

−1
0

100

Distance from primary vertex [cm]

−1
0

0

Neutron captures [event-1m-1sr-1]

20

0.1

0.8

Cosine of angle from z-direction

0.8

−1
0

1

Neutron captures [event-1m-1sr-1]

Selected signal + background0.1

Neutron captures [event-1m-1sr-1]

Cosine of angle from z-direction

Selected signal only
1

0.01

20

40

60

80

100

120

140

160

180

0

Distance from primary vertex [cm]Distance from primary vertex [cm]

0.1

0.8

0.09

0.6

0.08

0.4

0.07

0.2

0.06

0

0.05

−0.2

0.04

−0.4

0.03

−0.6

0.02

−0.8

0.01

−1
0

20

40

60

80

100

120

140

160

180

Neutron captures [event-1m-1sr-1]

Cosine of angle from z-direction

True neutron captures
10

Distance from primary vertex [cm]Figure 4.12: Neutron multiplicity as a function of distance d and angle θ. The observed
signal-only distribution is shown on the top left and the signal with background distribution shown on the top right. The background and efficiency corrected distributions
are shown in the middle, with the correction without unfolding on the left and the correction with unfolding on the right. All these distributions are given functions of thereconstructed distance and angle from the primary event. Also shown on the bottom isthe true distribution, as a function of the true distance and angle from primary event.

113

0.2

After correction

0.1
0
−0.1
−0.2
−0.3
0

20

40

60

80

100

120

140

160

180

0.25
0.2
0.15
0.1
0.05
0
−0.05
−0.1
−0.15
−0.2
−0.25
−1

−0.8

−0.6

−0.4

−0.2

0.006

0.6

0.004

0.4
0.002

0.2
0

0

−0.2

−0.002−0.4−0.004

−0.6
−0.8
−1
0

−0.006

20

40

60

80

100

120

140

160

0

0.2

0.4

0.6

0.8

1

Cosine of angle from z-direction

Cosine of angle from z-direction

1
0.8

corrected - true n-captures [event-1m-1sr-1]

Cosine of angle from z-direction

Distance from primary vertex [cm]180

Distance from primary vertex [cm]

1
0.006

0.8
0.6

0.004

0.4
0.002

0.2
0

0

−0.2

−0.002−0.4−0.004

−0.6
−0.8
−1
0

statistical error [event-1m-1sr-1]

Before correction(observed-true)/true n-captures [event-1sr-1]-1(observed-true)/true n-captures [event m-1]

0.3

−0.006

20

40

60

80

100

120

140

160

180

Distance from primary vertex [cm]Figure 4.13: The difference between the observed and the true distributions of neutron
capture distance (top left) and angle (top right), before and after the efficiency andbackground correction, with statistical errors, and the difference between the corrected
and true 2D distribution (bottom left) with its statistical errors (bottom right).
resolution on the neutron capture position, which is not accounted for in the unfolding
procedure used here. An extension of this procedure to account for these detector
effects on the distance and angle, unfolding these quantities as well as the multiplicity
at each distance and angle that is currently unfolded, should remove these effects andbring the overall difference to within the statistical errors. Already it can be seen thatexcept at the extremes of the distributions the difference after unfolding is consistently
smaller than the statistical errors; while the worst case is a 15 % reduction in the rate
at the shortest distance and a 5 % reduction of the most forward going neutrons, in allbins other than the shortest distance (0 cm to 25 cm) and the two extremes in angle(−1 < cos θ < −0.6 and 0.9 < cos θ < 1) the difference is within the statistical errors ofaround 2.5 %.

Conclusion
The intermediate water Cherenkov detector for Hyper-K aims to produce measurements
to help constrain the systematic uncertainties at the far detector. In the previous
chapter, the reduction in errors on oscillation analyses was seen when including thesemeasurements, with additional information provided through neutron tagged samples.

114

In this chapter, a proof-of-concept analysis has shown that very pure samples of Gadolinium neutron captures in the IWCD can be produced, despite the significant background rates, while still retaining a large portion of the signal. In addition to providing
neutron tagged samples, for which a demonstration of their effect on oscillation physics
was given in the previous chapter, the large neutron samples also provide distributionsof neutron capture distance, angle and multiplicity, giving additional input to various
analyses at the far detector of Hyper-K.
The distributions can help understand the neutron capture signal at the far detector,
to help improve the purity of samples produced there, as well as to enhance the ability to use neutron tagging at the far detector to reduce backgrounds for nucleon decay
searches. The detection of neutrons at the Hyper-K far detector – whether throughneutron captures on Gadolinium in doped water or on Hydrogen in undoped water –depends on understanding the positions the neutron are expected to capture. Understanding this through measurements like those presented in this chapter – the distanceof captures from the primary event and the angle the neutron travels relative to the incoming neutrino – will allow for efficient selection of neutrons in the far detector, which
will in turn provide purer samples for neutrino analyses, including separation of ν/νdiscrimination which is not otherwise possible in water Cherenkov. This efficient selection of neutrons will also help maximise the sensitivity to nucleon decay by identifying
and removing as much as possible of the main background coming from atmospheric
neutrinos, as described in Section 2.5.2.
Additionally, these distributions of distance and angle of neutron captures can provide
information related to the magnitude and direction of outgoing neutrons’ momenta.
Since neutrino interaction models currently have very high uncertainty in this area,
improvements could be made to these models using the new information from neutron
measurements. With the systematic uncertainties in the interaction models one of themost significant sources of error in oscillation analyses, any improvement to these models
will help to improve the sensitivities of these analyses.
Although the analysis presented here has only been performed for the E61 CCQE νµsample, the tools developed for this can be extended to other samples, to provide estimates of the ability to resolve the differences in the distributions for these other samples,
and therefore the ability to use neutron captures to increase separation of interaction
modes and neutrino flavours. The analysis can also be extended to the other off-axis
angle positions of E61, to study the ability to use linear combinations of these positions to effectively measure different desired neutrino energy fluxes; for example, acombination could be chosen to more closely match the atmospheric neutrino flux inorder to better understand the use of neutron tagging to reduce atmospheric neutrino
backgrounds.
In order to fully study the sensitivities to neutrons of E61, a thorough understanding of the systematic uncertainties involved in neutron captures is required, which has
not been included here. The number and type of these systematics depends on theanalysis being performed, but in all cases the most important sources of uncertainty
come from the modelling of neutrons propagating in the detector and their capture on
Gadolinium, which was briefly investigated in Section 4.1.1.1. Additional systematics
come from the methods used to separate signal from background; this includes both themodelling of backgrounds, or the ability to measure them in E61 independently through115

background-only measurements, as well as an understanding of the detector resolutions
and calibration which has a significant effect on the likelihood method used to remove
backgrounds and also the background and efficiency corrections applied later on. Other
sources of systematic errors are largely the same as other analyses of E61, such as theneutrino flux and interaction cross-sections, however since the overall primary event rate
only affects the size of the samples and is not part of the measurement itself, these areless critical.
While further study is required to fully understand the benefits of Gadolinium doping
and the neutron capture signal in the IWCD, the results of this chapter and the previous
chapter have shown that high-quality neutron measurements are feasible and that they
provide clear benefits to the Hyper-K project.

116

Chapter 5

Sensitivities and synergies of
T2HK and DUNE

117

Introduction
For any future experiment, it is important to understand its capabilities not just in
isolation, but also in the context of existing running experiment and future planned or
proposed experiments. Through previous and current oscillation experiments, existingknowledge of the neutrino sector has undergone a sea-change over the last decade. The
oscillation mechanism has been well established as the explanation of the anomalous
solar and atmospheric neutrino flavour ratios, and the paradigm has been subjectedto scrutiny from long-baseline accelerator and reactor experiments resulting in a measurement of the final mixing angle θ13 [13, 14, 114, 115]. Although some short-baseline
anomalies still remain unexplained [116–118], the oscillation mechanism has leapt many
hurdles to become a part of the new Standard Model (SM). However, some significant
unknowns remain: the ordering of neutrino masses, the existence and extent of CP violation (CPV) in leptonic mixing, and the precise value, including crucially the octant,of θ23 . In addition, the current precision on the oscillation parameters is insufficient torule out many theoretical models, for example those discussed recently in [2, 119–121].
With the intention of building on the progress of the oscillation programme, the international community has conceived of a range of future facilities with the potential
to explore the final unknowns in the conventional oscillation paradigm, and to hunt
for tensions in the data which might indicate that a richer extension of the SM isrequired. There are three major strands in the future experimental neutrino oscillation programme: short-baseline experiments such as those comprising the SBN programme [122], intermediate baseline reactor facilities, RENO-50 and JUNO [123–125],and long-baseline experiments such as LBNF-DUNE and T2HK [18, 126–130]. In thischapter the focus is on these latter two proposals for novel long-baseline facilities: LongBaseline Neutrino Facility-Deep Underground Neutrino Experiment (LBNF-DUNE, referred to subsequently as DUNE) and the beam programme of Hyper-Kamiokande (referred to here as T2HK). DUNE is the flag-ship long baseline experiment of the Fermilab neutrino programme [18, 130]. It consists of a new beam sourced at Fermilab
and a detector complex at Sanford Underground Research Facility (SURF) in South
Dakota separated by a distance of 1300 km. Over this distance, neutrinos producedin the decays of secondary particles from proton collisions at Fermilab will propagate,
undergoing oscillations and scattering processes in the matter of the Earth. The appreciable matter effects will modify the probability of detecting a given flavour of neutrino,
in a way that will ultimately make the facility highly sensitive to the mass ordering,parametrized by the sign of ∆m232 , while the broad spectrum of events arising from its
on-axis flux also allows for significant sensitivity to the unknown CPV phase δ. The
detector will use Liquid Argon Time Projection Chamber (LAr-TPC) technology, allowing for strong event reconstruction. As a result, a high signal to background ratio is
expected. In contrast, as described in detail in Chapter 2, T2HK [129] was conceived
with a smaller baseline of 295 km and a different detector technology, employing Water
Cherenkov technology at a significantly larger scale. Matter effects for this facility will
be smaller due to the shorter baseline (although non-negligible), and the significantly
enhanced event rate will allow for a high-statistics comparison between neutrino andanti-neutrino modes, searching for fundamental asymmetries due to the CP violatingphase δ. Another key difference between T2HK and DUNE is the narrow-band beamof T2HK versus the wide-band beam of DUNE. While the narrow-band beam provides
118

an energy spectrum which peaks in the oscillation maximum and at the energy where
the clean CCQE interactions are dominant, as described in Section 2.1, a wide-band
beam can provide a larger total neutrino flux with the possibility of measuring neutrino
oscillations at different energies from one beam, at the expense of a larger uncertainty
on the incoming neutrino energy. These two approaches can therefore be expected toprovide complementary information.
Much work has been done over the years assessing the physics reach of T2HK [129, 131,132] and DUNE [18, 133–136] (along with its predecessor designs LBNE [130, 132, 137–
139] and LBNO [132, 140, 141]). This chapter is based on the work published in [1],where the physics sensitivity of DUNE and T2HK is revisited for key measurements
relating to the mass ordering, δ phase and the mixing angle θ23 , focusing in particularon the combined reach of these designs. Recently, as the designs for T2HK and DUNE
have matured, both collaborations have been considering significant alterations to thebenchmark proposals in [129] and [18, 133]. The nuPIL (neutrinos from a PIon beamLine) design [142–144], developed by a working group of the DUNE collaboration, is a
novel beam technology building on accelerator R&D work done for the neutrino factory [145]. It foresees the collection and sign selection of pions from a conventional beam,
which are directed though a beam line and decay to produce neutrinos. This selection
and manipulation of the secondary beam forces unwanted parent particles out of thebeam resulting in a particularly clean flux. This screening process presents a particularadvantage over conventional neutrino beams, where the contamination of the flux due
to mesons of the wrong sign can limit the sensitivity of the antineutrino channel. In
this case, the contamination from intrinsic νµ is effectively enhanced by the cross-section
differences. This increases the relative number of wrong-sign events, and reduces the signal over background ratio. The simulated flux is also notably narrower than the DUNE
reference design (although this could be changed through modification of the design)
which will alter the sensitivity to the oscillation probability. In a parallel development,
T2HK has reconsidered the location of its second detector module. The current design
divides the detector into two modules installed at Kamioka following a staged implementation [17]: an initial data-taking period would use a single tank during which thesecond tank would be constructed and would start taking data after 6 years to further
boost the statistical power of the experiment. Instead of this plan, the suggestion has
been made to locate the second tank in South Korea at a baseline distance of between
1000 km to 1300 km from J-PARC [69]. This would allow T2HK + Korea (T2HKK)
to collect data from two different baselines and with two different off-axis angles (andconsequently energy spectra), crucially altering the phenomenology of the experiment.
Work on both of these proposals is ongoing.
Although the question of the combined sensitivity of DUNE and T2HK has been studied
before (most recently in [146]), this work brings three new elements to the discussion.
Firstly, the significant redesign and development work that has been performed in thelast few years on both designs has been incorporated. The simulation of T2HK is particularly noteworthy, departing significantly from those used in previous comparable analyses [146] by incorporating up-to-date information about detector performance from thecollaboration’s in-house simulation, and has been carefully calibrated against previously
published results. Secondly, the precision measurement of δ and its phenomenology isthoroughly addressed, often deemed a secondary question in earlier studies that focus
mainly on the discovery of CP violation, but one which is increasingly central to the119aims of the long-baseline programme. Finally, a detailed discussion is provided of thedifferences between the two designs as well as their possible redesigns (nuPIL, T2HKK)
and a quantification of their complementarity in an attempt to identify the optimal
choice from a global perspective.
This chapter begins with a brief recap of the relevant phenomenology of oscillation physics in Section 5.1. In Section 5.2, descriptions are provided of the details of DUNE andT2HK (including their alternative designs) taken into account in these simulations. Section 5.3 is devoted to the results of the simulations assuming the standard configurationsof each experiment which look at mass ordering sensitivity, CP violation discovery, theability to exclude maximally CP violating values of δ, the expected precision on θ23 andthe ability to resolve the octant. An analysis of the complementarity for precision on δis presented in Section 5.4, taking care to discuss the interplay of factors which influence
this measurement. In Section 5.5, these physics goals are reconsidered in light of thealternative deigns for DUNE and T2HK.

5.1

Oscillation phenomenology at DUNE and T2HK

Long-baseline experiments such as DUNE and T2HK aim to improve knowledge of the
PMNS mixing matrix U , as well as the atmospheric mass-squared splitting ∆m232 , by
the precision measurement of both the appearance νµ → νe and disappearance oscillation channels νµ → νµ , as well as their CP conjugates. In this section, the key aims of
the long-baseline program will be discussed, along with the important design featuresof these experiments which lead to their sensitivities. To facilitate this discussion, theapproximation of the appearance channel following [147] is used, which is derived by performing a perturbative expansion in the small parameter  ≡ ∆m221 /∆m231 ≈ 0.03 under
the assumption that sin2 θ13 = O()1 . The expression for the oscillation probability isdecomposed into terms of increasing power of , P (νµ → νe ; E, L) ≡ P1 + P 3 + O 2 ,(5.1.1)2where E is the neutrino energy, L the oscillation baseline, and the ordered terms
Pn = O(n ) are given by4(1 − rA )∆LP1 =sin2 θ23 sin2 θ13 sin2,
(1 − rA )22
(5.1.2)∆L
rA ∆L(1 − rA )∆L
P 3 = 8Jr
cos δ +
sin
sin,
(5.1.3)2
rA (1 − rA )2
22√
where Jr = c12 s12 c23 s23 s13 , rA = 2 2GF Ne E/∆m231 and ∆ = ∆m231 /2E. Using thesame scheme, the disappearance channel can be written at leading order as








P (νµ → νµ ; E, L) = 1 − sin2 (2θ23 ) sin2





∆L
2





+ O().

(5.1.4)For both channels, equivalent expressions for antineutrino probabilities can be obtainedby the mapping rA → −rA and δ → −δ.
1

For alternative schemes of approximation, see [148–151].

120

5.1.1Mass ordering, CPV and the octant of θ23

The sensitivity of long-baseline experiments to the questions of the neutrino mass ordering, the existence of CPV and the octant of θ23 , are by now well studied topics (fora recent review see e.g. [152]). To help clarify the role of the designs of DUNE and
T2HK, as well as their possible modifications, it is useful to understand how experimentson these scales derive their sensitivities using the approximate formulae expressed by
Equations (5.1.2) to (5.1.4).
The dependence on the sign of ∆m231 , and therefore the mass ordering, arises at longbaseline from the interplay with matter, where forward elastic scattering can significantly
enhance or suppress the oscillation probability. This is governed by the parameter rA
in Equation (5.1.1), which for the experiments of interest is small, and goes to zero in
the absence of matter. Changing from Normal Ordering (NO, ∆m231 > 0) to Inverted
Ordering (IO, ∆m231 < 0) requires the replacements ∆ → −∆ and rA → −rA . However,in vacuum (rA = 0) the leading-order term in Equation (5.1.1) remains invariant under
this mapping. This invariance is broken once a matter term is included (rA 6= 0), andthe oscillation probability acquires a measurable enhancement or suppression dependent
on the sign of ∆m232 . The size of this enhancement increases with baseline length, andthis effect is expected to be very relevant for appearance channels at a long-baseline
experiment νµ → νe and ν̄µ → ν̄e . However, the determination of the mass ordering
is further facilitated by the contrasting behaviour of neutrinos and antineutrinos. Dueto the dependence on rA , larger values of the matter density cause an enhancement in
the probability for νµ → νe oscillation at the first maximum for NO, whilst suppressing
the probability for ν µ → ν e . This behaviour is reversed for IO, with neutrinos seeing a
suppression and antineutrinos, an enhancement. Moreover, matter effects also affect theenergies of the first oscillation maxima for neutrinos and antineutrinos. Through precise
measurements around the first maxima, these shifts can be observed allowing longbaseline oscillation experiments to determine the mass ordering. It is worth noting thatan analogous analysis can be applied to the oscillations of atmospheric neutrinos, which
can come either directly from the atmosphere above or through the Earth below, with thevariable sized matter effect (and baseline) that this implies. Therefore the observation of
atmospheric neutrinos at the same detectors as used in T2HK and DUNE could provide
significant additional sensitivity to the mass ordering. In this study, however, only thelong-baseline accelerator neutrino analysis has been included.
To detect CPV in neutrino oscillations an experiment requires sensitivity to δ. Unfortunately, the leading order appearance probability is independent of the CP phase δ, withCP asymmetries between neutrino and antineutrino channels first appearing with thesubdominant term P 3 . In the presence of a background medium, CP violating effects are2
instead introduced in P1 ; however, these offer no sensitivity to the fundamental CP violating parameter δ, arising instead from the CP asymmetry of the background medium
itself via the parameter rA . As the sensitivity to δ is subdominant and masked by CP
asymmetry arising from matter effects, extracting the CP phase is a more challenging
measurement, requiring greater experimental sensitivity. Long baseline (LBL) experiments can obtain sensitivity to δ by looking not only at the first maximum but also at
the spectral differences between CP conjugate channels. In particular, an important roleis played by low-energy events in the sensitive determination of δ [139, 153–155]: aroundthe second maximum, CP dependent terms of the oscillation probability are more sig121

nificant. Although accessing these events can be a challenging experimental problem,and low statistics or large backgrounds could limit their potential [153], their benefit isclear from recent experimental work [156].
The atmospheric mixing angle is known to be large and close to maximal θ23 ≈ π/4, but
it is not currently established whether it lies in the first octant θ23 < π/4 or the second
octant θ23 > π/4. It is seen in Equation (5.1.2) that the appearance channel is sensitive
to the octant. However, it is also seen that changing the octant enhances or suppresses
the first maximum of the appearance channel in much the same way as the matterenhancement. For this reason, the sensitivity to these two effects can be expected to becorrelated and difficult to disentangle; however, this correlation will be reduced whendata from both neutrino and antineutrino is available as this effect is the same in both
CP conjugate channels. The determination of θ23 is also known to be beset by issuesof degeneracy with δ which can complicate its determination [152, 157, 158]. As both ofthese parameters enter the second-order terms in Equation (5.1.2), the freedom to varyδ can be used to mask the effects of a wrong octant, making their joint determination
more challenging. Fortunately, a precise measurement of sin(2θ23 ) is possible through its
leading-order dependence of the disappearance channel, seen in Equation (5.1.4). This
helps to break this degeneracy and, along with spectral information, is expected to help
mitigate this problem.

5.1.2

Precision on δ

Although the question of the existence of leptonic CP violation often dominates discussions about δ, the precision measurement of δ could prove to be the most valuable
contribution of the long-baseline programme. To determine the existence of fundamental
leptonic CP violation it suffices to exclude the CP conserving values δ = 0 and δ = π,those values corresponding to a vanishing Jarlskog invariant. Therefore the discovery
potential of a facility to CP violation is fundamentally linked to the precision attainable for measurements of δ in the neighbourhood of 0 and π. However, the question of
precision on δ goes beyond CP violation discovery. Many models of flavour symmetries,
for example, are consistent with the known oscillation data and make predictions forδ.2 No experiment on comparable time-scales will be able to compete with precision
measurements of δ from DUNE and T2HK.
An understanding of precision measurements of δ can be developed analytically following the approach of [164]. The assumption can be made that the precision is dominatedby events at a fixed energy, which could be due to a narrow beam or simply the predominance of the first maximum, and implies the number of events is proportional tothe oscillation probability P (να → νβ ; E). Under the assumption that the statisticaluncertainty dominates, and the number of events in that bin fluctuates in a Poissonian manner, the range of ∆δ which corresponds to this statistical uncertainty can beinferred,√P∆δ ∝ ∂P ,(5.1.5)∂δFor example, recent studies of mixing sum rules can be seen as predicting δ for long-baseline experiments [159–163]. For many more examples of predictive flavour symmetric models, see e.g. thereview [44].
2122

where the contribution from the higher order terms O ∆δ 2 has been neglected. Combining neutrino and anti-neutrino channels, a simple formula can be derived for the
precision on δ [164],1.
(5.1.6)∆δ ∝ p1 + cos (2δ)
This approximate formula well describes the qualitative features of the precision on δ
which are shown in Figure 5.1, and its general shape has been found in simulations ofspecific experiments [164].

Δ𝛿

1
Δ𝛿 ∝ √1+cos
2u�

-1

-0.5

0
True 𝛿/𝜋

0.5

1Figure 5.1: ∆δ against δ for the measurement of neutrino and antineutrino appearance
channels in vacuum. This is given by the approximation Equation (5.1.6) as discussed
in more detail in [164]. Although matter effects and finite beam width modifies thisshape, its characteristic peaks will still be discernible.
A notable feature of the analytic formula in Equation (5.1.6) is the total loss of sensitivity at values of δ which correspond to maximal CPV, δ ∈ { π2 , 3π2 }. These are seen
as vertical asymptotes in Figure 5.1. Clearly this marks a breakdown in the approximations that have been made, but the general fact that the maximal CPV (δ = ± π2 )corresponds to the worst precision in δ is true in more realistic simulations. This canbe understood qualitatively with reference to the approximate oscillation probability
given in Equation (5.1.3). Looking at the CP sensitive term at energies around the firstmaximum, where ∆L/2 ≈ π/2, the probability can be approximated byrA ∆L(1 − rA )∆LP 3 ≈ −8Jrsin δ sinsin.
2rA (1 − rA )
2
2


123







The highest sensitivity to δ is found when this function is most sensitive to changes in
δ, information naturally encoded in the function’s first derivative. Due to the sinusoidal
nature of the function, when the CP term has its largest effect (| sin δ| = 1), it is at a
maximum and consequently its gradient is at a minimum. Therefore, it is expected thatthe errors on δ will be small around 0 and π, when even though the absolute size of the
CP sensitive terms are small, they are most sensitive to parameter shifts. Taking matter
into account leads to a slight modification of the location of the worst sensitivity [164].
Assuming the measurement is close to the first oscillation maximum, and introducing
a dimensionless parameter ξ to describe the deviation from this point (where ξ = 0
corresponds to the first maximum), the relevant parameter governing the phase of thesinusoidal terms can be expressed by∆L = π1+ξ
.
1 − rA

(5.1.7)The value of δ for which the worst sensitivity is expected can then be found by minimising
the gradient of Equation (5.1.3), which occurs for the values
δ≈−

π 1+ξ
+ πn,
2 1 − rA

(5.1.8)for n ∈ Z. From this formula it is clear that the value of δ with the worst sensitivity
in a direction governed by the signs of rA and ξ. Specifically,
shifts away from (2n+1)π2
the dependence on rA means that the neutrino and anti-neutrino mode sensitivities atfixed energy have their worst sensitivity for different true values of δ. Running both
CP conjugate channels in a single experiment allows each channel to compensate forthe poorer performance of the other at certain values of δ, helping to smooth out the
expected precision. In this way, the multichannel nature of LBL experiments allows fora greater physics reach than a single channel experiment.
The argument above assumed that all events came from a fixed energy defined implicitlyby ξ in Equation (5.1.7). Due to the dependence on ξ in Equation (5.1.8), having
information from different energies will also be complementary, acting analogously to the
combination of neutrino and antineutrino data by mitigating the poorest performance.
Although all LBL experiments aim to include the first maximum, where event rates
are highest, none have a purely monochromatic beam and so-called wide-band beams
include considerable information from other energies. Therefore such experiments canbe expected to avoid the significant loss of sensitivity predicted by the simple analytic
formula. It can be inferred, however, that a narrow beam focused on the first maximum
in the presence of small matter effects should have a worse sensitivity at maximal valuesof δ compared to CP conserving values [164].
With reference to the traditional designs of T2HK and DUNE, from the above discussion
it can be inferred that T2HK can be expected to have a greater range of expected
precisions as δ is varied than DUNE. In particular, due to its narrower beam and small
matter effects, markedly poorer performance is expected for T2HK at δ ∈ {− π2 , π2 }than at δ ∈ {0, π}. DUNE on the other hand will be less variable as its broad band
mitigates the total loss of sensitivity at certain energies, and its large matter effect helps
to stabilise performance, but it can be expected to see its worst sensitivity at values of δslightly displaced from 0 and π, where the sensitivity at the first maximum is worst. Thissuggests a degree of complementarity of the wide-band and narrow-band beams when124

it comes to precision measurements of δ: a narrow-band focused on the first maximum
is optimal for precision around 0 and π (and by implication, for CPV discovery) whilea wide-band beam should perform better for precision measurements around δ = ± π2 .
This general behaviour will be relevant not only for the traditional designs of DUNE
and T2HK, but also their possible redesigns: nuPIL could lead to a narrowing of theneutrino flux, and T2HKK could see a wider-band component in its flux, or a narrowband component focused away from the first maximum. The interplay of these factors
will be explored in more detail in Section 5.4.

5.2

Simulation details

To better understand the sensitivities and complementarity of DUNE and T2HK (including their potential redesigns), simulation have been performed of the experiments
in isolation and in combination. The General Long Baseline Experiment Simulator(GLoBES) libraries [165, 166] have been used and the following sections describe thefeatures of the modelling of the two facilities and the statistical treatment.

5.2.1

DUNE

The DUNE experiment consists of a new neutrino source, known as Long Baseline
Neutrino Facility (LBNF), a near detector based at Fermilab and a LArTPC detector
complex located in SURF a distance of 1300 km away. Several variants of the LBNF
beam have been developed. In this work, three neutrino fluxes are studied: a 2-hornreference beam design [18], a 3-horn optimised beam design [167], and the neutrinos
from a PIon beam Line (nuPIL) [142–144]. All three fluxes used in these simulations
are shown in Figure 5.2.
The reference beam uses a front-end design based on NuMI, which uses 2 magnetic
horns, but with a thickened target to withstand the higher power beams of LBNF. In
the GLoBES simulation, the proton energy is taken to be 80 GeV, the beam power
1.07 MW, and 1.47 × 1021 protons on target (POT) are assumed per year. Recent development work by the DUNE collaboration has led to the design of an optimised beam.
This 3-horn design is based on the reference design, but has a stronger focus on producing lower energy events, leading to a substantial increase in flux between 0.5 GeV
and 4 GeV. This leads to a greater number of expected events from around the second
oscillation maximum, which is well-known to be particularly sensitive to the phase δ.
For this design, the proton energy is assumed to be 62.5 GeV and the POT per year istaken as 1.83 × 1021 . The nuPIL design is also considered, which has been studied bya working group of the DUNE collaboration as a potential alternative design. nuPIL
foresees the collection and sign selection of pions from proton collision with a target,
which are then directed though a beam line and ultimately decay to produce neutrinos.
This selection and manipulation of the secondary beam forces unwanted parent particles
out of the beam, resulting in lower intrinsic contamination of the neutrino (antineutrino)
flux by antineutrinos (neutrinos). In particular, this will improve the signal to background ratio of the antineutrino mode compared to a conventional neutrino beam. The
proton energy for this design is assumed to be 80 GeV, and the corresponding POT peryear is 1.47 × 1021 . Compared to the other two designs, nuPIL offers a lower intrinsic125

4
2 horn3 hornnuPIL

Flux [10-11/GeV/m2/PoT]

3.5
3
2.5
2
1.5
1
0.5
0
1

2

3
E [GeV]45

6

4
2 horn3 hornnuPIL

Flux [10-11/GeV/m2/PoT]

3.5
3
2.5
2
1.5
1
0.5
0
0

500

1000

1500
2000
L/E [km/GeV]

2500

3000

Figure 5.2: Top: νµ (ν µ ) flux component in ν-mode (ν̄-mode) shown as solid (dashed)lines for 2-horn reference, 3-horn optimised, and nuPIL beam designs. Bottom: Fluxesfor ν-mode shown as a function of L/E. In both panels, the shaded region shows theenvelope of the oscillation probability as δ is varied over its full range. The black lines
in the bottom panel show the probability for δ ∈ {0, π2 , π, 3π
2 }.
126

contamination from other flavours and CP states while maintaining low systematic uncertainties. It is also noted that nuPIL expects a smaller total flux, although this might
be avoidable through further design effort. Another characteristic of the nuPIL design
is its notably narrower flux. As events from the second oscillation maximum are expected to be highly informative about the true value of δ, this may impact the sensitivity
to δ. The coverage of first and second maxima is seen clearly in the bottom panelof Figure 5.2, where the fluxes are shown as a function of L/E. The first maximum(L/E ≈ 600 km GeV−1 ) is covered comparably well for all three flux designs, while theflux at the second maximum (L/E ≈ 1800 km GeV−1 ) varies significantly. The reference
2-horn design falls almost directly between the two alternatives: nuPIL offers far fewer
events at this value of L/E and the 3-horn design, far more.
Although alternative fluxes are considered, the same detector configuration is always
assumed of four 10-kiloton LArTPC detectors at 1300 km from the neutrino source.
The possibility of staging is neglected, assuming that all four tanks are operational at
the same time, and the expected improvement in performance throughout the lifetimeof the detectors is not accounted for. LArTPC technology has a particularly strong
particle identification capability as well as good energy resolution which are both crucial in providing high efficiency searches and low backgrounds. The LArTPC detector
response is modelled with migration matrices incorporating the results of Monte Carlosimulations undertaken by the collaboration. These matrices quantify the migration of
events from the true channel to the observed chanel, and from true to reconstructed
incoming neutrino properties, due to the detector resolutions, backgrounds and efficiencies. Fourteen migration matrices are used – seven each for the disappearance andappearance channels – describing the detection and reconstruction of all three flavours ofneutrino and antineutrino, as well as generic flavour blind NC events [168]. The channel
specific efficiencies are taken from [18].
Both appearance and disappearance searches are included in this study. The appearance
channel signal is taken as the combination of νe and ν̄e charged-current (CC) events. For
the disappearance channel, νµ and ν̄µ are studied for neutrino and antineutrino modes,
respectively. The backgrounds to the appearance channel are taken as neutral-current(NC) events, mis-identified νµ CC interactions, intrinsic νe CC events, and ντ CC events.
On the other hand, in νµ (ν̄µ ) disappearance NC events, ν̄µ (νµ ) CC events, and ντ CC
events are all considered. These considerations are the same as the study in [18]. The
rates of these backgrounds are governed by the migration matrices.
The same systematic errors are then assumed for all beam designs. The reduction of thesystematic errors is an ongoing task in the DUNE collaboration, and the values used are
based on the conservative end of the current estimates of 1 % to 2 % [18]. As such, an
overall normalization error is taken on the signal (2 % for appearance and 5 % for disappearance) and on the background rates (5 % for both channels). This accounts for fully
correlated uncertainties on the event rates in each bin, and uncorrelated uncertainties
are not considered. The nuPIL design could feasibly lower the systematic error withrespect to the conventional design, although the extent of this is unknown, and beating
1 % systematics will be challenging.

1275.2.2

T2HK

An up-to-date GLoBES implementation of T2HK has been developed, incorporating
the collaboration’s latest estimates for detector performance. The simulation is basedon the GLoBES implementation of T2HK [169] with comprehensive modifications tomatch the latest experimental design. The beam power and fiducial mass have been
updated to 1.3 MW and 187 kt per tank. For these studies, the staged design has beenused with one tank operational for 6 years followed by two operational tanks beyond
that time. In cases where results are shown against the run time of the experiment,
additional simulations with just a single tank operational throughout have been included,
to highlight the discontinuous nature of this design and the benefit gained from theintroduction of the second tank. The neutrino flux and channel definitions have been
updated to match those of [17], with separate channels for four interaction types (charged
current quasielastic, charged current with one pion, other charged current and neutral
current), for the νµ → νe and ν µ → ν e signals, and unoscillated νe , ν e , νµ and ν µbackgrounds. New tables of pre-smearing efficiencies and migration matrices have beencreated for each channel based on the full detector simulations used in [17]. New crosssections for interactions on water for the four interaction types have been generated using
version 2.8.4 of the GENIE Monte-Carlo neutrino interaction event generator [170].
The simulation determines the event rates for signal and background components foreach of νµ /ν µ → νe /ν e appearance and νµ /ν µ → νµ /ν µ disappearance measurements
in neutrino mode and antineutrino mode. The rates are determined for 12 energy bins,
given in Appendix C. For the appearance measurements, the energy range is restricted
to 0 GeV to 1.25 GeV, so only bins 1 to 8 are included. All bins are included in thedisappearance measurements. Separate uncorrelated systematic errors are assumed on
the total signal and background rates for each of the four measurements, where the sizeof the errors assumed, summarised in Table C.1, are the same as in the official Hyper-K
studies after an adjustment to account for correlations between systematics not includedin these simulations.
The design of T2HKK [69] and the location of the second detector module are still underdevelopment. As such, physics studies are being performed for a number of simulated
fluxes with varying off-axis angles, generally ranging from on-axis to 2.5° off-axis, whichis aligned with the first detector in Kamioka. The novelty of this design is not only
the longer baseline distance, which will enhance the role of matter effects, but also thefact that the energy profile of the flux remains similar to that at the detector at 295 km,
meaning that the oscillation probability is sampled at very different values of L/E. Thisis clearly seen in Figure 5.3, where the top panel shows how the flux aligns with the firstmaximum of the probability at Kamioka while the bottom panel shows that the fluxesalign around the second maximum for the Korean detector. When plotted against L/E,
as in Figure 5.4, it is seen that the T2HK flux has only minor coverage of the second
maximum in contrast to T2HKK. The fluxes used in these simulations were provided
by the T2HKK working group of the Hyper-Kamiokande proto-collaboration and were
produced in the same way as the fluxes used in [17] but with a baseline of 1100 km andoff-axis angles of 1.5°, 2.0° and 2.5°.

128Flux [10-6/50 MeV/cm2/1021PoT]

1.6

T2HK

1.4
1.2
1
0.8
0.6
0.4
0.2
0
0.2

0.4

0.6

0.8
E [GeV]1

1.4

T2HKK 1.5°
T2HKK 2.0°
T2HKK 2.5°

0.14
Flux [10-6/50 MeV/cm2/1021PoT]

1.2

0.12
0.1
0.08
0.06
0.04
0.02
0
0.2

0.4

0.6

0.8
E [GeV]

1

1.2

1.4

Figure 5.3: Top: T2HK’s flux plotted against neutrino energy for ν-mode (solid) and νmode (dashed). Bottom: The T2HKK fluxes plotted against energy for ν and ν modes.
The shaded region shows the envelope of the probability found by varying the true valueof δ. Due to T2HKK’s longer baseline but comparable energy range to T2HK, the fluxes
on the bottom sample a very different part of the probability.
129

PoT]

1.2

T2HK
(10×) T2HKK 1.5°
(10×) T2HKK 2.0°
(10×) T2HKK 2.5°-62Flux [10 /50 MeV/cm /10

1.4

21

1.6

1
0.8
0.6
0.4
0.2
0
0

500

1000

1500
2000
L/E [km/GeV]

2500

3000

Figure 5.4: The T2HK and T2HKK fluxes shown as a function of L/E. The shaded
region shows the envelope of the probability for L = 1100 km and the black lines indicate
the specific behaviour for δ ∈ {0, π2 , π, 3π2 }. Note that the T2HK flux actually samples
from the probability with a smaller matter effect corresponding to its shorter baseline
L = 295 km; however, on this scale the location of the first maximum does not deviate
much from what is shown here.

1305.2.3

Experimental run times and ν : ν ratios

The previous sections have discussed the models of the experimental details of DUNE
and T2HK. However, in the present study, a number of different exposures are considered for these experiments and their combination. This section is intended to clarify
the terminology and explain the choices of run time, neutrino–antineutrino sharing, anddetector staging adopted in the following analyses.
First, it is noted that although the ratio of the run time between ν and ν̄ beam modes
is also known to affect the sensitivities of long-baseline experiments, only the ratios
defined by each experiment’s official designs are used throughout this work. For DUNE
and T2HK, the ratio of ν to ν̄ are 1:1 and 1:3, respectively. The physical argument for
the asymmetric run-time of T2HK is that, as described above, the use of both neutrino
and antineutrino data is important in gaining precision on the measurement of δ, but
the expected event rates for antineutrino interactions is around a third as large as therates for neutrinos, due to the lower CCQE cross-section of antineutrinos and smaller
antineutrino beam flux. For DUNE, however, due to its more limited statistics andits wide-band beam reaching higher energies, the CCQE interaction channel and the
combination of neutrino and antineutrino data is less essential than obtaining a larger
overall event rate by using an equal ratio of neutrino and antineutrino run times. The
impact of changing these ratios was investigated, but found not to significantly impact
the results, and for both experiments the optimal ratio was close to those assumed here.
In the study for alternative designs, the same ratios as the standard configurations of
DUNE and T2HK are assumed.
Most of the results in this chapter deal with three configurations labelled as DUNE,
T2HK and DUNE + T2HK, and the sensitivities shown assume the full data taking
periods for these experiments have ended. These are the standard configurations, andare defined in terms of run times and neutrino–antineutrino sharing in the rows labelled“fixed run time” in Table 5.1. It is noted that, as the intention is to compare experimental
performance, the standard configuration of DUNE in these studies assumes 10 years ofdata taking, which differs for the 7 years considered in [18]. However, quantities arealso plotted against run time, and for these figures the sharing of run time between
components is defined in terms of a quantity defined here as the cumulative run time T ;these are shown in the rows labelled “variable run time” in Table 5.1. The cumulative runtime for the combination of DUNE and T2HK is defined to be the sum of the individual
experiments’ run times, i.e. if the two experiments were run back to back, with nooverlapping period of operation, then this definition of cumulative run time is identical tothe calendar time taken for the full data set to be collected3 . Of course, if the experiments
run in parallel, with identical start and end dates, this definition of cumulative run time
would be double the calendar time required to collect the data. To make clear when
this definition is used, the variable run time configuration is labelled as DUNE/2 +T2HK/2, as half of the cumulative run time goes to each experiment. Note also that,as per the official studies of each experiment, 107 s/year is assumed of active beamtime for T2HK (2.7 × 1021 POT/year at 1.3 MW with 30 GeV protons) while combined
In the interests of clarity, here the term calendar time is used to denote the actual time passed on
the calendar. This is highly dependent on staging and the relative placements of individual experimentschedules, and is only used later in the text as an informal means of comparison for certain staging
options.
3

131

accelerator uptime and efficiency of 56 % is assumed for DUNE (1.47 × 1021 POT/year
at 1.07 MW with 80 GeV protons).
Label
Fixed run time

DUNE
T2HK
DUNE + T2HK

Variable run time

DUNE
T2HK
DUNE/2 + T2HK/2

ν : ν at DUNE

ν : ν at T2HK

5: 5
0: 0
5: 5

0: 0
2.5 : 7.5
2.5 : 7.5

T /2 : T /2
0: 0
T /4 : T /4

0: 0
T /4 : 3T /4
T /8 : 3T /8

Table 5.1: The run times in years for each component of DUNE, T2HK, and their
combination (DUNE + T2HK) for both the standard full data taking period (top 3 rows)
and when considered with variable run times (bottom 3 rows). Plots with cumulative
run time T on the x-axis are for the “variable run time” configurations, whilst all otherplots are for the “fixed run time” configurations.
The possible staging options for the two modules of T2HK cause some added complication when plotting sensitivities against run time. In this study, it is assumed thatthe standard configuration of T2HK follows the staging suggested by the collaboration:
6 years of 1-tank running followed by 4 with an additional tank. In practice, this is
implemented as an effective mass for T2HK which depends on the run time assigned to
T2HK t defined by
t−6
M (t) = M0 1 + Θ(t − 6),twhere M0 is the mass of a single tank, defined above as 187 kt, and Θ(x) is the Heaviside
step function. As the definition of cumulative run time T would require 12 years to pass
before 6 years of data had been collected by T2HK in the combination of DUNE +T2HK, the discontinuity seen in sensitivity due to staging appear in two different placesin plots against run time: one for T2HK alone, and one for DUNE + T2HK. This canbe seen clearly in e.g. Figure 5.6, where the discontinuities are marked with verticaldashed lines. For comparison, also shown are the sensitivities against run time assuming
there is no second tank. In this case, M (t) = M0 for all values of t.
Finally, in Section 5.4 these configurations (and the labels in Table 5.1) will be deviated from as non-standard exposures are considered for the purpose of better exploring
the complementarity of DUNE and T2HK. This will be discussed in more detail in
Section 5.4.

5.2.4

Statistical method

The simulations of these studies use GLoBES [165, 166] to compute the event rates andstatistical significances for the experiments discussed in the previous section. The salient
details of the statistical model underlying the analysis are now described in this section.

132

Given the true bin-by-bin event rates, ni , for a specific experimental configuration, a χ2function is constructed based on a log-likelihood ratio,~ ξs , ξb ) = 2
χ2 (θ,X~ ξs , ξb ) − ni + ni ln
ηi (θ,ini~ ξs , ξb )ηi (θ,!

+ p(ξs , σs ) + p(ξb , σb ), (5.2.1)~ ξs , ξb ) is the hypothesis event rate for bin iwhere i runs over the number of bins, ηi (θ,and Ei is the central bin energy. The vector θ~ has six components, corresponding to eachof the three mixing angles, one phase and two mass-squared splittings of the hypothesis.
The parameters ξs and ξb are introduced to account for the systematic uncertainty ofnormalization for the signal (subscript s) and background (subscript b) components ofthe event rate, and are allowed to vary in the fit as nuisance parameters. For a given~ the event rate for bin i is calculated as
hypothesised set of parameters θ,~ ξs , ξb ) = (1 + ξs ) × ni + (1 + ξb ) × bi ,ηi (θ,
where ni and bi are the expected number of signal and background events in bin i,
respectively. The nuisance parameters are constrained by terms p(ξ, σ) = ξ 2 /σ 2 , representing Gaussian priors on ξs and ξb with corresponding uncertainties σs and σb . To test
a given hypothesis against a data set, unwanted degrees of freedom are profiled out. This
amounts to minimising the χ2 function given in Equation (5.2.1) over these parameterswhilst holding the relevant parameters fixed. The statistical parameters of interest will
be detailed for each analysis in the following sections; however, as an example consider
the study into how well different hypothesised values of δ fit a given data set. In thiscase, the quantity computed isχ2 (δ) =min~=δ,ξs ,ξb }
{θ6





~ ξs , ξb ) + P (θ)~ ,χ2 (θ,(5.2.2)where the notation θ~ 6= δ means all parameters other than δ. The function P (θ) acts
like a Bayesian prior, introduced to mimic the role of data from existing experiments
during fitting. In all fits that are performed, unless explicitly stated otherwise, true
values are used from the global fit NuFit 2.2 (2016) [171]. P (θ) comprises a sum of
the 1D χ2 data provided by NuFit for each parameter, except for δ, with either NO or
IO priors used depending on the mass ordering of the hypothesis. 4 This includes thecorrelations which are currently seen in the global data, and this treatment goes beyond
the common assumption of Gaussian priors, allowing for both the degenerate solution
and its relative poorness of fit to be more accurately taken into account. The values ofall parameters are permitted to vary, including the different octants for θ23 , the valueof δ and the mass orderings, subject to the global constraints. The choice of true values
depends on the mass ordering, and these are given explicitly in Table 5.2, unless statedotherwise.
Note that this treatment of the external data, which attempts to accurately model
the global constraints beyond the approximation of independent Gaussians, leads tosome differences between these results and those of previous studies [17, 18, 146]. The
differences can be traced to two key features: First, the significantly non-Gaussian
behaviour of the global constraints at higher significances is taken into account in these
Implementing more recent versions of the NuFit global fit has been completed for part of the analysis,
to verify that this does not make a significant difference for the results shown in this chapter.
4

133

Parameter

Normal ordering

Inverted orderingθ12 [°]
θ13 [°]
θ23 [°]∆m221 [×10−5 eV2 ]∆m231 [×10−3 eV2 ]33.72+0.79−0.76
8.46+0.14−0.15
41.5+1.3−1.1
7.49+0.19−0.17+2.526+0.039−0.03733.72+0.79−0.76
8.48+0.15−0.15
49.9+1.1−1.3
7.49+0.19−0.17−2.518+0.038−0.037

Table 5.2: The true values used in the simulations, unless otherwise stated explicitly,with their uncertainties (the 1 σ range of the priors used in the fit). These are based on
NuFit 2.2 (2016) [171], and are similar to the parameters found in other recent global
fits (see e.g. [172, 173]).
studies; this is particularly relevant for the prior on ∆m221 and this is commented on in
more detail in Section 5.3.1 and particularly in Section 5.3.1.1. The second importantfeature of the priors used in these studies is the strong correlation between mass orderingand the octant of θ23 . The current global data disfavours the combination of IO andfirst octant (or NO and second octant). This fact is reflected in the priors; although avisible local minimum is always present, it is never degenerate with the true minimum.
In previous studies, various treatments of this degeneracy have been employed, somewhich do not allow the alternative minimum, and some which do not penalise it atall. The method used here interpolates between these two extremes, and attempts tofaithfully describe the current global picture. More detail will be provided on the specificdifferences between these results and existing calculations of the sensitivity of DUNE,
T2HK and their variant designs on a case-by-case basis in the following sections.

5.3

Sensitivity to mass ordering, CPV, non-maximal CPV,and octant

In this section, the results are presented of simulations studying the sensitivity of thestandard configurations of DUNE and T2HK. This means using the 2-horn reference flux
for DUNE, discussed in Section 5.2.1, while for the T2HK detector both the staged implementation of the two detector modules at Kamioka, as discussed in Section 5.2.2, butalso the sensitivities for a single tank of 187 kt. The run time and neutrino–antineutrino
sharing for these configurations are discussed in more detail in Section 5.2.3. Afterconsidering these benchmark configurations and their complementarity, the potential of
alternative designs are returned to in Section 5.5.

5.3.1

Mass ordering sensitivity

The mass ordering is one of the central goals of the next generation of LBL experiments;
it is also one of the easiest to measure with this technology. The ability to determine
the mass ordering is quantified by computing the following test statistic,∆χ2MO = min~ s ,ξb }
{θ,ξ

h

i

χ2 (sgn∆m231 = true) − χ2 (sgn∆m231 = false) .

134(5.3.1)

That is to say, the smallest value of the χ2 function for any set of parameter values withthe wrong ordering. All parameters are allowed to vary during marginalisation whilst
preserving the ordering. Although this composite hypothesis violates the assumptions
pof Wilks’ theorem [174, 175], and therefore invalidates the mapping between ∆χ2 andσ-valued significance for discrimination of the two hypotheses, the work in this section
sticks to p
convention, reporting the expected sensitivities for the median experiment interms of ∆χ2 and discussing it inp terms of σ. For details on the precise formulationof the statistical interpretation of ∆χ2 , see e.g. [176].
The sensitivity found in Figure 5.5 is very strong. DUNE, with its large matter effects,
can expect a greater than 6 σ measurement of the mass ordering after 10 years for all
values of δ, with an average sensitivity of around 8 σ and a maximal sensitivity of around13 σ. T2HK alone has limited access to this measurement due to its shorter baseline,
but can still expect a greater than 3 σ measurement for around 25 % of the possible
values of δ after 10 years of data-taking. The combination of DUNE and T2HK runningfor 10 years each can reach sensitivities of at least 12 σ, with an average of around 14 σor 15 σ. Care should be taken when interpreting such large significances; however, it isclear that DUNE, and the combination of DUNE and T2HK, can expect a very strong
determination of the mass ordering. The strong complementarity is also noteworthy:
for the values of δ where DUNE performs the worst, the information from T2HK helps
to raise the global sensitivity by almost 8 σ to 9 σ. Despite this interesting interplay,the fact that this is such an easy measurement for DUNE means that the question of
optimising such a measurement further is not investigated further.
The sensitivities in Figure 5.5 deviate from previous published values for DUNE, generally reporting a worse ability for DUNE to exclude the ordering, with lower averagesensitivity and visibly discontinuous behaviour in the values of ∆χ2 . This is due to thepriors that have been imposed. Instead of a Gaussian approximation to the global data,
the global 1D χ2 functions have been implemented, as provided by NuFit [171]. The
true global data has strongly non-Gaussian behaviour at high significance, and there
exist non-standard parameter sets which are not excluded at greater than 6 σ. These
parameter sets sometimes become the best-fitting wrong-ordering solution, and must be
excluded to rigorously establish the mass ordering. This is discussed in more detail in
Section 5.3.1.1.
Another way to understand the complementarity
of DUNE and T2HK is in terms ofp2
minimal run time necessary to ensure a ∆χ > 5 measurement regardless of the true
value of δ. This quantity is plotted in Figure 5.6, for normal ordering (left) and invertedordering (right). The shaded bands take into account the variation in sensitivity due
to the true value of θ23 . DUNE alone takes between 2 years and 10 years to reachthis sensitivity, while the combination of DUNE and T2HK always takes less than 3
cumulative years (which if run in parallel is only 1.5 calendar years). T2HK runningalone cannot ensure a measurement of this significance over any plausible run time.
Shown explicitly is the difference in minimal sensitivity for T2HK with (dashed lines)
and without (solid lines) a second staged detector module at Kamioka. This leads to anegligible increase in performance for T2HK in isolation, but a modest increase in the
global sensitivity.

135

DUNE
T2HK
DUNE + T2HK

Normal ordering
Inverted ordering

20

√Δ𝜒 2

15

10

5

-1

-0.5

0
True 𝛿/𝜋

0.5

1Figure 5.5: The sensitivity to the mass ordering for DUNE and T2HK in isolationand combined for true normal ordering (solid) and inverted ordering (dashed). This
plot assumes the “fixed run time” configurations in Table 5.1 and the true oscillation
parameters given in Table 5.2.

136

DUNE
T2HK
DUNE/2 + T2HK/2

Inverted ordering

w/ staged T2HK
w/ 1-tank T2HK

w/ staged T2HK
w/ 1-tank T2HK

min √Δ𝜒 2

15

DUNE
T2HK
DUNE/2 + T2HK/2

Normal ordering

10

5
3
0

0

5
10
15
Cumulative run time [years]

20

0

5
10
15
Cumulative run time [years]

20

Figure 5.6: The least sensitivity for discovering mass ordering min ∆χ2 over values oftrue δ which can be reached by DUNE, T2HK and their combination as a function of
cumulative run time. The width of the bands shows the sensitivity for 40° ≤ θ23 ≤ 50°.
The left (right) panel assumes normal (inverted) ordering. These plots assume the“variable run time” configurations in Table 5.1 and the true oscillation parameters,
apart from θ23 , given in Table 5.2. The vertical lines mark the introduction of a staged
second detector for T2HK and leads to a notable discontinuity in sensitivity.
p137

5.3.1.1

Mass ordering sensitivity at high significanceThe sensitivity to mass ordering is conventionally reported as the difference between thevalue of a χ2 statistic for the true parameter set and the close degenerate set with theatmospheric mass splitting changed by the following mapping,∆m231 → −∆m231 + ∆m221 .
This local minimum becomes a worse and worse fit as data is collected, and reaches a∆χ2 value of above 8 σ within a few years of running DUNE. This method computes
the decreasing quality of a poor fit to the data; however, there are lots of parameter sets
which are poor fits to the current data, and many cannot be excluded with a significancegreater than 8 σ. Statistically speaking, to establish the mass ordering all possible
parameter sets with that ordering must be excluded regardless of the other parametervalues. In some circumstances, this may mean the local minimum identified above is
not the true global wrong-ordering minimum. This problem is found to be relevant
for DUNE as soon as the local minimum approaches a 5 σ exclusion. This is because
the global prior for the solar mass-squared splitting, ∆m221 has a second minimum ataround this significance. The long-baseline experiments considered in this chapter offer
no sensitivity to this parameter themselves, and rely on the priors to help constrain
it. The prior used in the simulations of this chapter has been plotted in the top panelof Figure 5.7, where the second minimum can be seen just above the global minimum.
For DUNE to exclude the wrong mass ordering at above 5 σ, it must be ensured thatit considers all values of ∆m221 allowed by the global data at this significance. It isfound that DUNE can often exclude this minimum only at lower significance than the
more obvious local minimum corresponding to the expected degeneracy. This causes
the lower significances, and discontinuous behaviour, that is reported in Section 5.3.1.
On average, this reduces the expected significance of the mass ordering measurement byaround 5 σ.
Of course, predicting any sensitivities at high significance requires good control over allother aspects of the statistical modelling, and it is not pretended that the method usedin the simulations of this chapter correctly models all uncertainties up to very small
fluctuations. However, this particular subtlety is pointed out as a concrete example of
how the oft quoted sensitivity is not quite what it seems; it is the confidence at which it
can be expected to exclude a particular local minimum, not to exclude the best-fitting
set of parameters with the wrong ordering. The difference in these quantities starts tobecome relevant for DUNE at very modest exposures. The bottom panel of Figure 5.7
shows the difference in ∆χ2 values for the local minimum (black) and the full set ofwrong ordering parameter sets (green), which starts to be visible after only 2 years run
time. It is hoped that this example helps to highlight some of the complexities of making
precise statements with high confidence sensitivities.

5.3.2

CP violation sensitivity

To fulfil the central aim of the LBL programme, the experiments must be able to rule out
CP conservation over a large fraction of the true parameter space. This would imply a
non-zero Jarlskog invariant and rigorously establish CP violation in the leptonic sector.
138

25

global minimum
local minimum

min √Δ𝜒 2

20

DUNE(2 yr)
DUNE(5 yr)
DUNE(10 yr)

15
10
5
0

-1

25

-0.5

0
True 𝛿/𝜋

0.51Normal ordering
Inverted ordering

√Δ𝜒 2

20
15
10
5
0
0.1

10
1
Δ𝑚u�2 /10−5 [𝑒𝑉 2 ]100

Figure 5.7: Top: The prior on ∆m221 provided by the NuFit global fit [171]. The second
local minimum with a significance around 5 σ (∆m221 ≈ 1.6 × 10−4 eV2 ) leads to theunusual behaviour in the reported mass ordering sensitivities of this chapter. Bottom:
Comparison of sensitivities based on the global minimum and more common publishedversions which show only a local minimum. This is due to the presence of additional
wrong-ordering minima at high significance in the global data.
139

Once again the conventional test statistic is followed, defining the quantity∆χ2CP = min ∆χ2 (δ),δ∈{0,π}(5.3.2)

which amounts to studying the composite hypothesis of CP conservation (δ = 0 or
δ = π) [177]. Although at low-significance this test statistic is known to deviate froma χ2 distribution [178], such effects are expected to be small for the experiments underconsideration in this study.
For the discovery of CP violation, the true value of the mass ordering and octant are
relevant. These values are not specified, and the sensitivity for all combinations of values
are studied. Shown in the top panel of Figure 5.8 is the significance for exclusion of CP
conservation for the standard designs of the two facilities, in isolation and combination.
T2HK is found to have a better overall sensitivity than DUNE, with at least a 3 σ (5 σ)discovery of CPV over 63 % to 71 % (30 % to 47 %) of the parameter space for DUNE
and 75 % to 81 % (55 % to 66 %) for T2HK. This can be understood by the discussion
in Section 5.1.2. Discovery potential for CPV is closely related to the precision on δ atthe CP conserving values; both rely on distinguishing between e.g. δ = 0 and othervalues. The best sensitivity to CP conserving values of δ is at the first maximum, where
the majority of T2HK events are found and consequently it sees a better sensitivity.
As mentioned in the last paragraph of Section 5.2.4, the prior used in these fits correlates
the allowed octant to the mass ordering, and this is responsible for differences between
these results and previously published work. In Figure 6 of [146], there is almost no CPV
sensitivity for 0 < δ < π for T2HK, which has not been found in the results presented
here, while the results for DUNE are similar. This feature is explained as being due tothe lack of MO sensitivity at T2HK, allowing for degeneracies to limit the sensitivity. In
the simulation use here, however, T2HK avoids this problem by its strong determinationof the octant and the correlation of the global data, which is included in this study but
not in [146]. This lifts the degeneracy, and allows a higher sensitivity to be obtained.
DUNE is found to perform slightly better in these simulation than is reported in theleft panel of Fig 3.13 in [18]. Around δ = π/2 (−π/2), their result shows the sensitivity
is about 5.4 σ (3.8 σ).5 However, this work’s simulations find a range of between 5.0 σ
to 6.5 σ for both δ = −π/2, π/2. There are two sources for this discrepancy. Firstly, alonger run time (10 years) is assumed, for the purposes of comparison between T2HK
and DUNE. Secondly, the degeneracy which impacted the T2HK simulation is also
relevant here, albeit only at higher significances. The CPV sensitivity for DUNE does
not peak around δ = −π/2 in the left panel of Fig 3.13 in [18] like these results, due
to the correlation described earlier. DUNE does not have as strong octant sensitivity
as for the mass ordering, and without this correlation DUNE data prefers the forbidden
minimum around δ = −π/2. Finally, agreement is found between these results andthose of Fig. 119 in [17]. This is largely because the mass ordering is fixed during fittingin [17], which mitigates the impact of the mass ordering degeneracy. This leads tosuperficial agreement between the two sets of results; although, these new results show
the sensitivity which is possible assuming only the current global data, whereas assuming
the MO is known would require new external data, perhaps from another long-baseline
experiment.
The range given in their work is for various beam designs.f For the design considered in the simulations presented here, the result is at the bottom of the range.
5140Shown in the bottom panel of Figure 5.8 is the fraction of values of δ for which a 5 σ
exclusion of CP conservation can be made as a function of run time. DUNE requires
between 7 years and 10 years of data-taking to reach at least a 5 σ measurement for
25 % of the possible values of δ, while T2HK alone expects to be able to make at leasta 5 σ measurement for more than 50% of the parameter space in less than 7 years.
The combination of DUNE and T2HK is shown as a function of cumulative run time,
the sum of the individual run times for each experiment, and as such interpolates thetwo sensitivities. However, if run in parallel, the combination of the two experiments
performs stronger than either in isolation, and expects a greater than 5 σ measurement
for more than 50 % of the parameter space after at most 5 years of parallel data-taking.

5.3.3

Sensitivity to maximal CP violation

Although the search for any non-zero CPV is the principle goal of the next LBL experiments, understanding the value of δ is also highly relevant. Current global fits [6,172,173]
point towards maximal values of δ, δ = ±π/2. Of course, these should be treated withsome scepticism; no single experiment can claim evidence for this at an appreciable level.
However, determining if a maximal CP violating phase exists will remain a high priority
for the next generation of long-baseline experiments. If established, it could be seen as
an “unnatural” value advocated as evidence against anarchic PMNS matrices. Indeed,
it is also one of the most common predictions in flavour models with generalised CP
symmetries.
This question is studied in Figure 5.9 where the quantity∆χ2MCP =minδ∈{− π2 , π2 }∆χ2 (δ)(5.3.3)is defined. This is analogous to ∆χ2CP defined earlier, and gives a measure of the compatibility of the data with the hypothesis of maximal CP violation. On the top panel,
the ability to exclude maximal CPV is shown as a function of the true value of δ. Thereis a similar sensitivity for both facilities, with T2HK being better for true values of δ
around the CP conserving points 0 and π, while DUNE can slightly improve on thissensitivity for those values of δ closest to maximal CP violation. In this way, the two
experiments once again exhibit a complementarity, and the combination of DUNE andT2HK inherits the best sensitivity of its two component parts, expecting a 3 σ exclusionof MCP for over 58% of the parameter space.
The bottom panel of Figure 5.9 shows the fraction of true values of δ for which a 5 σ
exclusion of maximal CP violation can be achieved. By running in parallel for 10 years,
DUNE and T2HK can expect a coverage at this significance of around 39 % to 48 % of
the parameter space with a second detector at Kamioka, or 36 % to 45 % of the spacewith a single tank. DUNE alone would struggle to compete on this measurement, after
10 years only covering a fraction of between 10 % to 23 %.

5.3.4

Octant degeneracy and the precision on θ23

Although it is known that θ23 is around 45°, the current global fit data allows for two
distinct local minima, one below and one above 45°. This ambiguity is known as the
141

16

DUNE
T2HK
DUNE + T2HK

40∘ ≤ 𝜃23 ≤ 50∘

14
12
√Δ𝜒 2

10
8
6

4
2

0
−1

−0.5

0
True 𝛿/𝜋

CPV fraction over 5𝜎

1.240∘ ≤ 𝜃23 ≤ 50∘

1

0.5

1

DUNE
T2HK
DUNE/2 + T2HK/2

0.8 (w/) staged T2HK(w/) 1-tank T2HK
0.6
0.4

0.2
0

0

5
10 12
15
Cumulative run time [years]

20

Figure 5.8: The sensitivity to CP violation for DUNE and T2HK in isolation and combined as a function of delta (top) and the fraction of δ parameter space for which greaterthan 5 σ CPV discovery is expected (bottom). A range of true θ23 is considered spanning
both octant solutions. The lower edge of the shaded regions corresponds to θ23 > 45°
due to a decrease in sensitivity arising from the relative suppression of the CP sensitive
terms in Equation (5.1.1). The top (bottom) plot assumes the “fixed run time” (“variable run time”) configurations in Table 5.1 and the true oscillation parameters, apartfrom θ23 , specified in Table 5.2.
142

14

DUNE
T2HK
DUNE + T2HK

40∘ ≤ 𝜃23 ≤ 50∘

12

√Δ𝜒 2

10
8
6

4
2

0
−1

MCP fraction over 5𝜎

0.8

−0.5

0
True 𝛿/𝜋40∘ ≤ 𝜃23 ≤ 50∘

0.7

0.5

1

DUNE
T2HK
DUNE/2 + T2HK/2

0.6(w/) staged T2HK
0.5 (w/) 1-tank T2HK
0.4

0.3
0.2
0.1
0

0

5 6
10 12
15
Cumulative run time [years]

20

Figure 5.9: Top: The significance at which maximal CP can be excluded for DUNE
and T2HK in isolation and combined as a function of true δ. Bottom: The fraction of
δ-parameter space for which maximal CP can be excluded as a function of run time. The
top (bottom) plot assumes the “fixed run time” (“variable run time”) configurations inTable 5.1 and the true oscillation parameters, apart from θ23 , specified in Table 5.2.

143

octant degeneracy and arises as the disappearance channel of νµ → νµ is sensitive at
leading-order only to sin2 2θ23 . However, the appearance channel breaks this degeneracy at leading-order, and future long-baseline experiments are expected to significantly
improve knowledge of θ23 . This section studies how well DUNE and T2HK will be able
to measure θ23 as well as settling two central questions: Is θ23 maximal, and which isits correct octant?
The ability to exclude the wrong octant for DUNE, T2HK and their combination isshown in Figure 5.10. Shown on the top is the sensitivity as a function of the true
value of θ23 . In these plots a fixed value of δ = 0 is assumed. The impact of varying
δ for these measurements is small, as the degeneracy is broken at leading-order in theappearance channel, and the subdominant effects of δ are less relevant. The ability
to exclude the wrong octant can reach up to 8 σ at the extremes of the current 3 σrange of θ23 , and it is seen that 3 σ determinations of the upper (lower) octant can be
expected for true values of sin2 θ23 less than 0.47–0.48 (greater than 0.54–0.56). This
corresponds to a 3 σ determination of the octant for all values of θ23 in the ranges
θ23 . 43.3° to 43.8° or θ23 & 47.3° to 48.4°. On the bottom, the true value of θ23 is fixed
while showing how the sensitivity depends on cumulative run time. The sensitivity
is seen to quickly plateau, and the staging options for T2HK make little difference.
Overall, the experiments expect to be able to establish the octant for this value of θ23
after only 2 years to 5 years. Although this plot assumes θ23 = 40°, changing the true
value of θ23 leads to a predictable change in sensitivity, as indicated in the top panel,but does not qualitatively change the behaviour against run time. Overall, T2HK isseen to perform better than DUNE for the determination of the octant. However, thedifference in performance is marginal, and their combination after 10 years of data foreach experiment, outperforms either experiment running alone for 20 years.
In this simulation, a prior has not been imposed on θ23 . This process differs from [18],in which they give a gaussian prior for θ23 . It also differs from the fitting method in [17],where they fit θ13 , θ23 and the value of ∆m231 without implementing any priors, but fix
θ12 , ∆m221 and the mass ordering. In [146], the details of the fitting process are notspecified. Despite these differences, qualitatively similar behaviour is seen between thethree sets of results. The regions of θ23 where the octant cannot be determined at 5 σ
are found to be θ23 ∈ [41.1°, 49.8°], θ23 ∈ [42°, 48.9°], and θ23 ∈ [42.3°, 48.9°] for DUNE,
T2HK, and their combination, respectively. In Fig. 3.18 of [18], the equivalent region
for DUNE is θ23 ∈ [41°, 50°], which is comparable to this work. In the middle panels of
Fig. 5 in [146], the authors estimate the region as 42.5° < θ23 < 48.5° for T2HK and the
combination of DUNE and T2HK, while for DUNE alone the range is slightly smaller
than in this work’s simulation at 42° < θ23 < 49°.
In Figure 5.11, the analogous plots are shown for the exclusion of maximal θ23 . It
is seen that maximal θ23 can generally be excluded at greater significance than theoctant. As before, T2HK dominates the measurement and can reach 5 σ sensitivity forsin2 θ23 . 0.47 as well as for sin2 θ23 & 0.55. DUNE in contrast can make a 5 σ exclusion
for sin2 θ23 . 0.43 and sin2 θ23 & 0.56. Due to its poorer sensitivity, DUNE plays less ofrole in the combination and DUNE + T2HK follows the sensitivity of T2HK. On thebottom, the sensitivity against cumulative run time is shown. Again, the combinationof DUNE + T2HK performs similarly to T2HK when the cumulative run time is divided
by two, while DUNE performs slightly worse. The staging of T2HK is seen to play anotable role, leading to significantly higher sensitivities.
144

10

DUNE
T2HK
DUNE + T2HK

𝛿=0

√Δ𝜒 2

8
6

4
2

0
0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58
True sin2 𝜃23

w/ staged T2HK
DUNE
w/ 1-tank T2HK
T2HK
𝜃23 = 40∘ ; 𝛿 = 0 DUNE/2 + T2HK/2

√Δ𝜒 2

10

5

0

0

5
10
15
Cumulative run time [years]

20

Figure 5.10: The sensitivity to exclude the wrong octant for DUNE, T2HK and their
combination, as a function of sin2 θ23 (top) and the cumulative run time (bottom).
These plots assume δ = 0 and normal mass ordering. The top (bottom) plot assumes
the “fixed run time” (“variable run time”) configurations in Table 5.1 and the true
oscillation parameters, apart from θ23 , specified in Table 5.2.

145

25

DUNE
T2HK
DUNE + T2HK

𝛿=0

√Δ𝜒 2

20
15
10
5

0
0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58
True sin2 𝜃23

w/ staged T2HK
DUNE
w/ 1-tank T2HK
T2HK
𝜃23 = 40∘ ; 𝛿 = 0 DUNE/2 + T2HK/2

40

√Δ𝜒 2

30
20
10
0

0

5
10
15
Cumulative run time [years]

20

Figure 5.11: The ability to exclude θ23 = 45° for DUNE, T2HK and their combination,
against the true value of sin2 θ23 (top) and the cumulative run time (bottom). Theseplots assume δ = 0 and normal mass ordering. The top (bottom) plot assumes the“fixed run time” (“variable run time”) configurations in Table 5.1 and the true oscillation
parameters, apart from θ23 , specified in Table 5.2.

146

The attainable precision on sin2 θ23 is studied in Figure 5.12, where ∆(sin2 θ23 ) is plottedagainst the true value of sin2 θ23 for normal mass ordering. For all configurations,the same behaviour is found: The uncertainty climbs up from about sin2 θ23 = 0.48
and falls down around sin2 θ23 = 0.54, peaking at sin2 θ23 ∼ 0.51. This is expected
for a measurement dominated by the disappearance channel, where the probability is
proportional to sin2 (2θ23 ) and a leading-order analytic treatment implies the relation∆(sin2 θ23 ) ∝ |tan(2θ23 )| ,which naively predicts a total loss of sensitivity at maximal mixing, analogous to ∆δ
at δ = π/2. This is mitigated by higher-order effects, as well as the information fromthe appearance channel, which becomes important around these values. The drop in
sensitivity seen in Figure 5.12 is quite sharp, and for values of sin2 θ23 away from maximal mixing there is only modest variation in precision. For DUNE, ∆(sin2 θ23 ) is about
0.01 at the boundaries, and peaks up to the value ∼ 0.045. T2HK has better performance, with ∆(sin2 θ23 ) ∼ 0.005 for sin2 θ23 = 0.43, 0.585. As with DUNE, the worst
performance for T2HK is near the peak at sin2 θ23 = 0.5 with ∆(sin2 θ23 ) ∼ 0.03. The
combination of DUNE and T2HK performs very similarly to T2HK; although the additional information from DUNE helps to improve this measurement, the sensitivity isseen to be dominated by T2HK. In these plots, the value δ = 0 is fixed, althoughqualitatively similar behaviour holds for other choices. There is, however, a correlation
between the precision on θ23 and δ.
An estimate of the joint precision on θ23 and δ attainable at DUNE and T2HK ispresented in Figure 5.13. In this plot, each ellipse shows the 1 σ allowed region for a setof true values inside its boundary taken from the sets δ ∈ {0°, ±90°, ±180°} and θ23 ∈{40°, 45°, 50°}. T2HK performs best for this measurement, however the combinationof additional data from DUNE helps to reduce the contours slightly. It can also beseen that the precision measurement of these two parameters are largely independent,
although DUNE exhibits a stronger correlation due to its reliance on data from boththe appearance and disappearance channels. The best measurements will be obtained
for large deviations from θ23 -maximality and values of δ close to the CP conserving
values, where DUNE (T2HK) can expect precisions on θ23 of ∆θ23 = 0.8° (∆θ23 = 0.5°).
Conversely, the worst precision comes from the values of θ23 near maximal mixing where
DUNE (T2HK) can expect larger uncertainties by a factor of around 3 with ∆θ23 = 2.3°(∆θ23 = 1.5°). Comparing the result in Figure 5.13 to Fig. 123 in [17], the value found in
this work for ∆ sin2 θ23 is very close to the official result for T2HK, despite the differencesin treatment of external data mentioned previously.

5.4

Complementarity of precision measurements of δ

For the reasons outlined in Section 5.1.2, an interesting interplay of sensitivities is expected for a narrow-band and wide-band beam for the determination of δ. In this section,
the complementarity of DUNE and T2HK is studied for precision measurements of δ.
Figure 5.14, shows the 1 σ precision on δ, which is attainable by the standard configurations of DUNE and T2HK and their combination. A range of true values of θ23 isconsidered as this significantly affects the ultimate precision. It is seen that for most of
the parameter space T2HK can attain a better precision, with values of ∆δ between 6°
147

0.05

DUNE
T2HK
DUNE + T2HK

Normal ordering

Δ sin2 𝜃23

0.04

0.03

0.02

0.01

0

0.44

0.46

0.48

0.5
0.52
2
True sin 𝜃23

0.54

0.56

0.58

Figure 5.12: The expected 1 σ precision on sin2 θ23 as a function of true value of sin2 θ23from 0.43 to 0.585 for DUNE, T2HK, and their combination, under the assumption of
normal ordering. This plot assumes the “fixed run time” configurations in Table 5.1 andthe true oscillation parameters, apart from θ23 , specified in Table 5.2.

148

DUNE

T2HK

DUNE+T2HK

150
100
50
𝛿[∘ ] 0

-50
-100
-150
0.35

0.4

0.45

0.5
sin2 𝜃23

0.55

0.6

0.65

Figure 5.13: The attainable 1 σ precision on sin2 θ23 and δ for DUNE, T2HK, and their
combination. In each case, the contours enclose the assumed true values for θ23 and δ,
marked with a point. This plot assumes the “fixed run time” configurations in Table 5.1
and the true oscillation parameters, apart from θ23 , specified in Table 5.2.

149

and 7° compared to between 9° and 11° for DUNE, for the CP conserving values of δ.
However, DUNE performs slightly better than T2HK for maximally CP violating valuesof δ. This leads to an effective complementarity between the two experiments, and their
combined sensitivity reduces ∆δ as compared to the two experiments in isolation by
between 1° and 4° depending on the value of δ.

25

DUNE
T2HK
DUNE + T2HK

40∘ ≤ 𝜃23 ≤ 50∘

20

Δ𝛿 [∘ ]

15

10

5

0
−1

−0.5

0
True 𝛿/𝜋

0.5

1Figure 5.14: The 1 σ precision on δ for DUNE and T2HK in isolation and combined. This
plot assumes the “fixed run time” configurations in Table 5.1 and the true oscillation
parameters, apart from θ23 , specified in Table 5.2.
An improvement is therefore seen when combining the data from the two experiments.
This was to be expected for a number of reasons: Firstly, there is a simple statisticalbenefit of combination – an increase in data reduces the statistical uncertainty andallows for a more precise measurement. On top of this, there is a synergistic benefit,
where the two experiments mutually improve the reconstruction of the parameter ofinterest. To try to understand the synergy between DUNE and T2HK, the statisticaladvantage can be mitigated in simulations using normalization procedures intended to
expose the complementarity shown by the information available in each data set. As
the experiments operate under such different assumptions, there is no universal way
to do this. There are many factors which influence an experiment’s sensitivity; forexample, the total flux produced by the accelerator, the effects of baseline distance on
the flux, the detector’s size, technology and analysis efficiencies, not to mention thepurely probabilistic effects of the oscillation itself, which occurs over different baseline
distances and at different energies. In the next two sections, different ways to normalise
the experiments are considered, which reveal different aspects of their sensitivities.
1505.4.1

Normalising by number of events

The statistical advantage of combining two experiments can be removed by fixing thenumber of events. Two ways of doing this are considered, with both based on the
total number of signal events S, composed of genuine appearance channel events andmis-reconstructed background events in the detectors.
The first of the two normalization methods fixes S. This is, of course, an unrealistic
goal in practice. However, it answers an interesting hypothetical question: Would afixed number of events be more informative if they came from DUNE or T2HK? The
simulations of T2HK and DUNE have been produced while fixing the number of eventsin the appearance channel. This number varies with δ, and so the effective run time has
been modified for each value of δ to keep the observed events constant. In the left-handpanel of Figure 5.15, the number of appearance events has been fixed to 8000 for eachconfiguration, roughly what is expected for the combination of events from 10 years eachat DUNE and T2HK. For a fixed number of observed events, DUNE is seen to dominate
the sensitivity. In this sense, events at DUNE are more valuable than events at T2HK.
This effect is quantitatively assessed in the right-hand panel of Figure 5.15. This plot
compares the performance of DUNE with a fixed 8000 events, with T2HK for increasing
numbers of events. To reach a comparable precision on δ T2HK needs between 2 and 3
times the number of events as DUNE. For T2HK to outperform DUNE for all values ofδ it needs more than 4 times the number of events collected by DUNE.
The second normalization scheme is designed to remove the effect of the probability
from the comparison with fixed event rates. The number of appearance channel events,
S, is to a good approximation proportional to the oscillation probability,
S ∝ P (νµ → νe ; hEi),
where hEi denotes the average energy of the flux, and a quantity N is introduced
denoting signal events with the probabilistic effects removed,
N (hEi) = S/P (νµ → νe ; hEi).

(5.4.1)

N can be thought of as the constant of proportionality between the number of signal
events and the oscillation probability, and it is affected by many factors, whose productis often referred to as the exposure of the experiment. These factors, such as run time, detector mass and power of the accelerator, describe technical aspects of the experimental
design and the exposure is often taken as a proxy for run time in phenomenologicalstudies of neutrino oscillation experiments. However, there are other factors affecting
the coefficient N such as the effects of cross-sections and detector efficiencies, which also
vary from experiment to experiment. This definition of N accounts for all of the factors
which affect the signal, apart from the fundamental effect of the oscillation probability.
Equating N assumes that all technical parameters are identical between the experiments,
allowing the effect of the oscillation probability to be studied alone. Whereas normalising by S puts the experiments on an equal statistical footing, normalising by N puts
them on more of an equal technical footing; the statistical advantage is still removed,
except where that statistical advantage is due to the actual physics (a higher oscillation
probability, as opposed to higher beam power or larger detector, for example).
The performance of DUNE, T2HK and their combination is studied here with a constantvalue of N . In practice, as the detector models have binned energy spectra, an analogous
151

25

DUNE: 8000
T2HK: 8000
DUNE + T2HK: 8000

Δ𝛿 [∘ ]

20

T2HK: 16000
T2HK: 24000
T2HK: 32000

15
10
5
0

-1

-0.5

0
True 𝛿/𝜋

0.5

1

-1

-0.5

0
True 𝛿/𝜋

0.5

1Figure 5.15: Left: The precision attainable by DUNE, T2HK and their combination
with a fixed number (8000) of appearance channel events. On the left, DUNE + T2HK
denotes the “fixed run time” configuration in Table 5.1, which expects around 8000
events. The DUNE configuration shown here is the corresponding “variable run time”
entry in Table 5.1. For T2HK, an average detector mass of 1.4 times a single tank’s mass
is assumed. In both cases, the cumulative run time for the experiment has been increased
until it has 8000 events. Right: The performance of T2HK with increasing numbers ofappearance events (in brackets) compared to DUNE with 8000 events. T2HK is seen toneed between 3 and 4 times the number of events to achieve a similar performance to
DUNE for all δ. In both plots, all unspecified parameters take the true values given in
Table 5.2.

152

quantity Ni is defined for each energy Ei and N is defined as the sum over Ni . An
additional advantage of this normalisation is that, unlike when fixing S, fixing thevalue of N for a given experiment corresponds to equivalent experimental run time;
the time required of an experiment to obtain a given number of events depends on
the true values of the oscillation parameters, whereas the time required to obtain agiven value of N is constant for a given experiment. Of course, the two experiments
investigated here require quite different run times in their standard configurations togenerate a comparable value of N . T2HK has considerably higher event rates and a
5+5 year run of DUNE contributes the same N as 0.96+2.88 at T2HK. In Figure 5.16,
the sensitivities are shown for the combination of 5+5 years of DUNE with 2.5+7.5
years of T2HK, compared to the two experiments running in isolation with run times
to ensure the same N : 3.46+10.38 years for T2HK, and 18+18 years for DUNE. The
figure shows that for a fixed N the combination of T2HK and DUNE always outperforms
T2HK running alone. However, for values of δ around the maximal CP violating values,
DUNE outperforms the combination by up to 3°. This comparison shows that given a
fixed N , DUNE generally outperforms T2HK and on average has a better sensitivity
than the combination of DUNE and T2HK, although not at CP conserving values ofδ. This suggest that events at DUNE are more informative about the value of δ thanat T2HK. However, it is worth pointing out that the run times assumed here lead toquite an asymmetric comparison: for T2HK to see the same N factor as DUNE it needs
to run only approximately 40 % of DUNE’s run time. Comparing this to the results in
Section 5.3.2, it is possible to conclude that a key advantage of T2HK is precisely thatit has larger event numbers despite an intrinsically less sensitive observable.
Comparing the expected precision on δ under the two different normalization conditions
gives an idea of the role played by the probability. Generally, the conclusions are seen tobe similar: when arranged to have equal normalizations, T2HK does worse than DUNE
for most of the parameter space. However, the behaviour around the CP conserving
values of δ depends subtly on the normalization condition, and therefore the probability
itself. When a fixed number of appearance events are considered (fixed S), DUNE
outperforms T2HK, yet for the probability independent normalization (fixed N), T2HK
performs best. This shows that the sensitivity of the probability itself is a key factor
in explaining DUNE’s performance. This section is concluded by noting that bothnormalization methods highlight the same aspect of the two experiments: events at
DUNE have more information on δ; however, it is already apparent that events are
harder to produce at DUNE. This will be studied in more detail in the next section.

5.4.2

Normalising by run time

Of course, one of the most pragmatic ways to normalise the experiments is by run time.
Would a decade of both experiments running in parallel be better than two consecutive
decades of DUNE or of T2HK? To make this comparison, the same cumulative run time
is assumed for the combination of experiments as the individual run time for each of
the experiments running alone. Figure 5.17 shows the results of this simulation. The
combination of DUNE and T2HK outperforms either experiment running for twice as
long, but T2HK reproduces the combination for most values of δ. There are some small
regions of parameter space where 20 years of DUNE outperforms not only T2HK but
also the combination of DUNE and T2HK, focused around the maximal CP violating
153

25

DUNE (36 yr)
T2HK (13.84 yr)
DUNE + T2HK

Δ𝛿 [∘ ]

20

T2HK (2×13.84 yr)
T2HK (3×13.84 yr)
T2HK (4×13.84 yr)

15
10
5
0

-1

-0.5

0
True 𝛿/𝜋

0.5

1

-1

-0.5

0
True 𝛿/𝜋

0.5

1Figure 5.16: Left: The expected precision on δ as a function of the true value of δ
for the standard configurations of DUNE, T2HK and their combination assuming all
experiments have the same value of N . See Equation (5.4.1) for a definition of N . The
DUNE + T2HK configuration is used as a reference, defined by the “fixed run time”
entry in Table 5.1. The DUNE configuration is the “variable run time” configuration in
Table 5.1 with the cumulative run time required to achieve the same N as the DUNE
+ T2HK reference, specified in brackets after the name. For T2HK, the number inbrackets also denotes the cumulative run time, the staging is simplified by assuming an
average detector mass of 1.4 times a single tank. Right: The precision on δ for T2HK
with an increasing run time up to the point where it consistently outperforms DUNE
(36 yr). In both of these plots, the true oscillation parameters are given in Table 5.2.

154

values of δ. At these values of δ, DUNE’s wide-band beam performs best by incorporating information from other energies. This benefit is also seen in the combination of
DUNE and T2HK, which outperforms 20 years of T2HK. This results indicates that the
combination offers two advantages. First, running the experiments in parallel allows thecollection of two decades of data in half the calendar time. This explains a significant
part of the sensitivity improvement; however, there is also a complementarity arisingfrom the different sensitivities of the two experiments. This is especially marked for thismeasurement about the maximally CP violating values of δ.

25

DUNE (20 yr)
T2HK (20 yr)
DUNE + T2HK

20

Δ𝛿 [∘ ]

15

10

5

0

-1

-0.5

0
True 𝛿/𝜋

0.5

1Figure 5.17: The 1 σ precision on δ as a function of the true value of δ for DUNE, T2HK
and their combination with the same cumulative run time of 20 years. The configurationof DUNE (10) is defined by the “variable run time” entry in Table 5.1, with T given
in brackets after the experiment’s name, whereas DUNE + T2HK is the corresponding“fixed run time” entry. For T2HK, the run time is also denoted in brackets, the scaling
is simplified by assuming an average detector mass of 1.4 times a single tank. This plot
assumes normal mass ordering and all other unspecified true parameters are given in
Table 5.2.
The behaviour of ∆δ for different experimental configurations as a function of run timeis shown in Figure 5.18. The staging of T2HK leads to a strong improvement in thesensitivity, which is comparable to that of DUNE for large run times at δ = 3π
2 andconsistently better for CP conserving values. This has been studied for two representative values of δ, around 0 and π/2, where the greatest difference in sensitivity for thetwo experiments is found. Running in parallel, the combination of DUNE and T2HK

155

expects ∆δ . 6° around CP conserving values, and ∆δ . 14° for maximally violating
values after 20 years.

𝛿=𝜋

20

40

DUNE
T2HK
DUNE/2 + T2HK/2

𝛿 = 1.5𝜋

35

DUNE
T2HK
DUNE/2 + T2HK/230

w/ staged T2HK
w/ 1-tank T2HK

Δ𝛿[∘ ]

15

w/ staged T2HK
w/ 1-tank T2HK

25

10

20

15

5

0

2

6
8
10 12 14 16
4
Cumulative run time [years]

18

0

20

2

6
8
10 12 14 16
4
Cumulative run time [years]

18

20

Figure 5.18: Left (right): ∆δ for δ = π (δ = 3π/2) at DUNE, T2HK and their combination as a function of run time. These plots assume the “variable run time” configurationsin Table 5.1 and the true oscillation parameters appropriate for normal ordering as given
in Table 5.2. Similar behaviour was also obtained for inverted ordering.
To end this section, the performance of the two experiments and their combinationis compared through the minimal exposures required to obtain certain physics goals.
Table 5.3, shows the value of N , see Equation (5.4.1), the number of signal events S andthe cumulative run time required to reach a precision on δ of 10° for a CP conserving
value of δ and a maximally CP violating value. It is clear from the study in thissection that to achieve a precision of 10° for maximally CP violating values of δ will
be a challenging measurement: around 30 years of data is necessary, requiring at least
15 years of both experiments running in parallel. For δ = π this is, however, a feasible
goal. DUNE expects a similar measurement after a full 10 years data-taking period,
while T2HK can achieve this goal in less than 2.5 years. The combination of DUNE andT2HK marginally improves on this, requiring only 1.8 years.
δ=πNS
Run time [years]

δ = 3π
2

DUNE

T2HK

Both

DUNE

T2HK

Both

50939
1981
10.33

31353
2814
2.44

31440
2378
3.54

1590857179
32.29

44953742113
35.07

26076123163
32.78

Table 5.3: Exposures required to reach a precision of ∆δ = 10° for a true value of δ of π
or 3π/2. T2HK has the best precision on reasonable time scales due to its very high event
rate especially at δ = π. DUNE marginally out performs T2HK for maximally violating
values of δ. The combination “Both” assumes a scaling of the standard configurationof DUNE/2 + T2HK/2, with the run time being the cumulative run time of these two
experiments, and can be seen to track the best performing experiment.
156

5.4.3

Impact of systematic errors

The previous section has looked at the precision on δ under a number of different assumptions. It has been seen that T2HK has a larger number of events, but that they
are relatively less valuable. However, the balance of these factors is in T2HK’s favour
for most of the parameter space, and it expects better precision on δ. This means thatthe relationship between statistical and systematic uncertainty will be quite different at
the different experiments and any assumptions about systematics, always a contentious
issue, may be significant. This section tries to understand these effects and explores theimpact on the expected precision on δ under differing systematics assumptions for the
combination of DUNE and T2HK.
A feel for the relevance of statistical versus systematic uncertainty can be obtained byseeing how the sensitivity scales with run time. In the model of the systematics usedin this study, only effective signal and background normalisation systematics are considered for both DUNE and T2HK. In Figure 5.19, the sensitivity to δ is shown fordifferent run times of the two experiments in isolation, with and without systematic
uncertainties. It is seen that there is little impact from the systematic uncertainty at
DUNE, and it continues to further its sensitivity as its run time is increased. This effectis quite different for T2HK where systematics clearly have a more important role; for
CP conserving values, there is no significant improvement in sensitivity after extensionsof the experiment run time by a factor of 4. This result neatly shows that DUNE isstatistically limited while T2HK has more reliance on its systematic assumptions. It isinteresting to note that in both cases, neither DUNE nor T2HK taken as a single experiment running for 40 years is comparable at CP conserving values with the combinationof DUNE and T2HK running for only 10 years each.
Due to the limiting effect of systematic uncertainties suspected at T2HK, it can be expected that its performance is quite sensitive to the assumptions made in the models
used. To understand how the combination of DUNE and T2HK can help reduce thissensitivity, simulations have been run while varying the value of the normalization systematics in T2HK. The case of 2 %, 4 %, 6 % and 8 % normalization uncertainty havebeen studied, in combination with DUNE and in comparison to T2HK running for twice
as long. The results are shown in Figure 5.20. It is seen that for 2 % systematic uncertainty, T2HK dominates the precision on δ and is limited strongly by the systematics,
meaning that doubling the run time leads to scant improvement. As the systematic
uncertainty on T2HK increases, there is more of an advantage of including DUNE. For
6 % systematics, the improvement in precision at δ = 0 is around 2° (around 25 % improvement). It can be concluded that as ∆δ by T2HK is systematically limited aroundδ = 0, π, including DUNE data can help to mitigate the effect of larger uncertainties.

5.5

Impact of potential alternative designs

As part of their continual optimisation work, both the DUNE and T2HK collaborations
have considered modifications of their reference designs, aiming to further the physics
reach of their experiments. As mentioned in Section 5.2.1, DUNE has considered an
optimised beam based on a 3-horn design as well as a novel beam concept, nuPIL. For
T2HK, the redesign efforts are focused on the location of the second tank. Originally
157

25

w/ sys.
w/o sys.

Δ𝛿 [∘ ]

20

DUNE (10 yr)
DUNE (20 yr)
DUNE (40 yr)
DUNE + T2HK

w/ sys.
w/o sys.

T2HK (10 yr)
T2HK (20 yr)
T2HK (40 yr)
DUNE + T2HK

15
10
5
0

-1

-0.5

0
True 𝛿/𝜋

0.5

1

-1

-0.5

0
True 𝛿/𝜋

0.5

1Figure 5.19: Left (right): The expected 1 σ precision on δ for DUNE (T2HK) with different run times with and without systematics (solid and dashed, respectively) compared
to a reference design of the “fixed run time” configuration of DUNE + T2HK from
Table 5.1. The DUNE configuration assumed here is defined in Table 5.1 with theircumulative run time T denoted in brackets after their names. For T2HK, the number inbrackets also denotes the run time, but these plots simplify the staging of the detectors
by taking a total mass of 1.4 times a single tank. All unspecified parameters are given
in Table 5.2.

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25

T2HK app. sys.= 0.02
0.04
0.06
0.08

T2HK (10 yr)
DUNE/2 + T2HK/2 (10 yr)

20Δ𝛿 [∘ ]

15

10

5

0

-1

-0.5

0
True 𝛿/𝜋

0.5

1Figure 5.20: ∆δ for T2HK and the combination of DUNE/2 + T2HK/2 each with
10 years cumulative run time for different normalization systematic uncertainties on theappearance channel in T2HK (2 %, 4 %, 6 % and 8 %). The normalization systematics
are held at 2 % for the appearance channels of DUNE. The configurations in this plot
are labelled “variable run time” in Table 5.1 with the cumulative run time denotedin brackets after their names. This plot assumes normal ordering, but all other true
parameters follow Table 5.2.

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foreseen as being installed at Kamioka 6 years after the experiment started to takedata, the possibility of installing the detector in southern Korea has been mooted. Thissection discusses the impact of these redesigns on the physics reach of the experiments,
both alone and in combination, via the results of the phenomenological discussion andsimulations in the previous sections of this chapter. The focus of this section is on themass ordering, CPV discovery, MCP and precision measurements of δ. Measurementsof θ23 are not discussed further, as it was found that there is little difference between
the alternative designs under consideration.

5.5.1

Experimental run times and ν : ν ratios

In all plots that follow, it is assumed that DUNE and its variants will run with equal
time allocated to neutrino and antineutrino mode, while T2HK and T2HKK will always
follow the 1:3 ratio of their standard configuration. It is also assumed that there is nostaged implementation of any of the variants of T2HKK, and that both detector modules
start collecting data at the same time.
The run time configurations for these alternative designs follow those of the “variable runtime” options in Table 5.1, albeit with variant fluxes for each experiment. All variantsof DUNE, T2HK and T2HKK when run on their own are assumed to have a cumulative
run time of 10 years. When a variant of DUNE is run in combination with a variantof T2HK, the cumulative run time is divided equally between the two experiments inthe same way as DUNE/2 + T2HK/2 in Table 5.1. This means that when not plottedagainst T , the combination of DUNE and T2HK will have T = 20, or 10 years running
time for each of the two experiments.

5.5.2

Mass ordering

As shown for the standard configurations in Section 5.3.1, identifying the mass orderingis almost guaranteed for experiments on this scale. However, a large difference is seen in
performance between DUNE and T2HK due to the difference in baseline distance. The
alternative beams of the DUNE collaboration do little to change this picture.
The results of the simulations are shown in Figure 5.21, which show the minimum
sensitivity to the mass ordering as a function of cumulative run time. The left columnof panels shows the performance of the alternative designs for DUNE (top) and T2HK
(bottom). It is seen that for DUNE, the 3 horn design improves the minimum sensitivity
by about 1 σ compared to the 2 horn design, whilst nuPIL performs almost identicallyto the 2 horn design. The 3 horn design is seen to reach greater than 5 σ significanceafter around 3 years run time, whereas the other two experiments require 4.5 years to
5 years.
For T2HK and its alternative designs the picture is quite different. Placing a secondtank in Korea will allow T2HKK to see large matter effects over the 1000 km to 1200 km
baseline, the sure-fire way to strong sensitivity to the mass ordering. Moreover, thepossibility of placing the second detector at a different off-axis angle, could produce
a wider beam, or a narrow beam whose peak is shifted away from the first maximum.
This interplay of factors could qualitatively alter the picture of mass ordering sensitivity160

9min √Δ𝜒 2

8
7

including T2HK data

8
7

6

6

5

5

4

4

3

3

2

2
9

T2HKK1.5∘
T2HKK2.0∘
T2HKK2.5∘
2-tank T2HK

4
min √Δ𝜒 2

92-horn DUNE
3-horn DUNE
nuPIL DUNE

including 2-horn DUNE data

8
7
6
5

3

4
3
2

1

3
2
4
Cumulative run time [years]

5

2

1

3
2
4
Cumulative run time [years]

5

Figure 5.21: Top (bottom) row: The minimum statistical significance of mass ordering
discrimination for DUNE (T2HK) with various beam designs. On both rows, the lefthand panels show the performance of the alternative designs in isolation, while the righthand panels show the impact of an alternative design on the combination of DUNE andT2HK by incorporating the standard T2HK and DUNE designs on the top and bottomrows, respectively. The configurations assumed here are described in Section 5.5.1 andthe true oscillation parameters are given in Table 5.2.

161

at T2HK(K). A greater variation in performance is seen as the fluxes are varied but,
as seen before, lower overall sensitivities. Due to the sizeable matter effects associatedwith the Korean detector, there is significantly increased sensitivity to the mass orderingover the standard T2HK design, but this does not strongly enhance the sensitivity.
This effect is understood to be due to the limited data collected by T2HKK at thelonger baseline. Fewer events associated with neutrinos travelling the longer baseline
are detected as the beam suffers significant suppression due to dispersion over the longer
distance6 . With WC technology, the advantage is known to come from scale, and such alimitation on event numbers means that longer baselines will not be competitive unlessoperated for a longer period of time. That said, the T2HK1.5° option does push thesensitivity above 3 σ after around 5 years. Although the full MO sensitivity against δ isnot shown in Figure 5.21, a limited comparison can be drawn between this work and Fig.
18 in [69]. The results presented here find slightly lower sensitivities: For T2HKK1.5°,
the difference is about 1 σ, while for off-axis angles of 2.5° and 2.0° the difference issmaller than 1 σ.
The sensitivity is seen to increase as the Korean detector is moved to smaller off-axis
angles. This can be explained by the different flux profiles of the T2HKK options. As thedetector is moved towards the beam axis, the events sample the oscillation probability
increasingly close to the first maximum. This is where the mass ordering is most visible
in the presence of matter effects and an accordingly stronger discovery potential is seen.
The right column of Figure 5.21 shows how the alternative designs impact the combination of the two experiments. As DUNE dominates the measurement of the ordering,these figures most closely resemble the top-left panel. Including T2HK data reduces
the difference in performance between the three DUNE beam designs, which all expect
a minimum sensitivity of 5 σ after 2.5 years. For T2HK, the inclusion of DUNE data,
pushes the overall sensitivity above 5 σ for the first time, with an extra Korean detector,
DUNE + T2HKK expects a greater than 5 σ measurement for all values of δ with lessthan 2 years run time.

5.5.3

CPV and MCP sensitivity

The sensitivity to CPV is understood to depend upon the energy of the events observed,
meaning that modifying the flux spectrum, for example with a narrower beam from
nuPIL or a beam located at the second maximum for T2HKK, could lead to significant
changes in the physics reach of the design. In the top-left panel of Figure 5.22 theperformance of the standard and alternative DUNE designs are compared. CPV andMCP sensitivities are shown for the three beam options as a function of δ in solid anddashed lines, respectively. The lower energy focus of the 3-horn optimised design is seen
to lead to an increase in sensitivity to both quantities, reaching a maximal sensitivity of
between 6.5 σ and 6.8 σ. This increase is as expected due to the heightened sensitivity ofthe second maximum to δ. The top-right panel shows how these sensitivities are changed
as information from the standard configuration of T2HK is included. It is seen that due
to T2HK’s strong sensitivity to the parameter δ, the impact of alternative designs for
6
The flux is dispersed by an inverse square law as baseline increases; subsequently, a Korean detector
sees around 11 % of the flux seen at Kamioka assuming the same off-axis angle.

162

DUNE is greatly reduced. Maximal sensitivities to CPV of above 11 σ are found for the
maximal values of δ ∈ { π2 , 3π
2 }.

10
√Δ𝜒 2

2-horn DUNE
3-horn DUNE
nuPIL DUNE

CPV
MCP

including T2HK data

5

3
T2HK
T2HKK1.5∘

√Δ𝜒 2

10

including 2-horn DUNE data

T2HKK2.0∘
T2HKK2.5∘

5

3
-1

-0.5

0
True 𝛿/𝜋

0.5

1

-1

-0.5

0
True 𝛿/𝜋

0.5

1Figure 5.22: The sensitivity to CPV (solid) and MCP (dashed) as a function of δfor various designs of DUNE (top row) and T2HK (bottom row). The configurations
assumed here are described in Section 5.5.1 and the true oscillation parameters are given
in Table 5.2.
For T2HKK three off-axis angles for the Korean detector are compared to the standard
configuration in the bottom row of Figure 5.22. The left panel, shows the performance of
these alternative designs in isolation. The experiments are seen to perform comparably,
but the best performance comes from the T2HKK2.0° flux. As can be seen in Figure 5.4,
this flux is the best aligned with the second maximum, suggesting that it is the accessto events which sample this part of the oscillation spectrum which lead to the increase
in sensitivity. Despite this, the increase remains modest. This is understood to beagain due to the suppression in event rates for a Korean detector: Although possessing
valuable information, they are seen in relatively small numbers, and their impact islimited. The bottom-right panel of Figure 5.22 shows the sensitivity to CPV and MCP
for combinations of DUNE and T2HKK. As T2HKK dominates this measurement,
there is little impact of including DUNE data aside from an overall improvement in thesensitivities by between 1 σ and 2 σ.
In Figure 5.23, the fraction of values of δ have been computed for which CP conservationor maximal CP violation can be excluded at great than 5 σ confidence. The top-leftpanel shows the performance of the alternative DUNE beam designs in isolation. The 3
163

horn design has a significantly higher sensitivity for all run times, with a CPV fraction
higher by between 10 % to 30 % and an MCP fraction higher by around 10 %. The 2horn design and nuPIL perform comparably for most run times. Considering 30 % to
be a benchmark CPV fraction, the 3 horn design expects to reach this sensitivity after
around 6 years, while the other two designs take 8 years. Excluding MCP is a hardermeasurement for all beam designs, and exposures of greater than 10 years would be
required to achieve a 30 % coverage of δ parameter space at 5 σ. The top-right panel
shows how the alternative DUNE designs are affected by the inclusion of T2HK data.
As it has already been seen that T2HK has a superior sensitivity to the phase δ, it isno surprise that the sensitivity is significantly higher for these combinations than the
DUNE variants in isolation. A relative suppression of the difference between variants is
also found; ultimately, DUNE offers less to this configuration and its precise design isless important. These combinations expect to reach a CPV fraction of 30 % (50 %) after
about 4 years (6 years). For the exclusion of MCP, these values will be approximately
reached after 10 years run time.

Fraction over 5𝜎

0.6

CPV
MCP

2-horn DUNE
3-horn DUNE
nuPIL DUNE

including T2HK data

0.4

0.2

Fraction over 5𝜎

0.0
0.8

T2HK
T2HKK1.5∘
2-tank T2HK

including 2-horn DUNE data

T2HKK2.0∘
T2HKK2.5∘

0.6

0.4
0.2
0.0

2

6
8
4
Cumulative run time [years]

10

2

6
8
4
Cumulative run time [years]

10

Figure 5.23: The fraction of true δ values where a CPV sensitivity (solid) and MCP
sensitivity (dashed) is expected to be over 5 σ, against cumulative run time. The configurations assumed here are described in Section 5.5.1 and the true oscillation parameters
are given in Table 5.2.
The bottom row of Figure 5.23 shows analogous plots for T2HK and T2HKK. On theleft, these alternative designs are considered in isolation. There is very little difference
between the T2HKK designs, although they all show an increase in CPV and MCP
fraction over the T2HK design. T2HKK expects a CPV fraction of over 50 % after164

less than 4 years, while T2HK requires around 6 years for the same sensitivity. MCP
fractions of greater than 30 % are possible after 5 years and 8 years for T2HKK and
T2HK, respectively. Compared with the results shown in the upper panels in Fig. 20
in [69], the same ranking of designs is found. However, sensitivities around 2 σ higher arefound near δ = ±π/2. This quantitative difference is suspected to be due to the priors
used, since in [69] priors for δ, θ23 and ∆m231 are not implemented. However, the studies
presented in this chapter use priors on all variables apart from δ, and the simulations
have slightly less leeway to accommodate degenerate solutions, and a correspondingly
improved ability to exclude CP conserving parameter sets.
The right panel shows the performance for the combination of DUNE data with theT2HK variants. As in the bottom-left panel, it is seen that the T2HKK designs perform
similarly, with T2HKK2.0 performing marginally the best. The inclusion of DUNE data
here makes little change to the sensitivities. In fact, as cumulative run time is definedas the sum of the individual DUNE and T2HKK run times, an apparent decrease inperformance is seen. Scaled appropriately for parallel data collection, DUNE + T2HKK
are found to expect a 5 σ CPV fraction of greater than 50 % after around 2.5 years
compared to 3.5 years for T2HKK alone.
To conclude this section, it is noted that the experiments running in isolation can expect
the exclusion of one of CP conservation or maximal CP violation for all values of δ at
3 σ and 5 σ for DUNE and T2HK, respectively. This can be seen clearly in Figure 5.22,
where the intersections between CPV and MCP lines are above the 3 σ or 5 σ horizontal
lines. This is true for all alternative designs, while the combination of DUNE and T2HK
ensures that one of these facts would be established with a significance greater than 6 σ.

5.5.4

Precision on δ

The difference in ∆δ is shown for the alternative designs in the left column of Figure 5.24. The top row shows that, as for CPV discovery, the 3 horn design performs
best, marginally improving ∆δ by between 0.5° and 1.5° over that of the 2 horn design.
These designs expect a precision on δ somewhere between 8° and 16° after their full
data taking period. The performance of the nuPIL design depends significantly on thetrue value of δ. For values near maximal CP violation δ = ± π2 , nuPIL performs worse
than the standard design. This can be understood due to the narrowing of the beam,
which when focused on first maximum, has insufficient events from other energies tomitigate the poor sensitivity around maximal CP violating phases. The top-right panelof Figure 5.24 shows the impact that the DUNE redesigns have on the combination of
DUNE and the standard configuration of T2HK. As shown in Section 5.4, T2HK dominates the significance for these measurements and a correspondingly small impact isseen of alternative beam designs for DUNE, but a significantly better precision than for
DUNE alone. Notably, the worsening of performance is however seen around maximal
CP violating values of δ for the combination of nuPIL and T2HK.
The expected sensitivity of ∆δ for the alternative designs for T2HKK are shown on thebottom-left panel of Figure 5.24. Here it is seen that all designs with a far detector allow
for a significant improvement in the precision on δ, generally seeing the best performancecoming from the 1.5° or 2.0° off-axis angle fluxes. A slight loss of performance is seen forlarger off-axis angles, which may be associated with the peak of the flux falling beyond
165

25

Diff.[∘ ]Δ𝛿 [∘ ]

20Δ𝛿 [∘ ]

20

15

15

10

10

5

5

0

0

3

1
0

0
-1
25
20

Diff.[∘ ]25

2-horn DUNE
3-horn DUNE
nuPIL DUNE

T2HK
T2HKK 1.5∘

25

T2HKK 2.0∘
T2HKK 2.5∘

20

15

15

10

10

5

5

0
0

0
0

-3
-4

-1

-1

-0.5

0
True 𝛿/𝜋

0.5

1

-2
-1

including T2HK data

including 2-horn DUNE data

-0.5

0True 𝛿/𝜋

0.5

1Figure 5.24: The 1 σ precision on δ for variants of DUNE (top row) wand T2HK (bottomrow). In the left column, these designs are considered in isolation while on the right,
variant designs of one experiment are combined with the standard configuration of theother. The configurations used are described in Section 5.5.1. These plots assume
normal mass ordering and the remaining true parameters are specified in Table 5.2.

166

the second maximum into a region of hard to identify, fast oscillations. The results for∆δ are very close to those shown in the upper panels of Fig. 23 in [69], with agreementon the ranking among alternative designs. This is notable, given the differences inducedby the priors in other variables of interest, but is explained by the fact that the priors
differ in their global structure more than in their local structure. It is this local structure
which dictates ∆δ, as at low significance the Gaussian approximation works well andmultiple minima are irrelevant. On the right panel of Figure 5.24, the combination isshown with different T2HKK fluxes and the standard DUNE configuration. Once again,
it is seen that T2HKK dominates the combination, and therefore the shapes of thesecurves closely follow those on the left panel.

5.5.5

Optimal configuration

The preceding sections have studied how the alternative designs of T2HKK and DUNE
could impact the physics reach for key measurements, considering both the experiments
in isolation and in certain combinations. It was seen that the best option for DUNE
is the 3 horn design, which excels at measurements of the mass ordering, and precision
measurements of δ for values around δ ∈ { π2 , 3π2 }. T2HKK in contrast performs best witha flux positioned between 1.5° and 2.0° degrees off axis. Here it maximizes its sensitivity
to CP violation, its ability to exclude maximal CP violation and to make precision
measurements of δ around CP conserving values. Whereas so far alternative designs
have only been considered for one experiment in combination with the standard designof the other, in this section the physics reach is reported of the optimal combination of
DUNE 3-horn and T2HKK1.5 (and T2HKK2.0).
Figure 5.25 shows the minimum sensitivity expected for the mass ordering for this optimal configuration of DUNE + T2HKK. A 3 σ measurement is expected after less thana year, which increases to 5 σ after around 1.8 years. Figure 5.26 shows the significanceat which the experiments are expected to exclude CP conservation (solid) and maximal
CP violation (dashed). These are expected to reach a maximal significance of 11 σ and
12 σ, respectively. The advantage of the combination is clearer when the performance isviewed in terms of the minimal run time required for the exclusions to be made at 5 σ.
The combination of DUNE + T2HKK expects to have greater than 5 σ exclusion of CP
conservation for more than 25 % (50 %) of the parameter space after 2.5 years (5 years).
For the exclusion of maximal CP violation, longer run times are required: about 6 years
ensures the exclusion for more than 25 % of values of δ. For the precision on δ, shown in
Figure 5.27, it is found that the optimal combination of DUNE + T2HKK could expect
a measurement around a CP conserving value with an uncertainty of only 4.5°. Thisworsens for maximally CP violating values of δ to around 11°.

Conclusion
DUNE and T2HK will lead the way in key measurements of the neutrino oscillation
parameters. These long-baseline experiments will make high statistics determinations of
the mass ordering, the first precision measurements of δ, and have an excellent chanceto establish the presence of fundamental CP violation in the leptonic sector. In thischapter, the expected performance of these two experiments has been studied, including167

3-horn DUNE + T2HKK1.5∘
3-horn DUNE + T2HKK2.0∘
2-horn DUNE + T2HK

9

min √Δ𝜒 2

8

7
6
5

4
3

1

2

3
Cumulative run time [years]

4

5

Figure 5.25: The minimum mass ordering sensitivity for the combination of DUNEwith the 3 horn flux and T2HKK1.5° (red) and T2HKK2.0° (blue) compared with thestandard configurations of DUNE with 2 horn flux and T2HK with a single tank at
Kamioka (green). The configurations assumed here are described in Section 5.5.1 andthe true oscillation parameters are given in Table 5.2.

168

CPV
MCP

3-horn DUNE + T2HKK1.5∘
3-horn DUNE + T2HKK2.0∘
2-horn DUNE + T2HK

√Δ𝜒 2

10

5

-0.5

-1

Fraction over 5𝜎

0.5

13-horn DUNE + T2HKK1.5∘
3-horn DUNE + T2HKK2.0∘
2-horn DUNE + T2HK

0.8
0.6

0
True 𝛿/𝜋

CPV
MCP

0.4

0.2
0.0

2

6
4
Total run time [year]

8

10

Figure 5.26: Top: The CPV and MCP sensitivity for the combination of DUNE with the
3 horn flux and T2HKK1.5°(2°) shown in red (blue). For reference, the combination of
the two standard designs are also shown: DUNE with 2 horn beam and T2HK (green).
Bottom: The fraction of δ parameter space for CPV (MCP) sensitivity over 5 σ for thesame configurations as on the top panel. The configurations assumed here are describedin Section 5.5.1 and the true oscillation parameters are given in Table 5.2.
169

25

3-horn DUNE + T2HKK1.5∘
3-horn DUNE + T2HKK2.0∘
2-horn DUNE + T2HK

20

Δ𝛿 [∘ ]

15

10

5

0

-1

-0.5

0
True 𝛿/𝜋

0.5

1Figure 5.27: The 1 σ error on δ for the combination of DUNE with the 3 horn flux and
T2HKK1.5°(2.0°) shown in red (blue). For reference, also shown are the combination of
the two standard designs: DUNE with 2 horn beam and T2HK with one tank (green).
The configurations assumed here are described in Section 5.5.1 and the true oscillation
parameters are given in Table 5.2.

170

possible alternative designs which have been recently suggested. It is seen that thanks
to their different designs, both the energy profiles of the beam and the different baseline
distances chosen, DUNE and T2HK have different sensitivities to the mass ordering andthe value of δ, leading to a natural complementarity.
DUNE, with its long baseline and significant matter effects, excels at measuring themass ordering. It can expect a greater than 5 σ measurement after between 2 years and7 years depending on the true value of θ23 . T2HK is limited in its sensitivity for thismeasurement, but the combination of data collected at T2HK with the DUNE data
reduces the impact of the worst-case scenario, significantly reducing the required run
times. DUNE+T2HK can expect the same measurement in less than 3 years regardlessof θ23 . The roles are reversed for measurements of the CP phase δ. T2HK dominates
this measurement, thanks to its high statistics. In isolation, T2HK expects to be able toexclude CP conservation at greater than 5 σ for more than 50 % of the parameter space
after around 7 years. DUNE alone would require between 10 years to 15 years of data
for the same measurement, but the combination of the two experiments, assumed tocollect data in parallel, would take at most 5 years. For the measurement of the octant
sensitivity, it is found that the large event numbers in the disappearance channel of
T2HK lead to better performance than at DUNE. For example, to exclude the upper
octant solution at 5 σ with a true value of θ23 = 40°, T2HK needs about 2 years, while
DUNE requires 5 years of data. This pattern is repeated for the exclusion of maximal
mixing, where for the true value θ23 = 40°, 5 σ exclusion at DUNE takes 2 years, while
T2HK can make this exclusion in only 1 year. For these measurements, T2HK dominates
the performance of the combination of DUNE and T2HK, although some small benefit
is found from the inclusion of extra data. Also studied is the precision on sin2 θ23 , where
there is a strong dependence on the true value of θ23 , with the worst precision close to
maximal mixing, as expected for a measurement driven by the disappearance channel.
At the peak, ∆(sin2 θ23 ) for DUNE is about 0.045 while T2HK can significantly improvethis, peaking around 0.032. Extending the study to the 1 σ joint precision on δ andsin2 θ23 , the measurement of these two parameters are seen to be largely independent,
due to the disappearance channel driving the fits to θ23 while the appearance channel
dictates δ. The precision gets worse at θ23 = 45°, as seen before, and improves uponmoving away from this maximal value. For θ23 = 40° or 50°, the precision on θ23 isaround 0.8° (0.5°) for DUNE (T2HK). However, near maximal mixing the value increasesto ∆θ23 = 2.3° (1.5°) for DUNE (T2HK).
In particular the sensitivity to δ has been stressed, studying the behaviour of the 1 σ
uncertainty on δ, ∆δ, in some detail. For most of the parameter space, T2HK outperforms DUNE if both are operated in isolation, expecting ∆δ to lie between 6° and 18°.
It was shown that T2HK is intrinsically less sensitive to δ, but increases its sensitivity
through large statistics. DUNE on the other hand, is limited by lower event rates. To
reach comparable precision on δ, T2HK requires between 2 and 3 times as many events
as DUNE. Beyond the question of statistics, the complementarity of the two experiments for precision measurements of δ was discussed. DUNE’s wide-band beam helps
to compensate for a loss of sensitivity at the first oscillation maximum, which hampers
T2HK’s performance. It was found that DUNE performs best for maximally CP violating values of δ and T2HK, in contrast, prefers CP conserving values. When combined,
these experiments complement each other, and the global sensitivity to δ is well covered171

by the two technologies: DUNE + T2HK is expected to reach ∆δ . 6° (14°) for CP
conserving (maximally violating) values of δ after 10 years of running in parallel.
Also considered were the potential alternative designs for T2HK and DUNE. T2HK may
locate its second detector module in southern Korea, while DUNE is considering two
beams beyond its standard 2 horn reference, a 3 horn optimised and the nuPIL design.
The ability of these designs to determine the mass ordering, to exclude CP conservationand maximal CP violation, and to measure δ was investigated. These alternatives arepromising extensions of the current physics programme, and lead to modest improvements in all measurements studied in this work. The combination of DUNE (3 horn)
and T2HKK with a flux between 1.5° and 2.0° off-axis was identified as the optimal
choice, based on the measurements considered in this work. This combination expects
to discover the mass ordering at 5 σ after only 1.8 years, to be able to exclude CP conservation at 5 σ for more than 50 % of the parameter space after 5 years, and to measureδ around CP conserving (maximally violating) values with an uncertainty of around 4.5°(11°) after its full data-taking period.
In conclusion, it has been shown that DUNE and T2HK have a natural complementarity,
thanks to key differences in their designs. Although design modifications, such as nuPIL
for DUNE or the location of T2HK’s second detector in Korea, have quite distinct
features which could upset the existing synergy, the combination of the two experiments
is found to be quite robust. Sensitivity to the mass ordering will come primarily from
DUNE, sensitivity to CP conservation is dominated by T2HK, but precision on δ is a bit
more nuanced with wider-band information being preferred for maximally CP violating
values of δ, and high statistics first maximum measurements preferred for CP conserving
values. Overall, the global physics program benefits from the breadth and variation indesign of these experiments.

172

Chapter 6

Testing Littlest Seesaw models of
lepton flavour

173

Introduction
While the immediate goals of the future long-baseline neutrino experiments were examined in the previous chapter, the broader motivation for these measurements is more
fundamental: Rather than simply increasing our knowledge of the properties of neutrinos, it is hoped that this will help further our understanding of these particles and themechanisms which underlie their physics, perhaps even paving the way to a successor to
the Standard Model of particle physics. The framework of neutrino masses and mixing
for explaining neutrino oscillations – the first direct experimental evidence for physics
beyond the Standard Model – is now firmly established [179]. All three mixing angles
together with the size of the two mass-squared differences have been measured, withexperimental efforts now focused on determining the final few unknowns: the orderingand scale of the neutrino masses, the value of the Dirac phase δ, and a precision measurement of the angle θ23 including, if non-maximal, its octant. Although there is someas yet inconclusive evidence for δ in the third or fourth quadrant, as well as for normal
ordering (NO) and non-maximal atmospheric mixing, the next generation of oscillation
experiments is relied upon to set these issues to rest. In Chapter 5 it was seen that,with the exception of only the scale of the neutrino masses, all of these measurements
are within reach.
On the theoretical side, however, the origin of neutrino masses and mixing remains unknown, with many possible models considered viable (for reviews see e.g. [38, 44, 180–182]). A large proportion of these models are based on the classic seesaw mechanism,
involving heavy right-handed Majorana neutrinos [183–188], providing both a mechanism for generating the neutrino masses and a natural explanation for their smallness.
However, in order to make predictions that can be probed experimentally, seesaw models
require additional assumptions or constraints [189].
To accommodate the three distinct light neutrino masses which drive the oscillation
phenomenon, the seesaw mechanism requires at least two right-handed neutrinos [190].
In order to reduce the number of free parameters still further to the smallest number
possible, and hence increase predictivity, various approaches to the two right-handed
neutrino seesaw model have been suggested1 , such as postulating one [191] or two [192]texture zeroes in the Dirac mass matrix in the flavour basis (i.e. the basis of diagonal
charged lepton and right-handed neutrino masses). However, such two texture zero models are now phenomenologically excluded [193] for the case of a normal neutrino mass
hierarchy. The minimal two right-handed neutrino model with normal hierarchy which
can accommodate the known data of neutrino mixing involves a Dirac mass matrix with
one texture zero and a characteristic form known as the Littlest Seesaw model [194].
The Littlest Seesaw model may be embedded in unified models of quarks and leptons
in [195, 196]. It leads to successful leptogenesis where the sign of baryon asymmetry isdetermined by the ordering of the heavy right-handed neutrinos, and the only seesaw
phase η is identified as the leptogenesis phase, linking violation of charge parity symmetry (CP) in the laboratory with that in the early universe [197]. The Littlest Seesaw
In seesaw models with two right-handed neutrinos, including those discussed in this chapter, ahierarchical spectrum of left-handed neutrino masses is obtained where the lightest left-handed neutrino
is massless. This is always the case since to the Dirac mass matrix connecting the two right-handed
and three left-handed neutrinos is 3 × 2, and so there can only be two non-zero eigenvalues in the 3 × 3
effective light neutrino mass matrix.
1

174

model can be understood as an example of sequential dominance (SD) [198,199] in which
one right-handed neutrino provides the dominant contribution to the atmospheric neutrino mass2 , leading to approximately maximal atmospheric mixing, while the secondright-handed neutrino gives the solar neutrino mass and controls the solar and reactor
mixing as well as the magnitude of CP violating effects via δ. SD generally leads tonormal ordering and a reactor angle which is bounded by θ13 . m2 /m3 [191], proposeda decade before the reactor angle was measured [179]. Precise predictions for the reactor(and solar) angles result from applying further constraints to the Dirac mass matrix, anapproach known as constrained sequential dominance (CSD) [200]. For example, keeping
the first column of the Dirac mass matrix proportional to (0, 1, 1)T , a class of CSD(n)models has emerged [194, 200–205] corresponding to the second column proportional
to√m2(1, n, (n − 2))T , with a reactor angle approximately given by [47] θ13 ∼ (n − 1) 32 m.
3
The Littlest Seesaw model corresponds to n = 3 with a fixed seesaw phase η = 2π/3.
It was recently realised that the alternative form of the Littlest Seesaw model withsecond column (1, 1, 3)T and seesaw phase η = −2π/3 (also proposed in [194]) may
be enforced by an S4 × U (1) symmetry, putting this version of the Littlest Seesaw
model on a firm theoretical foundation [48] in which the required vacuum alignment
emerges from symmetry as a semi-direct model [206]. In general the Littlest Seesaw
model is an example of trimaximal TM1 mixing [207–213], in which the first columnof the tri-bimaximal mixing matrix [46] is preserved, similar to the semi-direct modelof trimaximal TM1 mixing that was developed in [214]. To fix the seesaw phase, one
imposes a CP symmetry in the original theory which is spontaneously broken, where,
unlike [215], there is no residual CP symmetry in either the charged lepton or neutrino
sectors, but instead the phase η in the neutrino mass matrix is fixed to be one of the
cube roots of unity due to a Z3 family symmetry, using the mechanism proposed in [216].
As explained in more detail later in this chapter, the Littlest Seesaw model predicts all
neutrino masses and mixing parameters in terms of two or three parameters, and it has
been shown that the model is in agreement with all existing data, for a suitable range of
its internal parameters [205]. The model makes some key predictions about the neutrinomass spectrum, that the lightest neutrino is massless m1 = 0 and that normal ordering3
obtains ∆m231 > 0, which offer a means to exclude it via the observation of neutrinolessdouble beta decay4 , the measurement of the beta-decay end-point, or from cosmologicalmeasurements, as well as any measurement of inverted ordering from neutrino oscillation
searches. However, the model also provides a rich set of predictions and correlations forthe mixing angles and phases. This chapter covers the work published in [2], which
studies how the future long- and medium-baseline oscillation programme will be able toWith the lightest neutrino massless, m1 = 0, the two non-zero masses are referred to as the solar
neutrino mass and the atmospheric neutrino mass, corresponding to thepsquare roots of thepexperimentally measured solar and atmospheric neutrino mass splittings m2 = ∆m221 and m3 = ∆m231
respectively.
3
The prediction of NO is a general consequence of Sequential Dominance upon which the model isbuilt [198, 199]. In the case of the Littlest Seesaw models in this chapter, having only two RH neutrinos
leads to one massless neutrino, m1 = 0, as explained above, while the model’s Sequential Dominanceassumption of one RH neutrino being dominant for m2 and a second heavier RH neutrino dominant form3 leads directly to ∆m232 > 0, i.e. normal ordering [191].
4
The masslessness of the lightest neutrino leads to a neutrinoless double beta decay rate far smaller
than could foreseeably be observed [205].
2

175

test this model, under the assumption of Normal Hierarchy (m1 = 0 and NO), throughthe precision measurement of the oscillation parameters.
The layout of the chapter is as follows: in Section 6.1 the Littlest Seesaw models discussed above are defined and some of the predictions are expressed in terms of exact
sum rules of the neutrino oscillation parameters. In Section 6.2 the Littlest Seesaw models are confronted with existing oscillation data, showing the precise predictions madeonce this data is taken into account. Section 6.3 then covers how the predictions of
the models could be probed at future experimental facilities, showing the sensitivities ofexperiments to exclude the models and the combined measurements required to do so.

6.1

Littlest Seesaw models of neutrinos

Sequential dominance models of neutrinos arise from the proposal that, via the type-I
seesaw mechanism, a dominant heavy right-handed (RH) neutrino is mainly responsible for the atmospheric neutrino mass, a heavier subdominant RH neutrino for thesolar neutrino mass, and a possible third largely decoupled RH neutrino for the lightest
neutrino mass [198, 199]. This leads to the prediction of normal neutrino mass ordering
and, in the minimal case containing just the dominant and subdominant right-handed
neutrinos, the masslessness of the lightest neutrino. Constrained sequential dominance(CSD) constrains these models further through the introduction of family symmetry,with the indirect approach used to fix the mass matrix from vacuum alignments offlavon fields [200]. A family of such models, parametrised by n, either integer or real using the family symmetry groups S4 or A4 respectively, predicts the CSD(n) mass matrix
for left-handed neutrinos [47, 194]. Following the notation introduced in Section 1.1.3,
the RH neutrino Majorana mass matrix, for the case of just two RH neutrinos, is given
by
!
Matm
0
MR =
,
(6.1.1)0
Msol
where Matm and Msol are the masses of the RH neutrinos responsible mainly for theatmospheric and solar neutrino masses respectively. The form of the Dirac mass matrix
is determined by the family symmetry and is given by0ana  ,
mD = ee (n − 2)a

(6.1.2)

with the two additional complex parameters a and e. Following through the steps of theseesaw mechanism outlined in Section 1.1.3, this results in the left-handed light effective
Majorana neutrino mass matrix in the charged-lepton flavour basis given by
0 0 0
1
n
(n − 2)n2n(n − 2) ,mν = ma 0 1 1 + mb eiη  n
0 1 1(n − 2) n(n − 2) (n − 2)2








(6.1.3)where in addition to n there are three free real parameters: two parameters with the22dimension of mass ma ≡ Meatm and mb ≡ Masol , and a relative phase η determined byarg ae . A second version of this model has also been proposed, based on an S4 ×176

U (1) symmetry, where the second and third rows and columns of the mass matrix are
exchanged [48]. In this work, both these versions are discussed for the case where n = 3,
since it has been shown that n = 4 is disfavoured and other values of n are alreadyexcluded [48, 205]. These models are also known as Littlest Seesaw (LS) models since
they provide physically viable seesaw models with the fewest number of free parameters.
The two versions of the model used are denoted here as LSA and LSB;
1 3 1
0 0 0



iη 
ν
mLSA = ma 0 1 1 + mb e 3 9 3 ,
1 3 1
0 1 1

(6.1.4)

1 1 3
0 0 0




mνLSB = ma 0 1 1 + mb eiη 1 1 3 .
3 3 9
0 1 1

(6.1.5)

















Although, in the most minimal set-up, the relative phase η is a free parameter, it has
been shown that in some models the presence of additional Z3 symmetries can fix thephase eiη to a cube root of unity [215], with η = 2π/3 the preferred value for LSA andη = −2π/3 for LSB as determined by current data [205]. This restriction gives themodel greater predictivity by reducing the number of free parameters to two. These
cases are therefore given special attention, while also showing results for the case with
η left free.
Diagonalising the mass matrices above leads to predictions for the neutrino masses aswell as the angles and phases of the unitary PMNS matrix, UPMNS , which describes the
mixing between the three left-handed neutrinos
m1 0
0


T
ν
UPMNS m UPMNS =  0 m2 0  ,
0
0 m3




(6.1.6)

where UPMNS is defined by




β1

c12 c13s12 c13s13 e−iδ ei 2

c13 s23  
UPMNS = −s12 c23 − c12 s13 s23 eiδ c12 c23 − s12 s13 s23 eiδ 0s12 s23 − c12 s13 c23 eiδ −c12 s23 − s12 s13 c23 eiδ c13 c23
0

0
β2



0

0
1
(6.1.7)
with sij = sin θij and cij = cos θij . All of the parameters in this decomposition are therefore predicted in terms of the 2 (or 3) real parameters in Equations (6.1.4) and (6.1.5).
Due to the minimal assumption of only two right-handed neutrinos, the lightest neutrino
is massless m1 = 0 and the mass-squared differences, which are the only combinationsof masses accessible to neutrino oscillation experiments, are predicted to be ∆m221 = m22
and ∆m231 = m23 . Of the remaining parameters, θ12 , θ13 , θ23 and δ, are also experimentally accessible via neutrino oscillation, while the Majorana phases β1 and β2 are
not.
ei 2
0



As will be seen in more detail in the next section, due to their similar forms, LSA
and LSB make similar predictions. However, the process of diagonalisation reveals thatthe octant of θ23 is reversed, along with the sign of δ, while all other parameters areunchanged. Changing the sign of η, however, also reverses the sign of δ, with no othereffect, and so with the sign of η not fixed by the model the only physical difference
between LSA and LSB is the octant of θ23 .
177

6.1.1

Sum rules of LS

Before probing the viability of the LS models using experimental data, it is useful tosummarize the hard predictions of the model irrespective of the predictions made once
existing observations are enforced. In addition to the predictions of normal ordering andthe lightest neutrino being massless, the form of the PMNS matrix given above enforces
relationships, known as sum rules, between the neutrino mixing parameters. While theexact values of the parameters are not predicted, the relationships described in thissubsection give testable predictions of the model that will be investigated throughout
the remainder of this chapter.
It has already been shown that, since the first column of the LS mixing matrix UPMNS isequal to that of the tri-bimaximal mixing matrix, LS (both LSA and LSB for all valuesof η) obeys the TM1 sum rules [47, 48]
1 qtan θ12 = √1 − 3s213 ,2

1
sin θ12 = √3q

1 − 3s213
c13,

cos θ12 =r

cot 2θ23 (1 − 5s213 )q
cos δ = − √,2 2s13 1 − 3s213

2 1
,
3 c13

(6.1.8)(6.1.9)where sij = sin θij and cij = cos θij , and the forms in Equation (6.1.8) are equivalent.
2π
For LSA with η = 2π
3 or LSB with η = − 3 , there are several additional sum rules,
which have been newly derived and are presented here. A set of these additional sum
rules can be derived using the fact that the only two remaining input parameters ma andmb have dimensions of mass, so all the mixing angles and phases must depend only onmb
the (unitless) ratio r ≡ m. The exact expressions for the mixing angles and Dirac phaseaas a function of r can be found in Appendix D, along with new exact sum rules derived
using these expressions. These results make clear the difference between predictions of
LSA and LSB; while θ13 and θ12 remain unchanged, cos 2θ23 and cos δ differ by a changeof sign.

An exact expression for the Jarlskog invariant J has previously been given as [47, 48]
J = s12 c12 s13 c213 s23 c23 sin δ = ∓24m3a m3b (n − 1) sin η.
m23 m22 ∆m232(6.1.10)with the negative sign taken for LSA and positive for LSB. For both LSA with η = 2π
3 ,and LSB with η = − 2π
,
the
new
relation3
m2 m3 = 6ma mb(6.1.11)

is found. Using this relation and inserting n = 3 into Equation (6.1.10) leads to the new
relation for the Jarlskog invariant Jq

J =−∆m221 ∆m231√
3 3∆m232(6.1.12)and hence the new sum rule,q∆m221 ∆m231
sin δ = − √,3 3∆m232 s12 c12 s13 c213 s23 c23
178(6.1.13)2πwhich is valid for both LSA with η = 2π
3 and LSB with η = − 3 .

6.2

Probing LS with existing data

Existing measurements of the neutrino mixing parameters have been shown to be ingood agreement for CSD(n) for the n = 3 case [205]. The best-fit value of η is found to
be close to ± 2π
3 , with the positive sign for LSA and the negative sign for LSB, which
has been theoretically motivated as one of the cube roots of unity required due to an
additional Z3 symmetry as part of a larger GUT model [47]. This section, studies boththe case where η is fixed by symmetry and the case where it is left as a free parameterof the theory.

6.2.1

Predictions of oscillation parameters with fixed η = ±2π/3

2π
In the n = 3 case of LSA with η = 2π
3 (or LSB with η = − 3 ), all neutrino masses,
mixing angles and phases are fully determined from the two remaining parameters ma
and mb and the three most precisely measured of these parameters, θ13 , ∆m231 and∆m221 , currently provide the strongest test of the LS model. Exact expressions are used
for all mixing angles and neutrino masses, derived in [47, 48], to study these predictionsof the LSA and LSB models. Figure 6.1 shows how these parameters vary in the ma −mbplane, along with the regions corresponding to the 1 σ and 3 σ ranges for these parametersfrom the NuFit 3.2 (2018) global fit [6], assuming normal mass ordering and a lightest
neutrino mass of m1 = 0. The SD proposal requires ma to be significantly larger than
mb and for this portion of the parameter space the approximate proportionality relationsof m2 ∼ mb and m3 ∼ ma can be seen, verifying the approximations previously derived
in [47].

θ13 [°]
4

6

8

10

m2 [meV]
12

14

16

4

4.0 (a)68

m3 [meV]
10

12

(b)

30 35 40 45 50 55 60 65(c)mb [meV]

3.5
3.0
2.5
2.0
1.5
1.0
15

20

25

ma [meV]

30

35 15

20

25

ma [meV]

30

35 15

20

25

30

35

ma [meV]2π
Figure 6.1: Predicted values from LSA with η = 2π
3 (or LSB with η = − 3 ) of oscillation
parameters depending on the input parameters ma and mb . Regions corresponding to
the experimentally determined 1 σ (solid lines) and 3 σ (dashed lines) ranges for eachparameter are also shown.

179

Even at 1 σ the three allowed regions coincide at a single point, as can be seen in
Figure 6.2, and so this benchmark point can be used to make predictions of the remaining
angles θ12 and θ23 and the Dirac phase δ. As described in Section 6.1 these parameters,
along with θ13 , depend only on the ratio r = mb /ma ; this dependence, given by the
relations in Equation (D.0.1), is shown in Figure 6.3, with the 1 σ and 3 σ NuFIT 3.2
ranges and reference point at mb /ma = 0.1. For θ23 and δ, the predictions of both LSA
and LSB are shown. At this point it can be seen that all mixing angles θ13 , θ12 and θ23
lie within their 1 σ ranges and a prediction on the value of the Dirac phase is made ofδ ' −90°.

4.0

3.5

mb [meV]

3.0
★

θ13

2.5

Δm221
Δm231

2.0

1.5

1.0
15

20

25
ma [meV]30

35

Figure 6.2:
Regions in the ma -mb plane with fixed η = 2π/3 (η = −2π/3) for LSA (LSB) corresponding to the experimentally determined 1 σ and 3 σ ranges for θ13 , ∆m221 and ∆m231 .
Combining these results for all parameters that have been experimentally measured,displayed together in Figure 6.4, the predictions for θ12 and θ23 are seen to lie within
the current 1 σ bounds for both LSA and LSB.

6.2.2

Predictions of oscillation parameters with η as a free parameterIn the versions of the LS models with η as an additional free parameter, the mixing
angles and phases now depend on both the ratio r = mb /ma and η. The masses m3 andm2 depend on all three input parameters; however, their ratio m2 /m3 (and therefore theratio ∆m221 /∆m231 ) will depend only on r and η. As previously, the strongest constraints
come from the very precise measurements of θ13 and the mass-squared differences ∆m221
and ∆m231 . Figure 6.5 shows the regions corresponding to the 1 σ ranges for all the
180

0

70
60

-50
θ23

40

θ12
30

δ [°]

θ [°]

50

δ

20

-150

10
0
0.0

-100

θ13

0.1

0.2

0.3

0.4

0.0

0.5

0.1

0.2

0.3

0.4

0.5

mb /mamb /maFigure 6.3: Predicted values from LS with fixed η = 2π/3 for LSA (solid) and η = −2π/3
for LSB (dot-dashed) of the mixing angles and delta as a function of the ratio mb /ma .
Note that only θ23 and δ differ between LSA and LSB. Horizontal bands show theexperimentally determined 1 σ and 3 σ ranges for each parameter. A reference point
giving a good prediction for all parameters is shown at r = mb /ma = 0.1.

LSA

4.0

LSB

3.5

mb [meV]

3.0
★

★

2.5

2.0

1.5

1.0 16

18

20

22

24

26

28

30

32

34

16

18

20

ma [meV]

22

24

26

28

30

32

34

ma [meV]
θ12

θ23

θ13

Δm221

Δm231

Figure 6.4: Regions in the ma -mb plane with fixed η = 2π/3 (η = −2π/3) for LSA(LSB) corresponding to the experimentally determined 1 σ ranges for solar and reactor
mixing angles and mass-squared differences. The θ23 regions shown are in tension withother measurements, however, extending to 2 σ these regions become far larger, covering
the entire parameter space shown in these plots.

181

mixing angles, δ and m2 /m3 , where it is seen that all the five regions overlap at a region
around η = ±2π/3 for LSA and LSB, respectively. That two input parameters should
give a good description of five observables, within their one sigma errors, is ostensiblya remarkable achievement, indeed perhaps better than might be expected on statisticalgrounds.
LSA

1.0

LSB

★

η/π

0.5

0.0

-0.5
★

-1.0

0.025

0.050

0.075

0.100

0.125

0.150

0.175

0.025

0.050

mb /ma

0.075

0.100

0.125

0.150

0.175

mb /ma
θ12

θ13

θ23

m2 /m3δFigure 6.5: Regions in the (mb /ma ) - η plane corresponding to the experimentally
determined 1 σ ranges for all mixing angles, δ and the ratio of neutrino masses m2 /m3
for LSA (left panel) and LSB (right panel).

6.2.3

Fitting LS models to global fit data

In order to provide a more concrete measure of the agreement between the predictionsof the model and existing data, as well as to make further predictions of the less wellmeasured parameters, a χ2 fit is performed to the four cases discussed above: LSA and
LSB with η fixed and free. As a proxy for the full data sets of previous experiments,
these fits use the results of the NuFIT 3.2 global analysis [6]. This analysis combines
the results (as of early 2018) of solar, atmospheric, long baseline accelerator, and long,
medium and short baseline reactor neutrino experiments, to obtain a combined fit to
the six standard neutrino oscillation parameters. The χ2 data provided by NuFIT5 isused, for the case where normal mass ordering is assumed, combining both the 1D χ2data for each mixing parameter with the 2D χ2 data to include correlations between
parameter measurements
χ2Fit (Θ) =

X
θi ∈Θ

χ21D (θi ) +

X





χ22D (θi , θj ) − χ21D (θi ) − χ21D (θj ) ,(6.2.1)

θi 6=θj ∈Θ

where the first sum in this expression combines each of the 1D χ2 data into a first
approximation of the full 6D χ2 while the second sum provides corrections to this coming
The full data used for one- and two-dimensional ∆χ2 projections of the NuFIT analysis, available
at http://www.nu-fit.org
5

182

from the 2D correlations between each pair of parameters. This procedure is used since
the full 6D χ2 data is not available and so using the 1D and 2D data together in thisway provides the best possible approximation. The more simple (and more commonly
applied) method of using only the 1D χ2 data for each parameter would be insufficientfor our case, since it is often the correlations between parameters, rather than theirvalues independently, that are testing the model predictions.
This result is then applied first to the standard mixing case, then to the LS model caseas follows:• For the case of standard mixing Θ = ΘPMNS ≡ θ12 , θ13 , θ23 , ∆m221 , ∆m231 , δ andthe NuFIT 3.2 results are simply combined as shown above, in order to include
correlations, and use this to calculate χ2 (ΘPMNS ) ≡ χ2Fit (Θ) for this case.
• For the LS model Θ = ΘLS ≡ {ma , mb , η} (or ΘLS = {ma , mb } when fitting with η
fixed) is used instead, which is then minimised over the LS parameter space, using
the analytic relations to calculate standard mixing parameters from LS parametersand hence calculate χ2 (ΘLS ) ≡ χ2Fit (Θ) for this case.
The test statistic used for a particular LS model is then given by:
q

∆χ2 =

r

min [χ2 (ΘLS )] − min [χ2 (ΘPMNS )].
ΘLS

ΘPMNS

(6.2.2)

While a number of the assumptions in Wilks’ theorem [217] are broken by the parameter
space of the LS models (and indeed standard neutrino mixing), Monte-Carlo simulationsof randomised experimental data were used to confirm that Wilks’ theorem holds well forthis statistic, i.e. it is approximately distributed according to a chi-squared distribution.
This allows the statistic to be used to quote results in terms of a number of σ in theconventional way.
The best fit LSA and LSB points for fits with η left free or with η fixed at 2π
3 are given
in Table 6.1 together with errors on the predicted parameters corresponding to the 1 σ
allowed ranges. The number of degrees of freedom (d.o.f.) is either 3 or 4, which is justthe difference between the number of observables (which is taken to be the parametersin ΘPMNS ) and the number of LS parameters (namely the parameters in ΘLS , which iseither 3 or 2, depending on whether η is free or fixed). For LSA a best fit is found with
∆χ2 = 1.6 (3 degrees of freedom) with η free and ∆χ2 = 1.9 (4 degrees of freedom)2
fixing η = 2π
3 , while slightly better fits are found for LSB, with ∆χ = 1.5 (3 degrees of2
freedom) and ∆χ = 1.7 (4 degrees of freedom) for η free and η = − 2π
3 respectively.
Figure 6.6 shows the best fit points with 1 σ and 3 σ contours of the fits in the ma − mb
plane for fixed η and in the r − η plane for free η. The significance at which a LS modelis allowed is determined from the distribution of the ∆χ2 test statistic, where N σ hasbeen calculated assuming that Wilks’ theorem applies.
The fit can also be used to identify the regions of standard neutrino mixing parameter
space predicted by LS, once existing data has been taken into account. This corresponds
to mapping the regions of LS input parameter space allowed by the fit onto the standard
mixing parameter space.
Figure 6.7 shows the predictions of LS (for the fixed η case) in the planes made from eachpair of mixing angles and δ. Since these values all depend only on the single parameter
183

LSA

NuFIT 3.2

η freeη fixedη freeη fixedglobal fit

26.80
2.666
0.672π

26.61
2.670
2π/3

26.74
2.669
−0.670π

26.64
2.673
−2π/3

—

34.34+0.03−0.03
8.57+0.13−0.16
45.46+0.52−0.56−88.0+2.2−2.4
7.396+0.203−0.2092.485+0.029−0.02934.33+0.02−0.02
8.61+0.11
−0.10
+0.11
45.72−0.10+0.4−86.9−0.4+0.1557.311−0.136+0.0302.485−0.027+0.04
34.33−0.03+0.13
8.59−0.16+0.60
44.45−0.52−92.4+2.6−2.2
7.379+0.204−0.1992.485+0.030−0.02834.33+0.02−0.03
8.61+0.11−0.10
44.28+0.10−0.11−93.1+0.4−0.4
7.328+0.150−0.1452.490+0.027−0.03033.62+0.78−0.76
8.54+0.15−0.15
47.2+1.9−3.9−126+43−31
7.40+0.21−0.20
2.494+0.033−0.031

1.6 / 3

1.9 / 4

1.5 / 3

1.7 / 4—ma [meV]mb [meV]η [rad]θ12 [°]
θ13 [°]
θ23 [°]
δ [°]∆m221 [10−5 eV2 ]∆m231 [10−3 eV2 ]LSB∆χ2 / d.o.f

Table 6.1: Results of the fit of existing data to LSA and LSB with η left free and for2π
η = 2π3 for LSA and η = − 3 for LSB. The best fit values are given for both the input
parameters, ma , mb and η, and the output parameters, θij , ∆m2ij and δ, together withthe ∆χ2 / degrees of freedom for the best fit. For the output parameters, errors arealso given corresponding to the 1 σ ranges of the fit. The results of the NuFIT 3.2(2018) global fit to standard neutrino mixing for the normal ordering case are shown forcomparison.

LSA★

Best fit

1σ

3σ

LSB

★

Best fit

1σ

3σ

2.80

0.74

2.78

0.72

2.76

0.70

2.74

0.68
0.64

2.70

0.62

2.68

η/π

mb [meV]

★

0.66

2.72

★★

2.66

-0.60
0.60
-0.62

2.64

-0.64

2.62

-0.66

2.60

-0.68

2.58

-0.70

2.56

-0.72

2.54
26.0

26.2

26.4

26.6

26.8

27.0

27.2

27.4

ma [meV]★

-0.74
0.080 0.085 0.090 0.095 0.100 0.105 0.110 0.115 0.120

mb /maFigure 6.6: Results of the fits to LS of the NuFIT 3.2 (2018) global neutrino oscillation
data. Left: LS fit with fixed η = 2π/3 (η = −2π/3) for LSA (LSB). Right: LS fit withη as a free parameter.

184

r, the predictions of LS form lines of allowed solutions in each plane, corresponding tosum-rules between the oscillation parameters. For example, Figure 6.7a corresponds to
the TM1 sum rule in Equation (6.1.8), while Figures 6.7b to 6.7f correspond to those in
Equation (D.0.6) or to combinations of these sum rules. It can be seen that very strong
restrictions are placed on the allowed values of the less well measured parameters, θ12 ,θ23 and δ. For the remaining angle, θ13 , the majority of the NuFIT 3.2 range remains
viable in LS.
0.026

(a)

sin2 θ13

0.024

LSA
3σ

All

LSB
3σ

All

NuFIT
1σ

3σ

0.5

0.6

0.022
0.020
0.018
0.7 (b)(c)sin2 θ23

0.6
0.5
0.4
0.3
0

(d)

(e)

(f)

-30

δ [°]

-60
-90
-120
-150
-180

0.25

0.30

sin2 θ12

0.35

0.018

0.020

0.022

sin2 θ13

0.024

0.026 0.3

0.4

0.7

sin2 θ23

Figure 6.7: Allowed values for LSA (red) and LSB (blue) with η = 2π/3 and η = −2π/3
respectively. The dotted lines labelled "All" show all possible values allowed by themodel, corresponding to the solutions of the sum rules in Equations (6.1.8) and (D.0.6))
or combinations thereof without any constraints from experimental data, with the solid
segments showing those values allowed experimentally at 3 σ. Also shown are the 1 σ(solid) and 3 σ (dashed) regions from the NuFIT 3.2 2018 global fit (grey).
Figure 6.8 shows the allowed regions of parameter space for pairs of variables including
the mass-squared differences. In these plots, as the mass-squared differences can dependon both ma and mb independently, regions of allowed values are seen instead of lines. For
each of these planes, any point will fully determine both input parameters ma and mb ,and so these contours correspond exactly to the equivalent regions shown in Figure 6.6.
In addition to the tight constraints on θ12 , θ23 and δ already mentioned, in Figures 6.8b
and 6.8e it can be seen that the allowed range of θ13 is correlated with that of both ∆m221
and ∆m231 , suggesting that combining future measurements of these parameters could
provide a better probe of LS than the individual parameter measurements alone. The
ability of future experiments to exclude the model then depends on both the predictions
185

of the model seen here, combined with the sensitivity of experiments to measurementsof the parameters in the region of interest predicted by LS, which is the focus of thenext section.
LSA

1σ

LSB

3σ

1σ

3σ

NuFIT

1σ

(a)

(b)

(c)

(d)

(e)

(f)

3σ

Δm221 [10-5 eV2 ]

8.5
8.0
7.5
7.0
6.5
2.8

Δm231 [10-3 eV2 ]

2.7
2.6
2.5
2.4
2.3
2.2

0.26 0.28 0.30 0.32 0.34 0.36 0.38

0.018

0.020

sin2 θ12
0

0.022

0.024

0.026 0.35 0.40 0.45 0.50 0.55 0.60 0.65

sin2 θ13(g)sin2 θ23
2.8

(h)

-30

Δm231 [10-3 eV2 ]

δ [°]

-60
-90
-120
-150
-180

(i)

2.7
2.6
2.5
2.4
2.3
6.5

7.0

7.5

8.0

Δm221 [10-5 eV2 ]

8.5

2.2

2.3

2.4

2.5

2.6

2.7

2.8

Δm231 [10-3 eV2 ]

2.2

6.5

7.0

7.5

8.0

8.5

Δm221 [10-5 eV2 ]Figure 6.8: Allowed 1 σ (solid) and 3 σ (dashed) regions for LSA (red) and LSB (blue)
with η = 2π/3 and η = −2π/3 respectively. Also shown are the current allowed regions
from the NuFIT 3.2 2018 global fit (grey).

6.3

Sensitivity of future experiments

In order to understand the potential for future experiments to exclude the LS models,simulations are performed of a combination of accelerator and reactor experiments,
modelling the experimental data expected over the next two decades. The General
Long Baseline Experiment Simulator (GLoBES) libraries [165, 166] are used to simulate
future experiments and to fit the simulated data to both standard mixing and the LS
models. In all these simulations it is assumed that the mass ordering is known to benormal ordering, as this is a requirement of the LS models; a measurement of invertedordering would immediately exclude the models.

186

6.3.1

Future neutrino oscillation experiments

The combination of experiments included in the simulations of this section use the samedetailed models of the T2HK and DUNE long-baseline accelerator experiments as usedin Chapter 5, to provide precision measurements of ∆m231 , θ23 and δ, together with basic
constraints on θ13 from the Daya Bay short baseline reactor experiment and on θ12 and∆m221 from the JUNO and RENO-50 medium baseline reactor experiments. Details of
the models for T2HK and DUNE are provided in Section 5.2; the remaining experiments
and the treatment of them in these simulations are now described below.
6.3.1.1

Short baseline reactor experiments

By observing the oscillations of the ν̄e produced in nuclear reactors, short baseline reactor
neutrino experiments are able to measure the mixing angle θ13 with particularly highaccuracy. The Daya Bay experiment [218] currently has the most precise measurementof this parameter with the aim to achieve a precision on sin2 θ13 of better than 3 % [219].
The experiment measures anti-neutrinos produced in six nuclear reactors in south China.
A total of eight 20 t liquid scintillator detectors are used; two are located at each of two
near detector sites and four at a far detector site L =1.5 to 1.9 km from the reactors
near the first atmospheric neutrino oscillation maximum for ∆m231 ∼ 2.5 × 10−3 eV2 ,given the low nuclear energy of the neutrino beam E ∼ few MeV. Results of the Double
Chooz [220] and RENO [125, 221] short baseline reactor experiments also contribute to
the precision obtained on θ13 combined with the Daya Bay result. Although DUNE andT2HK will also measure this parameter with high precision, the measurement of theshort baseline reactor programme by that time is expected to be at least as precise, andwill provide a measurement independent of the other parameters which influence theappearance channel at long-baseline accelerator experiments.
6.3.1.2

Medium baseline reactor experiments

The Jiangmen Underground Neutrino Observatory [124] (JUNO) and the future plans of
the Reactor Experiment for Neutrino Oscillation (RENO-50) [125] are medium baseline
reactor neutrino experiments which, like the Daya Bay experiment, will observe the oscillations of electron anti-neutrinos produced in nuclear reactors. The JUNO experimentwill use a 20 kt liquid scintillator detector approximately L =53 km from two planned
nuclear reactors in southern China, while RENO-50 will use an 18 kt liquid scintillator
detector approximately L =50 km from a nuclear reactor in South Korea. Given the low
nuclear energy of the neutrino beam E ∼ few MeV, these longer baselines correspond to
the first solar neutrino oscillation maximum for ∆m221 ∼ 7.5×10−5 eV2 , where the higher
frequency atmospheric oscillations appear as smaller fluctuations in the probability as
L/E is varied. Thus the longer baseline than at Daya Bay gives greatest sensitivity toa different set of oscillation parameters, in particular θ12 and ∆m221 . The precision on
the measurements of both sin2 θ12 and ∆m221 is expected to reach 0.5 % [124, 125].

187

6.3.1.3

Details of experimental simulation

Complete simulations of the latest designs for both DUNE and T2HK are used with theassumption that both experiments run for 10 years. For the short and medium baseline
reactor experiments, basic constraints are included on the values of sin2 θ13 , sin2 θ12 and∆m221 ; since these measurements are expected to be approximately independent of otherparameters these are implemented as simple Gaussian measurements with a mean of thetrue simulated value and error as given in Table 6.2.
ExperimentParameter

Precision

Short baseline reactor
Medium baseline reactor
Medium baseline reactor

sin2 θ13
sin2 θ12
∆m221

3%
0.5 %
0.5 %

Table 6.2: Precision on oscillation parameter measurements expected by future reactor
experiments, which have been used as constraints in these simulations.

6.3.2

Statistical method

To determine the statistical significance with which the LS model could be excluded
based on simulated data, a minimum-χ2 fit is performed to both standard three neutrino
mixing andto the LS model. As in Section 6.2.3, for the case of standard mixing Θ =ΘPMNS ≡ θ12 , θ13 , θ23 , ∆m221 , ∆m231 , δ is used, while for LS Θ = ΘLS ≡ {ma , mb , η} (orΘLS = {ma , mb } when fitting with η fixed) is used. The test statistic for the significanceto exclude the LS model is then given by
q

∆χ2 =

r

min [χ2 (ΘLS )] − min [χ2 (ΘPMNS )].
ΘLS

ΘPMNS

(6.3.1)The significance at which LS is excluded is then determined from the distribution of the∆χ2 test statistic; where sensitivities are given in terms of N σ, this quantity has been
calculated assuming the that Wilks’ theorem applies. Wilks’ theorem states that whencomparing nested models, the ∆χ2 test statistic is a random variable asymptotically
distributed according to the χ2 -distribution with the number of degrees of freedom
equal to the difference in number of free parameters in the models. In this case theLS models, with two or three free parameters, are treated as sub-models of standard
neutrino mixing with six free parameters, leading to a χ2 -distribution with 4 degrees of
freedom when η is kept fixed or 3 degrees of freedom when η is left as a free parameter.
As for the fit in the previous section, Monte-Carlo simulations were used to verify thatthe distribution of this ∆χ2 test statistic is well approximated by these distributions.
In applying the above formula, the χ2 (Θ) is minimised over the parameters Θ in the fits
and is built from three parts;
χ2 (Θ) = χ2LB (Θ) + χ2R (Θ) + P (Θ),(6.3.2)

with χ2LB (Θ) for the full simulations of the long-baseline experiments DUNE and T2HK,
χ2R (Θ) for the constraints from reactor experiments Daya Bay and JUNO, and P (Θ)
188for a prior intended to include information from the results of existing experimental
measurements.
For the long-baseline experiments the statistical model of the GLoBES library [165,166]
is used, where the χ2LB (Θ) is a sum of contributions from each of the experiments’channels. The individual contributions are constructed as"

χ2c (Θ) =min

ξ={ξs ,ξb }

2

X
i

ni
ηi (Θ, ξ) − ni + ni ln
ηi (Θ, ξ)#+ p(ξ, σ) ,(6.3.3)where χ2c denotes the contribution from a given channel of a given experiment. The
sum in this expression is over the i energy bins of the experimental configuration, withsimulated true event rates of ni and simulated event rates ηi (Θ, ξ) for the hypothesis
parameters Θ and systematic error parameters ξ. The systematic errors of the experiments are treated using the method of pulls, parametrised as ξs for the signal error andξb for the background error. These parameters are given Gaussian priors which form theterm p(ξ, σ) = ξs2 /σs2 + ξb2 /σb2 , where σ = {σs , σb } are the sizes of the systematic errors
given by the experiment.
For the reactor experiments independent Gaussian measurements are assumed such that


χ2R =

sin2 θ13 − sin2 θ13
σθ213

2



+

sin2 θ12 − sin2 θ12
σθ212

2



+

∆m221 − ∆m221
2
σ∆m
2

2

,

(6.3.4)21

where θ13 , θ12 and ∆m221 are the true parameter values and σθ13 , σθ12 and σ∆m2 the21
corresponding experimental measurement uncertainties.
The prior P (Θ) provides information from existing experimental measurements and is
calculated using the results of the NuFIT 3.2 global fit in the same way as the fit in
Section 6.2.3, so that P (Θ) = χ2Fit (Θ) as defined in Equation (6.2.1).
In all these simulations, the true parameters are taken to be the best-fit values fromthe appropriate LS fit results given in Table 6.1, except where stated otherwise. During
minimisation of ∆χ2 , the test parameters are allowed to vary with no restriction on
their values. This means that ma and mb for LS and the mass-squared differences forstandard mixing can have any value greater than 0, with η for LS and the mixing angles
and phase δ for standard mixing taking any value from −π to π. However, in all casesin these simulations the parameter values at the ∆χ2 minimum remained close to thespecified true values (for standard mixing parameters) and the physically viable ranges
found in the fits of the previous section (for LS parameters).

6.3.3

Results

The sensitivity to exclude either version of the LS model is shown as a function of thetrue value of each parameter in Figure 6.9, where the range along the horizontal axes
has been set to the range given by the currently allowed at 3 σ by the NuFIT 3.2 global
fit. In each case, the parameters not shown are assumed to take their best-fit values
from the fit to LS described in Section 6.2.3.
In the upper panels in Figure 6.9, θ12 , θ23 and δ are seen to provide the strongest tests ofthe model, with there only being a relatively small portion of the presently allowed true189

parameter space where the model would not be excluded. This is due to the strong predictions of these parameters by the LS models, as discussed in Section 6.2.1. Note thatthese parameters are those that will be measured most precisely by the next-generation
experiments used in the simulations, θ12 by JUNO and RENO-50, θ23 and δ by DUNE
and T2HK. For these three parameters, allowing η to vary does not significantly change
the sensitivity other than the additional solution (currently disfavoured by experiment)with δ = +90°, which occurs when changing the sign of η. For θ12 in particular there
is no effect of allowing η to vary. This is due to the sum rule in Equation (6.1.8) which
relates θ12 with θ13 independently from the value of η; the precise measurement of θ13
then fixes the value of θ12 to a narrow range such that a measurement of θ12 outside ofthis would exclude the LS model regardless of the LS parameter values. Similarly theprecise measurements of θ13 , ∆m221 and ∆m231 strongly constrain the magnitude (butnot sign) of η, so that the LS allowed regions of the other variables are not significantly
changed when η is allowed to vary, with the noted exception that changing the sign of
η allows the sign of δ to also change.
In the lower panels in Figure 6.9, it is seen that the sensitivity to exclude LS frommeasurements of θ13 , ∆m221 or ∆m231 is much less than for the other three parameters
and the sensitivity is also significantly reduced when allowing η to vary. By the converse
argument to that used above, this is due to these three parameter measurements driving
the fit to ma and mb (and η), and so a measurement of these parameters will tend tomove the fitted LS parameter values rather than exclude the model, particularly when
fitting the extra free parameter η. However, a particularly small measurement of θ13
or particularly large measurement of ∆m221 , relative to their current allowed range ofvalues, may still exclude the fixed η version of the models.
The results shown in Figure 6.9 show only the dependence of the significance to exclude
LS on the true value of each variable individually. However, the sensitivity will generally
have a strong dependence on the true values of the other parameters. The significanceto exclude the LS models depending on the true values of each pair of variables, for thecases where η is kept fixed, is shown in Figures 6.10 and 6.12 for LSA and in Figures 6.11
and 6.13 for LSB.
Each panel of Figures 6.10 and 6.11 includes two dimensionless variables (i.e. angle orphase) which both depend only on the ratio of LS input parameters r = mb /ma , andso, in a LS model, a measurement of any one of these parameters corresponds to ameasurement of r = mb /ma (see Figure 6.3). Combining two of these parameter measurements therefore gives two measurements of r = mb /ma , with any conflict between
them providing strong evidence to exclude the model. For this reason the significanceto exclude the models is close to being simply the combined significance from individual
measurements implied by Figure 6.9.
By contrast, each panel of Figures 6.12 and 6.13 shows the results for the pairs ofvariables including at least one dimensionful mass-squared difference. Here it can beseen in Figures 6.12b, 6.12e and 6.12i for LSA, and in Figures 6.13b, 6.13e and 6.13i
for LSB, that there is a strong correlation between the measurements of θ13 , ∆m221 and∆m231 . This shows clearly that, although individual measurements of these parameters
cannot exclude a LS model (since the parameters of the LS model could be adjusted toaccommodate any of them individually) a combined measurement of two of them could

190

fixed η=2π/3

LSA
12

(a)

free η

fixed η=-2π/3

LSB

(b)

free η

(c)

Nσ to exclude LS

10

8

6

4

2

0

0.28 0.29 0.30 0.31 0.32 0.33 0.34

0.45

True sin2 θ12
12

(d)

0.50

0.60 -180

0.55

-90

0

90

180

True δ [°]True sin2 θ23
(e)

(f)

Nσ to exclude LS

10

8

6

4

2

0
0.020

0.021

0.022

0.023

True sin2 θ13

0.024 6.8

7.0

7.2

7.4

7.6

7.8

True Δm221 [10-5 eV2 ]

8.0

2.45

2.50

2.55

True Δm231 [10-3 eV2 ]Figure 6.9: The predicted sensitivity of future experiments to excluding LSA (red) andLSB (blue), shown as a function of the true value of each parameter. Solid curves
2πcorrespond to the case with η fixed at η = 2π
3 for LSA or η = − 3 for LSB, while dashedcurves correspond to the case with η left free. The ranges of true parameters shown in
the plots corresponds to the current three sigma allowed NuFIT 3.2 regions.

191

0.024(a)sin2 θ13

0.023

Nσ to exclude LSA
0.022

1 2 3

5

10

0.021
0.020
0.60 (b)(c)

sin2 θ23

0.55

0.50

0.45
180
135

(d)

(e)

(f)

90

δ [°]

45
0
-45
-90
-135
-180

0.28 0.29 0.30 0.31 0.32 0.33 0.34 0.020

sin2 θ12

0.021

0.022

sin2 θ13

0.023

0.024

0.45

0.50

0.55

0.60

sin2 θ23

Figure 6.10: The predicted sensitivity of future experiments to excluding LSA, with η
fixed at η = 2π
3 , shown as a function of each pair of true parameters. The ranges of true
parameters shown in the plots corresponds to the current three sigma allowed NuFIT
3.2 regions.

192

0.024(a)sin2 θ13

0.023

Nσ to exclude LSB
0.022

1 2 3

5

10

0.021
0.020
0.60 (b)(c)

sin2 θ23

0.55

0.50

0.45
180
135

(d)

(e)

(f)

90

δ [°]

45
0
-45
-90
-135
-180

0.28 0.29 0.30 0.31 0.32 0.33 0.34 0.020

sin2 θ12

0.021

0.022

sin2 θ13

0.023

0.024

0.45

0.50

0.55

0.60

sin2 θ23

Figure 6.11: The predicted sensitivity of future experiments to excluding LSB, with η
fixed at η = − 2π
3 , shown as a function of each pair of true parameters. The rangesof true parameters shown in the plots corresponds to the current three sigma allowed
NuFIT 3.2 regions.

193

Nσ to exclude LSA
1 2 3

Δm231 [10-3 eV2 ]Δm221 [10-5 eV2 ]

8.0

5

10

(a)

(b)

(c)

(d)

(e)

(f)

7.8
7.6
7.4
7.2

2.55

2.50

2.45

0.28 0.29 0.30 0.31 0.32 0.33 0.34 0.020

0.021

sin2 θ12
180

0.022

0.023

0.024

0.45

sin2 θ13

(g)

0.50

0.55

0.60

sin2 θ23(h)(i)Δm231 [10-3 eV2 ]

135
90

δ [°]

45
0
-45
-90

2.55

2.50

2.45

-135
-180

7.2

7.4

7.6

7.8

Δm221 [10-5 eV2 ]

8.0

2.45

2.50

2.55

Δm231 [10-3 eV2 ]

7.2

7.4

7.6

7.8

8.0

Δm221 [10-5 eV2 ]Figure 6.12: The predicted sensitivity of future experiments to excluding LSA, with η
fixed at η = 2π
3 , shown as a function of each pair of true parameters. The ranges of true
parameters shown in the plots corresponds to the current three sigma allowed NuFIT
3.2 regions.

194

Nσ to exclude LSB
1 2 3

Δm231 [10-3 eV2 ]Δm221 [10-5 eV2 ]

8.0

5

10

(a)

(b)

(c)

(d)

(e)

(f)

7.8
7.6
7.4
7.2

2.55

2.50

2.45

0.28 0.29 0.30 0.31 0.32 0.33 0.34 0.020

0.021

sin2 θ12
180

0.022

0.023

0.024

0.45

sin2 θ13

(g)

0.50

0.55

0.60

sin2 θ23(h)(i)Δm231 [10-3 eV2 ]

135
90

δ [°]

45
0
-45
-90

2.55

2.50

2.45

-135
-180

7.2

7.4

7.6

7.8

Δm221 [10-5 eV2 ]

8.0

2.45

2.50

2.55

Δm231 [10-3 eV2 ]

7.2

7.4

7.6

7.8

8.0

Δm221 [10-5 eV2 ]Figure 6.13: The predicted sensitivity of future experiments to excluding LSB, with η
fixed at η = − 2π
3 , shown as a function of each pair of true parameters. The rangesof true parameters shown in the plots corresponds to the current three sigma allowed
NuFIT 3.2 regions.

195

serve to exclude the model. This is the reason for presenting these combined sensitivity
plots.
Of the three parameters for which such combined measurements provide the strongest
test of the model, it is interesting that each pair includes measurements from different
experiments, with θ13 coming mainly from the short-baseline reactor measurement suchas Daya Bay, ∆m221 from a medium-baseline reactor measurement of JUNO or RENO50, and ∆m231 from a long-baseline accelerator measurement of DUNE and T2HK. This
demonstrates a strong synergy between all these experiments in attempts to exclude theLS models.

Conclusion
In this chapter, the ability to probe one of the most predictive viable neutrino mass andmixing models, the Littlest Seesaw, with future neutrino oscillation experiments, hasbeen investigated. The LS models work within the framework of the Type I seesaw mechanism, using two right-handed neutrinos to generate the left-handed neutrino masses.
Combined with constraints from family symmetries, the neutrino mixing angles andphases can be predicted from a small number of parameters; in its most constrained
form all neutrino masses, angles, and phases are determined from just two input parameters. In fact, it was shown that while the neutrino masses depend on the two mass
parameters independently, the mixing angles and phases depend only on a single dimensionless quantity, the ratio of these two input parameters.
Two versions of this model (LSA and LSB) have been studied, which use different
family symmetries to enforce constraints which result in different permutations of thesecond and third rows and columns of the neutrino mass matrix, leading to different
predictions for the octant of θ23 . Using the results of a recent global fit of neutrino
oscillation experiments, it was found that both versions can well accommodate theparameter values as measured by experiment, with no tension between the model andexperimental measurements even at the 1 σ level. The LSB version, predicting a valueof θ23 in the lower octant, was found to be slightly preferred.
The ability of future experiments to exclude these models then comes from a convolutionof the strength of the predictions of the model with the sensitivity of the experiments
in measuring those parameters. Through fitting the models to current global neutrino
oscillation data, it was seen that the LS models make strong predictions for the valuesof θ12 , θ23 , and δ, the three parameters for which current measurements are weakest. In
addition it was found that, for certain combinations of the remaining observables, θ13 ,∆m221 and ∆m231 , the LS models predict strong correlations.
With future experiments expected to improve precision on all six parameters measured
through oscillations, the simulations presented here have shown that the LS models
can be thoroughly tested through future precise individual measurements of θ12 , θ23 ,and δ. This can be readily understood since the free parameters of the LS models are
currently most constrained by the precise measurements of θ13 , ∆m221 and ∆m231 , leading
to predictions for the currently less well determined parameters θ12 , θ23 , and δ.

196

The predictivity of the LS models means that an even higher precision measurementof those parameters which currently drive the fit of the input parameters, namely θ13 ,∆m221 and ∆m231 , could still exclude the LS models when considered in combination witheach other. For example, the combination of any two of them could require a region of
LS parameter space already excluded by the third.
These above results all highlight the strong complementarity between different classesof oscillation experiment. While the long baseline accelerator experiments DUNE andT2HK are expected to provide the strongest measurements of θ23 and δ (two of those that
can individually test the model’s viability) the third, θ12 , will come from medium baseline
reactor experiments such as JUNO and RENO-50. The strongest complementarity,
however, comes from combining precision measurements of ∆m221 , ∆m231 and θ13 , where
any pair of these measurements relies on the results from all the different experiments:
long-baseline accelerator experiments for ∆m231 , medium-baseline reactor experimentsfor ∆m221 , and short-baseline reactor experiments for θ13 .
In summary, the work presented in this chapter shows that the most straightforward
way to exclude the LS model is to provide a better individual determination of the threecurrently less precisely measured parameters θ12 , θ23 , and δ, which requires both medium baseline experiments such as JUNO and RENO-50, and long baseline experiments
such as DUNE and T2HK, where the synergy between the latter two experiments isthoroughly explored in the previous chapter. In addition, the LS model could be constrained by combined measurements of the three remaining parameters ∆m221 , ∆m231 andθ13 , where an even higher precision of the latter reactor parameter at the short baseline
Daya Bay experiment can also play an important role.
Note that, although the above conclusions have been established for the LSA and LSB
models, similar arguments could be expected to apply to any highly predictive flavour
models which determine the oscillation parameters from a smaller number of input
model parameters. In any such model, the input parameters will tend to be tuned
to fit the strong constraints from the most precisely measured parameters, leading totestable predictions of the other parameters. If the models can accommodate individual
measurements in this way, distinguishing between them using those parameters which
drive the fit is still possible, if those models are highly constrained, but this requires theparameter measurements to be considered in combination.
In conclusion, the need for future reactor and accelerator experiments to measure individually θ12 , θ23 and δ, plus combinations of θ13 , ∆m221 and ∆m231 , may be considered tobe general requirements in order to probe predictive flavour symmetry models. Therefore a broad programme of such precision experiments seems to be essential in order totake the next step in understanding neutrino oscillations in the context of the flavour
puzzle of the Standard Model.

197

198

Conclusions
This thesis has explored the future of neutrino oscillation physics through studying thenext-generation long-baseline neutrino beam experiments. Since providing the first andonly observation of particle physics beyond the Standard Model, neutrino oscillations
have been thoroughly examined through a broad programme of experiments detecting
the interactions of a diverse spectrum of neutrino types and energies. The panoramicview of neutrino oscillations provided thanks to these efforts leaves only a few aspectsof the phenomenon still unknown, with the next generation of experiments expected to
begin to probe all of these in the next decade.
The main focus of this thesis has been to examine the measurement of the Dirac phaseof neutrino oscillations by the Hyper-Kamiokande experiment, both alone and in combination with other measurements and experiments, in the search for CP violationin the leptonic sector. The design of Hyper-K and its intermediate water Cherenkov
detector has been studied to investigate possible improvements in sensitivities of its
measurements, followed by studies into where these measurements of Hyper-K stand incombination with other experiments. The complementarity of these experiments has
been identified both in determining oscillation parameter values themselves and theresulting ability to probe theoretical models introduced in attempting to gain a more
fundamental understanding of neutrino mass and mixing.
Chapter 3 has documented the development of a new high-energy reconstruction package used to aid the development of the intermediate detector of Hyper-K. This software,
which is essential in allowing the positions, directions, energies and types of particles tobe determined, also met the additional requirements of speed and adaptability that thechanging detector design demanded. The construction of selection criteria for samples ofneutrino interaction events seen in the intermediate detector was also detailed, culminating in the use of these selections in producing improved sensitivities of the oscillation
analyses of Hyper-K.
With the intermediate detector having progressed to the later stages of its design process,the use of advanced detector-specific software has taken over, however the techniquesdeveloped are still applicable to other water Cherenkov detector design projects; thatthe reconstruction software developed in this work has also been used in the design stageof the ANNIE detector provides a case in point.
Chapter 4 has then described a full analysis of a possible measurement of neutron captures with a Gadolinium doped intermediate detector. This measurement was motivatedby the benefits it can bring to various analyses at the far detector of Hyper-K. Measuring neutrons in the intermediate detector, with its large event rate, will provide two199main benefits: Firstly, the additional information related to the number, energies anddirections of neutrons produced in neutrino interactions allows the interaction crosssections to be constrained, thereby reducing systematic uncertainties in analyses that
rely on these models. Secondly, a thorough understanding can be gained of the ability tomeasure neutrons in the far detector, where the neutrino flux is greatly reduced, allowing
the use of neutron tagging to improve the purity and reconstruction of neutrino interaction events, as well as significant reduction in the atmospheric neutrino backgroundof nucleon decay searches.
The analysis of Chapter 4, saw that despite the large rate of external backgrounds, including incoming neutrons capturing in an identical way to that of the signal neutrons,
the measurement can provide a high level of detail on the signal neutron captures’ properties: Through the development of a likelihood-based selection method, backgrounds
were separated from signal neutrons allowing very pure samples to be produced while
retaining the vast majority of signal neutron captures. A method developed to correct
for the remaining background and selection efficiency was then seen to work as expected, demonstrating the ability to successfully reproduce the true joint distributions ofmultiplicities, distance and angle of the neutron captures following neutrino interactionsin the detector. Although this analysis has begun to demonstrate the potential of sucha measurement, extensions to provide a total neutron capture multiplicity measurement
and to correct for additional detector effects seen in the results should provide further
improvements, while a thorough study of the effects of systematic uncertainties not currently included in the analysis is required to truly prove the measurement’s feasibility.
In Chapter 5, a full analysis of the neutrino beam oscillation measurements of Hyper-K
is then presented in combination with the other planned next-generation accelerator
neutrino experiment, DUNE. Due to the significant differences in design of these two
experiments, a strong synergy between them was seen to emerge. Most significantly,
the longer baseline of DUNE provides it with strong matter effects to enhance its sensitivity to determining the mass ordering, to which the beam programme of Hyper-K
has little sensitivity. Conversely, Hyper-K’s high statistics and narrow-band beam focused around the oscillation maximum give it the strongest sensitivity in measuring the
Dirac phase. The complementarity of these two strengths of the experiments is striking when combined; the precision constraint on δ from Hyper-K allows an even greater
sensitivity to the mass ordering at DUNE, by helping to exclude degeneracies between∆m232 and δ, while this also works in reverse where the mass ordering measurement at
DUNE resolves these same degeneracies and greatly improving the sensitivity to δ atHyper-K. The potential redesigns of both Hyper-K and DUNE, with a second detector
placed in Korea and with an optimised 3-horn beam, respectively, was also investigated;the best combination of these two experiments was found to be able to determine themass-ordering in under 2 years and to have the potential to discover CP violation at 5 σ
for 50 % of the parameter space after just 5 years. This analysis has demonstrated theclear benefits of both of these experiments and their synergistic nature in resolving theremaining unknowns of neutrino oscillations, however only the beam programme of theseexperiments has been included and the two experiments were assumed in simulations
to have entirely uncorrelated errors. Extending the analysis to include other aspectsof the experiments, such as the atmospheric neutrino programme of Hyper-K that can
provide sensitivity to the mass ordering, and to include a full analysis of the correlations

200

between systematic error sources such as the use of common interaction models, could
result in modifications to these conclusions.
Finally, in Chapter 6, the Littlest Seesaw models of neutrino flavour, mass and mixing was investigated in the context of measurements of current and future oscillation
experiments. The highly predictive models were found to be in good agreement withexisting data; no tension is found even at the 1 σ level between experiments’ measurements and the models’ predictions. With these models’ predictions fitting well withcurrent data, the ability of future experiments to probe the models, by attempting to exclude them through oscillation parameter measurements, was investigated. It was foundthat the three strands of neutrino oscillation experiments – long baseline beam experiments, medium baseline reactor experiments and short baseline reactor experiments –each provide unique tests of the models both in isolation and when combined with eachother: The measurements of the three less-precisely known parameters, δ and θ23 at thelong-baseline beam experiments and θ12 at the medium baseline reactor experiments,
can all individually provide tests of the LS models, while more precise measurementsof the already well-known parameters, θ13 at short baseline reactor experiments, ∆m221
at medium baseline reactor experiments and ∆m231 at long baseline beam experiments,
can also test the model when at least two of any of these measurements are combined.
This is due to the few free parameters of the models being highly constrained by fitting
to the precisely measured oscillation parameters, effectively providing predictions of theless constrained parameters as well as making the fit fragile to more precise measurements of the other parameters. While the analysis has shown the impressive ability ofthe various experiments to potentially exclude these models, a plethora of other models
exist making different predictions. Further work to investigate the ability to exclude
other sets of models, and to discriminate between the various models, will be required
to meet the goal of moving towards a more complete fundamental understanding of theorigin of neutrino mass and mixing.
In conclusion, the state of neutrino oscillation physics has come a long way in a relatively
short time, with only a handful of unknowns left, all expected to be resolved with thenext generation of experiments. This thesis has documented some of the essential work
going towards the optimisation of the sensitivities of these experiments. But despite
the experimental progress, on the theoretical side there are many potential explanations
for what the mechanism is that determines the parameters of neutrino oscillations, andwhere the neutrino mass comes from to enable the phenomenon to occur. To makeprogress in this area, with the eventual hope of finding a successor to the Standard
Model of particle physics, the measurements of the next generation of experiments willbe essential.

201

202

Appendices203

Appendix A

Further details of likelihood
distribution fits
The details of the fitted functions of the distributions used in the likelihoods of Section 4.2 are summarised below, for the fitted variables listed in Tables 4.2 and 4.3.
Table A.1 describes the fitted distributions and the fitted parameter values with thedefinitions of the probability distributions given in Table A.2. For sand and neutral current backgrounds, the timing distribution was fitted separately for those coming fromneutron or from non-neutron sources, due to the significantly longer times for neutrons
to be observed in the detector, while the position distributions will be approximately
similar. Where the distributions are described as convolutions, this is a convolution of
two probability density functions physically implying the sum of two random variables,for example the time for a high energy neutron to thermalise plus the time for thethermal neutron to capture. Where the distributions are described as mixtures, this is aweighted sum of probability density functions physically implying the event could havecome from different possible sources with different distributions.
Quantity

Distribution

Signal t

Convolution of Exp(λ)Γ(k, θ)Fitted parametersλkθχ2 / d.o.f

Signal d

Convolution of Exp(λ)Γ(k, θ)λkθ
χ2 / d.o.f

Signal cos θ

Mixture of U(−1, 1) weight w
B(α, β) weight 1 − ww
αβχ2 / d.o.f

205

0.039 61
1.16482.750
2.00.011 80
2.714
8.219
1.7
0.50593.162
0.9166
1.4

Quantity

Distribution

Sand tb(neutron)

Convolution of Exp(λ)Γ(k, θ)Fitted parametersλkθχ2 / d.o.f

Sand tb(other)Mixture of Exp(λ1 ) weight w
Exp(λ2 ) weight 1 − ww
λ1
λ2
χ2 / d.o.f

Sand yMixture of B(α1 , β1 ) weight w
B(α2 , β2 ) weight 1 − w

w
α1
β1
α2
β2
χ2 / d.o.f

Sand r2

2 )1 weight w w
Mixture of U(0, rmax
1 2
Tri(c)
weight (1 − w1 )w2

LN(µ, σ 2 ) weight (1 − w2 )

w1
w2
c
µ
σ2
χ2 / d.o.f

Sand ϕCos(α)αχ2 / d.o.f

NC tb(neutron)

Convolution of Exp(λ)Γ(k, θ)λkθ
χ2 / d.o.f

NC tb(other)Convolution of Exp(λ)
U(0, Tb )2

NC y

Mixture of B(α1 , β1 ) weight w

λχ2 / d.o.f

B(α2 , β2 ) weight 1 − w

w
α1
β1
α2
β2
χ2 / d.o.f

1
2

rmax = 371 cm is the radius of the tank.
Tb = 0.058 µs is the length of each bunch

206

0.038 16
1.632
2.032
1.3
0.666019.95
0.57790.9
0.086 86
10.58
1.005
0.87131.363
2.0
0.44630.9351
118 9673.287
0.60321.0
0.5431
1.3
0.039 90
1.3524
2.433
1.0
164.0
1.0
0.084 00
2.731
11.18
1.332
1.365
2.9

Quantity

Distribution

Fitted parameters

NC r2

Mixture of Quad(a, b) weight w1
LN(µ, σ 2 ) weight (1 − w1 )

w1
a
b
µ
σ2
χ2 / d.o.f

NC ϕLin(a)aχ2 / d.o.f

Cosmic y

Mixture of B(α1 , β1 ) weight w
B(α2 , β2 ) weight 1 − w

w
α1
β1
α2
β2
χ2 / d.o.f

Cosmic r2

Mixture of Lin(a) weight w
B(α, β) weight 1 − wwaαβχ2 / d.o.f

0.9819−1.00810.14833.362
0.70201.5−0.036 01
1.4
0.38343.051
2.013
0.89660.45062.7
0.8357−4.949 × 10−6
86.17
1.574
2.8

Table A.1: Fitted distributions and parameter values for for quantities used in likelihoods for the E61 neutron selection.

207

Distribution
Exponential

Exp(λ)

Gamma

Γ(k, θ)UniformU(a, b)Beta

B(α, β)Triangular

Tri(c)

Logit-normal

LN(µ, σ )CosineCos(α)

Probability density functionRangef (x; λ) = λe−λx

0<x

x
1
xk−1 e− θΓ(k)θk1
f (x) =a+b
Γ(α + β) α−1f (x; α, β) =x
(1 − x)β−1
Γ(α)Γ(B)f (x; k, θ) =f (x; c) = 2(x−a)(b−a)(c−a)
 2(b−x)(b−a)(b−c)

0<x
a<x<b
0<x<1
a<x<c
c<x<b

(log(x/(1−x))−µ)2

Quadratic
Linear2

Quad(a, b)
Lin(a)

2σ 2
1 e−f (x; µ, σ ) = √x(1 − x)σ 2π
1 + α cos(x)f (x; α) =πa bf (x; a, b) = ax2 + bx + 1 − −3 2af (x; a) = ax + 1 −
2

2

0<x<1
0<x<π
0<x<1
0<x<1

Table A.2: Definitions of the probability density functions used in the distributions ofTable A.1. For bounded distributions, the ranges are adjusted according to the relevant
range of each fit; for the vertical position from −ymax to ymax = 521 cm, for the radial
position from 0 to rmax = 371 cm.

208Appendix B

Obtaining total multiplicity
distributions
Section 4.4 provides results of an example neutron measurement at E61, including afull 3D distribution of neutron capture multiplicities as a function of distance from theprimary event and angle to the beam. This distribution has been corrected for efficiency
and background using an unfolding method. However, a total neutron multiplicity
distribution across all angles and distances is difficult to obtain either through combining
the unfolded rates at each angle and distance or through applying an efficiency andbackground correction to the observed total multiplicity distribution.
The total multiplicity is accumulated over all bins in d and/or θ, but the unfolding
method must be applied with a dependence on d and θ, i.e. it must be applied to eachbin individually, due to the model dependence, as described in Section 4.3.
The difficulty comes from the fact that the rates of neutrons observed at each distanceand angle are highly inter-dependent; The number of neutrons produced in an interaction can be correlated with their energies, and therefore the distance they travel, forexample. The most significant dependence between rates at different angles and distances simply comes from the fact that fundamentally a number of neutrons is produced
in an interaction that then capture at some position; once a neutron is observed at one
position, however, this drastically reduces the likelihood of another neutron being observed at some other position, since this would require that another neutron was actually
produced by the original interaction in addition to it capturing at the new position. This
results in large anti-correlations between the probabilities of positive neutron capture
multiplicities in different bins of distance and angle, and positive correlations between
a positive multiplicity in one bin and zero multiplicity in other bins.
In the absence of this inter-dependence, to obtain an unfolded total neutron multiplicity
distribution, the unfolding matrix Pn (Ti |Oj ) in Equation (4.3.3) for each d and α bin(determined when producing the unfolded distribution of each bin for the full 3D distribution) could be applied to each event individually. This gives a probability Pi (ni ), fora given event, of the true number of neutrons ni in the bin i, where i = 1 . . . N numbers
the N bins of d and α. The probability of the total multiplicity P (T ), for a given event,209

can then be determined as
P (T ) =

−n1
T TX
X
n1 =0 n2 =0

PN −1
ni
i=1
X

T−

···

P1 (n1 )P2 (n2 ) · · · PN (nN ),(B.0.1)nN =0which sums over all possible combinations of multiplicities in each bin that sum to T .
This could then be summed over each event, to give the total unfolded multiplicity
distribution.
To account for the dependence between the Pi (ni ), the above method would need toQ
be modified, replacing the product of independent probabilities N
i−1 Pi (ni ) in Equation (B.0.1) with the full joint probability P (n1 , n2 , . . . , nN ) of observing ni neutrons in
the i bins. This full joint probability is not simple to calculate taking into account all
correlations between bins; there are a total of N M terms in the above sum (where M is
the maximum multiplicity in a given bin), which cannot be determined as the productof independent probabilities.
Alternatively, a more feasible approach may be to attempt to simply unfold the observed
total multiplicity distribution. In this case, the distance and angle dependence of the
background rates and efficiencies would need to be taken into account in the constructionof the response matrix. Fundamentally, the response could be constructed in exactly theway that was done for unfolding individual bins, through Equation (4.3.4), with the µ
now being the total background rate across all distances and angles, and  now being theoverall efficiency of all neutron captures at all positions. In practice, although the total
background rate can be determined simply by integrating the background estimates overall positions (i.e. summing over the bins of distance and angles), determining the overall
efficiency is more complex; a weighted average would need to be taken over the distanceand angle bins, weighted by the likelihood of signal neutrons capturing in that position.
The distribution of signal neutrons’ positions is another part of the measurement, andthe unfolding of this true distribution was found to work well, as presented in Section 4.4.
These results – specifically the 2D unfolded distribution of signal neutrons distance andangle, given in Figure 4.12 – could be used to provide the weighting of each distance andangle to perform the weighted average of the efficiencies described above. The response
matrix can then be constructed in the same way as for the unfolding of individual bins,
for the same unfolding procedure to be applied to the total multiplicity distribution.
However, attempts at using this method have so far not been successful at reproducing
the true total multiplicity distribution from the observed distribution.

210

Appendix C

Further details of T2HK GLoBES
simulation
The model of the T2HK detector used in Chapter 5 significantly deviates from previous
work using the GLoBES software and is much closer to the models used in official HyperKamiokande simulations. In this appendix, some further details of its implementation
which where glossed over in the main text and a comparison with the collaboration’s
simulation.

C.1

Energy bins

This model of the T2HK detector(s) features 12 energy bins. Bin 1 collects all events
below 0.35 GeV. The next 5 bins are 0.1 GeV wide, collecting events from 0.35-0.85
GeV. The next two bins are 0.2 GeV wide, followed by a single bin of 0.25 GeV width.
There are then 3 increasingly broad bins, from 1.5 to 3.5, 3.5 to 6 and an overspill bin
from 6 to 10 GeV.

C.2

Channel systematic uncertainties

The model of the systematic uncertainty at T2HK uses two general normalisation systematics for the signal and background of each channel. The precise systematic errors
used in this simulation are given, channel by channel, in Table C.1.

SignalBackgroundνµ → νeνµ → νµνµ → νeνµ → νµ

2.4 %
2.4 %

2.7 %
2.7 %

2.925 %
2.925 %

2.7 %
2.7 %

Table C.1: Systematic errors used for T2HK simulation.

211

C.3

Comparison with published event rates

Figure C.1 compares the event rates from this simulation to the official rates publishedby T2HK. The official simulation does not use GLoBES, and the reproduction is a nontrivial check to show that the signal and background modelling in this GLoBES simulation is faithful. Additional checks have also been made to ensure that these simulations
are able to reproduce the final sensitivities of official simulations, after modifications tomatch the priors and chosen fitted parameters of the official simulations.
300

Signal νµ → νe
(HK official)
Background νe + ν̄e(HK official)
Background νµ + ν̄µ(HK official)

Events per 50 GeV

250

200

150

100

50

0
0

0.2

0.4

0.6

0.8

1

1.2

Reconstructed energy [GeV]

Figure C.1: T2HK appearance spectrum from the GLoBES simulations used in
Chapter 5 compared to official event rates [64].

212Appendix D

Exact expressions for LS sum
rules
The following expressions represent relations between the neutrino mixing parametersin the Littlest Seesaw models, derived to be used in the work of Chapter 6. The angles
and Dirac phase can be written as
sin2 θ13 = s(r),tan2 θ12 = t(r),cos 2θ23 = ±c(r),cos δ = ±d(r), (D.0.1)

with positive signs taken for LSA and negative for LSB and where1
55r2 + 4(1 − 4r)s(r) =
1− p
6((11r)2 + 4(1 − 7r)) ((11r)2 + 4(1 − r))!

1
55r2 + 4(1 − 4r)t(r) =
1+ p
4((11r)2 + 4(1 − 7r)) ((11r)2 + 4(1 − r))!

c(r) =(D.0.2)(D.0.3)

2r(11r − 1) 55r2 − 16r + 4 − 5 ((11r)2 + 4(1 − 7r)) ((11r)2 + 4(1 − r))p((11r)2 + 4(1 − 7r)) ((11r)2 + 4(1 − r)) + 4r2 ((11r)2 + 2(2 − 11r))c(r)(1 − 5s(r)).
d(r) = − p
2 2s(r)(1 − c(r)2 )(1 − 3s(r))(D.0.4)(D.0.5)

Similar expressions for the Majorana phases also possible. Combining these, expressions
relating any two of the angles and/or phases can be found. The first such relation,
relating θ13 and θ12 , is the same as Equation (6.1.8), which is general for all CSD(n).
New exact relations between θ13 and θ23 or θ12 and θ23 , as well as the relation between
2π
δ and θ12 , true for LSA with η = 2π
3 or LSB with η = − 3 , are found of the formf± (θ13 , θ23 ) = 0,g± (θ12 , θ23 ) = 0,

213

h± (δ, θ12 ) = 0,

(D.0.6)

where again the positive (negative) sign is used in the functions valid for LSA (LSB).
Exact expressions are given asq

f± (θ13 , θ23 ) =44s213 1 − 3s213

c213 cos 2θ23

4(1 − 6s213 ) ∓ 3c213 cos 2θ23

± q

1 − 3s213

s−

8s213 c413 cos2 2θ23,−
3
3(1 − 3s213 )(D.0.7)qg± (θ12 , θ23 ) =h± (δ, θ12 ) =22s212 1 − 3s212±2(5s212 − 1) ∓ cos 2θ23
5s212 − 1qs12 1 − 3s212

cos 2θ23q√

1 − 3s212

s−

4s212 −

cos2 2θ23,
3(1 − 3s212 )(D.0.8)q

11 1 − 12(s212 − 3s412 ) sin2 δ√.
±q+2(6s212 − 1) sin δ ∓ 2 3 cos δ
1 − 12(s2 − 3s4 ) sin2 δ
3 cos δ12

12

(D.0.9)214

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