S T U D I E S O F T H E E N E R G Y, A Z I M U T H A L , A N D T I M E S P E C T R A
O F T H E AT M O S P H E R I C N E U T R I N O F L U X AT
SUPER-KAMIOKANDE
euan richard

ABSTRACTThis thesis presents three direct measurements of the atmospheric νe +
ν̄e and νµ + ν̄µ neutrino fluxes at Super-Kamiokande: the directionallyintegrated energy spectra, the azimuthal spectra, and the modulationof the fluxes with time over the 11-year solar cycle.
In particular, the energy spectra in the sub-GeV to TeV range wasmeasured and compared to the predictions of various published flux
models. While none of the models were strongly inconsistent withthe data, some preference was seen for the HKKM11 model as themost realistic model. The neutrino and antineutrino fluxes were also
examined separately, and found to be consistent with the models.
The azimuthal analysis measured the east-to-west asymmetry in
the neutrino flux, caused by the geomagnetic field, for both flavours
at > 5 σ. Measurements were made of the strength of the effect asfunctions of energy and zenith angle. There was also an indicationthat the alignment of the asymmetry was dependent on zenith angle,
seen at the 2.2 σ level.
A search for a long-term correlation between the atmospheric neutrino flux and the solar magnetic activity cycle was performed, however the expected effect based on the HKKM model was calculated tobe relatively small. An indication of a correlation was seen at the 1.1 σlevel. During particularly strong solar activity events, known as Forbush decreases, no theoretical prediction is available, but a deviationfrom the expected neutrino event rate is seen at the 2.4 σ level.

iii

ACKNOWLEDGMENTS

I am extremely grateful to my supervisor Takaaki Kajita, especially for
his friendly and welcoming support of international students, andfor giving me the fantastic opportunity to study and work at SuperKamiokande (and several other experiments based in Japan).
I would also like to express warm thanks to Kimihiro Okumura andMorihiro Honda for many helpful, interesting, and enjoyable discussions over the years.
Thanks are also most definitely due to Ed Kearns and Masato Shiozawa for their tireless efforts in coordinating the high-energy analysis group, and to Roger Wendell, without whom I’m fairly certain most
students at Super-Kamiokande would be considerably more confusedthan we already are.
Many more people have left me with good memories of the last
three years, and I sincerely hope they would forgive me in not thanking them all, but time is short, and there’s science to be done.
So gladly we march to mines, tunnels or seas
New physics or not, we get PhDs –Condensed to a preprint we’ll mail to our mothers
Of one page of physics (two pages of authors).
— B. Nolty and K. ScholbergvCONTENTS

i introduction
1
1 neutrino physics
3
1.1 Motivation: The Atmospheric Neutrino
3
1.2 Detection of the Neutrino
5
1.2.1 Theoretical basis
5
1.2.2 Experimental history
7
1.3 Neutrino Oscillation
8
1.3.1 The lepton mixing matrix
8
1.3.2 Oscillation10
1.3.3 Matter effects
11
1.3.4 Current experimental status
14
1.4 Massive neutrinos
15
1.4.1 Mass terms
15
1.4.2 The seesaw mechanism
17
2 the atmospheric neutrino flux
19
2.1 Origin
19
2.2 Atmospheric Neutrino Flux Simulations
20
2.2.1 Historical approach20
2.2.2 Rigidity cutoff
21
2.3 The HKKM Flux Model
23
2.3.1 Current model
23
2.3.2 Three-dimensional calculation
23
2.3.3 Systematic error estimations
25ii the experiment27
3 the super-kamiokande detector
29
3.1 Principles of Cherenkov Detectors
29
3.1.1 Cherenkov radiation29
3.1.2 Photomultiplier tubes
30
3.2 Structure of the Detector
32
3.2.1 Detector phases
33
3.2.2 Monitoring systems
34
3.3 Electronics and Trigger System
34
3.3.1 SK-I to SK-III
34
3.3.2 SK-IV upgrade35
3.3.3 High voltage supply
37
3.4 Calibration
37
3.4.1 Calibrations of basic detector properties
37
3.4.2 Calibrations using physics events
42
4 data reconstruction and monte carlo
45
4.1 Overview
45

vii

viii

contents

4.2

4.3

4.4

4.5

4.6

4.1.1 Event topologies
45
4.1.2 Reduction and reconstruction
45
Fully Contained Sample
47
4.2.1 Reduction
47
4.2.2 Reconstruction
56
4.2.3 Neutrino / antineutrino separated selections
Partially Contained Sample
73
4.3.1 Reduction73
4.3.2 Reconstruction
75
Upward-Going Muon Sample
76
4.4.1 Reduction and reconstruction
76
4.4.2 Sub-sample categorization
79
Monte Carlo80
4.5.1 Neutrino interaction simulation
81
4.5.2 Detector simulation
84
4.5.3 Event weights
84
4.5.4 Systematic Error Estimation
85
Data and Monte Carlo Sets
86

67

iii the analyses
89
5 measurement of the energy spectra of the neutrino flux
91
5.1 Introduction
91
5.2 Event Classification
92
5.3 Flux Unfolding
92
5.3.1 Overview
92
5.3.2 Bayesian unfolding
94
5.3.3 Binning definitions for the output spectra
96
5.3.4 Calculation of the absolute flux values
96
5.4 Statistical and Systematic Uncertainties
97
5.4.1 Overview
97
5.4.2 Reconstruction and cross-section errors
98
5.4.3 Oscillation parameter uncertainties
98
5.4.4 Regularization uncertainties
99
5.4.5 Propagation of the statistical and systematic uncertainties
99
5.4.6 Error source comparisons 101
5.5 Results and Discussions 101
5.5.1 Measured flux spectra 101
5.5.2 Comparison to flux models 104
5.5.3 Cross-check by SK period 106
5.5.4 Comparison to other measurements 108
5.6 Measurement of Neutrino and Antineutrino Fluxes 109
5.6.1 Data selection and binning 109
5.6.2 Systematic errors 110
5.6.3 Results and discussions 114

contents

6

measurement of the azimuthal spectra of theneutrino flux119
6.1 Introduction 119
6.2 Event Classification and Analysis Method 119
6.2.1 Selection and binning 119
6.2.2 Asymmetry parameter 120
6.2.3 Zenith dependency of asymmetry shape 120
6.3 Systematic Uncertainties 122
6.3.1 Azimuthally-dependent detector systematics 122
6.3.2 Second-order systematics due to azimuthal flux
asymmetries 124
6.3.3 Implementation of the systematics 125
6.3.4 Azimuthal angle reconstruction 126
6.4 Results and Discussions 126
6.4.1 Azimuthal distributions 126
6.4.2 Zenith dependency of asymmetry shape 129
6.4.3 Flux model comparisons 133
7 measurement of the correlation between the solar cycle and the neutrino flux
135
7.1 Introduction 135
7.1.1 HKKM flux model predictions 136
7.2 Event Classification and Analysis Method 137
7.2.1 Super-K data selection 137
7.2.2 Neutron monitor data 137
7.2.3 Parameterization of the effect 139
7.2.4 Systematic errors 141
7.2.5 Fitting method 143
7.2.6 Toy MC 145
7.2.7 Search for effects during Forbush decreases 146
7.3 Results and Discussions 147
7.3.1 Continuous correlation search 147
7.3.2 Event rate during Forbush decreases 150
8 conclusions
151
iv appendix
153
a sk systematic error database
155
a.1 Flux 155
a.2 Cross Section 156
a.3 Detector 156
a.4 Oscillation 158
b poisson likelihood method
159
b.1 Poisson likelihood ratio 159
b.2 Wilks’ theorem 159
bibliography

161

ix

LIST OF FIGURES

Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6
Figure 7
Figure 8
Figure 9
Figure 10
Figure 11
Figure 12
Figure 13
Figure 14
Figure 15
Figure 16
Figure 17
Figure 18
Figure 19
Figure 20
Figure 21
Figure 22
Figure 23
Figure 24
Figure 25
Figure 26
Figure 27
Figure 28
Figure 29
Figure 30
Figure 31
Figure 32
Figure 33
Figure 34
Figure 35
Figure 36
Figure 37
Figure 38
Figure 39
Figure 40
Figure 41x

Neutrino oscillation evidence.
4
Proton decay search at SK.
5
SM neutrino interactions.
6
Neutrino mass hierarchies.
14
Cosmic ray particle shower.
20
The east-west effect.
22
Rigidity cutoff map at SK.
22
HKKM15 predicted flux spectra at SK.
23
HKKM model estimated systematic errors.
25
Cherenkov light.
29
PMT operation diagram.
31
PMT QE distribution.
31
PMT charge distributions.
31
Super-Kamiokande.
32
An SK data event.
33
SK PMT single-photoelectron distribution.
39
SK “TQ map”.
40
MC Eν distributions by SK sub-sample.
46
FC reduction cuts.
49
FC reduction flasher cut.
50
FC reduction pattern-matching flasher cut.
52
Cherenkov ring fitting.
57
Illustration of the Hough transform.
59
Hough transform on SK data.
60
Ring-counting likelihood separation.
61
PID graphical examples.
61
Muon track Cherenkov emission.
62
PID likelihood separations.
64
Energy reconstruction graphical examples.
640
π mass peak.
65
SK sub-sample purities.
67
ν/ν̄ ratios from the HKKM15 model.
68ν/ν̄ separation by Michel electron tagging.
70
UPMU background reduction fitting.
78
Neutrino interaction cross-sections by type.
81
Energy resolution of sub-samples.
88
Number of events in the flux unfolding bins.
93
Unfolding matrix.
96
Unfolding regularization systematic error. 100
Unfolding error correlation matrix. 102
Systematic error effects on the flux unfolding. 102

Figure 42
Figure 43
Figure 44
Figure 45
Figure 46
Figure 47
Figure 48
Figure 49
Figure 50
Figure 51
Figure 52
Figure 53
Figure 54
Figure 55
Figure 56
Figure 57
Figure 58
Figure 59
Figure 60
Figure 61
Figure 62
Figure 63
Figure 64
Figure 65
Figure 66
Figure 67
Figure 68
Figure 69
Figure 70
Figure 71
Figure 72

Measured flux energy spectra. 103
Measured flux compared with models. 104
Fitted flux normalization and spectral indices. 107
Unfolding separated by SK period. 107
Measured flux compared with other experiments. 108
Unfolding matrix for ν/ν̄. 111
Regularization errors for ν/ν̄. 113
Measured flux values for ν/ν̄. 117
Measured ν/ν̄ fluxes compared with models. 118
East/west separated ring counting and PID. 122
Michel decay spectrum. 123
SK azimuthal energy reconstruction error. 124
Energy spectra for east and west-going events. 125
CCQE ν and lepton directional correlation. 128
Azimuthal plots. 130
A parameter. 131
Azimuthal MC truth plots. 131
Optimized azimuthal plots. 132
B parameter. 132
HKKM solar modulation reweighting. 136
NM conversion functions. 138
Solar activity during SK. 139
Solar activity effect on SK data. 140
Parameterization using α. 140
Observation time by NM count. 143
e-like up-going sample predicted solar modulation effect. 144
Solar modulation toy MC example. 146
Solar modulation data. 148
Solar modulation test statistic. 148
Yearly changes in the event rate. 149
Solar modulation results by sub-sample. 150

L I S T O F TA B L E S

Table 1
Table 2
Table 3
Table 4
Table 5
Table 6
Table 7
Table 8
Table 9Oscillation parameter values.
SM field charges.
16
Cherenkov energy thresholds.
SK trigger types.
36
FC daily event rates.
56
ν/ν̄ separated sample purities.
PC daily event rates.
75
UPMU daily event rates.
79
Neutrino interactions in NEUT.

15
30

72

81

xi

Table 10
Table 11
Table 12
Table 13
Table 14
Table 15
Table 16
Table 17
Table 18
Table 19
Table 20

SK detector livetimes by period.
86
Data and MC numbers by SK period.
87
Energy binning for the flux unfolding.
93
Measured flux energy spectra. 105
χ2 results for the energy spectra. 106
Energy binning definitions for ν/ν̄. 112
Measured flux values for ν/ν̄. 116
Systematic effects on the azimuthal analysis. 127
SK sub-GeV data summary. 141
Systematic effects on the solar cycle analysis. 142
Forbush decrease events. 147

ACRONYMS

Analogue Timing Module
AFT After Trigger
CC
Charged Current
CCQE Charged Current Quasi-Elastic
DIS
Deep Inelastic Scattering
DOF Degrees of Freedom
FC
Fully Contained
FV
Fiducial Volume
GUT Grand Unified Theory
HE
High Energy
HVHigh Voltage
ICRR Institute for Cosmic Ray Research
ID
Inner DetectorLE
Low Energy
MCMonte CarloNC
Neutral Current
NM
Neutron Monitor
OD
Outer Detector
PC
Partially Contained
PDG Particle Data GroupPDF Probability Distribution Function
PID
Particle Identification
PMT Photomultiplier Tube
QE
Quantum Efficiency
QE
Quasi-Elastic
RES
Resonant
SK
Super-Kamiokande
SMStandard Model
SLE
Super Low Energy
SHE Super High Energy
ATM

xii

acronyms

Time and Charge
TOF Time of Flight
UPMU Upwards-going Muon
UTC Coordinated Universtal Time
TQ

xiii

Part I
INTRODUCTION
I think, therefore I am.
— René Descartes

1

NEUTRINO PHYSICS

1.1

motivation: the atmospheric neutrino

Atmospheric neutrinos are generated from the decay of mesons produced in cosmic-ray interactions in the Earth’s atmosphere, and are
one of the main experimentally available neutrino sources. Since thefirst detection of the atmospheric neutrino in underground experiments in the 1960s [1, 2], further measurements of them brought thediscovery of neutrino oscillation (and thus finite neutrino masses)
in 1998 [3]. Since then the understanding of the oscillation parameters has improved quickly, by using the atmospheric [4, 5] datain combination with solar [6, 7], reactor [8, 9, 10, 11], and accelerator [12, 13, 14, 15] sourced neutrinos.
The atmospheric neutrino sample spans a wide energy region, peaking at O(MeV) but so far measured up to O(PeV), and atmospheric
neutrinos can be detected up to ∼ 13 000 km away from their initial
creation point (i.e. after crossing the diameter of the Earth). They arethus a useful source as a varied and high-statistics neutrino beam,
however to understand and use them as a sample, an accurate prediction of the expected flux depending on neutrino flavour, energy,
direction, and time is required. For example, the discovery of neutrino oscillation was by comparison of their expected zenith angle
distribution with and without oscillation applied, as shown in Fig. 1(although an up-down asymmetry can be predicted using some simple flux assumptions and geometrical considerations [16], a more detailed flux model is necessary for an accurate quantitative comparison). Furthermore, the atmospheric neutrino is not just a signal but
also a background source to many other experiments, such as astrophysical neutrino, proton decay, and dark matter searches – one example is given in Fig. 2. An accurate prediction of the flux is thus ofparamount importance to many current and future experiments. As
one example, the planned Hyper-Kamiokande [17] experiment could
use the atmospheric sample to uncover the neutrino mass hierarchy.
Current atmospheric neutrino flux predictions are given by Monte
Carlo simulations by several groups [18, 19, 20, 21]. These models
are based on data such as the primary cosmic ray proton flux andthe secondary muon flux, measured in the atmosphere by particle
detectors on balloons [22, 23] and spacecraft [24, 25], and hadron production measured in accelerator experiments [26, 27]. Direct experimental measurements of the flux energy spectra were made by the
Frejus [28] collaboration (before neutrino oscillation was known), and3

neutrino physics(a) Graphical illustration
number of events

4

100

200

80

150

60

100

40
20
0
-1

electron-like
cos Θ

50
0

muon-like

-1

cosΘ

(b) Zenith distributions

Figure 1: The first strong evidence for neutrino oscillation found by theSuper-Kamiokande detector. Zenith angle distributions show earlydata reported in 1998 for contained lepton events, where cos(Θ) =1 represents down-going leptons. Dashed (hatched) values show
the predictions based on neutrino flux models for an oscillated (unoscillated) flux. In particular, it appears that the up-going νµ + ν̄µflux has reduced on its way through the Earth, matching the modelof oscillations to tau or sterile neutrinos.

more recently by the AMANDA-II [29, 30] and IceCube [31, 32, 33]
collaborations at higher energies (up to the 100 TeV range).
The estimated uncertainties on the simulations, especially at lower
energies, were historically much smaller than the uncertainties on the
direct experimental measurements of the flux. However, as the current generation of neutrino detection experiments improve statistics
and reduce systematic uncertainties, measuring the atmospheric neutrino flux by direct data measurements becomes increasingly useful –allowing more accurate cross-checks and feedback to the future development of the simulations, and discovery of several new effects thathave been predicted by the simulations but not yet observed, such as
directional asymmetries and modulation over time. Specifically thisthesis presents three direct measurements of the atmospheric neutrino flux at Super-Kamiokande: the directionally-integrated energy
spectrum of the flux (with high statistics in the 100 MeV to 10 TeV
range), the azimuthal spectrum, and the modulation of the flux withtime over the eleven-year solar cycle.

Number of Events

1.2 detection of the neutrino

100

80

60

40

20

0

200

225

250

275

300

Pµ(MeV/c)

Figure 2: An example of the atmospheric neutrino flux as a backgroundsource – data from the search for proton decay via the modep → ν + K + at Super-Kamiokande. The reconstructed muon momentum distribution is shown. The red boxes correspond to theatmospheric neutrino simulation, and the blue line histogram isthe expected proton decay signal.

1.2
1.2.1

detection of the neutrino
Theoretical basis

In the Standard Model (SM), the neutrino is a spin- 12 lepton which
participates only in the gravitational and weak interactions. Depending on the neutrino’s cosmological energy density, their gravitational
effects may be observed on the large scale, e.g. as a minority component of dark matter [34]. Detection of individual neutrinos can beaccomplished via the weak force, which is carried by the charged andneutral vector bosons W ± and Z0 with masses of 80 GeV and 91 GeV
respectively, and thus has an effective range restricted to ∼ 10−17 m.
Although the intrinsic strength of the weak force is much higher thanthe electromagnetic force, weak interactions are thus comparatively
infrequent.
The possible interactions of a neutrino by the neutral current (NC)are simple ones of the form Fig. 3a, where the Z0 boson facilitatesa transfer of momentum and spin. Charged current (CC) interactions are more complicated, and involve changes in weak isospin (i.e.
up-type vs. down-type quarks, or charged vs. neutral leptons) andflavour. The flavour changes originate due to the difference between
propagation and interaction eigenstates; this will be considered indetail in Sec. 1.3. One such example is shown in Fig. 3b, where it
can be observed that contrary to NC interactions, in CC interactions
the flavour of the interacting neutrino is preserved in the outgoing
charged lepton, and may be experimentally observable in a practicalsense.

5

6

neutrino physics

να

να

lα−

να

Z0
f

W+

f
(a) Neutral current

νβ (qu′ )lβ− (qd )(b) Charged current

Figure 3: Examples of Standard Model interactions for a neutrino να (α =e, µ, τ). In (a), f represents any weakly-charged fermion such asa charged lepton, left-handed neutrino, or quark. In (b), l β− is anegatively charged lepton of flavour β, qd represents a down-type
quark, and q0u an up-type quark after a flavour basis rotation (i.e. asuperposition of u, c, and t).

Mathematically speaking, the propagator for the W ± and Z0 bosons
is given by−i ( gαb − qα q β /m2 )q2 − m2(1)

where gαβ is the Minkowski metric, m is the mass of the W ± or Z0 boson, and q is the momentum associated to the boson. When q2  m2 ,this reduces to igαβ /m2 , which is a pointlike interaction as postulated
in Fermi’s original theory of weak interactions. In other words, whenthe energy transfer is less than the mass of the W ± and Z0 bosons, we
can ignore the effects of their propagation. The full matrix element is
then


GF 
MF = √ ϕ̄γα (1 − γ5 ) ϕ ϕ̄γα (1 − γ5 ) ϕ2(2)

where the definition of the overbar operator is ϕ̄ = ϕ† γ0 , GF is the
Fermi coupling constant, and 21 (1 − γ5 ) is the left-handed chiral projection operator. The use of the Fermi theory, although we now know
as being simply an effective theory, is a useful simplification overa wide energy range. The presence of the operator 12 (1 − γ5 ) however bears close examination – it essentially represents the important
experimentally-determined fact [35] that the weak interaction violates
parity; only left-chirality fields (or right-chirality antiparticle fields)
participate in it. This is often referred to as the vector – axial vector(or V − A) structure of the weak interaction, referring to the objectsϕ̄γα ϕ and ϕ̄γα γ5 ϕ, which transform respectively as a four-vector andan axial vector under a Lorentz transformation (where an axial vector,
or pseudovector, gains an additional factor of -1 under a parity flip).1
1 The other possible Lorentz-invariant currents are the scalar, pseudoscalar, and tensor
currents, which are experimentally found to not be involved in the weak interaction.

1.2 detection of the neutrino

From an experimental point of view, we observe neutrino interactions with hadronic targets such as water or iron, and a range of
hadronic effects must be additionally considered; this is discussed ingreater length in Sec. 4.5.1 when the Monte Carlo event generators are
considered. Depending on the neutrino energy and target nuclei, inparticular for neutrino energies of roughly 1 GeV, some CC reactions
still take a simple form as in Fig. 3b and are termed “Quasi-Elastic”(CCQE) interactions. In this case, the energy transfer q2 tends to besmall, and the outgoing lepton can roughly preserve the momentum
vector of the incoming neutrino; providing that the outgoing lepton
can be well measured, we can thus reconstruct all of the neutrino’s
properties.2
1.2.2

Experimental history

The initial motivation for postulating the existence of the neutrino, by
W. Pauli in 1930 [36], was to explain the “missing” momentum in thebeta-decay process that we now know to proceed as n → p + e− + ν̄e .
The discovery, by Cowan and Reines in 1956 [37], used a similar but
inverted reaction ν̄e + p → e+ + n which could take advantage of an
intense ν̄e flux produced by a nuclear reactor. The actual detection
method used was to monitor for a 2γ signal from positron annihilation, proceeded a few microseconds later by a γ signal emitted when
the neutron is captured by a nucleus.
The discovery of a naturally-occurring neutrino flux came arounda decade later, in the form of atmospheric neutrinos, by plastic scintillation detectors several kilometers underground [1, 2]. The depth ofthe experiments reduced the expected background of cosmic-ray secondary muons, allowing the signal events of neutrino-induced muons
created in the surrounding rock to be more easily observed (althoughsome care still had to be taken to avoid contamination by the remaining down-going background muons).
A few years later, the Homestake experiment [38] used a radiochemical detector in a mine to identify neutrinos produced in the
Solar fusion process, which created a precisely-measurable numberof Argon isotopes in the reaction νe + 37 Cl → 37 Ar + e− . However, theflux was approximately a third of that expected by theoretical solarmodel calculations – due to the as-yet-undiscovered neutrino oscillations (modified by matter effects inside of the Sun). It was not until
three decades later, with evidence from the water Cherenkov detectors Super-Kamiokande (SK) [3] and the Sudbury Neutrino Observatory (SNO) [39] that enough evidence was gathered to prove the exis2 Strictly speaking, even in the CCQE case the momentum of outgoing leptons canvary considerably compared to the neutrino’s original momentum, but as the process
is theoretically relatively simple to study much information can be extracted in astatistical sense.

7

8

neutrino physics

tence of neutrino oscillations. These detectors observe the Cherenkov
light emitted by charged leptons, which originate in neutrino collisions in large fiducial-volume water tanks (in the case of SNO, heavy
water), and unlike previous detectors were able to accurately to reconstruct the momentum and flavour of individual neutrinos in nearreal-time (the SK case will be examined in detail in Sec. 4.1).
Several recent experiments have returned to reactor sourced neutrinos [8, 9, 10, 11], or constructed artificial neutrino beam generators [12, 13, 14, 15] in order to make precision measurements of theneutrino oscillation parameters.
The field of neutrino astronomy also received a kick-start with thedetection of neutrinos originating from the supernova 1987A [40], located in the Large Magellanic Cloud, in agreement with models suggesting that 99% of a supernova’s energy is released in the form ofneutrinos. The discovery of a regular astrophysical neutrino flux has
as of this year been confirmed by the gigaton-volume Cherenkov detector IceCube [41], which due to its huge volume is able to reconstruct events up to PeV energies.
1.3

neutrino oscillation

The oscillation of neutrino flavour during propagation is a comparatively recent and influential discovery, implying neutrino mass andseveral new parameters, and providing clear evidence for a necessary
extension beyond the SM. A detailed description of the oscillation
mechanism, parameters, and uncertainties is necessary for the measurements in this thesis, as we will be comparing the measured neutrino flux at the SK detector with the unoscillated predictions fromthe flux simulations.
1.3.1

The lepton mixing matrix

Neutrino oscillation measurements so far are consistent with the threeflavor lepton mixing scheme described by a unitary 3×3 PMNS matrix [42, 43]

 


νe
Ue1 Ue2 Ue3
ν1

 


 ν , ν = U
(3)
U
U
µ2
µ3   2 
 µ   µ1
ντUτ1 Uτ2 Uτ3

ν3

which represents a rotation in field space between the weak-flavour
basis να (α = e, µ, τ) and the mass basis νi (i = 1, 2, 3), or in otherwords the interaction and propagation bases. Anti-particles can be
described by the conjugate matrix. Crucially, if the masses mi of eachstate νi are different, the propagation of each mass state will be different, and then the presence of any off-diagonal elements in the PMNS

1.3 neutrino oscillation

matrix U can cause the flavour of a neutrino to change during propagation.3
In principle the choice of “flavour basis” is somewhat arbitrary, andthe matrix could be defined instead in the charged lepton sector; theabove notation is however common since charged lepton oscillation
is in most cases not experimentally observable, essentially due to thelarge mass differences for the charged leptons [44]. The PMNS matrix is analogous to the CKM quark mixing matrix, and describes thesame underlying physics for the charged and neutral leptons, eventhough the probing of the matrix elements takes different approaches
in the two cases.
This matrix introduces 18 free parameters (9 complex numbers),
which the unitarity condition reduces to 9 free parameters. Considering that Dirac fields can be freely redefined by an unobservable complex phase, we could naively further reduce 6 parameters by redefining the charged lepton and neutrino fields – however examining thecharged-current Lagrangian LCC , it is seen that one complex phasein the lepton sector must be kept as physically observable [45]. The 3flavour mixing matrix can thus be fully described by four parameters.
A common parameterization uses the Euler mixing angles








cos θ12sin θ12
0
1
0
0

 




U =  sin θ12 − cos θ12 0  ×  0 cos θ23 sin θ23 001
0 − sin θ23 cos θ23


−
iδ
CP
cos θ130 e
sin θ13



,
×
(4)
0
1
0


−e−iδCP sin θ13 0

cos θ13

where the phase δCP controls the size of the imaginary part of the matrix, and thus controls the level of charge-parity violation (originatingin the fact that antiparticle interactions contain the conjugate matrix).
In the case of Majorana neutrinos, the neutrino field phases may not
be freely redefined due to the condition ν = νc (where C represents
charge conjugation), and two extra phases are needed; these are commonly parameterized as an additional rotation to the above matrixby
UMajorana = U × eiα1 /2
0
0

0

0




eiα2 /2 0 
.
0

(5)

1

3 It is then clear that the lepton family number symmetries present in the SM (Le ,Lµ and Lτ ) are not conserved, while the total lepton number L T is still conserved,
assuming a unitary matrix.

9

10

neutrino physics

1.3.2

Oscillation

Oscillation experiments generally measure neutrino interactions in
the flavour basis, at some distance L from the initial creation of theneutrino, and we are therefore interested in the probabilityP(να → νβ ) =νβ |να ( L)

2

,

(6)

where νβ is the state vector for a flavour β and |να ( L)i represents
the propagated state vector that was initially a flavour α. To calculate|να ( L)i, we first consider the propagation of a neutrino mass state νiunder the assumption of a plane wave solution of definite momentum
p and energy E, which can be simply described by the phase factor|νi ( L)i = e−i(Et− pL) |νi i .

(7)

A state of initial flavour α is a weighted superposition of three massstates, as given by the PMNS matrix U, and can thus be propagatedas3|να ( L)i = ∑ Uαi e−i(Ei t− pi L) |νi i .

(8)

i =1

As a simplifying assumption we note that neutrinos are generally
relativistic, so we may approximate t ≈ L and furthermore assume
that the mass states have similar kinetic (andq therefore total) energies
Ei ≈ E. In this case we can expand pi =and we obtain3

E2 − m2i ≈ E(1 − m2i /2E2 ),|να ( L)i ≈ ∑ Uαi e−i(mi L/2E) |νi i .
2

(9)

i =1

Finally, explicitly expressing the above in terms of the flavour basis βgives|να ( L)i ≈
=

∑

"
νβ

νβ

β=e,µ,τ

∑β=e,µ,τ

3∑ Uαi e#−i (m2i L/2E)

i =1

"
νβ

3

∑ Uαi Uβi∗ e

|νi i

(10a)

,

(10b)

#
−i (m2i L/2E)

i =1

and any oscillation probability Eq. (6) for a chosen flavour of β may
be read off from the bracketed coefficients of Eq. (10b) in terms of mi ,E, L, and U.
In principle, a plane-wave approximation, coupled with an equalenergy (or equal-momentum) approximation for the mass states, may1.3 neutrino oscillation

not be a particularly accurate model. A wave-packet treatment of theoscillation was first suggested as necessary by [46] and performed in[47] by Giunti, Kim, and Lee, who later showed that the single-flavour
state vectors are not even themselves well-defined, except in the relativistic limit [48]. Further development of the theoretical frameworkof oscillation is in fact still under active discussion; a good overview
is given in [49]. However, the oscillation probabilities calculated withthe simple plane-wave model are consistent with the more comprehensive calculations in the domain E  m, where neutrino interactions are actually observable, so we may proceed using the formulae
as shown above.
Oscillation probabilities are often given depending on the parameterization of the mixing matrix U, and in the three-flavour case canresult in quite long formulae. In the two-flavour case for Dirac neutrinos however, the parameterization of U is reduced to only one freeparameter θ:

Utwo−flavour =

cos θ

sin θ

− sin θ cos θ!
,and the probability can be simply calculated as∆m2 [eV2 ] L [km]22P(να → νβ ) = sin (2θ ) sin 1.27
E [GeV]

(11)

(12)

where ∆m2 ≡ m2β − m2α and α 6= β. This simplified yet illustrative
case shows the principle of neutrino oscillation measurements; for aknown neutrino beam of flavour α and path-length to energy ratio
L/E, we may study the neutrino oscillation parameter(s) θ and sizeof the mass-difference(s) ∆m2 , if not the ordering or the absolutevalue of the neutrino masses. It also illustrates why the atmosphericsample’s wide range of path-lengths and neutrino energies make itsuch a useful neutrino source.
1.3.3

Matter effects

The previous section discussed neutrino oscillations in vacuum; thesituation will be different in the presence of normal matter, where
almost all charged leptons are electrons. While the NC interactions
are identical between neutrino flavours, electron neutrinos may undergo CC coherent forward scattering reactions with electrons, of the
leptonic form in Fig. 3b where we set α = β = e; muon neutrinos cannot undergo such a process, and thus the oscillation scheme will be
modified (of course, muon neutrinos with Eν > mµ may still undergo
the process where α = µ and β = e, but this is not a coherent process). This modification will be shown to be equivalent to a change
in the effective masses of the propagating neutrinos, and is generally

11

12

neutrino physics

called the matter effect [50]; a somewhat appropriate analogy would
be to consider the dispersion of a light beam inside a material with arefractive index.
The coherent scattering interaction for Dirac neutrinos introduces
a term of the form
GF
α55
Hmat = √ ν̄e γ (1 − γ )νe ēγα (1 − γ )e,2(13)which can be simplified by applying the chiral projection operator,
notating left-handed chirality by the subscript L, to√
Hmat = 2GF ν̄e,L γα νe,L ē L γα e L .
(14)Further considering an interaction in the rest frame of unpolarized
normal matter, only the γ0 contribute for the electrons, which gives√
Hmat = 2GF ν̄e,L νe,L ē L e L(15)
which is simply a potential energy term for electron neutrinos, withthe potential√
Vmat = 2GF Ne(16)
where Ne is the electron density.
Looking back to Sec. 1.3.2, putting our description of neutrino oscillation into matrix form, and differentiating with respect to time (fort = L) we have





2
ν
m1 0
0
ν
 1 
 1 
1 





2
i∂t  ν2  =(17)
0 m2 0   ν2 

2E 
ν3
ν3
0
0 m3
which has the same form as the time-dependent Schrödinger equation i∂t |νi i = Hvac |νi i, where Hvac can be interpreted as the vacuum Hamiltonian. However, we must be careful to not over-interpretthe similarity, as we are explicitly dealing with relativistic neutrinos.
Nonetheless, under the same approximations used in Sec. 6, it is seenthat [51, 52] the effective Hamiltonian for propagating vacuum andmatter effects is simply H E = Hvac + Hmat and the propagation equation is then



νe





m21


  1  i∂t  νµ  = U 2E  0
ντ
0





0

0

m22

 † 

0 
U +
m23

0

Vmat 0 0
0
0

 


νe

 



0 0 
  νµ  .
0 0
ντ
(18)

Here, the mass matrix was moved to the flavour basis, while the matter effects are of course native to the flavour basis; it is then clear that1.3 neutrino oscillation

the MSW effect can be seen as a change of the effective masses of
the neutrinos. In general this matrix can be simplified to absorb an
unobservable change to the overall phase of all three flavours, for example a subtraction involving m21 , leaving only ∆m212 and ∆m213 , andin fact the NC potential has already implicitly been absorbed. After re-diagonalization of the matrix, the oscillation formulae such as
Eq. (12), can be given in matter in terms of two effective mass differences ∆m2ij,eff and three effective mixing angles θij,eff (plus the complex
phase).
For a concrete example, we assume ντ does not exist, and take thetwo-flavour case of νe and νµ with notation as in Eqs. (11) and (12).
Here, the effective Hamiltonian becomes!
1−∆m2 cos(2θ ) + 2EVmat∆m2 sin(2θ )HE =, (19)
4E∆m2 sin(2θ )∆m2 cos(2θ ) − 2EVmat
where a constant term was removed and the Vmat split across the diagonal by a redefinition of the overall phase. We can always diagonalize
this matrix to!
1−∆m2eff
0
HM =
(20)
4E
0
∆m2
eff

by finding a transformation H M = U M H E U†M , where!
cos θeff sin θeff,
UM =− sin θeff cos θeff(21)

and thus we have the new effective parameters. Compared to thevacuum parameters, the new mixing angle will be given by
2EVmattan(2θeff ) = tan(2θ )/ 1 −.
(22)∆m2 cos(2θ )Considering the various cases of whether or not cos(2θ ) > 0 and (or)m2 > m1 , the effective mixing strength can be increased or decreased
given an appropriate material Vmat . In particular, if both are positive
we have the possibility of a resonant enhancement, where the bracketed term becomes zero and the effective mixing angle becomes maximal (θeff = 45°); considering Eq. (12), depending on the path-lengthand energy, complete conversion between flavours would be possible
for any intrinsic value of θ.
Above, we assumed that Ne was constant. In cases where the observed neutrinos pass through areas of varying electron density, suchas in the Sun or Earth, generally division of the density profile into
various zones or a numerical treatment can be used. A more carefulanalytical treatment of propagation through the Sun, during which
the resonant enhancement point is gradually crossed, showed that total conversion of the νe flux (created in the fusion process) to νµ is

13

14

neutrino physics

Figure 4: The “normal” (left) and “inverted” (right) possible mass hierarchies for neutrinos. The three colors show the flavour componentsof each mass state (e, µ, τ). [55]possible for all neutrinos above a certain energy; this enhancement is
known as the MSW effect, after the first published authors Wolfenstein(on matter effects) [50] and Mikheyev and Smirnov (on the resonant
enhancement) [53]. A particularly clear description of the effect, including a correction of the sign of the potential, is given by Bethein [54].
Finally, it should be noted that importantly for anti-neutrinos, thepotential Vmat will have the opposite sign and thus quite different
matter effects. The extra phases of Majorana neutrinos however can
not change the effective parameters.
To summarize, unlike the vacuum oscillation probabilities, probing
neutrino oscillations in matter with neutrinos and anti-neutrinos canin principle show different observable effects depending on the neutrino mass hierarchies and mixing parameter octants.
1.3.4

Current experimental status

Currently all the mixing angles θ12 , θ23 , and θ13 are experimentallymeasured, and are considerably larger than the mixings seen in thequark sector; however there is still ambiguity as to the true octant ofθ23 (greater or less than 45°). The charge-parity violation phase has
not yet been significantly measured as non-zero. Both mass splittings∆m212 and ∆m223 are measured, with ∆m223  ∆m212 . The mass hierarchy (i.e. ordering of the masses) is known at least for ∆m212 wherem2 > m1 , however the sign of ∆m223 is not yet known, as illustrated inFig. 4.
Constraints on the absolute neutrino mass scale have been obtainedby methods including direct travel-time measurements from the Supernova 1987 [40], cosmological energy-density constraints and doublebeta decay experiments [56]. The discovery of Majorana neutrinos(the existence of which is widely believed in the theoretical commu-

1.4 massive neutrinos

parameter∆m221|∆m232 |sin2 θ12
sin2 θ23
sin2 θ13

nominal value

uncertainty

7.50 × 10−5±0.2 × 10−5

2.32 × 10−3
0.310
0.500
0.0240δCP

0sign of ∆m32+

±0.12 × 10−3−0.014 + 0.023±0.112±0.0023
0 → 2π+ or −

Table 1: Nominal value and 1 σ uncertainty (or for the CP violation phaseand mass hierarchy, possible values) of the oscillation parameters
used in this thesis, based on [59]. The unit of ∆m2ij is eV2 .

nity) may be possible by detection of neutrinoless double-beta decay [57]. The existence of Dirac neutrinos is currently believed to be
not directly provable, as it would require an infinitely stringent limit
on such a decay.
It is predicted that the sign of ∆m223 and the θ23 octant may beprobed in the next generation of neutrino detectors [17] via atmospheric neutrinos passing through the Earth and consideration of
their matter effects, as such detectors are expected to have a stronger
separation of neutrinos and anti-neutrinos. The δCP phase is similarly
expected to be measurable, for a significant fraction of the possible
range, by long baseline neutrino beam experiments. The existence ofmatter effects, although an integral part of the standard oscillation
theory and confirmed with high significance (if somewhat indirectly)
in the solar neutrino model, should also be observable inside of the
Earth [58] as a further check on the consistency of the theory.
A summary of current measurements is given in Tab. 1.
1.4

massive neutrinos

Since the discovery of neutrino oscillations, the understanding of theorigin of neutrino mass has become an important task in theoreticaland experimental physics. Here the principles are discussed briefly,
as while essential for a general understanding of neutrino physics,
the results here are not directly used in later parts of this thesis.
1.4.1

Mass terms

In order to incorporate neutrino masses into the SM, we will see thatsome new fields must always be postulated. The standard Dirac massterms for neutrinos (assuming one generation) take the form−m D ν̄ν = −m D (ν̄L νR + ν̄R νL ).

(23)

15

16

neutrino physics

field


ν
 L 
eL
eR
νR


uL




uR
dR



φ+
φ0

1
2

1
2

− 12

Y

−1

Q
0
1

0

−2

−1

0

0

0

0

1
2

1
2

− 12

1
3

0

0

4
3

0

0

- 32




I3

0


dL



I

1
2

1
2

− 12

1

2
3

− 13
2
3

- 13
1
0

Table 2: Charges of the SM fields (first generation), and the necessary
charges of a sterile neutrino singlet νR . Here φ is the Higgs doublet, and the convention is Q = I3 + Y2 , where Q is electric charge,
I is isospin, and Y is weak hypercharge. The leptons also carry lepton number L T = 1, except in the case of Majorana neutrinos where
L T = 0.

The immediate problem is that these mass terms are chiral, whereas
in the SM no right-handed neutrinos (or left-handed anti-neutrinos)
exist by default, since they were not observed to participate in the
weak interaction. It is of course possible to add a right-handed neutrino field, as long as it belongs to an isospin singlet, i.e. a field that
does not experience any forces except gravity – often referred to bysuch terms as “sterile neutrino” or “neutral heavy lepton”. The termson the right side of Eq. (23) then have I3 = ± 12 and Y = ±1 (seeTab. 2), and can be written into the SM by coupling with the standard
Higgs’ doublet Φ such that the SM charges are conserved. The total
lepton number is also explicitly conserved by the ν̄ν pairing.
Another possible mass term, though not used in the SM, are theMajorana mass terms with the form
1
− m LM (ν̄L νLC + ν̄LC νL )2
1 R
− m M (ν̄R νRC + ν̄RC νR )
2

(24a)
(24b)

where the charge conjugation operator C is essentially similar to overbar, but maintains the necessary row or column structure. Majorana
mass terms were originally designed to be used in the case of Majorana fermions – particles that are their own anti-particles, i.e. ν = νC .
Naively this kind of term could avoid the problem of extra neutrino
fields, if we take only the terms in Eq. (24a) as physical; however,1.4 massive neutrinos

these terms have I3 = ±1 and Y = ±2, and thus would require an
extension4 of the Higgs sector to include a field with Y = 2. Lepton number conservation is also explicitly broken for a Dirac fermion,and even for a Majorana fermion we see that there is no U(1) symmetry νL → eiϕ νL , and by Noether’s theorem no associated conserved
current.5
1.4.2

The seesaw mechanism

It is also possible that both types of mass terms are actually present,
and write all of the above terms into a matrix!
!
 mL mC
1
ν
D
M
L
Lm = −+ h.c.,(25)ν̄L ν̄RC
2
mD mR
νR
M

in which case it seems that the right-handed fields (νR and νLC ) andleft-handed fields (νL and νRC ) no longer have definite masses individually. The matrix can be diagonalized to
!
!
 m
1
0
νa
a
LM = −+ h.c.,(26)ν̄a ν̄b2
0 mb
νb
where the a and b neutrinos must have the structure e.g. νa = νa,L +C ; in other words, if both mass terms are present the neutrinos must
νa,Lbe Majorana fermions.
The observable masses m a and mb then depend on the physicallymotivated choices we make for m LM , m RM , and m D . As discussed inSec. 1.4.1, m LM is forbidden in the SM. The Dirac mass m D is forbidden before symmetry breaking but allowed after symmetry breaking,thus should take its value from the Higgs mechanism with the vacuum expectation value of 246 GeV. Finally, m RM is generally allowedin the SM, and could be generated by any mechanism, including new
physics at the Grand Unified Theory (GUT) scale, which is plausibly
around 1015 GeV.
Then, assuming the Yukawa couplings are of the order unity, we
can takem LM = 0,m D ∼ 102 GeV,m RL ∼ 1015 GeV,

(27)

which by examination of the matrix diagonalization givesm a ∼ 10−4 eV,

mb ∼ 1015 GeV.

(28)

This choice of masses is known as the seesaw mechanism [60], which
may provide some justification for the comparatively tiny observed
4 This could also be expressed by a non-renormalizable dimension 5 operator within
the SM, although this explicitly turns the SM into an effective field theory.
5 However, the violations of L T in the overall Lagrangian will be small; this does
however allow the previously-mentioned double-β decay.

17

18

neutrino physics

neutrino masses mi ∼ m a . The name is also applied to various othermodels and choices of the parameters that can result in the observed
hierarchy.
In the above case, as can be seen from the mixing of the doublet
νL and singlet νR into νa and νb , the sterile state participates in theneutrino flavour mixing.6 The matrix U must then be extended toinclude the sterile flavour(s); at the present time, the measurementsof the matrix parameters in the three-flavour scheme are consistent
with the unitarity of the matrix, and there is no conclusive evidence
for the existence of steriles (although there is some tension between
experimental results [61]). In any case, the oscillation effects of sterile
neutrinos must be small, and while the discussion of neutrino mass isof great interest, the effects will be ignored for the rest of this thesis.7

6 In the three-flavour case this is generally true for Majorana neutrinos, but not Dirac
ones.
7 For a full discussion see e.g. [51].

2

T H E AT M O S P H E R I C N E U T R I N O F L U X

2.1

originThe atmospheric neutrino flux originates from the cosmic ray flux
colliding with the Earth’s atmosphere. The cosmic ray flux consists
mostly of protons and Helium nuclei in the ratio 9 : 1, with heavier
nuclei at the ∼1% level, and is isotropic within 0.1% [62]. Electrons,
positrons and electromagnetic radiation make up an even smaller percentage, negligible from the neutrino physics perspective. The cosmic
rays are believed to originate mostly from outside the Solar System,from sources such as supernova remnants in our galaxy and the active
nuclei of other galaxies; the proportional contribution of the sources,and exact nuclei composition across the wide energy range, is however still under study (e.g. by [63]).
In general cosmic ray collisions in the atmosphere create particle
showers which are dominated by light mesons such as pions andkaons. A graphical representation of such a shower is shown in Fig. 5.
The dominant decay chain (with branching ratio > 99%) of charged
pions involves the creation of neutrinos as
π ± → µ± + νµ (ν̄µ )(29)

and the muons further decay via the “Michel decay” asµ± → e± + νe (ν̄e ) + ν̄µ (νµ ).

(30)

Charged kaons undergo the identical decay as Eq. (29) with a branching ratio of 64%, with the remainder of decays being either hadronic(to pions) or semileptonic as
K ± → π 0 + e± + νe (ν̄e )

K

±

0

±

→ π + µ + νµ (ν̄µ )

(31a)
(31b)

with branching ratio 5% and 3% respectively. The neutral pion decays
generally as π0 → γ + γ, and the neutral kaon (via the short and long
weak eigenstates) decays primarily to pions. Therefore, consideringthat the majority of decays to neutrinos will be by charged pions orkaons as in Eq. (29), then by muons as in Eq. (30), we can make thefirst-order approximations for the neutrino flux
νµ + ν̄µνe + ν̄e
νµ
ν̄µ
νe
ν̄e

∼ 2

(32a)

∼ 1

(32b)

∼

µ+
µ−

(32c)

19

20

the atmospheric neutrino flux

Figure 5: A schematic representation of a primary cosmic ray collision andshower in the atmosphere. [65]assuming that all decay chains reach their final states. More accurate
determinations of the flux, such as the overall normalization, must
be based on a more complete study of the primary cosmic ray flux,
the hadronic processes in the atmosphere, and the geomagnetic field
effects.
At energies above 100 TeV, a different mechanism than the above
is expected to dominate, whereby neutrinos are primarily createdfrom the decay of heavier mesons involving charm quarks; these aretermed “prompt” neutrinos [64]. While the prompt flux is expected
to be low, it is in principle distinguishable by large-volume neutrino
telescopes [41]. However, the Super-Kamiokande detector does nothave significant power to observe such a flux, and this thesis focuseson the “conventional” neutrino flux from pions and kaons.
2.2
2.2.1

atmospheric neutrino flux simulations
Historical approach

As mentioned in the above chapter, any simulation of the atmospheric
neutrino flux must generally take into account the primary cosmic
ray spectra, all hadronic effects and interactions in the atmosphere2.2 atmospheric neutrino flux simulations(in other words, the yield of secondary particles), and the effects ofthe geomagnetic field. At the simplest level the neutrino flux as afunction of energy φν ( E), for an infinitesimal solid angle, could be
produced by a simple convolutionφν ( E) = ∑
N

Z ∞
0

φN ( E0 − E) × YN ( E0 , E) × R N ( E0 ) dE0(33)

where N is a sum over nuclei (from protons to helium and heavier nuclei), φN ( E0 ) is the primary cosmic ray spectrum, YN ( E0 , E) represents
the average yield of neutrinos reaching the ground for a nucleon Nof energy E, and finally R N ( E0 ) is a simple energy-filtering Heaviside
step function (i.e. zero or one). The energy cutoff of the step function
is calculated based on the incident location and direction of the primary cosmic ray, and decides whether a nucleon could have reached
the atmosphere or not depending on the magnetic field effect known
as the rigidity cutoff (explained in Sec. 2.2.2).
In this method, the distributions are calculated choosing a particular point on the Earth’s surface, and integrating around a 4π solid angle. While this may be possible analytically, the disadvantage of sucha method is obvious; while the geomagnetic cutoff information is included for the primary particle, all secondary particles are assumed
to travel in a perfectly straight line, ignoring geomagnetic effects, andeven conservation of momentum of the decay products. Furthermore,
only the average neutrino yield is used. In order to recreate the neutrino flux accurately, a fully three-dimensional Monte Carlo simulation, with individual tracking of all particles, seems necessary.
2.2.2

Rigidity cutoff

To understand the meaning of the rigidity cutoff, consider a random
incident location and direction of a cosmic ray collision on the atmosphere. After solving the equations of motion in the geomagnetic
field in the backwards time direction, we may find that the tracks below a certain energy range actually originated on the Earth’s surface;such tracks are forbidden as they imply that the cosmic ray would
have originally had to travel through the Earth. An example tracingof two tracks for a fixed energy is shown in Fig. 6, where only one
track passes the rigidity cutoff. For a given ground location, we candraw a map of the rigidity cutoff as a function of direction and primary cosmic ray energy, such as the one shown in Fig. 7; the irregularshape originates in the fact that the geomagnetic field is not perfectly
symmetric, but contains some complex structure.

21the atmospheric neutrino flux

Figure 6: Illustration of the rigidity cutoff effect, for two primary cosmic ray collisions that could result in neutrino events at SuperKamiokande. After tracing backwards in time, only one cosmic
ray passes the rigidity cutoff test. [66]

cos θ

22

1
10
10
20

0

50
10

20

30
5
5
20

10

5

5

10

-1

180

90

0

270
180
Azimuth ϕ [deg]Figure 7: Contour map of the rigidity cutoff, seen from the perspective of the
SK detector. The labeling of the azimuth angle φ and zenith angle
θ is the same as in Sec. 6.2. The units represent the rigidity cutoff
in GV (e.g. for a proton with a charge of +e, this is the requiredenergy in GeV to pass the cutoff). [67]

2.3 the hkkm flux model10

3

φ ν x E 3ν (m−2s−1sr−1GeV 2 )

νµ

10

νµ

2

νe

νe

10

1

0

10

−1

10

0

10

10

1

10

2

10

3

10

4

E ν (GeV)

Figure 8: The energy spectra of the neutrino fluxes at the Super-Kamiokandesite as predicted by the HKKM15 model. [68]

2.3
2.3.1

the hkkm flux model
Current model

The default flux model used at Super-Kamiokande is produced by the
HKKM group [18, 19, 68], and has been in continuous development
since 1988. The model tracks the neutrino flux at specific locations
on the Earth corresponding to the current neutrino detectors, such as
Super-Kamiokande, INO, IceCube, and so on. For each location, theenergy and directional spectra of the neutrino flux are published intable format, along with instructions for accurate interpolation. An
example of the final energy spectrum of the fluxes is shown in Fig. 8.
In this section, the current status and approach of the most recent flux
model (2015, [68]) will be discussed.
2.3.2

Three-dimensional calculation

The calculation is fully three-dimensional, in general tracking every
necessary particle from its generation to interaction within the atmosphere. The Earth is assumed as a sphere of radius Re = 6378.140
km (while this is not exactly true, the approximation is expected tohave a negligible effect on the final neutrino flux), and furthermore
the injection sphere is defined at Re + 100 km, and the escape sphereat 10Re . For each cosmic ray simulation, the energy and nuclei size issampled from the data-based primary cosmic ray distributions takenfrom the AMS [24, 25] and BESS [22, 23] groups, and a random position and direction on the injection sphere is chosen. A rigidity cutoff
test, as explained in Sec. 2.2.2, is performed by tracing the equations

23

24

the atmospheric neutrino fluxof motion in the IGRF [69] magnetic field model backwards in time,
until the track hits either the escape sphere or returns to the injection
sphere. If the particle passes this test of extraterrestrial origin, it is fed
into an atmospheric propagation simulator.
The propagation simulator follows the traditional cascade model,
tracking the motion of all particles step-by-step, accounting for theforces applied by the magnetic field. For example, the bending ofeach muon’s path before decay has a significant effect on the flux at
a given location, compared to a simple one-dimensional model.
All appropriate particle decays are implemented. The hadronic interactions with air nuclei are handled by a combination of the JAM
model (a subset of the PHITS Particle and Heavy-Ion Transport System code [70]) at E < 32 GeV, and a modified DPMJET-III [71] code
above that, which was updated to give better agreement to the atmospheric muon spectra of secondary cosmic rays measured by the BESS
group. The probabilities for interactions are primarily based on the
air density, and to a lesser degree temperature, both taken from the
NRLMSISE-00 [72] model (recently updated from the US-standard ’76
model [73]). The model is based on empirical measurements and contains a full positional (latitude, longitude, and altitude) and seasonal
dependence from the Earth’s surface far into space. The seasonal dependence is largest at the poles with a maximum ∼ 10% variancein the νe and ν̄e fluxes (less for νµ and ν̄µ ) at around Eν = 3.2 GeV,
although quickly decreases moving towards the equator where thevariance is < 1%.
Finally, the simulation records the neutrino flux arriving at particular areas of the Earth’s surface, including of course the flux that passes
through from opposite sides of the Earth, corresponding to the position of the various neutrino detectors. However, the actual size of thedetectors is tiny compared to the size of the Earth, and thus when using the fully three-dimensional calculation, even with extremely highstatistics simulations it is unlikely that many neutrinos will directly
impact the detectors. The concept of “virtual detectors” is then introduced, which are areas covering the Earth’s surface up to a radius of1113.2 km from the detector positions (which corresponds to a change
in longitude of 10° at the equator), and any neutrino passing throughthese areas is registered. This of course requires a correction factor
depending on the zenith angle, with a higher weighting for events
arriving horizontally at the detector. While the virtual detector canintroduce some bias due to essentially averaging the fluxes over awide area, the strength of this bias can be examined by testing virtual detectors of various size at a given location. After application ofvarious correction factors, the bias is reduced to the 1% level.

2.3 the hkkm flux model

Figure 9: The estimated systematic errors on the HKKM07 model. The error sources are as follows: atmospheric density profile uncertainties δair , mean free path (cross-section) uncertainties δσ , kaon flux
uncertainties δK , pion flux uncertainties δπ , and total uncertainty
δtot . [75]2.3.3Systematic error estimations

An estimation of the systematic uncertainties on the HKKM07 simulation, produced by the HKKM group, is shown in Fig. 9. It can beseen that the flux is estimated very accurately in the 1 ∼ 10 GeV
range, but with increasing uncertainties at energies down towards
the MeV scale or up towards the TeV scale. The π and K flux uncertainties mainly come from experimental uncertainties on variousµ flux measurements, from which the π/K ratio and normalizations
are extracted. In the HKKM11 model, the uncertainty is expected tobe reduced by ∼ 5% at < 1 GeV due to the updates to the hadronicinteraction simulation, and for the HKKM15 model some reductionin the atmospheric density profile uncertainty is also expected [74].

25Part II
THE EXPERIMENT
I think I’m thinking, therefore I possibly might be.
— Arnold J. Rimmer

3

THE SUPER-KAMIOKANDE DETECTOR

3.1
3.1.1

principles of cherenkov detectors
Cherenkov radiation

The basic principle of Cherenkov detectors is to observe the Cherenkov
radiation produced when a charged particle moving faster than thephase velocity of light, in a particular medium, emits electromagnetic
radiation around the visible range. This emission is due to the polarization and relaxation of the material close to the particle track, generating wavelets at each point, which cause constructive interference
in a forward direction as visualized in Fig. 10.
Purified water can provide at low cost a massive target volume,
while having a long light attenuation length, allowing its use in largevolume Cherenkov detectors. The speed of light in a medium is given
by vγ = c/n, where the refractive index of water n is approximately
1.33. Therefore, considering the relative mass of a given type of particle, the minimum energy required for them to exceed vγ and thus
become observable by Cherenkov light may be easily calculated, withsome commonly observed particles shown in Tab. 3 (although in practice, the observational threshold of a given detector will be somewhat
higher than these values, considering light collection efficiency andbackground noise).
The Cherenkov light emission occurs at the Cherenkov angle ϑ, depending on the velocity of the charged particle v p as cos ϑ = c/(nv p ).
For particles with an energy precisely at the Cherenkov threshold, we
have ϑ = 0°, and then ϑ increases with higher particle energies; forrelativistic particles with v p /c ≈ 1 in water, we have ϑ = 42°. The
Cherenkov angle is thus useful to differentiate low-energy particles,
but loses its effectiveness once particles have total energy just a few
times larger than their rest mass, where v p /c approaches unity.

Figure 10: Cherenkov light production by a charged particle track. Dotted
angles represent right angles. [55]

29

30

the super-kamiokande detector

particle typecherenkov threshold [mev]

e±

0.77

µ±

159.8

π±

210.7

p±

1419.0

Table 3: The Cherenkov thresholds (minimum total energy required to emit
Cherenkov light) for various particle types in water with n = 1.33.

Considering a particle traveling on a short straight track, Cherenkov
light will be emitted in a hollow forward-facing cone, that when projected onto a flat surface (e.g. at the edge of the target volume) produces a ring shape. The longer the particle track, the thicker the ring
will be, where a track that reaches to the projection surface producing a completely filled ring. For minimally ionizing particles, such as
muons with energy O(GeV), the track length and thus particle momentum will be well correlated with the thickness of the ring. For
particles such as electrons, which can undergo multiple scatteringand electromagnetic showering, the ring shape will be more diffuse
and the energy is better correlated with a simple count of the emitted
photoelectrons [4].
3.1.2

Photomultiplier tubes

Considering the physics of Cherenkov light emission as discussed in
the previous paragraph, it is clear that the projection of the Cherenkov
light at the boundary of the interaction medium in principle contains
information that allows us to reconstruct the direction, energy, andparticle type of a charged particle track. A common method of sensitive detection of light is by the use of photomultiplier tubes (PMTs),
the basic operating principle of which are shown in Fig. 11. They use
the energy of an incident photon to liberate an electron via the photoelectric effect, where the material of the photocathode is chosen as tobe photosensitive to the light wavelengths of interest. The probabilityof emission depending on the absorbed wavelength is known as thequantum efficiency (QE), and an example of a QE distribution for a
PMT optimized for detecting Cherenkov light is given in Fig. 12. The
electron is then accelerated in an electric field, and amplified via collision with a series of dynodes, eventually producing a current large
enough to feed into standard electronics systems. If multiple photons
impact the PMT over a short time frame, the output will be a higher
total charge, which after calibration can be converted to a count ofphotoelectrons npe . The conversion usually cannot be perfect, due toa non-zero width of the output charge distributions for a true impact

3.1 principles of cherenkov detectors
Photocathode

AnodeElectrons

Electricalconnectors

Incident
photonFocusing
electrode

DynodeQuantum efficiency

Figure 11: Basic operating principle of the photomultiplier tube. [55]
0.2

0.1

0

300

400

500

600

700

Wave length (nm)Figure 12: Example distribution of the quantum efficiency of a PMT photocathode (used in Super-Kamiokande). [4]of a particular number of npe ; an example of PMT charge outputs isgiven in Fig. 13 for a high quality modern PMT.

Figure 13: Example raw charge outputs from a modern PMT, showing “darkrate” data (green), charge outputs during exposure to a lightsource (blue), and fitted npe distributions (red). The dark rate
refers to random outputs in the absence of a light source, comingfrom e.g. thermionic emission. Shown for illustrative purposes
only (not used in Super-Kamiokande). [76]

31

32

the super-kamiokande detector

Figure 14: 3D model of Super-Kamiokande. [66]3.2

structure of the detector

Super-Kamiokande (SK) is a water Cherenkov detector located in the
Mozumi mine of Gifu prefecture, Japan [77]. A model of the detector is shown in Fig. 14. It is located within Mt. Ikenoyama, with anaverage rock overburden of ∼ 1000 m and density of 2.7 g cm−3 , inorder to minimize the cosmic ray muon flux reaching the detector.
About 11,000 20 inch PMTs are mounted on the wall of the detector
facing inwards, covering the region known as the inner detector (ID).1
An optically-separated region on the outer side contains about 2,000
PMTs, known as the outer detector (OD).2 The main purpose of the
OD is to act as a veto for incoming cosmic ray events, however it canalso provide further information on charged particle tracks that exit
the ID, and muon tracks that enter the detector from below (which
may be identified as originating from a neutrino interaction).
Both ID and OD regions are filled with ultra-pure water, approximately 50 kton total, and the region within a boundary 2 m inside
the inner wall is considered as the fiducial volume (FV), which contains 22.5 kton of water. This region is defined so that there is an extra
buffer against low-energy radioactivities in the surrounding rock andentering cosmic ray particles, which may be falsely reconstructed as
contained events, and to exclude tracks passing close by or throughthe ID PMTs, which are difficult to reconstruct accurately.
The continuous purification of the water is necessary in order to remove radioactive isotopes which may act as a background (especiallyfor solar neutrino measurements, which have energies Eν within tensof MeV), and to maintain the longest possible light attenuation length.
1 The ID uses 20-inch diameter PMTs developed by Hamamatsu Photonics [78].
2 The OD also uses Hamamatsu PMTs, of a different and smaller type, originally takenfrom the IMB experiment and later partially replaced with a new model [77].

3.2 structure of the detector

Figure 15: A three dimensional projection of a multi-ring data event. The
color scale represents time over an 80 ns period, and charge isrepresented by the size of the square placed at each PMT position. [66]A statistic commonly used inside the collaboration is the “light left after 15 m”, which for Cherenkov wavelengths is kept at the 90% level.
Above the OD at the very top of the tank, there is a cushion of air
between the water and the ceiling, which is also filtered in order toreduce the concentration of radon isotopes that seep into the mine airfrom the rock (despite the rock surfaces being coated with a protective
substance). As PMTs depend on an electric potential to accelerate andcollect electrons, it is clear that a magnetic field can affect the collection efficiency; in fact, even an 10 mG field would alter the efficiency
by approximately 1% [79]. Thus the geomagnetic field must be compensated for to reduce such an effect, which is done by Helmholtz
coils placed around the tank.
3.2.1

Detector phases

SK has four experimental phases so far. These are designated as SKI (1996-2001), SK-II (2002-2005), SK-III (2006-2008), and SK-IV (2008present). The major detector changes that distinguish these periods
are as follows.
• SK-I: The initial period, ended when an implosion accident destroyed about half of the ID PMTs.

33

34

the super-kamiokande detector• SK-II: The remaining PMTs were rearranged for even coverage(a reduction in total photo-sensitive surface coverage from 40%
to 19%), and protective PMT shieldings were introduced.
• SK-III: Coverage was restored to the SK-I level with newly produced PMTs.
• SK-IV: An improved electronics system was installed, allowing
for continuous data recording and thus better reconstruction ofsub-events [80].
3.2.2

Monitoring systems

The SK detector is ideally in operation 24 hours per day, and several
automated and manual processes regularly check the status and condition of the detector. At least one physicist is on duty at all times, ineither the mine itself or the nearby campus. Physical properties suchas the electronics temperature, water level and temperature, highvoltage power supply levels, and so on, are automatically monitored,
as is the data acquisition and processing stream. Any problems areautomatically reported to the shift physicist, who also performs additional tasks such as visually monitoring event triggers in real time,and checking for any problems such as variations in the trigger rateor strange behaviour from individual PMTs.
On a less immediate basis, daily checks are performed consideringvarious physics parameters obtained during the data reduction (to be
described later, in Sec. 4.2 and onwards), which generally check for
unusual rates of certain classes of data, unusual PMT charge distributions, and so on. On a weekly basis, events that reach the final stagesof the reduction (that is, events classified or nearly classified as neutrino events) are manually scanned and investigated, to ensure thatthe physics abilities of the detector generally remain constant.
3.3

electronics and trigger system

Here we discuss both the ID and OD electronics and trigger system.
The OD has a similar approach but generally different hardware to
the ID, due to the historical fact that it was initially funded and maintained by the American members of the SK collaboration.
3.3.1

SK-I to SK-III

The SK-I to SK-III phases used custom built analogue timing modules(ATMs), where each ATM measures the output of 12 PMTs, and converts the analogue charge and timing information to a digital signal.
Each PMT channel registers a “hit” if the charge passes a backgroundnoise threshold equivalent to npe > 0.25 (where the exact meaning of

3.3 electronics and trigger systemnpe for the SK PMTs will be explained in Sec. 3.4), at which point thetime is digitized, and the charge is integrated for the following 400 ns
period and then digitized. The digital information is then sent to acentral electronics system, and while this is happening the channel
is free to be triggered again if necessary.3 Various “global triggers”
are defined, which are issued depending on the number of hit PMTs
in a 200 ns time window (which is the approximately the time for
Cherenkov light to traverse the detector across the longest possiblepath). If a global trigger is issued, the integrated charge and time ofevery PMT hit is saved for a period of ±1.3 µs around the trigger.
The global triggers initially defined at the beginning of SK-I were
the high energy (HE), low energy (LE), and from May 1997 the super low energy (SLE) triggers, which corresponded to requiring 31,
29, and 13 hits respectively,4 and had trigger rates of approximately
5 Hz, 11 Hz, and 110 Hz. The SLE trigger is defined in order to record
solar neutrino events, which have energies > 10 MeV, as 13 hits corresponds roughly to observation of a 4.6 MeV electron (assuming 50%
of true hits were detected). From the high frequency it can be seenthat the SLE trigger requires extra computational power, and would
be heavily dominated by radioactive background events. Work has
continued throughout the operation of SK to improve the computer
processing techniques, and reduce the background by improving water and air purification techniques, which has allowed the SLE energy
threshold to be lowered – the current threshold corresponds to observation of a 3.5 MeV electron, and further tests to lower the threshold
are ongoing [81].
The OD trigger operated on the same principle, requiring 19 hits
in 200 ns, where again a hit is defined as npe > 0.25 in the OD PMTs.
In the OD case however, data is saved for a longer time, from −10 µs
to +6 µs, in order to catch external events such as stopping cosmic
ray muons that later produce a Michel electron. An ID trigger will
instantly cause an OD trigger, however an OD trigger will cause an
ID trigger only after a 100 ns delay (assuming an ID trigger has notoccurred naturally in the meantime), which may avoid cutting off
information at the end of the ID event.
3.3.2

SK-IV upgrade

The SK-IV upgrade replaced both the ID ATM and equivalent OD
boards with new custom built components known as QBEEs5 , anda new central electronics system was installed. The QBEEs take 24
3 This is achieved by having a pair of converter systems for each channel, such thatthe secondary system can take over while the primary system is discharging.
4 After subtraction of an average “dark rate”, or noise background, of ∼ 18 hits per
200 ns.
5 Standing for QTC (charge and time converter) Based Electronics with Ethernet, andare described in [82].

35

36

the super-kamiokande detector

trigger namehit thresholdrecording period [µs]OD

22 (OD)SLE

34− 5 → + 35

LE

47

HE

50

SHE

58

AFT

–

− 0.5 → + 1− 5 → + 35

− 5 → + 35

− 5 → + 35

+ 35 → + 535

Table 4: The current trigger types used in SK-IV. “Hit threshold” represents
the number of hits required in a 200 ns time window (with no background subtraction). “Recording period” is given relative to the trigger time. The AFT trigger occurs if an SHE trigger has completed
without an OD trigger being detected.

inputs each, and have improved charge and timing resolution compared to the ATM boards [79], however the main improvement is
the ability to quickly digitize each hit and have them individually reported to a central software-based customizable trigger. In contrast
to the inbuilt hardware trigger of SK-I to SK-III, this allows essentially constant monitoring, and when required software recording for
longer periods of time. Dead time after initiation of a hardware trigger is also eliminated, so Michel electrons from muon decay can bereconstructed more often.
Different time periods may also be recorded for each type of trigger, as shown in Tab. 4.6 In particular, low energy events are generally
short single-particle events and do not need a large time range, and so
the data output may be reduced using a short LE trigger, and a new
special high energy (SHE) trigger is also introduced. If a SHE trigger
is activated and there is no OD trigger, which is common in contained
neutrino events, a further after-trigger (AFT) is issued which records
a long time period after the initial event. This is intended to monitor for the γ-ray emission that occurs during neutron capture, whichallows for an improvement in the ν/ν̄ separation [83, 84].7 Other parameters, such as the npe threshold and PMT charge integration time,
were left unchanged to maintain consistency with the SK-I to SK-III
period.
3.3.2.1 Supernova triggers
The SK-IV electronics can process up to around 6 × 105 events eachsecond. This is generally enough, even in the case of a supernova
neutrino burst within our galaxy, and such an event is monitored for
6 in fact, data is recorded in fixed-time blocks such that the recorded data time may beslightly longer than the time windows shown in the table; the recorded data outside
the period are however marked as “out of gate” and are generally not used.
7 As anti-neutrino events are more likely to produce neutrons in the final state.

3.4 calibration

by a dedicated process. Candidate events are reported in real-timeon the SuperNova Early Warning System (SNEWS, [85]). Supernovae
within ∼ 7 kpc however, for which there are some possible candidatestars [86], would cause enough neutrino events to overflow the system, and so a dedicated system to record such an event is currently
being designed [76].
3.3.3High voltage supply

Each ID PMT requires a high voltage (HV) power supplied, to generate the potential difference over which photoelectrons are accelerated;
this was provided by CAEN [87] modules until late 2013, when a new
system by iSeg [88] was installed, due to the ageing of the original
system. The HV supplies are physically separated into boards, crates,
and huts, where in general 24 PMTs are connected to each board, 10
boards are in a crate, and 12 crates are contained within one climatecontrolled electronics hut, of which there are four placed on top of the
SK detector. The PMT connections to each board generally matches
the same set of PMTs connected to a given QBEE (while not particularly interesting from a physics point of view, this structure must be
considered when processing data, in the case of electronics downtime
or noise events).
3.4
3.4.1

calibration
Calibrations of basic detector properties

Several types of continuous calibration are performed on the detector, to ensure reliable operation even under changes in the water
quality, hardware effectiveness, and so on, over time. Both ID and
OD PMTs need extensive calibrations in order to understand physically the charge and timing distributions measured by the electronics.
Low energy events such as solar neutrinos generally produce only
a few PMT hits with a single photoelectron each, for which an accurate knowledge of each PMT’s quantum efficiency distribution isparamount, whereas high energy events tend to have multiple photoelectrons per PMT, and the conversion between output charge andnumber of induced photoelectrons npe must be well-understood (thisis often referred to as the gain factor of each PMT).
3.4.1.1 ID PMT gain calibration
An example distribution of the charge outputs for a range of npe wasgiven in Fig. 13, and it is important to note that this kind of distribution can vary on a per-PMT basis. Unfortunately, the electronics usedin SK-I→III did not allow discrimination of these distributions on a

37

38

the super-kamiokande detector

per-PMT basis. As such it was necessary to calibrate the detector in
the way described below.
The first step in the calibration, most recently performed between
SK-II and SK-III, was to determine the correct HV value for 420 reference PMTs. This was done by placing each reference PMT inside alight-proof box with a “scintillator ball” designed to diffuse light uniformly, with an optical input from a wavelength-filtered Xenon flash
tube; such a setup allowed an extremely consistent light intensity, andthe HV inputs were tuned such that the output charge was the samefor each PMT. These reference PMTs were distributed throughout thedetector, and the scintillator ball was positioned in the center of thetank. The HV values of the non-reference PMTs were tuned such that
their output matched those of the reference PMTs positioned withan equivalent geometrical relationship to the scintillator ball (thismethod was used as it was previously found difficult to predict the
number of npe expected at a given position mathematically, considering all water quality effects, surface reflections and so on).
After the above procedure, the charge outputs at a particular lightintensity and frequency are well-calibrated, however the factor that
determines these outputs for each PMT is a combination of both the
QE and gain factors. To extract the gain factors, we use a dye laser
with variable intensity, injected into a diffuser ball with variable position inside the SK tank. Both high-intensity events (with npe  1
for every PMT) and low-intensity events (where only a few PMTs arehit) are generated. After many high-intensity flash events, the total
charge qi output by a PMT i is assumed to be given byqi = Ii × f i (ϑ ) × Qi × Gi ,(34)

where Ii is the totaled intensity of the arriving light, f i (ϑ ) is a function
accounting for angular acceptance differences, Qi is the QE, and Gi
is the gain factor of each PMT. For a low-intensity event however,
PMTs are generally hit by a single photon, and the true npe will be 0
or 1 with a probability depending on the QE. The output charge for
npe = 1 should be noticeably above the threshold independently of
the gain calibration; that is to say, for a large number of low-intensity
flashes the average number of hits ni is given byni = Ii0 × f i (ϑ ) × Qi ,(35)

where Ii0 is the totaled intensity of the arriving light, in the lowintensity case. Comparing the above two equations, we findGi =Ii0 qi
,
Ii ni

(36)

3.4 calibration

Figure 16: The single photoelectron distribution at SK, using “nickel source”
data. The green part of the curve was measured by specially lowering the charge threshold of the ATM modules (see Sec. 3.3), andthe red part of the curve is a simple extrapolation. [79]and assuming Ii0 /Ii is roughly constant for each PMT measurement8
we may define the relative gain factor
qi D q E
÷
,
(37)
gi =
ni
n
where the angular brackets denote an average over all PMTs. The
relative gain factors gi are thus the fractional difference in gain fromthe average, and are distributed around unity, with their 1σ width
found to be approximately 7%.
Finally, we can use these relative gain distributions gi to uncovera more accurate measurement of the average absolute gain h Gi i asfollows. A “nickel source”, which is a radioactive source inside of anickel ball that causes γ-ray emission after neutron capture, is used
to generate a large number of events, > 99% of which are events withsingle photoelectron emission. The gain corrections 1/gi are applied
for each PMT, then the average single photoelectron distribution forevery PMT in the ID can be plotted, and is shown in Fig. 16. The
fitted value of the single photoelectron peak (the second peak in the
figure) then gives the absolute value of the average gain conversion
factor, from which individual PMT gains can be measured.
From the above calibrations, it is estimated that the gains and quantum efficiencies of each PMT are known accurately at the 1% level;
while this does not directly translate to an equivalent accuracy in themeasured light intensities, it allows an accurate Monte Carlo simulation of the detector which shows that we can perform accurate energy
reconstructions for contained events from a few MeV up to a few GeV(to be discussed in Sec. 4).

8 Which should be true as the light source remains in the same position during eachi, and the same power settings are used for the high and low intensity flashes forevery i.

39

the super-kamiokande detector

T (ns)

40

3102

0 1 2 3 4 5 6 7 8 9 10
1230

10

Q (pC)

1.0
0.9

1220

0.8
0.7

1210

0.6
0.5

1200

0.4
1190

0.3
0.2

1180

0.1
1170
0

20

40

60

80

100

120

140

160

180

0.0

Bin Number

Figure 17: Example time vs. charge distribution map (TQ map) for an SK
ID PMT. The upper x-axis showing the actual charge is a combination of linear and log scales, the black line represents a seriesof fitted polynomial functions, and the color scale is arbitrarily
normalized. [79]3.4.1.2

ID PMT timing calibration

The arrival time of the PMT signals must also be calibrated, considering small differences in the physical construction of each PMT, thelength of cabling for each PMT, and also the fact that the PMT trigger
activation time is itself a function of the total charge; the charge integral will pass the threshold more quickly for a larger current, whichis generally correlated with a larger total charge.
The timing calibration again uses a diffuser ball in the center of
the tank, of a type that introduces a minimum time spread during
diffusion, with light supplied by a nitrogen laser with a short pulse
width of ∼ 0.4 ns and a variable intensity. The laser output is directly
monitored by a fast-response PMT with a time sensitivity of better
than 1 ns. Using this system the trigger time of each PMT is recorded
as a function of the total charge, as shown in Fig. 17, and a series ofpolynomial functions are fitted. These functions are used to correct
for the variations in timing of each PMT.
After correction, the timing resolution was measured to be on average 2.1 ns at the single photoelectron charge level, improving to 0.5 ns
at the ∼ 100 photoelectron level. This can be roughly seen to be a satisfactory level, if one considers that Cherenkov light takes on average∼ 100 ns to cross the tank.
3.4.1.3

ID water properties

The optical properties of the water are measured using laser light
injection from various positions in the detector, and at several wavelengths in the λ = 337 → 473 nm range, which contains the blue
part of the visual spectrum where Cherenkov light emission is concentrated. The PMTs directly hit by the laser can be used to measure
the amount of light absorbed across the laser’s path, while the other3.4 calibration

PMTs may measure the amount of light reflected. The intensity oflight remaining in the path of the laser I ( x, λ) at a distance x is expected to follow a simple exponential lawx
I ( x, λ) = I0 exp −(38)
αabs (λ) + αsym (λ) + αasym (λ)
where αabs represents the wavelength-dependent absorption coefficient, and αsym and αasym are the “symmetric” and “asymmetric” scattering coefficients. The “symmetric” coefficient represents Rayleighscattering and symmetric Mie scattering, and is not actually symmetric but by approximation takes the form 1 + cos2 ϑ, where ϑ is theangle between the laser direction and the scattered light. The “asymmetric” coefficient represents forward Mie scattering, and takes thesimple form cos ϑ for |ϑ | < 90°. The parameters are estimated regularly by an automated system, by fitting the laser flash ID events with
PMT charge distributions obtained from a laser event Monte Carlothat was generated with various values of the parameters.
3.4.1.4 OD calibrations
The OD must also undergo PMT calibrations, although the requirements for charge and timing accuracy are not as stringent as the ID,
as the OD data does not generally play as large a role in the fine reconstruction of physics events.9 In SK-IV, charge calibration is constantly
done via comparisons of the dark rate with cosmic ray muon data,
where selecting early hits from muon events generally gives single
photoelectron triggers, from which the average npe = 1 charge is directly taken. There is also a fiber-optic system with outputs at various
parts of the ID, and the nitrogen laser (used for the ID PMT timing
calibration) is also used to provide a set of single photoelectron triggers and give a secondary cross-check. The gain is found to be in therange 1 → 6 pC per npe , and the accuracy of the measured charge
was estimated as between 10% and 20%, depending on the particular
PMT. The timing offsets are estimated considering the cable lengthand the laser events, and also by cross-checking timing offsets withthe cosmic ray muon data, and after correction the resolution was estimated as better than 10 ns. The water properties of the OD are notmeasured by direct light injection, but their effects are simulated ina Monte Carlo and tuned to give the best agreement to the physics
data.

9 For events contained inside the detector, the OD is essentially a veto counter. Evenconsidering uncontained events used in the analysis, such events cannot be accurately reconstructed even in principle, since information is lost once the charged
particles are completely outside of the detector.

41

42

the super-kamiokande detector

3.4.1.5

Other calibrations

There are several other calibration procedures that provide finer adjustments to the electronics and detector model, however a detailed
discussion is left to the published papers [77, 79]. These include accurate geometric measurements, water convection measurements, fine
QE and gain measurements by comparison with Monte Carlo simulations, testing non-linearity of charge10 using a high intensity lightsource, and measurements of the light reflection properties of the various materials used in the tank.
3.4.2

Calibrations using physics events

In principle, from the detector calibrations listed above, we can reconstruct the total intensities of the Cherenkov light observed. Then,
while considering the geometry of the detector and physics predictions from Monte Carlo simulation (see Sec. 4.5), we can reconstruct
the physics parameters of particle tracks. In order to test the accuracyof these reconstructions (themselves explained in detail in Sec. 4.2.2),
and do final calibrations to the reconstructed energy variables, we use
several physical processes with known parameters.11
3.4.2.1 Low energy events
A linear accelerator (or LINAC) is installed above the SK tank, which
can produce a beam of electrons of specific energy chosen across the
few MeV to tens of MeV range.12 This is complimented by a neutron
generator, which can be lowered into the detector to produce 16 N,
the decay of which is dominated by an electron and a gamma ray
with energies of a few MeV. Using these techniques, the energy resolution after reconstruction was shown to be better than 20% and thevertex resolution within 2 m on an event-by-event basis, even for theabsolute lowest energy events. For events with Eν ≈ 10 MeV, theseparameters improved to ∼ 10% and < 1 m respectively.
3.4.2.2 High energy events
For higher energies, statistical tests of cosmic ray muon and neutrinoinduced pion events are performed. As mentioned previously, theobserved energy in photoelectrons is not generally expected to correlate exactly with the energies of the observed particles, and recon10 In other words, a charge-dependent gain factor in the PMTs or electronics; this isfound to occur sometime before the saturation point (highest possible charge observed by the electronics), which is npe ≈ 300 for SK-I to SK-III and npe ≈ 1200 for
SK-IV.
11 All of these methods are repeated for each of the SK-I to SK-IV periods.
12 While the LINAC was used every few months in early stages of the SK experiment,
it was a highly manpower intensive task and was not used often in the later phasesof the experiment.

3.4 calibration

structing the physical parameters of a track requires a comparison
with the MC simulation. However, we can at least confirm that theobserved charge and reconstructed energy distributions match whatis expected by the MC for certain types of high energy events, which
gives us confidence in our simulation.
In particular, muons that stop inside of the tank decay to electrons,with an energy given by the well-known Michel spectrum, extending
to around 50 MeV. Such events are selected by requiring a stoppingmuon, identifiable by a clear entering hit cluster in the OD and nocorresponding exit cluster, followed by a comparatively low energy
second event a few microseconds later.13 The reconstructed mean energy of these selected events agrees with the mean of the predicted
spectrum at the 1% level, and is stable over time.
The cosmic ray muons themselves can also be used; after identifying the muon stopping point using the fitted Michel electron vertex, the track end point can be accurately reconstructed. Furthermore, the expected Cherenkov angle (useful for muons with energy
Eµ > 400 MeV) or the expected energy loss per unit distance (e.g.
∼ 2 MeV cm−1 for 1 GeV muons in water) are theoretically well known,
and the muon true energy can be quite accurately predicted fromthe above parameters (denoted as Epred ), and compared with the observed total charge qobs . Comparing the above parameters for data
events with the equivalent parameters simulated in MC events,
qMC
qobs
obs
÷ MC
Epred
Epred

(39)

gives a high-statistics check across a wide range of binned muon energies Epred , and is consistent with unity to within 2%.
A final check comes in the form of the decay of neutral pions, which
proceeds by π 0 → γ + γ. From events with two reconstructed electromagnetic showers fitted (discussed further in Sec. 4.2.2), representing
the two γ-rays, the invariant mass of the pion can be reconstructed
and compared with the true value of 135 MeV. The position of themass peak is found to be correct within 2%.
Considering all of the above procedures, across all SK periods andincluding a wide range of particle energies from tens of MeV up to afew GeV, we estimate the accuracy of the average reconstructed event
energies to be better than 2%, and the stability of this value over time
as better than 1%.14

13 The fact that the selected events are Michel electrons is confirmed by an accurate fitof the expected decay time of the muon.
14 The event-by-event energy resolution is discussed for various event types inSec. 4.2.2.

43

4

D ATA R E C O N S T R U C T I O N A N D M O N T E C A R L O

4.1
4.1.1

overview
Event topologies

The high energy atmospheric neutrino data events are selected andreconstructed into three main samples with different event topologies, in principle containing only neutrino induced events,1 divided
as follows. In the fully-contained (FC) sample, the vertex is inside
the fiducial volume and all visible secondary particles are contained
inside the inner detector. In the partially-contained (PC) sample thevertex is also inside this fiducial volume, but outgoing particles areallowed to exit the inner detector. PC events have longer charged particle tracks, which are therefore mostly muons induced by νµ (or ν̄µ )CC interactions. In the upward-going muon event sample (UPMU),
neutrinos interacting with the rock create muons which enter the detector from below. Down-going muons are ignored, as these are overwhelmingly produced by cosmic rays. The UPMU sample is also a
predominantly νµ (ν̄µ ) induced sample.
The detector in its current configuration has only limited power
to distinguish neutrinos from anti-neutrinos. Some separation techniques exist, and will be described and used for one sub-section of
the energy spectrum analysis. In most cases however, we discuss andmeasure only the combined ν + ν̄ fluxes, therefore further referencesto νµ or νe should be taken to imply inclusion of ν̄µ and ν̄e respectively,
unless explicitly stated.
An overview of the main samples indicating their energy ranges
and topologies is shown in Fig. 18 (anticipating some results fromlater chapters, which provide definitions for dividing each sample
into the various sub-samples).
4.1.2

Reduction and reconstruction

The recording of high-energy data events occurs based on some simple trigger criteria as defined in Sec. 3.3. The selection of physics
events requires first a removal of many background events such ascosmic-ray muons, radioactive decays in the tank, electronics noise
1 Assuming the event selections are perfect, and there are no special events such as
proton decays. To be clear, this thesis discusses the “high energy” neutrino data, anddoes not consider the separate processes that are used to select “low energy” (solar)
neutrino data, or other specialized samples.

45

data reconstruction and monte carlo

Number of Events

46

1600
1400
1200
1000
800
600
400
200

sub-GeV single-ring e-like
multi-GeV single-ring e-like
multi-ring e-like

1600
1400
1200
1000
800
600
400
200

sub-GeV single-ring µ -like
multi-GeV single-ring -µ -like
multi-ring µ -like

300
250

PC stopping
PC through-going

200
150
100
50

300
250
200
150

UPMU stopping
UPMU non-showering
UPMU showering

100
50
0 -1
10

1

10

2

10

3

10

4

10

Neutrino Energy [GeV]

Figure 18: MC neutrino true energy distributions for each sub-sample. From
top, the FC e-like, FC µ-like, PC and UPMU samples, divided intosub-samples as described in Sec. 4.2→Sec. 4.4. Pictorial detector
representations show the ID and OD sections, and event topologies.

4.2 fully contained sample

events, and so on, which is done by a series of increasingly complex
cuts and is known as the data reduction. Events passing the data reduction have various processes applied to measure the physics parameters of the events, such as the interaction vertex and particle momenta,
and is known as the data reconstruction.
Each of the three main samples (FC, PC and UPMU) have a distinct set of reduction and reconstruction processes, which will be explained in order below. In general the final sub-sample definitions
are evolving, and various SK analyses may use different definitions.
Those used in this thesis for flux analysis purposes are similar to
those used in the SK atmospheric neutrino oscillation analysis [5],with a few small changes and updates, and without separation of νand ν̄ events.
4.2
4.2.1

fully contained sample
Reduction2

The FC reduction is divided into five steps, denoted as FC1 → FC5,
where events that do not pass each step are not processed further
in order to optimize the processing time. Several reconstruction programs (or event “fitters”) are gradually applied during the reduction,
which are mentioned briefly here, and explained further in Sec. 4.2.2
on the FC reconstruction. After the final step, fiducial volume andvisible energy cuts are performed.
The reduction cuts were defined by comparing raw data with expectations from the SK atmospheric neutrino MC (Sec. 4.5), aiming
to keep at least 99% selection efficiency of true high-energy neutrino
events for each step, while reducing background to the level of  1%compared to the neutrino events. Here, MC “true high-energy events”
are defined as those with a true vertex inside the FV, with visible energy Evis > 30 MeV (which is a parameter that will be defined inSec. 4.2.2), and with less than 16 clustered hits in the OD (less than10, in the case of SK-II).
Some short notes on reading the below cuts: many are given interms of estimated number of photoelectrons (npe ), which are converted from PMT charge outputs as explained in Sec. 3.4, where tubes
that are not “hit” do not contribute to npe totals; secondly, the trigger
timing is explained in Sec. 3.3, and if cuts do not specify a time range,
they should be assumed to cover the standard range of ±1.3 µs; finally, all cuts are given here in terms of the requirements for passing
each cut.

2 The author of this thesis was in charge of the FC data reduction process, and so thissection goes into extensive details (and was thoroughly checked) for the purposes of
future reference.

47

48

data reconstruction and monte carlo

4.2.1.1

FC1

The first reduction implements very basic cuts:
1. total ID npe in a sliding 300 ns time window > 200 (for SK-II,> 100)
which allows only events with significant ID activity, cutting low energy events such as radioactive background and solar neutrinos. The
SK-II criteria is relaxed, in this and many other cuts, considering that
number of active PMTs was approximately half that of the other periods. Then,2.

a) number of OD tubes hit > 0b) and number of OD tubes hit within a time window −500 ns
to +300 ns is ≤ 50 (for SK-IV, ≤ 55)

which avoids events in the rare case of no OD information (i.e. OD
dead-time), and events with obvious OD activity, such as cosmic raymuons (entering before the trigger) and PC events (exiting after thetrigger). The criteria is slightly higher in SK-IV, due to the replacement of the OD electronics boards and changes in the PMT gains,
and the cut is tuned to reject similar muon energies as the earlier
periods.
After FC1, the event rate is reduced from ∼ 106 events per day toapproximately 2,500 events per day. The efficiency for selecting truehigh-energy neutrino events is estimated by applying the reduction
to the atmospheric neutrino MC and is found to be 100.0% for FC1.3
4.2.1.2 FC2The second reduction continues with more simple cuts:
3. highest npe for an ID PMT divided by the total ID npe < 0.5
which removes electrical noise events whereby the majority of themeasured charge comes from a single tube, then4.

a) number of OD tubes hit within a time window −500 ns to+300 ns is ≤ 25 (for SK-IV, ≤ 30)b) or the total npe of the ID is >100,000 (for SK-II, > 50,000)

which avoids events with OD activity in a stricter sense than cut 2b,
while allowing extremely energetic events in the ID, which are expected to leak some charge into the OD even if the particle is fullycontained (e.g., by electrical cross-talk between channels). The effectsof cuts 3 and 4a are illustrated graphically in Fig. 19, showing theatmospheric MC and raw data distributions.
3 The event rates and efficiencies given in this section refer to the SK-IV period, butother periods are similar.

Number of events

4.2 fully contained sample

104

rejected
103MCdata × 0.01

102

10

10

0.2

0.4

0.6

0.8

1

Number of events

Highest ID PMT charge / total ID PMTs charge
rejected
103MCdata × 0.01

102

10

0

10

20

30

40

50

Number of OD tubes hit within -500 ns to +300 ns

Figure 19: Distributions of the atmospheric MC and data before the FC2
cuts 3 (top) and 4a (bottom), where MC events are true highenergy FC neutrino events in the FV, and data represents the raw
detector output before each cut. Note the logarithmic y-axis andthe scaling factor of the data distribution. [89]After FC2, the event rate is reduced to approximately 500 events
per day, and the efficiency of FC2 to select true neutrino events isagain estimated as 100.0%.
4.2.1.3 FC3The third reduction features more complicated cuts, which are designed to pick out particular types of background events. Several ofthese cuts target cosmic-ray muons; the first cut targets “hard” muons,
which are muons with Eµ ? 1 TeV whose energy loss by processes
such as pair production and bremsstrahlung is greater than their ionization energy losses, and can thus cause many hits in a short time:
5. number of OD tubes hit within a sliding 500 ns time window< 40.
The next cut targets through-going muons, by applying a simple fitterthat can find two clusters of hits in the OD, corresponding to an entering and exiting muon with an appropriate travel time. The fitter isonly applied if the ID contains more than 231 photoelectrons spreadover 1,000 hit tubes, however once the fitter is applied events may still
pass as follows:
6.

a) through-going muon goodness of fit < 0.75b) or number of hits in both the entrance and exit OD hit
clusters < 10.

A similar fit for stopping muons is also applied, which finds an entering but not an exiting cluster, and events may pass by having

49

data reconstruction and monte carlo

200

Number of hits50

FC neutrino
ﬂasher

150
100
50
0

-200

Scanned region
0
200
400
600
Time of ID PMT hits [ns]800

Figure 20: Comparison of a nominal FC neutrino event and a nominal
flasher event, both found in SK-I data. The scanned region shows
the timing region tested by the FC3 cut 10. [90]7.

a) stopping-muon goodness of fit ≤ 0.0b) or number of hits in the entrance OD hit cluster < 10,
and for SK-I only, a further finely-tuned cut is applied as
8.

a) stopping-muon goodness of fit < 0.5b) or number of hits in the entrance OD hit cluster < 5.

The next cut targets remaining low energy events, using a simple
time-of-flight (TOF) fitter “point-fit” to obtain an event vertex basedon the timing information of each PMT and the speed of light inwater. Assuming all photons were generated at this vertex, the TOF
of each hit is subtracted, and events must pass
9. number of time-of-flight corrected hits in the ID within a sliding
50 ns time window ≥ 50 (for SK-II, ≥ 25).
The next cuts remove so-called “flasher” events, which are events
whereby electrical discharge inside a PMT causes some emission oflight, which may look like a contained event. In general the timing
distribution of hits is longer for flashers compared to real particle
track events; an example is shown in Fig. 20. For SK-II to SK-IV a cut
is thus made as
10. minimum number of hits in the ID within a sliding 100 ns time
window between +200 ns and +700 ns is < 20which disallows flasher events with a long tail in the hit timing. The
SK-I cut is slightly more complicated, as it was tuned in a tighter
sense. For SK-I only,
11.

a) minimum number of hits in the ID within a sliding 100 ns
time window between +200 ns and +700 ns is < 10b) ori. the above count is < 15
ii. and total number of hits in the ID is > 800

4.2 fully contained sample

which essentially places a tighter constraint, unless the event was particularly energetic, then a medium constraint is placed. Low-energy
flashers are also cut by a goodness-of-fit cut on the point-fit TOF fitteras
12. TOF fitter goodness-of-fit > 0.4
since for wide time distributions, a simple time-of-flight vertex fit
searching for a single point will not work well. The next event typetargeted are the “coincidence muon” events, which are due to thehigh frequency of SLE triggers, during which cosmic-ray muons sometimes enter the detector by coincidence. This requires either no late
OD activity, or an ID activity lower than most cosmic-ray muons, as
13.

a) number of OD hits between +300 ns and +800 ns is < 20b) or total ID npe count is ≤ 5, 000 (for SK-II, < 2, 500).

The final third reduction cut removes “cable hole muons”. The cabling for the PMTs enters the SK superstructure through twelve holes
in the top of the tank, and (due to the need to insulate the highvoltage power supply) the radius of the cabling bundle is quite wide
at around 1 m. Down-going muons could theoretically pass throughthe cable holes and be reconstructed as FC events; to test for such
events, four of the cable holes are covered with plastic scintillator
veto counters. In conjunction with the stopping-muon fitter, the cut
14.

a) number of veto-counter hits = 0b) or stopping-muon best fit entry-point is > 4 m from thecable holes (in both x and y directions)is applied, which ends the third reduction.
After FC3, the event rate has been significantly reduced to around50 events per day, and all events passing the third reduction have the“APFIT” software applied; this is the complex high-energy fitter andapplies the fitting processes described in Sec. 4.2.2. The efficiency of
FC3 to select true neutrino events is estimated to be 99.8%.
4.2.1.4 FC4The fourth reduction specifically targets flasher events, that were notremoved by the above cuts, by use of a pattern-matching algorithm.
Such an algorithm is useful because flashing tubes often emit lightin a specific way, and create similar hit patterns within the tank repeatedly. The algorithm divides the detector into many 4 m2 patches,sums the npe in each patch, and compares the similarity of the pattern to all other events in that SK period by the parameter r, which isdefined between two events A and B as
1 N QiA − Q A × QiB − Q B
r=
(40)
N∑
σ A σBi51

data reconstruction and monte carlo

No. of Events

52

Neutrino data
Neutrino MC

Cut
Flasher

Prob
Figure 21: Comparison of the likelihood parameter “Prob” for neutrino data,
MC, and hand-selected flasher events, showing the cut value. [76]where N is the number of patches, Qi is the summed npe in eachpatch i, and σ is the RMS of the Qi .
A further parameter dKS is also introduced, comparing the time distribution of the charge by a Kolmogorov-Smirnov test,4 and a thirdparameter is defined depending on whether or not the PMT withthe highest npe (which is often the flashing tube) matches between
events. These parameters are combined into a single likelihood parameter “Prob”, and if two events score higher than a cut value (determined based on a comparison of Monte Carlo neutrinos against
hand-selected real flasher events), they are both removed as flasher
events, as shown in Fig. 21.
The fourth reduction removes on average only a few events per day,with a neutrino efficiency of 99.6%.
4.2.1.5

FC5

The fifth reduction again targets several specific background types
with tailored cuts, generally targeting muons in a stricter sense than
the third reduction, along with other cuts on particular types of electronic noise events and calibration-related events. The first cut targets“invisible muons”, which are cosmic-ray muons below the Cherenkov
threshold that are undetectable, but stop in the tank and decay via theMichel decay. This can create an electron above its Cherenkov threshold originating inside the fiducial volume, which could be mistaken
as an FC event. These events are removed by requiring
15.

a) total ID npe > 1,000 (for SK-II, > 500)b) or number of OD tubes hit within a sliding 200 ns time
window from -9000 ns to -200 ns is < 5

4 The “K-S” test uses as a statistic the largest difference in the integrated charge distributions over time, and is essentially sensitive to shifts in position or shape of thetime distributions of hit PMTs.

4.2 fully contained samplec) or the above count plus the number of OD tubes hit within-200 ns to +300 ns is < 10 (applied only if the distance between the hit clusters in the two time periods is within
0.5 m)
where the first cut allows all events above the maximum energy of theMichel decay,5 and the next two cuts allow events below this energy
only if they have insignificant early OD activity.6 The second fifthreduction cut targets again coincidence muons, and requires
16.

a) the number of OD tubes hit within a sliding 200 ns time
window from +300 ns to +2100 ns is < 20b) or total ID npe > 300 (for SK-II, > 150)

which scans over a longer time range than cut 13a, but allows events
if they were not caused by SLE triggers (and thus statistically unlikely to produce coincidence events). Several cuts to remove stopping muons, in a tighter sense than previous cuts, are then performed.
A clustering algorithm is run on the OD data, identifying OD tubes
that are hit close together, and the following cut is performed for SK-I
only:
17. number of OD PMTs found in a cluster < 10.
The third reduction cut 8a is then applied on SK-II to SK-IV data at
this stage. Furthermore, a tighter cut on cable-hole-muons focusing
on the cable holes which do not have veto counters is performed for
SK-IV only:
18.

a) stopping muon fitter goodness-of-fit < 0.4b) or total ID npe < 4000c) or the fitted muon’s directional unit vector has cos(z) >−0.6d) or the distance between the stopping muon’s best-fit entrypoint and the closest cable hole is >2.5 m
and a cut based on APFIT fitted muon rings is introduced, where
19. number of OD tubes hit within 8 m from the APFIT fitted entry point within a sliding 200 ns time window from −500 ns to+300 ns is < 4.
The next cut is another flasher cut, targeting wide hit-timing distributions in a stricter sense than cuts 10 and 11. However, this cut alsouses a more advanced TOF-fitter, and allows events if the timing distribution corresponds well to the fitted particle tracks:
5 The parent muon has almost certainly come to rest in the lab frame.
6 Even “invisible muons” would be expected to have some OD activity, as if they could
reach the fiducial volume, they would have had to have been above the Cherenkov
threshold during passage through the OD.

53

54

data reconstruction and monte carlo

20.

a) the minimum number of ID tubes hit within a sliding 100 ns
time window from +200 ns to +700 ns is < 6b) or TOF fitter goodness-of-fit > 0.4 ,then and a stricter criteria for the TOF fitter is applied on events witha wide hit-timing distribution (applied for SK-II to SK-IV only):
21.

a) the minimum number of ID tubes hit within a sliding 100 ns
time window from +200 ns to +700 ns is ≥ 6b) and TOF fitter goodness-of-fit > 0.3.

The rest of the cuts remove fake events occurring due to various particularities of the SK electronics (and so are less interesting from aphysics perspective, but nonetheless important). The first such cut
targets electronic noise events coming from signal leakage in the highvoltage systems or electronics boards, which tend to result in a large
number of fake tube hits at low npe values. We require
22.

a) number of ID tubes hit with less than a single photoelectron is < 250 (for SK-II, < 125)b) or the above count minus the number of ID tubes hit withgreater than a single photoelectron is < 100 (for SK-II, <50).

The next cut removes events where any of the electronics huts (eachaccounting for one quarter of the detector) were temporarily off forthat event.7 A dead hut can be assumed if all tubes associated withthe hut report no hits.
23. Number of dead electronics huts = 0.
The next cut targets events where the OD timing is out of sync with
the ID timing, which was a problem identified in the early stages ofthe experiment in some high-energy events, and is applied for SK-I to
SK-III only. It can be identified by large early-timing clusters, or thetrigger counting an unusual number of sub-events:
24.

a) total ID npe < 1000 (for SK-II, < 500)b) or
i. number of OD tubes hit within a sliding 200 ns time
window from −9000 ns to −200 ns is < 20

ii. or number of sub events ! = 7.

The final two cuts remove laser and xenon calibration events (seeSec. 3.4). Although these events should be tagged and removed at calibration time, some events remain in the data, and are thus removed
based on signals from dedicated PMTs directly monitoring the calibration sources:
7 Monte Carlo studies showed that smaller problems, such as deadtime for individualboards, do not strongly compromise the ability of the detector to correctly reconstruct events.

4.2 fully contained sample

25. laser and Xenon monitor PMT hits = 0
which ends the fifth reduction.
Again, only a few events per day are removed by the fifth reduction,and the estimated neutrino efficiency is 99.9%.
4.2.1.6

Final Reduction

Finally, although not directly applied in the reduction software, thefollowing cuts are required for events to be classified into the highenergy dataset:
26. the distance from the neutrino interaction best-fit vertex and thewall is > 2 m,
27. the number of OD PMTs found by a clustering algorithm is < 16(for SK-II, < 10),
28. the visible energy Evis > 30 MeV,
where the fiducial volume cut 26 and the visible energy cut 28 (where
Evis will be defined in Sec. 4.2.2, and corresponds roughly to the energy scale of the event) are expected to remove the majority of remaining low-energy background events, such as radioactivities.
After the above cuts, the event rate is approximately 8 events perday. The neutrino efficiency of these cuts is estimated to be 99.2%.
4.2.1.7 Summary
After all of the above steps, the final FC dataset has daily event rates
as shown in Tab. 5, for data and MC for SK-I to SK-IV. We see that thepredicted MC rates are slightly lower than the data rates – howeverthese numbers are generated using a simplified two-flavour oscillation scheme, and do not take any systematic errors into account. In
the final oscillation analysis, the agreement is marginally improved
after applying three-flavour oscillation, and the data rates are within
the estimated systematic error range. [5]The efficiency in the table is estimated (as described above) using
the atmospheric MC, by the surviving percentage of FC high-energy
events inside the FV. The uncertainties on these numbers were estimated by a study [89] where each cut parameter was varied slightly;
these uncertainties are later considered as a systematic error. The
background, on the other hand, is estimated by direct eye-scanningof all events passing FC5, during which events such as flashers andcosmic-ray muons are tagged (but not removed from the analysis, inorder to avoid a bias in the data and MC selection processes). In thetable, we show the percentage of such tagged events in the dataset

55

56

data reconstruction and monte carlo

event rate [day − 1 ]
sk-i
sk-ii
sk-iii
sk-iv

data

mc

8.26 ± 0.06

7.75

8.24 ± 0.10

7.73

8.12 ± 0.06

7.94

8.41 ± 0.13

7.91

efficiency [%]background [%]97.75 ± 0.20–

98.31 ± 0.20

0.30

98.47 ± 0.30

0.02

97.96 ± 0.80

0.39

Table 5: The final daily event rates of events passing the FC reduction, withstatistical error (negligible for MC). The meanings of “efficiency”
and “background” are explained in the text.

after the final reduction.8 No eye-scanning data is available for SK-I,
but the background rate is expected to be similar to the other periods.
4.2.2

Reconstruction

The job of the FC reconstruction is to estimate the neutrino’s flavour
and momentum as closely as possible. Fully-contained events canhowever range from simple single-lepton-track events, to complicatedmulti-track hadronic events. As the accuracy of the neutrino reconstruction depends on the type of event, sub-samples are defined whichgroup together similar events. For example, single-ring events haverelatively higher purity in neutrino flavor, while multi-ring events
tend to cover a higher neutrino energy range.
This section will describe the various processes used to assign event
properties, and then the division of events into final sub-samples.
While the important principles of the fitting procedures are discussed,
more extensive technical details can be found in [91]. Furthermore,while various final state products of neutrino interactions are mentioned here, a detailed discussion of the neutrino interactions which
create them will be given later in Sec. 4.5.1.
4.2.2.1

Vertex fitting

The neutrino interaction vertex is reconstructed in a multiple step process: firstly, a “point-fit” time-of-flight algorithm roughly estimates
the vertex position, then the edge of the primary Cherenkov ring emanating from that vertex is fitted, after which corrections due to finite
particle track length are precisely fitted and the vertex is iteratively
improved. Thus, the vertex fitting is closely interrelated with the primary ring fitting.
The point-fit algorithm assumes that Cherenkov light is emitted
from a single point inside the detector at an unknown time. Taking
8 Depending on the particular reconstruction and selection procedure of a given analysis, sometimes more conservative systematic errors on the backgrounds are defined.

npe

4.2 fully contained sample

150
100
50

d2npe / dθ2

0
10

0

20

0

20

40

60

80

40

60

80

0

-10

ô

Cherenkov angle θ [degrees]

Figure 22: An example distribution of observed charge depending on the
Cherenkov angle θ, and its second derivative, for a given ring
position and direction. The dotted line indicates the fitted edgeof the ring. [66]an initial test point x, the time-of-flight between that point and each
hit PMT can be calculated, and by subtracting the expected time forlight to travel those distances we define a set of “residual timings”
ti (x) for each hit PMT i. If the observed light in the event was indeed
generated at that point, the ti (x) should be narrowly distributed at theoffset t0 (the interaction time), with some Gaussian smearing given by
the PMT timing resolution. The Gaussian parameterG p = ∑ e−(ti (x)−t0 )

2

(41)

i

is largest when the distribution is narrow, so maximising G simultaneously for the point x and time t0 gives the “point-fit vertex”.
The second step is to identify the primary Cherenkov ring, whichshould be the highest charge ring in a multi-ring event. The directionof the charged particles leaving the vertex is established in a general
sense by defining a vector from the point-fit vertex to each hit PMT,
weighted by the observed charge at that PMT, and summing all suchvectors. Using this vector as an ansatz, we can plot the charge distribution npe (θ ) as a function of the Cherenkov angle to the ansatz
vector θ, making careful corrections for the geometry of the detector,
the light attenuation in water, and the angular acceptance functionof the PMTs. An example distribution is shown in Fig. 22. To findthe outer edge of the ring θedge , which is to say the estimated initial
Cherenkov angle, we find the first turning point
d2 npe (θ )
=0
dθ 2

(42)

57

58

data reconstruction and monte carlothat occurs after the peak value of npe (θ ), as illustrated in Fig. 22. A
goodness-of-fit parameter Q(θedge ) for the fitted ring is defined as

R θedgenpe (θ )dθQ(θedge ) = 0
×sin θedge



!
!2

(θedge − θexp )2
dnpe (θ )× exp −dθ2σ2θ =θedge(43)

where θexp is the theoretically expected Cherenkov angle based on the
total charge in the cone, and σ is the expected resolution of the edge
search (based on the distance between ID PMTs). This parameter isessentially designed to prefer fitted cones containing a large numberof photoelectrons (first term), with a sharp edge (second term), anda fitted edge near to the expected value (third term). To penalize unphysical Cherenkov cones, in the case of θedge > 43° the second termis removed. The parameter Q(θedge ) is then maximised by shifting
θedge , and also allowing shifts in the ansatz vector, which may improve the fitting. The PMTs intersecting the edge of the best-fit ring
are tagged for use in the next step.
Final adjustments are performed by the fitter TDCFit, which considers a finite track length for the primary Cherenkov particle, and finds
the best-fit start and end points. The track length is initially estimatedby counting the observed npe and assuming the energy deposited perunit length is dE/dx = 3 MeV cm−1 (which is at least a reasonablevalue for electrons or muons of some energies, and was empiricallyfound to result in good fits for many cases without getting stuck in local minima). New timing residuals are calculated based on emissionof photons at the same Cherenkov angle along the particle track, andfurthermore the PMTs outside of the tagged ring edge are also considered, taking into account the scattering parameters estimated during
water quality checks. The track length, vertex position, and ring direction are then iteratively tuned until a local minimum is found. The
performance of the vertex fitting procedure is tested with the atmospheric MC events, and the resolution9 of the vertex fit at this stage
is found to be better than 0.7 m in position, and a few degrees in
direction, even for complicated multi-ring events.
Finally, in events where secondary rings are not found, and theflavour of the lepton is established (as to be described in the next
sections) a further precise ring fitting procedure called “MS fit” isapplied. Various parameters, such as dE/dx, can be better predicted
depending on the lepton type, and a more precise fitting is possible. After the improvements by MS fit, the resolution for single-ring
events is found to be better than ∼ 0.3 m in position and ∼ 3° indirection.

9 Defined as the distance from the true vertex within which 68% of fitted vertices arefound.

4.2 fully contained sample(a) Image space59(b) Hough space

Figure 23: Illustration of the principle of the circular Hough transform,
where blue represents image intensity, and black represents theHough parameter intensity. The recorded image is shown in (a),and the transformation to circular Hough space in (b), where thelocation of the darkest point is the estimated centre of the fixedwidth circle (the image itself is overlaid in the Hough space, forillustrative purposes).

4.2.2.2 Ring counting
After the vertex fitting, it is expected that the primary (i.e. brightest)
Cherenkov ring has been well fitted; the next step is to search forpossible secondary rings.
A common class of method of searching for a given pattern or
shape in an image is known as the Hough transform; by moving
to a new parameter space based on the original image, for example
intensity or color as a function of some different co-ordinate system,features of the image may be extracted. The circular case is particularly simple, as illustrated in Fig. 23, where the Hough space can bethought of as a convolution of the original image with a circular function with the expected radius, and has the same co-ordinate spaceas the original image. The dark points in the Hough space are likely
locations for the center of a circle.
The SK case follows essentially the same principle, with of course
corrections for the geometry of the detector and the behavior of Cherenkov
light in water. The observed npe fitted as belonging to the primaryring are also subtracted, to search for new rings only, originatingfrom the previously fitted vertex. The initial convolution pattern usedis the expected Cherenkov distribution for a 500 MeV electron. An example Hough search at SK is shown in Fig. 24, where the peaks represent likely Cherenkov rings. Peaks found close to the primary ring
however (within 15°) are discarded as remnants of the primary ring,which may have come from e.g. a kink in the particle track.
Once candidates are identified from the Hough peaks, they are eachtested by a likelihood method to decide if they are true rings or not.
The likelihood L( N ) for testing each new peak (in addition to theN − 1 peaks already fitted) essentially tests the observed charge distri-60

data reconstruction and monte carlo

600
400
200
0
150

Θ ( 100degr
e e 50
s)

00

100

200

Φ (d e300

g re e s)

Figure 24: An example Hough-space for a ring search in SK, showing thevalue of a Hough parameter depending on the zenith Θ andazimuth φ angles from the fitted vertex, where two rings areclearly found (in this figure, the npe of the primary ring is not
subtracted). [66]expbution nobs
pe,i for each PMT i against the expected distribution npe,i (η ),which is a distribution that can be varied by changing the estimatedphysics parameters η (such as particle type and energy) of the new
Nth ring. The likelihood is thus writtenexp
L( N ) = ∑ ln P nobs
(44)
pe,i | npe,i (η )iwhere the probability function P comparing the charge of each PMT
is defined as a Poisson distribution convoluted with the single-photoelectron
exp
exp
distribution for npe,i (η ) < 20, and simplified to a Gaussian for npe,i (η ) ≥exp

20. The maximal L( N ) is determined by finding the best-fit npe,i (η ).
Due to the fact that extra peaks must represent at least some energy
measured by the PMTs, there is always some charge that must be
accounted for, which means the momentum physics parameter inη has some minimum value. If the particle track is well fitted thenL( N ) > L( N − 1), however due to the minimum momentum requirement a bad fitting will cause a decrease in L( N ), so we may have
L ( N ) < L ( N − 1).
The final choice of whether or not a ring is fitted as true is not
actually directly based on the L( N ) − L( N − 1) parameter, but by amore detailed processing of the two charge distributions obtained in
the L( N ) and L( N − 1) cases, summarized in a final obtained parameter denoted L0 .10 Extra rings are searched for, until none canbe found that increase the best-fit likelihood L0 . The final parameter
L0 ( N ) − L0 ( N = 1), which denotes the single-ring vs. multi-ring likelihood, is shown for SK-IV in Fig. 25, divided into two samples basedon the energy. It can be seen that true CCQE events, in which the initial neutrino interactions are free from complicated hadronic effects,

10 L0 uses a weighted L( N ) − L( N − 1) combined with extra shape fitting of the charge
distributions and timing corrections; details in [91].

4.2 fully contained sample

Super Kamiokande IV 1987.6 days : Monitoring

Super Kamiokande IV 1987.6 days : Monitoring

700
single ring

multi ring

600

600

single ring

multi ring

CCQE

CCQE

CC Single π

500

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500

CC DISCC DIS

CC Coherent π

CC Coherent π

NC

NC

400

400
300

300

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15

Ring counting likelihood sub-GeV(FC), stack

(a)

0
-15

-10

-5

0

5

10

15

Ring counting likelihood multi-GeV(FC), stack(b)

Figure 25: Ring-counting likelihood separations for the sub-GeV and multiGeV samples for SK-IV, showing the number of MC (normalized
stack plot by true interaction type) and the data (points with statistical error bars) event counts. In this and other figures in thissection, the MC is shown without systematic error pulls and using a simple two-flavour oscillation model.

(a) e-like(b) µ-likeFigure 26: An example comparison between an e-like and a µ-like ring pattern of roughly the same energy, where each dot represents a hit
PMT, colored corresponding to the detection time. [66]are often identified as single-ring events (approximately 90% of thetime) as expected.
Multi-ring events undergo a further processing to cleanly separate
and assign the npe contributions of overlapping rings. Also, after theparticle flavour identification has been later performed, the rings are
re-fitted using this new information, which can result in further corrections to the final number of fitted rings.
4.2.2.3 Particle identification
The detector has excellent particle identification (PID) capability byusing the Cherenkov ring pattern and opening angle. Electrons orhigh energy photons undergo electromagnetic showering and multiple scattering, and create fuzzy Cherenkov rings; in contrast meson tracks, such as muons or pions, are generally straight and create
Cherenkov rings with sharply-defined edges, as shown in Fig. 26.

61

dx

r⋅d
θ
⋅sin
θ

data reconstruction and monte carlo

dθ

62

r
θ
dx

μ±
Figure 27: Diagram showing the calculation of Cherenkov light emission
from a muon track (variables are explained in the text). [66]The electron and muon hypotheses are compared for each ring using a likelihood parameter, which takes as input the difference between the measured and expected charge distributions (as in the caseeof ring counting). For electrons, the expected charge distributions npe,iare obtained by interpolating tabulated distributions at various fixedenergies, taken from high-statistics SK MC simulations. For muons
however, the expected distribution can be obtained analytically eachtime. Essentially the muon is propagated a distance x from the initial vertex, during which the energy losses cause a decrease in the
Cherenkov angle θ, as shown in Fig. 27 for a propagated distance dx.
Each PMT then receives photons from roughly a single position of
the muon track, and the expected charge observed at each PMT i isgiven (to first order, without normalization) by
sin2 θiµnpe,i =

|ri | sin θi + ri · [∂ x θ ] x=xi

 e−|ri |/α f i (ϑ )(45)

where xi is the distance along the track to the point from which the
PMT i receives Cherenkov light, ri is the vector from that point to the
PMT, and θi is the Cherenkov angle at that point. In the first term, thenumerator is a factor accounting for the dependence of emitted lightintensity as a function of the Cherenkov angle, and the denominatoraccounts for the light intensity at the PMT i, as can be calculated byconsidering Fig. 27. The second term controls the attenuation of lightin water, where α is the attenuation coefficient, and the third termf i (θ ) is the angular acceptance function of the PMTs. In fact, extra
terms are also included to account for scattered light and knock-on
electrons created in collisions, but we do not discuss these terms indetail here.
µ
e
The expected distributions npe,iand npe,i are both fit the observed
distribution nobs
pe,i as closely as possible by optimizing the physics parameters, such as the track direction and the initial Cherenkov angle.

4.2 fully contained sample

The likelihood parameter LPID of each ring is then calculated using
these best-fit distributions asµP nobs|npe,i pe,i
LPID = ∑ ln e
P nobs|nipe,i pe,i(46)

where, similarly to Eq. (44), the distribution P is defined as a Poisson
distribution convoluted with the single-photoelectron distribution for
obs
nobs
pe,i < 20, and a Gaussian for npe,i ≥ 20.
For the single-ring case, events are denoted as e-like or µ-like depending on the type with the highest likelihood. Figure 28 shows the
LPID separation overlaid with MC truth information, from which we
can see that the e-like and µ-like samples correspond very well to νeand νµ CCQE interactions respectively, with a tiny leakage between
each flavour at the < 1% level. Some backgrounds however remain
whereby the flavour is difficult to reconstruct, including neutral current and single pion events, at the few percent level for the sub-GeV
and somewhat worse for the multi-GeV events.
Multi-ring events can be coarsely separated based on the PID of
the most energetic ring, however depending on the energy range
and interaction type, this separation is generally not as good as the
single-ring case (especially for sub-GeV events), with wrong-flavour
background at the 30% level. We thus use a further procedure, as
described in [5] but extended to include SK-IV, to produce a reduced
multi-ring sample that can be separated into e-like or µ-like categories
more accurately. Using the energy reconstruction definitions to be explained in the next subsection, events with Evis > 600 MeV, and where
the primary ring is µ-like with Erec > 600 MeV, are selected as µ-like.
To select e-like events, we require Evis > 1330 MeV, and the event topass a likelihood-based selection that considers the type, momentum,
and timing of all fitted rings. This reduces background contaminationfrom hadronic showers, produced by NC or νµ CC events, which areoften dominated by π 0 and mis-reconstructed as e-like. These multiring samples have a flavour purity of 96.6% for µ-like and 77.8% fore-like, in SK-IV.
4.2.2.4 Energy reconstruction
From the above vertex and ring fitting processes, we have establishedfor each Cherenkov ring a PID and a tagged list of PMTs (and thefraction of npe registered in each PMT which is estimated to belong
to that ring specifically). The reconstruction of the momentum of the
lepton that created the ring then uses a different method dependingon whether the lepton is e-like or µ-like. In the electron case, themomentum is taken as being correlated with a simple count of thenpe associated with the ring, whereas for a muon, measurement of

63

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data reconstruction and monte carlo

Super Kamiokande IV 1987.6 days : Monitoring350

e-like 4409

Super Kamiokande IV 1987.6 days : Monitoringmuon-like 4366

Super Kamiokande IV 1987.6 days : Monitoring

350

350

e-like 4409

e-like 4409

muon-like 4366

300

300

300

250

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250

muon-like 4366
CCQE

CCQE electron

200

CC DIS
CC Coherent π

CCQE muon

200

200

150

150

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100

100

100

50

-8

-6

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10

PID likelihood sub-GeV 1ring (FC)(a) True CCQE electrons

NC

50

50

0
-10

CCMEC
CC Single π

0
-10

-8

-6

-4

-2

0

2

4

6

8

0
-10

10

PID likelihood sub-GeV 1ring (FC)(b) True CCQE muonsSuper Kamiokande IV 1987.6 days : Monitoring

-8

-6

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-2

0

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10

PID likelihood sub-GeV 1ring (FC)(c) All interaction types

Super Kamiokande IV 1987.6 days : Monitoring

Super Kamiokande IV 1987.6 days : Monitoring

200
160

e-like 1056

muon-like 988

e-like 1056

160

muon-like 988

180
e-like 1056

140

140

160

120

120

140

100

CCQE electron

80

80

60

60

40

40

20

20

CCQE
CC Single π
CC DIS
CC Coherent π

120

CCQE muon

100

muon-like 988

NC

100
80
60

0
-30

-20

-10

0

10

20

30

PID likelihood multi-GeV 1ring (FC)(d) True CCQE electrons

40
20

0
-30

-20

-10

0

10

20

30

PID likelihood multi-GeV 1ring (FC)(e) True CCQE muons

0
-30

-20

-10

0

10

20

(f) All interaction types

Figure 28: PID likelihood separations for the sub-GeV (top) and multi-GeV(bottom) single-ring sample for SK-IV, showing the MC (red line),
and the data (points with statistical error bars). Three plots areshown overlaid with various aspects of the MC truth information.

(a) 405 MeV(b) 1.1 GeV30

PID likelihood multi-GeV 1ring (FC)(c) 4.2 GeV

Figure 29: An example comparison between ring patterns for a muon ofvarying energy, where the size and color of the dots represent the
hit charge of each PMT. [66]

4.2 fully contained sample

Super Kamiokande IV 1987.6 days : Monitoring

200

CCQE
CC Single π

180

CC DIS
CC Coherent π

160

NC

140
120
100
80
60
40
20
0
0

50

100

150

200

250

300

Mass(2 e-like rings) MeV/c2

Figure 30: The π0 mass peak after reconstruction, from the π0 -like (twofitted electron ring) sample. Both the MC with interaction typebreakdown, and data points with statistical error are shown.

the ring width and Cherenkov angle gives a better measurement of
the true momentum (as illustrated in Fig. 29). For single-ring events,
by examination of the MC the resolution of the muon momentum
on an event-by-event basis is found to be ∼ 4%, and for electrons
is found to be energy-dependent at ∼ 10%, ∼ 5%, and ∼ 3% for a
100 MeV, 500 MeV, and 2 GeV electron respectively.11 In the case ofmulti-ring events, due to ring overlap the momentum resolution foreach ring is decreased by roughly a factor two.
An important variable used to classify events is the “visible energy”
Evis , which is defined assuming that all fitted rings are electrons andsumming their estimated momentum. This variable is used due to thesimple correlation between the electron energy and the total npe , andit gives a general energy scale in units of eV which can be assigned toeach event. Events are classified into so-called “sub-GeV” or “multiGeV” samples, which are actually divided at the threshold of exactly
1.33 GeV in Evis .12
We also define the variable Erec , which conversely does take into
account the fitted PID of each ring, and adds the reconstructed momenta and masses appropriately.
4.2.2.5

π0 -like event identification

Neutral pions are a common product of neutrino interactions in water(as described in Sec. 4.5.1). Events whereby only pions are detectedcannot discriminate the neutrino flavour (although they do carry useful information on the neutrino’s other properties), and thus are categorized differently. Since we are interested in samples with accurate
11 These errors are in addition to the overall energy scale error, as discussed in Sec. 3.4.
12 This is partially for historical reasons relating to the Kamiokande experiment, although is also the energy around which the leptons produced in CCQE interactions
start to have good directional correlation with the incoming neutrino (which is asomewhat arbitrary statement, but one can compare e.g. the ∼ 15% resolution at
1330 MeV to the ∼ 30% and ∼ 60% resolutions at 1000 MeV and 700 MeV respectively).

65

66

data reconstruction and monte carlo

neutrino flavour, we do not plan to use such events, but by identifying them they can be removed and the flavour purity of other samples
may be increased.
Fortunately, neutral pions can in principle be easily identified bytheir dominant decay π 0 → γ + γ, whereby the photons produce twoelectromagnetic showers and thus two e-like rings in the detector, theinvariant mass of which is the pion mass. Fig. 30 shows the reconstructed invariant mass of such events with two reconstructed rings;
these are classified into a “π0 -like” sub-sample.
The reconstruction of two e-like rings is however not always easy;
one or both rings may have low charge and be difficult to reconstruct,or the two rings may be overlapping due to a Lorentz boost between
the lab frame and the decay frame. Therefore, for sub-GeV singlering e-like events, an extra π 0 finder algorithm [92] is also applied,which performs a forced search for a second γ ring that was originally
missed. The algorithm identifies events using a likelihood method,comparing the single-ring best-fit result to the two-ring best-fit, whileconsidering also if the reconstructed invariant mass is close to theπ0 mass. These events are classified into a “single-ring π0 -like” subsample.
4.2.2.6 Final division into sub-samples
For all three of the analyses in this thesis, a strong requirement is theselection of separate νe and νµ induced samples with high flavour purity. We give here the final definitions of the FC sub-samples that will
be used in all of the analyses (with the exception of Sec. 5.6, whichuses instead the neutrino / antineutrino separated FC sub-sample
definitions described in Sec. 4.2.3). Firstly, all π0 -like sub-samples are
discarded, due to their high NC and wrong-flavor CC backgrounds,
as seen in Fig. 31. All single-ring events with Erec > 100 MeV for elike events, and Erec > 200 MeV for µ-like events, are selected, andmulti-ring events are selected as described in the previous section.
The remaining FC events are thus generally divided by three variables (also as described in the previous section): sub-GeV or multiGeV, e-like or µ-like, and single-ring or multi-ring. Multi-ring eventsare mostly multi-GeV and are not further divided by their visible energy, leaving six final FC event categories, which are:
• FC e-like single-ring sub-GeV,• FC e-like single-ring multi-GeV,• FC e-like multi-ring,• FC µ-like single-ring sub-GeV,• FC µ-like single-ring multi-GeV,• FC µ-like multi-ring.

ν e CC
ν µ CC
NC

UPµ

2500

PC

FC µ-likeFC e-like

Number of Events

4.2 fully contained sample

2000
1500
1000

18
UPµ shower

UPµ non-s

16
UPµ stop

PC thru

14
PC stop

MG MR-µ12
MG 1R-µSG 1R-µ 2dcy

10

SG 1R-µ 1dcy

SG 1R-µ 0dcy

8
MG MR-ν e

MG MR-ν e

6
MG 1R-ν e

4
MG 1R-ν e

SG 1R-e 1dcy

SG 1R-e 0dcy

2

SG π0 -like00

SG 1R-e π0 -like

500

Figure 31: The flavour purity of various SK event samples, described inSec. 4.2→Sec. 4.4, estimated by MC. Here SG and MG stand for
sub-GeV and multi-GeV, while 1R and MR stand for single-ringand multi-ring, and ndcy refers to the number of Michel electrons
detected.

4.2.3

Neutrino / antineutrino separated selections

As previously mentioned, the detector does not have an ability to distinguish ν from ν̄ on an event-by-event basis, however recent work inthis area has defined and improved several statistical methods of ν/ν̄
separation based on the different kinematics of their interactions. In
this section, we discuss the creation of separate particle and antiparticle sub-samples for the FC dataset; since the background of wrongsign neutrinos remains relatively high, we refer to these samples as νor ν̄ enriched samples. We abandon for this section the abridged notation where e.g. νe ≡ νe + ν̄e , and discuss νe , ν̄e , νµ , and ν̄µ separately.
The difference in the number of neutrino and antineutrino interactions in the detector obviously depends on two factors: the relative
normalization of the two fluxes, and their interaction probabilities, asa function of energy. The true ν/ν̄ ratios estimated by the HKKM
group, with very small systematic errors of O(1%), are shown in
Fig. 32. We make use of these theoretical ratios when defining ourneutrino and antineutrino samples. Since ν are more abundant thanν̄, it is generally harder to make ν̄ samples with high purities. Many ofthe discussions below refer to the various neutrino interaction types,
making use of the discussion in Sec. 4.5.1.
4.2.3.1

Neutron tagging

Free neutrons in water can be captured by hydrogen nuclei to create
deuterium, with a half-life of 205 µs, by the interaction n + p → d + γ,

67data reconstruction and monte carlo

5
All Direction Averaged Flux Ratios68νµ + νµνe + νeνµ + νµx0.2
νe + νe

2

νµ

νe
νe

x1.5

νµ

1

−1

10

0

10

1

10
10
Eν (GeV)

2

10

3

10

4

Figure 32: The ν/ν̄ ratios at the SK site, as estimated in the HKKM15 simulation. [68]

where the emitted γ-ray has an energy of precisely 2.2 MeV.13 Whilethe detection of a low energy γ-ray in a large time window is not aneasy task for a detector such as SK, as mentioned in Sec. 3.3.2 the SKIV electronics allowed for the introduction of an AFT trigger, which
records data up to 535 µs after a high-energy event. Since the primaryvertex is known from the high-energy interaction, and the expectedenergy of the γ-ray is precisely known, it is nonetheless possible topick out some candidates from the expected radioactive background.
An initial selection is done by first selecting candidates whereby
the number of ID PMT hits n within a sliding 10 ns time window
is 7 < n < 50, where the time distribution uses the time-of-flight
subtracted timing based on the APFit best-fit vertex. It is also requiredthat the total number of ID hits within ±10 ns of the candidate is lessthan 200, in order to avoid cosmic-ray muon background. Over the
535 µs period, this cut has a 33.2% efficiency to detect true neutrons,
whereas an average of 4.5 false events will be detected (estimatedbased on dummy trigger data). These initial candidates are then fed
through a neural network trained to pick out true neutron events,using 16 parameters which generally describe the spatial and timing
distribution of the selected hit PMTs (mean opening angle, RMS of
the time distribution, etc.). The output of the neural network has,averaging over all positions in the FV, an efficiency of 20.5% to detect
true neutrons for a much-reduced background of 0.018 events per
AFT trigger. More details of the algorithm can be found in [83, 84].

13 Neutron capture by oxygen is also possible, although capture by hydrogen is approximately 103 times more common.

4.2 fully contained sample

Low-energy CCQE interactions, considering charge conservation,must proceed by one of the following:ν + n → l− + pν̄ + p → l + + n

where the left-hand nucleons are bound within a nucleus, but theright-hand nucleons are often freed. Here, the benefit of neutron tagging in neutrino / antineutrino separation is clear. At high energies
however CCQE interactions are less common and the discrimination
power of neutron tagging is lower.14
4.2.3.2

Decay electron tagging

As previously mentioned, it is also possible in SK to tag Michel electrons15 originating from the decay of µ± . In fact, as Michel electrons
have an average energy of a few tens of MeV, this is clearly mucheasier than tagging a 2.2 MeV γ-ray, and is reliably achieved with thestandard FC reconstruction tools, although with some inefficiency forvery early Michel electrons particularly for the SK-I to SK-III electronics.
To understand the benefits of electron tagging in ν/ν̄ separation,
an important fact to note is that while µ− can be captured in water
to form muonic atoms (approximately 20% of µ− in SK are capturedon oxygen nuclei), µ+ has no such capture, and thus has a longer
lifetime and a higher possibility to create a Michel positron. Thus,
considering µ-like single-ring events (which are mostly CCQE events,
especially at low energies) those with a single decay electron taggedshould contain more µ+ and thus a higher fraction of true ν̄µ .
However, considering medium-energy neutrino interactions thatcreate meson showers, we have the following possible interactions:ν + p → l − + N + P++ν + n → l − + N + P+ν̄ + p → l + + N + P0

ν̄ + n → l + + N + P−
where N is the modified outgoing nucleus, and P represents a meson shower (generally mostly pions) with the superscript indicating
the total charge. Considering the decay chain π ± → µ± → e± , asgiven in Eq. (29) and Eq. (29) on page 19, it is seen that for neutrino
events causing hadronic showers, νe and νµ events would tend to have
14 As a final note on the topic of neutron tagging, studies evaluating the possibility to
add small amounts of gadolinium in SK and future detectors [93, 94] are recently
nearing completion; this greatly increases the capture and identification rate of neutrons, and is expected to be a major upgrade for neutrino water Cherenkov detectors.
15 And of course positrons, however with no power to separate between the two.

69

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data reconstruction and monte carlo

e

μ

e

μ

Figure 33: The true neutrino purities of the multi-GeV single-ring e-like (left)
and µ-like (right) samples, as a function of the number of tagged
Michel electrons (or positrons). The histograms are normalized tounity, and the wrong-flavour background of each sample is notplotted. [84]more decay electrons. This effect should be expected in the multi-ringsample, however it can leak into the single-ring sample if the meson
shower is below the Cherenkov energy threshold; in this case, it competes with the opposite effect described in the previous paragraph(where ν̄µ events are more likely to have a decay electron). The prediction from MC is shown in Fig. 33, where we see the result of thetwo effects for multi-GeV single-ring events.
4.2.3.3

Hadronic multiplicities

Experimental data indicate [95] that ν interactions generally have ahigher multiplicity than ν̄ interactions, which is to say that they tend
to create more particles in hadronic showers. Thus, events with ahigher number of fitted rings have a slightly higher chance to be atrue ν.
There are also secondary effects of the increased multiplicity for νinteractions. As the deposited energy is shared amongst more rings,
the fraction of Erec in the primary ring to the total Erec tends to beslightly smaller. The distribution of decay electrons within the tank isalso affected; as more particles are involved in the shower, the chancesof a decay electron to be observed further from the primary vertex(both spatially and temporally) is increased. Finally, the PID variable
is often shifted slightly towards e-like instead of µ-like, originating in
the fact that there is a higher chance for unidentified mesons to interfere with the primary ring identification (i.e., to increase the “fuzziness” of the primary ring).
4.2.3.4 Final division into sub-samples
As in the previous section, we divide events as single-ring or multiring, and for the single-ring samples also as sub-GeV or multi-GeV.
The π 0 samples are also separated in the same way as before. All

4.2 fully contained sample

sub-GeV single-ring events with Erec > 100 MeV for e-like events, andErec > 200 MeV for µ-like events, are selected – however unlike theprevious section, they are also divided by the number of decay electrons nd that were detected. For single-ring multi-GeV and multiring events, a more complicated procedure is used in to divide bothgroups into four νe , ν̄e , νµ , and ν̄µ -like samples, as described below.
Single-ring multi-GeV e-like events are classified as νe -like if nd > 0
and nn = 0 (where nn is the number of tagged neutrons, and is only
available and checked in SK-IV), otherwise they are classified as ν̄e like.
Single-ring multi-GeV µ-like events are classified as νµ -like or ν̄µ -likebased on an artificial-neural-network likelihood method,16 which considers the parameters nd , nn , the PID likelihood, the maximum Michelelectron distance from the primary vertex, and the longest time between the primary event and a Michel electron. This neural network,and others mentioned in this section, were defined in [89] and further
developed in [84]; these two PhD theses contain detailed informationsuch as the output shape of the network, the choices of final cut parameters, and so on.
Multi-ring e-like events are classified as νe -like or ν̄e -like based on aneural network considering nd , nn , the PID likelihood, the number ofrings, and the primary ring’s Erec compared with the total Erec .
Multi-ring µ-like events are classified as νµ -like or ν̄µ -like based on aneural network considering all of the parameters mentioned above.
Finally, any multi-ring multi-GeV event that fails the likelihood selections is classified into a “multi-ring other” sample. From the above
definitions, using the SK MC (discussed in Sec. 4.5) we show the estimated purities of each of these sub-samples in Tab. 6. We see thateach sample has a distinctive shape in the distribution of its true neutrino content. For νe and νµ , it is possible to obtain several samples
with purity > 70%, whereas for ν̄µ we can manage at best 50%, andfor ν̄e only around 30%. In Sec. 5.6, when these samples are used foranalysis, we will discuss a sub-selection and binning of these events.

16 An artificial neural-network is a multivariate analysis method, which can be trained
to discriminate between a set of possible outputs given certain input parameters.
The end result is somewhat similar to a likelihood method, except that the weights
contributed by the individual parameters are automatically tuned by the neural network, and can depend on the values of the other parameters.

71

data reconstruction and monte carlo72

1. single-ring sub-GeV e-like nd = 0
72.4

73.4

νe

4.3

2.1

23.2

ν̄e

74.4

1.5

13.0

0.3

νµ

29.2

8.6

0.2

3.9

0.1

ν̄µ

1.8

12.8

81.0

8.5

3.1

NC + ντ + ν̄τ

4114

829

183

344

3107

Events2. single-ring sub-GeV e-like nd ≥ 1
12.9
1.0

68.9

Estimated fractional purity

3. single-ring π 0 -like
3.2
0.0

Sub-sample

4. single-ring sub-GeV µ-like nd = 0
0.1

102

5. single-ring sub-GeV µ-like nd = 1

5.1

833

314

2.6

9.0

670

2.0

9.7

0.6

0.2

600

4.0

7.9

2.7

26.4

0.1

328

94.0

7. two-ring π 0 -like

74.7

32.9

72.9

49.6

20.6

235

0.0

8. single-ring multi-GeV νe -like

54.8

0.2

50.3

0.9

11.4

645

0.0

9. single-ring multi-GeV ν̄e -like

0.3

0.0

11.5

0.6

6.2

447

573

10. single-ring multi-GeV νµ -like

0.0

12.2

4.3

12.4

1.0

575

89.2

11. single-ring multi-GeV ν̄µ -like

54.9

29.5

77.8

34.7

29.3

0.1

12. multi-ring νe -like

54.2

0.5

63.5

4.8

1.6

13. multi-ring ν̄e -like

3.1

0.0

39.0

2.3

14. multi-ring νµ -like

0.7

2.7

6.9

15. multi-ring ν̄µ -like

24.2

6. single-ring sub-GeV µ-like nd ≥ 2

16. multi-ring other

Table 6: The fractional purities of each sub-sample defined for the ν/ν̄ separated analysis and expected number of events, estimated by MC, where nd
is the number of decay electrons. In this table, samples that will later be discarded are also included, in order to examine the relative puritiesof each sample. This table refers to SK-IV only, however selections in other periods are similar within 2 ∼ 3%. The bold text shows the primary
component of each sample.

4.3 partially contained sample

4.3

partially contained sample

4.3.1

Reduction17

The PC reduction is (like the FC sample) divided into five steps, andis denoted PC1→PC5. Considering that PC events by definition have
OD activity, which is expected to be generated by a single exiting
muon track, the reduction process must be fundamentally quite different, even at the early steps. Cuts are described in the same way as
in Sec. 4.2.1.
4.3.1.1 PC1
The first PC reduction consists of some basic cuts:
1. total ID npe > 1, 000
which is roughly equivalent to the energy deposited by a muon of
initial momentum 310 MeV traveling ∼ 2 m, which is used as it is theminimum energy for a single muon created in the FV to penetrate
into the OD. Next,
2. width of the OD timing distribution < 260 ns
which references the RMS of the OD hit timings, and should remove
events with two event clusters (such as a through-going muon thatenters and later exits the detector). This cut is supported by a simple positional cluster detection algorithm searching for nearby hits,
requiring
3. number of hit clusters (simple algorithm) in the OD = 1.
4.3.1.2 PC2The second reduction depends on a more advanced clustering algorithm, which divides the ID PMTs into a grid of 441 patches, andthe OD into 121 patches, and searches for clusters by finding steep
gradients between neighbouring patches. The similar cut
4. number of hit clusters in the OD ≤ 1
is performed, and if an OD cluster is found it is also required that
5. total ID npe in the ID cluster closest to the OD cluster (within
200 cm of the highest-charge PMT) is > 1, 000.
17 This section describes the reduction process as used in SK-I, in order to show abroad understanding of the PC data processes. Several tunings and updates werelater applied, particularly when OD segmentation was installed at the beginningof SK-III, optically separating the top, center, and bottom areas of the OD. For athorough description of the SK-IV process, see [89], and for more detail on SK-I and
SK-II please see [96].

73

74

data reconstruction and monte carlo

Next, it is required that
6. least active detector region has < 7 hit PMTs
where the “detector region” refers to either the top, side, or bottom
surfaces of the cylindrical SK detector. These cuts are again designed
to find PC events that have a single exit point in the detector, and are
related to an ID hit cluster.
4.3.1.3 PC3The third reduction targets the specific backgrounds of flasher eventsand stopping muons. The long time distributions of flasher events are
cut by requiring:
7. minimum number of OD hits in a sliding 100 ns time window
from +0 ns to +500 ns is < 10 (this criteria is relaxed to < 15, if
more than 800 ID PMTs are hit).
Stopping muons are then rejected using the point-fit algorithm, whichroughly gives the ID vertex and direction. By extrapolating backwards, a rough entry-point can be found in the OD, which should
contain an early hit cluster if the event was a stopping muon:
8. number of hit OD PMTs within 8 m of the estimated entry-point< 11.
4.3.1.4 PC4Some stricter cuts on through-going cosmic-ray-muons, which by chance
had low OD activity at the entry point, are still necessary in the fourth
reduction. This is done using a specialized through-going muon fitter,
which uses the most likely early hit ID cluster. Events with
9. goodness-of-fit of through-going muon fitter < 0.85
are accepted, whereby the entry cluster’s fitted position and timing
will correspond poorly to the projected through-going muon track. A
similar-purpose cut is made using the point-fit information, comparing the fitted direction to the earliest hit ID PMT, requiring
10. inner product of point-fit vector and earliest-hit vector > −0.85,
i.e. these two vectors should not be back-to-back.
4.3.1.5 PC5After the fourth reduction, the event rate is down to ∼ 20 events perday, and the application of advanced fitters18 to these events is feasible. The fifth reduction then uses several finely-tuned criteria based
18 Including apfit, stmfit, pfdofit, and msfit.

4.3 partially contained sample

event rate [day − 1 ]
sk-i
sk-ii
sk-iii
sk-iv

data

mc

0.67 ± 0.02

0.69

0.63 ± 0.03

0.67

0.66 ± 0.02

0.69

0.66 ± 0.04

efficiency [%]background [%]85 ± 2–96 ± 12

0.69

80 ± 5–95 ± 1

2

Table 7: The final daily event rates of events passing the PC reduction, withstatistical error (negligible for MC). The meaning of the “efficiency”
and “background” columns is the same as in Tab. 5, as was explained at the end of Sec. 4.2.1. In the PC case however, eye-scanning
records (for estimating the background rate) are only available for
SK-III and SK-IV. Values are taken from [76].

on these reconstructed tracks to target the remaining backgroundsources. This includes many types of through-going and stopping
muon cuts; targeting for example muons that pass through the detector superstructure and PMTs, or corner clipping muons. The fifth reduction also targets veto-counter events and electronics noise events,
in a similar way to the FC reduction.
4.3.1.6 Final reduction
The final requirements to pass the PC reduction, in SK-I, are:
11. the distance from the neutrino interaction best-fit vertex and thewall is > 2 m,
12. the number of OD PMTs found by a clustering algorithm is≥ 16,
13. the visible energy Evis > 350 MeV.
The final PC sample has a somewhat lower efficiency for true neutrino
events than the FC sample, as shown in Tab. 7. Of course, this should
not bias the analysis, since the MC events are selected with the same
efficiency.
4.3.2

Reconstruction

The PC sample uses the same software and essentially the same techniques as the FC case, with some modifications to account for the extra charge deposited in the OD. Most importantly, events are divided
into two samples:• PC stop,• PC through,

75

76

data reconstruction and monte carlo

based on whether the muon particle stops inside of the OD or leaves
the detector completely (where the PC through sample has muchlarger statistics). This separation is done because the momentum of
the muon can be determined accurately for events in the PC stopsample, but for the PC through sample the energy deposited by theparticle after it has left the detector cannot be measured, and the momentum resolution is much worse.
The selection of PC stop events is made in a simple fashion [97], by
testing the observed charge in the OD, requiring
1. maximum OD npe in a sliding 500 ns time window < 32 nexp ,2. at least one of the two highest energy rings in the ID is classified
as µ-like,
where nexp is the expected deposited npe of a muon track, assuming
it exited the OD (which varies as a function of the fitted exit point, totake into consideration the geometry of the detector). All other eventsare categorized into the PC through sample.
4.4
4.4.1

upward-going muon sample 19
Reduction and reconstruction

The UPMU reduction has the task of distinguishing upwards-going
muons that enter the detector from below, originating from a neutrino
interaction in the rock, and either stop inside the detector or exit
through the top or sides. Like the FC and PC reductions, backgrounddata are removed by a series of automated cuts, however unlike the
FC and PC reductions, the UPMU reduction also uses subtractions of
the estimated backgrounds, and manual eye-scanning to select final
events. The reduction consists of four automated steps (UM1 → UM4)
and a final eye-scanning step.
4.4.1.1 UM1 to UM4
The first reduction keeps only events with a total ID n pe > 6000 (forSK-II, > 3000), which corresponds roughly to a 1 GeV muon traveling a few meters. It also saves by default all ultra-high energy eventswith total ID npe > 1.75 × 106 , which occur only once every few days,
and are used for astrophysical studies [101, 102, 103, 104]. These “astrophysical” events require a completely different reconstruction process, and are not used in this analysis.
Events are then passed to a series of fitters of increasing complexity. UM1 uses the fitter stopmu1st, which does some simple cluster
searching, and muboy, which provides a direction and goodness-of-fit
19 more details on the upmu sample can be found in the theses [98, 99], or the paper[100].

4.4 upward-going muon sample

for the muon track. Events are passed through with some fairly relaxed criteria on the goodness-of-fit and direction; allowing up-goingand near-horizontal single-track events, but cutting events that areclearly down-going or multi-track events. Events that pass close by
the seams between the top or bottom cap and the side of the detector (“corner clipping” events) are also cut, as they are difficult toreconstruct and can cause false UPMU events (corner clipping eventsshould in principle not make the final sample, due to a requirementthat the track length be at least ∼ 7 m, described later).
Next, UM2 applies a series of more complicated fitters, classifying
events as either “stopping” (where the muon comes to rest insidethe detector) or “through-going” (where the muon exits the detector).
Some tighter cut criteria target background events using the fitter
information. UM3 applies the most precise and processing-intensive
fitter, called precisefit. For SK-IV only, an additional UM4 step wasadded, which make use of the additional OD segmentation installedat the beginning of SK-III (optically separating the top, bottom, andside regions). OD events that cross this region are examined carefully,
to further reduce the number of corner clipping events.
An event need not be reconstructed perfectly as an UPMU by all
fitters, but their outputs are considered together to reach a decision,and establish a final entry point, momentum, directional vector, andtrack length. Events that pass the selection are also flagged into two
sub-samples, “stopping” or “through-going”, depending on the fitter
information..
4.4.1.2 Eye scanning
After UM4, the UPMU data still contains ∼ 20% background events.
In many of these cases, the fitting software has landed in a local minima or otherwise produced a poor fit; for example due to events near
the detector walls or corners (causing difficult timing distributions),or large regions of PMTs reaching their charge saturation point, or
mis-selection of PMT hit clusters. A visual examination of the ID andOD charge patterns and the fitted Cherenkov ring shapes can often
quickly reject such badly fitted events,20 and every UPMU event ischecked by two expert scanners who tag background events; disagreements between scanners are rare, and the tagging of true UPMUs as
background is estimated to be negligible or even zero.
4.4.1.3 Final reduction
Events passing the eye-scanning stage have a single fitted muon track,and are flagged as stopping or through-going. The final selection criteria for the stopping sample requires
20 Thanks in general to the evolutionary process, and the superior pattern-matchingability of the human.

77

cosθ

data reconstruction and monte carlo

cosθ

78

0

0

-0.25

-0.25

-0.5

-0.5

-0.75

-0.75

-1

0

100

200

300

φ [degree](a) Stopping muons

-1

0

100

200

300

φ [degree](b) Through-going muons

Figure 34: Fitted stopping and through-going muon data for SK-I, where φis the azimuthal angle and θ is the zenith angle. Mostly up-going(cos θ < 0) neutrino-induced events are shown, although in the
0 < cos θ < 0.1 range some clusters of the cosmic-ray backgroundare clearly visible.

1. fitted muon momentum > 1.6 GeV,
2. number of hit OD PMTs within 8 m of the best-fit exit point is< 10 (for SK-II, < 16),
where the first cut eliminates short tracks (less than ∼ 7 m) which arebackground rich, and the second cut introduces a tight requirementthat only selects stopping events. The through-going sample events
are similarly required to have
1. distance between the best-fit entry and exit points of the ID> 7 m,
2. number of hit OD PMTs within 8 m of the exit point is ≥ 10 (forSK-II, ≥ 16).
The final efficiency of the UPMU reduction is calculated based on
MC, considering only true up-going events with a vertex outside the
ID and a path length of > 7 m, and is shown in Tab. 8.
4.4.1.4

Background estimations

The UPMU sample has a non-removable background, due to cosmicray muons coming from near horizontal directions that scatter before
entering the detector and become up-going, which cannot be distinguished from neutrino-induced up-going muons. These deflections
are rare, but the backgrounds are non-negligible considering that thecosmic-ray muon flux (even at the SK detector ∼ 1 km underground)is higher than the true neutrino UPMU flux by a factor of ∼ 105 .
Due to the shape of the mountain, the rock overburden from thedetector varies strongly depending on the azimuth and zenith angles;4.4 upward-going muon sample

event rate [day − 1 ]
sk-i
sk-ii
sk-iii
sk-iv

data

mc

1.41 ± 0.03

1.25

1.49 ± 0.05

1.24

1.33 ± 0.04

1.23

1.38 ± 0.09

1.23

efficiency [%]background [%]99.0 ± 1.0

2.7 ± 0.5

99.1 ± 1.0

3.0 ± 0.6

97.8 ± 1.0

99.0 ± 1.0

4.6 ± 0.8

2.4 ± 0.3

Table 8: The final daily event rates of events passing the UPMU reduction,with statistical error (negligible for MC). The meaning of the “efficiency” column is the same as in Tab. 5. The meaning of the “background” column is described in the text of this section. The values
are taken from [89] for SK-I to SK-III, and [76] for SK-IV.

in Fig. 34, some cosmic-ray muon clusters can be clearly seen justabove the horizontal, at certain azimuthal angles. This azimuthal distribution (in the 0 < cos θ < 0.1 range) is extrapolated downwards
in zenith by an exponential function, and fitted against the expected
neutrino-induced muon distributions. The final data set is then given
as a distribution, defined as the observed data minus the fitted backgrounds, as a function of zenith angle and reconstructed momentum.
The final UPMU sample efficiencies are shown in Tab. 8. The data
rates are given before background subtraction, and the “background”
column represents the subtracted estimated background, and the systematic errors on the background subtraction process. These errors
are generally 10 ∼ 20% of the subtracted background amount, andcome primarily from the statistical uncertainties in the backgroundfitting procedure.
4.4.2

79

Sub-sample categorization

So far, UPMU events were selected, reconstructed, and divided into
stopping and through-going samples. The through-going events arefurther divided into two sub-samples depending on whether or notthey undergo intense electromagnetic showering, which occurs whenthe muon’s energy approaches 1 TeV, due to increased pair production and bremsstrahlung emission relative to simple ionization atthese energies.
The showering sample is thus selected by requiring a greater energy deposited per unit range along the fitted muon track. The parameter used is a χ2 -inspired statistic, based around the observed
charge Qi associated with each 50 cm ID track segment i, defined as

!2 
2
N −2 
Qi − Qexp
Qi − Qaveχ2 = ∑ 
+
(47)
σ
σ
Q
Q
exp
ave
i =3

80

data reconstruction and monte carlo

where Qave ≡ h Qi i, and Qexp is a constant representing the expected
Qave for a non-showering muon. Furthermore, σQave and σQexp are theestimated errors on those parameters.21 This parameter thus prefers
tracks with the sudden bursts or generally high values of energy loss
associated with a showering muon track.22 The sum ignores the firstand last two track sections, which are within 1 m of the entry and exit
points, where a small error in the reconstructed track could cause alarge change in the associated values of Qi .
The final sub-sample definitions of the UPMU events are thus:• UPMU stop,• UPMU showering,• UPMU non-showering.
Stopping events are clearly lower energy (considering that they do
not have enough energy to fully traverse the detector) – thus the threesamples provide some energy discrimination for the UPMU events,
as was shown in Fig. 18. Of course, the amount of energy depositedby the muon before entering the detector is completely unknown, so
while we can reconstruct a muon momentum at the point of entry,
an estimation of the true energy spectrum of UPMU neutrinos can bedone only in a statistical sense by comparison to MC.
4.5monte carlo

In the SK Monte Carlo (MC), neutrino interaction events are generated from an interpolation of the energy and azimuthal distributionsof the HKKM07 [18] atmospheric flux model, and the NEUT [105] neutrino interaction simulator. NEUT calculates the results of the primary
neutrino interaction with a water molecule and all secondary nuclear
effects, and then the list of exiting particles is handed off to the detector simulation SKDetsim, which handles all propagation and physicaleffects necessary in water Cherenkov detectors, and fully simulates
the detection and digitization processes of the PMTs and SK electronics. After simulation of the events, reduction and event reconstruction
processes are applied, which are essentially identical to those appliedon the data. The MC is generated in all four detector configurations,corresponding to 500 years of livetime for each of the SK-I to SK-IV
periods.

21 For example, we expect dE/dx = 2 MeV cm−1 for a 1 GeV muon; considering the
average expected non-showering muon energy, we can estimate the average dE/dx
and its spread, and thus Qexp and σQave . The error σQave , on the other hand, is astatistical one based on the Qi .
22 The actual selection uses an additional parameter accounting for detector tunings,
and fitted to a MC prediction (see [98]).

4.5 monte carlo

Figure 35: Cross-sections for the different types of neutrino interaction,
showing νµ (left) and ν̄µ (right). The solid lines are predictions
are from the NUANCE neutrino interaction generator, and thedata points are taken from various experiments. The meanings of
QE, DIS, and RES are described in the text of this section. [106]interaction typefinal products(quasi-)elastic scatteringν + N → l + N0

single π production
single γ production
single K production
single η production
coherent π production
deep inelastic scatteringν + N → l + N0 + π
ν + N → l + N0 + γ
ν+N → l+Λ+Kν + N → l + N0 + η
ν+O → l+X+π

ν + N → l + N 0 + hadrons

Table 9: Primary neutrino interactions considered by NEUT. Here, N and N 0
are the initial and final nucleons (e.g. n or p), X is the final nucleus,
and other notations are standard. Each interaction can proceed byboth charged or neutral currents, which is to say that the outgoing
lepton l can be either a neutrino or a charged lepton (e± , µ± , τ ± ), if
the appropriate choice is also made for N 0 .

4.5.1

Neutrino interaction simulation

4.5.1.1 Primary interactions
The neutrino interaction generator NEUT considers several types ofprimary neutrino interaction, as listed in Tab. 9, with cross-sections
estimated based on theoretical models with free parameters fitted bya wide array of experimental data, such as bubble chambers and otherneutrino experiments. Several target materials such as carbon, argon,
iron, oxygen, and hydrogen are implemented in NEUT (the latter two of
which are of interest in SK). The cross-section of a neutrino interaction
with an electron is several orders of magnitude smaller than thosewith nuclei, and is neglected.23

23 An excellent overview of neutrino interactions is given in [107].

81

82

data reconstruction and monte carloquasi-elastic scattering (qe) is the dominant process up toaround 1 GeV, whereby a neutrino scatters off an entire nucleon, usually ejecting one or even several nucleons from the nucleus. This isoften referred to as “elastic scattering” in the NC case, and “quasielastic” in the CC case. The cross-sections in NEUT are based on the
Llewellyn-Smith model [108], for free nucleon scattering. To obtain
the QE cross-sections for particular nuclei, modifications by the theSmith and Moniz [109] model are used, where a neutrino is taken tointeract with a bound nucleon described within the nucleus by a relativistic Fermi gas model (although simple free nucleon interactions
can still occur, at a ratio predicted by the model). These models contain both vector and axial-vector form factors (which are necessary
due to the V − A structure of the weak interaction), both of which
are modeled as “dipole” type,24 which is to say the form factors F
depend on the total four-momentum transfer q2 as
F (q2 ) = F0 1 − q2 /M2

 −2

(48)

where F0 is a base value predicted by the model, and M is the mass
parameter, which is not predicted but must be determined by experimental measurements. For the vector form factor, the mass parameter
V is well determined by electron scattering experiments, whereas
MQE
A is set at 1.21 GeV based on data
the axial-vector mass parameter MQEfrom the K2K [112] and MiniBooNE [113] neutrino experiments (here,
the subscript QE is added in order to differentiate this parameter withother axial-vector mass parameters that are used in e.g. the single
meson production interactions, which are based on different models
and thus may have different values). The systematic errors on the
QE scattering cross-section are estimated using the errors on the experimental results measuring the mass parameters, and the difference
between available theoretical models (for example, between the Smith& Moniz model, and the Nieves [114] model of nuclear interactions).
As mentioned before, the QE process is particularly useful for measuring neutrino interactions due to the simple two-body kinematics,
as the recoil of the final nucleon is generally far below the Cherenkov
threshold in SK.
resonant productions (res) are interactions producing a single meson (or photon), whereby at specific energies excited baryonicstates of nucleons are created within the nucleus, which then proceed to decay into nucleons and mesons. Of these interactions, single
π production is the dominant process. In NEUT the cross-sections for24 In fact, this type of form factor disagrees with experimental measurements for thevector coupling [110]. It is nonetheless maintained in NEUT, partially for reasonsof comparability with other generators and older experiments. Recent measurements also indicate that the axial vector coupling may need to be updated in the
future [111].

4.5 monte carlothese interactions are mostly based on the Rein and Sehgal model [115],
which is a calculation of all possible baryonic excited states up to an
invariant mass of W = 2 GeV along with their decay mode probabilities. Similar to QE interactions, the RES model also depends on
experimentally-measured mass parameters, mostly fitted according
to the MiniBooNE data. So-called “pionless delta decay”, where theenergy of the baryonic resonance may be adiabatically absorbed by
the nucleus, is also modeled based on [116] (accounting for ∼ 20%
of excited resonances). These resonance interactions are importantaround the few-GeV range.
In coherent π production, the neutrino interacts with the nucleus
as a whole and produces a forward-going pion (or very occasionaly,
a heavier meson) without transferring a large amount of momentum
to the nucleus, such that no excited nuclear states or ejections of nucleons occur within the modified nucleus (denoted X in Tab. 9). The
angular distribution of the outgoing lepton and meson are strongly
peaked in the forward direction. Coherent π production is predicted
to occur in both CC and NC interactions by the Rein and Sehgal
model [117], and while both have been seen in many high energy
experiments [107], measurements by K2K [118] and SciBooNE [119]are consistent with the absence of such CC interactions at the 90%
confidence level at low energy (below 2 GeV), and a conservative systematic error of 100% is assigned to the predicted cross-section. For
NC interactions, a systematic error of 30% is assigned based on the
SciBooNE [120] measurement.
deep inelastic scattering (dis) interactions are important
at higher energies, where neutrinos can resolve the quark structureof the nuclei, and cross-sections must be calculated considering thisstructure [121, 122]. These interactions create complicated hadronic
outputs, where the particle multiplicities for a given energy are estimated by a study of bubble chamber data [123, 124] for W < 2 GeV,and simulated by the package PYTHIA/JETSET [125] above that. The
fractional systematic error on the total cross-section for all DIS interactions is estimated, by comparison to the CC inclusive cross-section
measurement by MINOS [13], as 0.4 × ( Ev [GeV])−1 , e.g. 40% at 1 GeV
but decreasing to 4% at 10 GeV.
4.5.1.2 Final state interactions
It is known that the mesons and hadrons produced in a “primary”
interaction generally interact within the nucleus before escaping, andNEUT also tracks these nuclear effects (known as final state interactions or FSI) until all generated particles have been reabsorbed into
the nucleus, or have traveled far enough to be considered to haveleft the nucleus. This is done by the Monte Carlo cascade model, be-

83

84

data reconstruction and monte carlo

ginning all particles at the vertex of their generating interaction,25
and stepping each particle by a certain distance while calculating theprobability to undergo various interactions at each step. The nucleon
density is based on the Woods-Saxon distribution.
In general the implementation of all possible FSI effects is an extremely difficult and ongoing task, based on theoretical models but
requiring many re-tunings to fit the experimental data. While NEUT
is often updated to keep consistent with the current data, the uncertainties in the final outputs are relatively high, and the errors on thevarious cross-sections and FSI effects must be thoroughly considered
as systematics in the final neutrino analyses.
4.5.2

Detector simulation

The detector simulation SKDetsim is based on a customised version
of GEANT3 [126]. It contains a full model of the detector, and incorporates all calibrations and known effects discussed so far in this thesis– including precise measurements of the PMT and detector geometries, water properties, reflection properties of the various materials,
electronics, and PMT response (such as angular acceptance, quantum
efficiency, and charge-timing response).
Some models of the physical processes relevant at SK have also
been enhanced or updated. All charged particles in the simulation
emit Cherenkov radiation, and all standard electromagnetic effects
such as ionization, pair production, Compton scattering, Bremsstrahlung,and so on, are included. Particle decays, such as the Michel decay formuons, are fully simulated. Upon further nuclear collisions, hadronicinteractions are simulated by the CALOR package [127] above an energyof 500 MeV, whereas below this particles are handed back to NEUT andtreated by the cascade model, the combination of which gives the best
reproduction of the current SK data. Recent updates to SKDetsim (andNEUT) are described in [128].
4.5.3

Event weights

Each neutrino event has several associated reweighting factors. Firstly,
events are reweighted from the HKKM07 atmospheric flux modelto the more recent HKKM11 [19] model on an event-by-event basis.
Weights due to neutrino oscillation are also calculated and applied
per event; SK analyses (except the oscillation analysis itself) generally use the three-flavor oscillation parameterization taken from theyearly PDG average [59], under the assumptions of normal hierarchy(∆m232 >0) and CP symmetry (δCP =0). Finally, weights are applied toaccount for the expected neutrino flux changes due to changes in the
25 Except in the case of an initial DIS interaction, where hadrons may be generated
away from the neutrino interaction vertex.

4.5 monte carlosolar activity. The HKKM11 model provides a complete dataset forthree particular degrees of low, medium and high solar activity. The
average degree of solar activity for each SK period was estimatedfrom cosmic neutron count data [129], and all events for a given SK
period are assigned the average solar weighting taken from a linear
combination of the three datasets.
Although the HKKM11 model is taken as the default one in mostSK analyses, it is possible to reweight each event instead based on
the published energy spectrum of other flux models, such as the
Fluka [20] or Bartol [21] models. In general however, these groups
have not released detailed directional distributions for the neutrino
flux at the SK site, and those are always taken from the HKKM07
model.
4.5.4

Systematic Error Estimation

While not strictly a part of the MC simulation, we discuss here themethod of calculation of systematic errors on the SK data, comingfrom uncertainties on the detector reconstruction procedures, and thecross-section and oscillation models. A systematic error database ismaintained by the SK collaboration, which is a codebase from which
the effects of the systematics may be calculated on an analysis-byanalysis basis. This database is the result of thorough systematic error
studies over the > 20 year operational history of the detector, and isconsidered to give an accurate estimation of all known systematic
error sources on the SK data. All analyses in this thesis use the above
database, although in some cases additional systematics that were notpreviously considered are checked and estimated where necessary.
The actual method used by the software is as follows. Firstly, thesoftware requires a defined output binning of reconstructed energy
Erec and reconstructed zenith angle θ, for each sub-sample being used;
we denote all bins in this three-dimensional structure by the index j.
Importantly, this binning can differ depending on the needs of a particular analysis. Each MC event is then input to the software, which
carries the information of the neutrino energy Eν , true zenith angle,
and which sub-sample the event was reconstructed into by the fitting procedure. The software then calculates, for each event and eachsystematic error, the estimated change in each output bin j for a 1 σchange in the given systematic.
As a simple example, we can consider a single-ring µ-like eventwith no identified Michel electron (nd = 0). This event would have a
few-percent chance to migrate into the nd = 1 sample, when considering the “Michel electron tagging efficiency” systematic error source.
It would also have few-percent chance to migrate into a different Erec

85

86

data reconstruction and monte carlo

livetime [days]
sk-i

sk-ii

sk-iii

sk-iv

total

fc and pc

1489

799

518

1993

4799

upmu

1646

828

636

1993

5103

Table 10: Livetimes for each SK period. UPMU data livetimes are longer due
to less strict conditions for good run selection.

bin, considering the energy scale error26 . Thus, the value of the output bin j in which the event is classified would be decreased by a few
percent, and some other bins would gain a value equivalent to the decrease, for those two systematic errors. After looping over all events,
the total effect of all errors (indexed as k) in all bins j is given by theset of coefficients f jk , which represent the fractional shift in each bin
for the 1 σ effect of each error.
A full list of all systematics included in the database is given in Appendix A, with some descriptions of the meaning of each systematicwhen necessary. For the detector related errors, 29 sources are considered and estimated for each SK period (a total of 116 errors), thecross-section uncertainties are accounted for by 17 errors, and finally
there are 6 oscillation parameter uncertainties. The detector errors
have been estimated by a combination of MC studies, and physics
studies on control samples such as atmospheric muons or calibrations as described in Sec. 3.4. The neutrino cross-section errors aretaken from the model uncertainties as discussed in Sec. 4.5.1, andthe oscillation parameter uncertainties are based on the results fromother experiments - in this thesis, we update the oscillation errors to
use the latest parameterizations from the PDG [59].
4.6

data and monte carlo sets

Tables 86 and 11 summarize the detector livetimes and event numbersin the data and MC sets used in the analyses in this thesis (except inthe ν / ν̄ separated analysis), including the interaction mode fractions
estimated by MC, for all reconstructed sub-samples described in thissection. We see that all selected e-like and µ-like sub-samples have ahigh purity of ν e and ν µ CC interactions, respectively.
In terms of energy coverage, looking back to Fig. 18 in Sec. 4.1,
the MC true neutrino energy distributions of the sub-samples indicate that each has a different sensitive range. The FC µ-like sample
extends above 10 GeV, while the e-like sample extends up to 100 GeV(indicating that µ-like events in the fiducial volume beyond 10 GeV
tend to exit the inner detector and be classified as PC events). The
26 That is to say, the uncertainty of Erec in the MC based on e.g. the PMT models, notthe variance of Eν .

856

449

multi-GeV single-ring

multi-GeV multi-ring

712

603

multi-GeV single-ring

multi-GeV multi-ring759

through-going

1564.4

271.7

non-showering

showering( 291.6)(1352.4)(433.7)(783.8)(145.3)(656.5)(795.4)(3235.6)(470.1)(842.8)(3104.7)

110.1

726.3

206.4

350

77

337

400

1684

267

396

1745

(107.0)(697.5)(215.7)(383.0)(73.2)(343.8)(423.9)(1731.8)(252.1)(443.7)(1632.8)

sk-ii

110.0

612.9

193.7

290

54

228

238

1139

140

274

1209

(126.0)(504.1)(168.3)(308.8)(53.3)(237.9)(273.9)(1152.0)(161.9)(299.5)(1100.7)

sk-iii

350.1

1960.7

492.7

1093

237

863

989

4379

634

1060

4251

(274.4)(1690.3)(504.1)(1146.7)(229.0)(927.8)(1051.5)(4394.7)(654.9)(1080.0)(4072.8)

sk-iv

841.9

4864.31324.8

2492

511

2031

2339

10386

1490

2586

10493

(799.0)(4244.4)(1321.8)(2622.3)( 500.8)(2166.0)(2544.7)(10514.0)(1539.0)(2666.0)(9911.0)

total

0.1

0.2

1.0

0.8

12.7

3.4

0.4

0.9

73.0

86.3

94.1

ν e cc

99.8

99.7

98.7

98.6

82.7

91.1

99.5

94.2

10.9

4.9

1.6

ν µ cc

0.1

0.1

0.3

0.6

4.6

5.5

0.1

4.9

16.1

8.8

4.4

nc

int. type [%]Table 11: Event numbers for each SK period for each sub-sample. Numbers in brackets are the MC expectation, based on the HKKM11 [19] flux model,
which are calculated separately for each SK period and then scaled to the data livetime. The oscillation and solar activity weights are included.
UPMU has fractional data event numbers due to the subtraction of horizontally-going cosmic muon backgrounds. The relative frequency ofinteraction type in each sub-sample, estimated from the MC datasets averaging over all SK periods, are shown in the last three columns.

432.0

stopping

upmu

143

stopping

pc

3184

sub-GeV single-ringfc µ-like

3288

sub-GeV single-ring

fc e-likesk-i

number of events

4.6 data and monte carlo sets87

data reconstruction and monte carlo∆ Log10(Eν)

88

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3

SG 1R e-like
MG 1R e-like
MG MR e-like
SG 1R mu-like
MG 1R mu-like
MG MR mu-likePC stop
PC thru
UPMU stop
UPMU non-sh
UPMU shower

0.2
0.1
0
-1 -0.5

UPMU
PC
FC
0

0.5

1

1.5

2

2.5

3

3.5

4

Log10(Eν[GeV] )Figure 36: The energy resolution in the true neutrino energy Eν for eachsub-sample, estimated by MC events.

FC e-like sample also has a lower neutrino energy threshold than theµ-like sample (100 MeV as opposed to 200 MeV), due to the differencein Cherenkov threshold between e± and µ± . The PC and UPMU samples approximately range from 1 GeV to 10 TeV, split across several
sub-samples which again provide some discrimination of the true
neutrino energy range.
We show in Fig. 36 the energy resolution of each sub-sample, takenfrom the MC. We see that for the FC sample our estimation of thetrue neutrino energy per event is distributed around the true energyEν with a 1σ width of 10∼20% in log10 ( Eν ), but slightly less accurate
for PC (20∼30%) and somewhat worse for UPMU (45∼70%) due tothe missing energy deposited outside of the detector.

Part IIIT H E A N A LY S E S
I may not be very good at what I do, but at least I’m slow.
— FuzzDad

5

MEASUREMENT OF THE ENERGY SPECTRA OF
THE NEUTRINO FLUX

5.1

introduction

As explained in Sec. 1.1, knowledge of the atmospheric neutrino flux
density is of vital importance to several current and future generation
neutrino experiments, and to other rare event searches such as thosefor dark matter, proton decay, or astrophysical neutrinos. In particular, the flux as a function of neutrino energy should be well-known,
as the neutrino oscillation effects depend strongly on the neutrinoenergy, and many rare event searches need to know the atmospheric
neutrino background at particular energy scales.
In this section, we aim to reconstruct the energy spectrum of theνe + ν̄e and νµ + ν̄µ fluxes at the Super-Kamiokande detector, to ahigh precision that should be useful for testing to the current flux
simulations, over a wide energy spectrum. After this reconstruction
has been explained in detail, we further attempt to reconstruct theνe , ν̄e , νµ , and ν̄µ fluxes separately. The ν/ν̄ separated samples havemuch smaller statistics and lower purity, such that the number ofdata points is expected to be smaller and the systematic errors larger;
such a study can nonetheless add new information, which may help
to understand e.g. any systematic bias seen in the νe + ν̄e and νµ + ν̄µmeasured fluxes. Although theoretically the systematic errors on thepredicted atmospheric ν/ν̄ ratios are at most a few percent, this measurement is the first direct test of these predictions using atmospheric
neutrino data.
Hereafter, until the section on the ν/ν̄ separated measurement, antineutrinos are not mentioned explicitly and should be assumed to be
included. Also, while ντ induced events have been observed at SuperKamiokande with a high significance [130] compared to the no ντhypothesis, the detector cannot yet make an accurate measurementof their flux, and they are not considered here.

Authorship note: this chapter (5) presents work performed in participation with Prof.
Kimihiro Okumura (ICRR). With regards to the ν + ν̄ study, while the author of thisthesis made contributions such as calculating the systematic errors and performing
cross-checks of the unfolding procedure, Prof. Okumura was the originator of thestudy and primary contributor to the analysis. The ν/ν̄ separated analysis, on theother hand, is an original contribution of this author.

91

92

measurement of the energy spectra of the neutrino flux

5.2

event classification

We use all sub-samples and events shown in Tab. 11. Noting that theFC sample may be well separated by its νe and νµ components, andthat the PC and UPMU samples generally contain νµ interactions, wedefine two data samples for the energy spectrum measurement:
1. a νe sample containing FC e-like events,
2. a νµ sample from all FC µ-like, PC, and UPMU events,
which gives us a generally high-purity selection of νe or νµ induced
CC interactions, across a wide energy range. For each sample, thebackgrounds are defined as CC interactions of other flavour neutrinos, and NC interactions of all flavour neutrinos (non-neutrino backgrounds are assumed to be negligible).
In each sub-sample, events are binned by the reconstructed energy
Erec , as defined in Sec. 4.2.2. Essentially for single-ring events, Erec is
calculated with the reconstructed lepton momentum, and for multiring events the linear sum of each particle’s reconstructed momentum is taken. The binning is defined as shown in Tab. 12, in a unit oflog10 ( Erec [GeV]). These bin widths were determined considering theenergy resolutions of the true neutrino energy Eν for each sub-sample,
which are typically ∼ 20% in log10 ( Eν [GeV]) for the FC and PC stopping events, and larger at 40 ∼ 60% for the PC through-going and
UPMU events (as the energy deposited outside the detector cannot
be observed). No energy binning is performed for UPMU showering
or non-showering events, since there is little separating power of thetrue neutrino energy from the observed energy. We refer hereafter
to these defined bins simply as the “data bins”, and the number of
events in each data bin j as M j . Figure 37 shows the values of M j for
all SK-I to SK-IV data.
5.3
5.3.1

flux unfolding
Overview

To measure the atmospheric neutrino energy spectrum, we employ
an unfolding method. This is a class of method where a true spectrum is deconvolved from an experimentally measured one, basedon a knowledge of the experimental reconstruction. This class of
methods is in contrast to forward-fitting methods, where the reconstructed spectrum is compared to model predictions that have also
been passed through the simulated experimental reconstruction. A
benefit of unfolding methods is that they allow direct comparisonsof the unfolded spectra between different experiments, in a modelindependent way. In our case, the reconstructed energy spectrum M j

5.3 flux unfolding

sub-sample

bin edgesjbins

log10 ( Erec [GeV])

fc e-like
1→5

sub-GeV single-ring

5

multi-GeV single-ring

4

multi-GeV multi-ring3

10 → 12

0.0, 0.4, 0.7, 3.0

sub-GeV single-ring

5-0.8, -0.6, -0.4, -0.2, 0.0, 0.2

multi-GeV single-ring

13 → 172

multi-ring

4

20 → 23-1.0, 0.0, 0.4, 0.7, 2.0

stopping

2-1.0, 0.4, 2.0

through-going

24 → 254

26 → 29-1.0, 0.0, 0.4, 0.7, 3.0

stopping

30.0, 0.4, 0.7, 3.0

non-showering

30 → 32

1

33–

showering

1

34

–

fc µ-like-1.0, -0.6, -0.4, -0.2, 0.0, 0.2

6→90.0, 0.4, 0.7, 1.0, 3.0

18 → 19

0.0, 0.4, 2.0

pc

upmu

Number of Events

Table 12: Energy binning definitions for the data bins M j for the 11 subsamples.

5000

ν µ sample

ν e sample

4000
SK I-IV dataν e CCν µ CC
NC

3000
2000

UPµ stop

PC thru

PC stop

MR µ

MG 1R µ

SG 1R µMR e

MG 1R eSG 1R-e0UPµ non-sh
UPµ shower

1000
jFigure 37: Number of events for each reconstructed energy bin M j for eachsample, for the entire SK-I to SK-IV period. The filled histograms
are the MC truth predictions, and red represents the data counts
with statistical error. Some inconsistency between the MC anddata is seen, which can be accounted for within the estimatedsystematic errors (not plotted here).

93

94

measurement of the energy spectra of the neutrino flux

(with binning defined in Tab. 12) will be unfolded into a true neutrino energy spectrum, of correct flavour CC interactions only, which
we denote NiCC (with binning to be defined later, but note that thedefinition may differ from the reconstructed energy binning).
The relationship between the reconstructed data and the true data
is in general expressed by the experimental response matrix A ji as
M j = A ji NiCC ,(49)

where A ji can be estimated by the detector MC, and accounts forthe inability of the detector to reconstruct perfectly the true neutrinoenergy (e.g. energy smearing or inefficiency). We can write the inverse
relationship using the unfolding matrix Uij as
NiCC = Uij M j .

(50)

1
However, taking Uij as A−
ji is in principle a poor approach, as the
response matrix will not have been estimated perfectly (and may
not even be invertible). Various more advanced techniques or algorithms are available, allowing for a more reliable unfolding to be performed [131].

5.3.2

Bayesian unfolding

We adopt an iterated Bayesian method [132], using the RooUnfold
library [133] for the practical implementation. The method is known
as Bayesian due to its use of Bayes’ theorem in the construction of
the unfolding matrix at each iterative step.1 The RooUnfold library is
chosen as it is known to be reliable and easy to implement.
The specific details of the unfolding procedure are as follows. The
first estimation of the detector response matrix A ji is made using the
SK MC dataset, but using only correct flavour CC events reaching thefinal samples. The background events (i.e. events that were actuallywrong flavour or NC interactions) are considered as an extra possibletrue cause bin i, and filled into an extra column of the matrix. The
contribution of background events in the data can be seen in Tab. 11
to be small, and can be essentially subtracted during the unfolding
procedure to leave a negligible impact on the final results. We also
apply the normalization condition ∑ j A ji = 1, which means that detector inefficiencies are not accounted for in the matrix at this stage –but will be accounted for later, when converting from the measured
number of events to the actual neutrino flux values. Figure 38 shows
the initial estimation of A ji (not including the background events column).
To construct the unfolding matrix, we first define the notation P( j|i )as the probability for an event in true energy bin i to be detected
1 Although the final result is not strictly related to the input data in a Bayesianway [131].

5.3 flux unfolding

in reconstructed energy bin j, which are exactly the elements of the
response matrix A ji . We then define initial prior probabilities P0 (i ) fora final sample event to fall in true energy bin i as
P0 (i ) =

CC
NMC,i
CC
∑k NMC,k,(51)

CC normalized towhich is simply the MC default true spectrum NMC,ione. We can then state Bayes’ theorem

P (i | j ) =P( j|i ) P0 (i ),P0 ( j)(52)

where the normalization constant P0 ( j) is calculated as
P0 ( j) = ∑ P( j|i ) P0 (i ).

(53)

i

Now that we have the estimated inverted probabilities P(i | j), we canuse our data M j to make a first estimation of the true flux values N̂iCC
as
N̂iCC = ∑ P(i | j) M j .

(54)

j

As this estimation takes inputs from both MC and real data, it generally lies somewhere between the prior probabilities and the true
number of events. Thus, it is useful to proceed in an iterative way, byinserting the normalized N̂iCC as an updated set of prior probabilities
P0 (i ) in (52) to generate new P(i | j), and in turn iteratively update
N̂iCC . The final iteration is denoted NiCC , where the operation of theabove procedure on the data essentially took the place of the unfolding matrix Uij in Eq. (50).
This method is seen to cause a variable degree of regularization (orsmoothing) depending on the number of iterations; a low number of
iterations may be too close to the statistically smooth prior values andnot fully reflect the information input by the data, while a high number of iterations will lead statistical fluctuations in the data to distortthe shape of the unfolded spectrum. In general any prior (including aflat prior) will thus cause some bias in the unfolded spectrum whichcannot be perfectly corrected for, but importantly can be accurately
estimated and included as a systematic error; this will be shown in
Section 5.4. In principle, the prior distribution should contain ourbest knowledge of the true spectrum, and the bias is minimized bytaking the nominal MC distribution as the initial prior. In practice,the iterative Bayesian method is strongly data-driven and produces
excellent results compared to other unfolding methods after only a
few iterations; we set the number of iterations to five based on MC
optimizations.

95

1

100 MeV...100 GeV 200 MeV........10 TeV

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2

UPµ non-sh
UPµ shower

UPµ stop

PC thru

PC stop

MR µ

MG 1R µ

SG 1R µMR e

SG 1R-e

0.1
MG 1R eµmeasurement of the energy spectra of the neutrino flux

e

96

0

Super-Kamiokande Event Samples

Figure 38: The initial estimation of the detector response matrix A ji for the
SK-I period, where the x-axis represents the M j bins as definedin Tab. 12, and the y-axis the NiCC bin as defined in Tab. 13. (If
we take the arrangement of the axis definitions seriously however,the figure actually shows Aij−1 .)5.3.3

Binning definitions for the output spectra

The energy range of the unfolded atmospheric neutrino flux spectrum
NiCC is defined in units of log10 ( Eν [GeV]), for the νe (νµ ) flux in therange -0.8→2.0 (-0.6→3.0), and divided into 11 (12) bins of variable
size. The binning is shown later, together with the results, in Tab. 13
on page 105. This energy range, and the bin widths, are again determined considering the coverage and resolution of the true neutrinoenergy in the reconstructed data, as was shown in Fig. 36 on page
88. Smaller bins of width 0.2 are adopted below 10 GeV due to therelatively finer energy resolution of the FC sample, and wider bins
are adopted for higher energies due to the deterioration of the energy
resolution in the PC and UPMU samples.
5.3.4

Calculation of the absolute flux values

We obtain the measured neutrino flux values at the detector positionναΦiνα (where α = e, µ) by using the predicted flux values ΦMC( Ēi ), theCC
predicted number of CC interactions NMC,i , and the unfolded numberof CC interactions NiCC , asαΦiνα = ΦνMC
( Ēi ) ×CC
∑SK Ni,SKCC∑SK NMC,i,SK(55)

where ∑SK is a sum over the four SK periods, as considering thedifference in detector configurations an unfolding must be performed

5.4 statistical and systematic uncertainties

separately for each period (denoted by the subscript SK). Here, theναpredicted flux values ΦMC( Ēi ) are calculated at the mean energy ofναeach bin i, by integrating the predicted flux spectrum ΦMC( E) acrossthe energy range of each bin (according to [134]) as1ναΦMC( Ēi ) =Ei+1 − Ei

Z Ei+1EiναΦMC( E)dE(56)

where Ei and Ei+1 are the lower and upper edges of i-th energy bin.
Here, the predicted flux spectrum is itself the integral of the differential atmospheric neutrino flux model by solid angle (including neutrino oscillation) as follows:ναΦMC( E) =

Z
4π

 να
ν̄α
dΩ φMC( E, θ, φ) × Oνα ( E, θ ) + φMC( E, θ, φ) × Oν̄α ( E, θ ) ,(57)

α ,ν̄αwhere φνMC
( E, θ, φ) is the differential neutrino or anti-neutrino flux,
as a function of neutrino energy E and the zenith and azimuthal angles of arrival direction θ and φ, which is taken by default from theHKKM11 model with the solar activity parameter set to its middle
degree. Furthermore, Oνα ,ν̄α ( E, θ ) is the neutrino or anti-neutrino oscillation probability from the production point to the detector; as discussed in the introduction, this is calculated with the standard threeflavor oscillation model with the oscillation parameters as shownin Table 1 (including the matter effect when propagating inside theEarth). Finally, for shorthand, we also define Φi (with no superscript)
νas the final binning containing both the Φiνe and Φi µ values.

5.4

statistical and systematic uncertainties

5.4.1

Overview

In addition to the statistical uncertainties, the systematic errors in theanalysis can be considered to relate to one of four categories:
a. event reconstruction by the detector,
b. neutrino cross-sections,c. neutrino oscillation parameters,d. systematic bias in the unfolding procedure.
As will be described in this section, the uncertainties on A and B
are estimated by the SK collaboration and are treated by modifying
the data bins M j . Next, the uncertainties on C are estimated by the
PDG [59] and treated by modifying Oνα ( E, θ ) and Oν̄α ( E, θ ) in Eq. (57).

97

98

measurement of the energy spectra of the neutrino flux

Finally, the effect of D is estimated specifically for this analysis, andis also treated by modifying M j .2
While it may seem more natural to think of the detector uncertainties A as affecting A ji and the neutrino uncertainties B and C as
CC , our treatment of modifying M combined with theaffecting NMC,ijestimation of D should be equivalent, and much less computationallyintensive. The equivalence was checked and confirmed with a few
sample systematics.
5.4.2

Reconstruction and cross-section errors

The reconstruction and cross-section systematic error effects are calculated from the SK systematic database, which was described inSec. 4.5.4. For this analysis all errors in the database, as summarizedin Appendix A, are considered. Their effects are calculated as systematic error coefficients f jk , which represent the fractional shift ineach data bin M j resulting from a 1 σ shift of the k-th systematic error
source.
For this analysis, the modified expectation of the number of eventsin each bin M̃ j considering the effects of all errors would then become
Nsys

M̃ j = M j ×

1 + ∑ f jk gk

!
,

(58)

k

where Nsys is the number of systematic error sources, and the gk represents the applied strength of each systematic in units of σ. The
estimated error sizes on the final measured flux values Φiνα will be
propagated by a toy MC method, which is described later in this section, using randomly-generated Gaussian sets of gk .
5.4.3

Oscillation parameter uncertainties

The uncertainties of the oscillation parameters are shown in Tab. 1,
and are assumed to be Gaussian (except for δCP where uniform probability is considered, and for the mass hierarchy where both possible
values are considered). The correlation among these parameters are
not taken into account. By directly shifting the values of Oνα ( E, θ )αand Oν̄α ( E, θ ) in Eq. (57), the error on the final values ΦνMCin Eq. (57)α ,tmay be directly estimated. We generate toy flux spectra ΦνMC
withrandomly shifted oscillation uncertainties.
2 We will see that although unfolding methods are sometimes disfavoured in the highenergy physics community (due to inaccurate error reporting in previous cases), if
the particular case under study is well-understood then the errors can be accurately
estimated. Then, the bias originating due to inaccuracies in the MC or theoreticalmodel should be no worse than in commonly-used forward-fitting methods.

5.4 statistical and systematic uncertainties

5.4.4

Regularization uncertainties

Another contribution to the uncertainty in the values NiCC is the regularization error; although the iterative procedure is data-driven, someinfluence from the MC shape still remains. In other words, the regularization error is due to inaccuracy in the initial estimation of the
CC , but must be applied
response matrix A ji and Bayesian prior NMC,iin addition to the systematic errors on those estimations (which wereapplied directly on the data bins M j ). Any bias due to regularization
should be relatively small, but it may be noticeable in unfolded bins
with low statistics. This bias cannot be corrected for exactly, but wemay estimate an associated error by unfolding pseudo-data sets withan energy spectrum reasonably far from the default MC prediction,and observing the resulting difference.
Such pseudo-data sets were produced from the MC data by reweighting the true neutrino energy spectral index α and overall normalization γ, defining a new binned spectra Φ0MC,i according to0ΦMC,i( Ēi ) = (1 + ∆α)



Ēi
1 GeV

∆γ

ΦMC,i ( Ēi ).

(59)

A modification range of ±0.05 in α and ±20% in γ was considered,
which corresponds roughly to a 1 σ deviation in the predicted spectra,
according to the estimated systematic uncertainties in the flux modelthat were previously shown in Fig. 9 on page 25.
Figure 39 shows the comparisons of several input pseudo-data sets
and their unfolded flux spectra, along with the fractional deviation
between the input and unfolded output in each case, for both νe andνµ . The difference becomes largest in the highest energy bins, andthe error sizes are ±6% and ±8% at most in the case of a ±0.05 and±20% change. The RMS of the differences are taken as the estimateduncertainties due to regularization, and implemented as a systematic
error coefficient f jk .
5.4.5

Propagation of the statistical and systematic uncertainties

For propagation of both the systematic and statistical uncertainties
from M j to the flux values Φi , we employ a toy MC method. 3 In
this method, pseudo-data sets are generated by randomly fluctuating
MC data according to their systematic and statistical error probability distribution functions (PDFs), as follows. First, the systematicallyshifted pseudo-data M̃ j are calculated by Eq. (58) in Sec. 5.4.2, where
the gk are generated randomly each time, assuming that their PDFs
3 While RooUnfold provides an accurate theoretical calculation of the statistical errors [133], simultaneous treatment of systematic uncertainties within the software is
not implemented.

99Φ / ΦMC

measurement of the energy spectra of the neutrino flux

1.8
1.6
1.4
1.2
1
0.8
∆ α = +0.1, ∆ γ = -0.025∆ α = -0.1, ∆ γ = +0.025∆ α = +0.2, ∆ γ = -0.05∆ α = -0.2, ∆ γ = +0.05

0.6
0.4

Deviation

0.2

0.15-1
0.1
0.05
0
-0.05
-0.1
-0.15
-1

-0.5

0

0.5

1

1.5

2

2.5

-0.5

0

0.5

1

1.5

2

2.5

3

3

3.5

4

3.5

4

3.5

4

3.5

4

Log (Eν /GeV)
10Φ / ΦMC(a) νe

1.8
1.6
1.4
1.2
1
0.8
∆ α = +0.1, ∆ γ = -0.025∆ α = -0.1, ∆ γ = +0.025∆ α = +0.2, ∆ γ = -0.05∆ α = -0.2, ∆ γ = +0.05

0.6
0.4
0.2

Deviation

100

0.15-1
0.1
0.05
0
-0.05
-0.1
-0.15
-1

-0.5

0

0.5

1

1.5

2

2.5

-0.5

0

0.5

1

1.5

2

2.5

3

3

Log (Eν /GeV)
10(b) νµFigure 39: Pseudo-data input and the unfolded energy spectra (upper) andtheir ratios (lower) for νe and νµ . The vertical axis is the ratio tothe HKKM11 [19] model ΦMC . Four sets of pseudo-data inputs
are tested, where ∆α and ∆γ are the deviations of the flux normalization and spectral index respectively.

5.5 results and discussions

are Gaussian-distributed and uncorrelated. This includes the regularization error, as described in Sec. 5.4.4. Then, the statistical errors
are applied by generating the final toy data Mtj using Poisson distributions with means M̃ j . These data Mtj are unfolded, using themethod as described in the previous section, into toy measurements
να ,tNiCC,t . Toy flux spectra ΦMC( E) are also generated as was describedin Sec. 5.4.3.
να ,tTwo thousand sets of toy data NiCC,t and ΦMC( E) (for t = 1...2000)
are generated, from which we calculate the set of toy flux values Φitby Eq. (55). The 1 σ deviation of these toy measurements are taken
as the final uncertainties ∆Φi , including all systematic and statisticalerror sources.
The error covariance matrix Cij , which is a measure of the correlation of the effect of systematic errors in each pair of flux bins Φi , isalso estimated at this stage using the toy MC sets asih
Cij = E Φit − E[Φit ] Φtj − E[Φtj ](60)
where E denotes the expected value, i.e. the mean, across the toy sets
t.4 The diagonally-normalized covariance matrix Ĉij , which will be
used later for the χ2 calculation, is defined asĈij = Cij /∆Φi ∆Φ j

(61)

and is shown in Fig. 40.
5.4.6

Error source comparisons

Figure 41 shows the breakdown of the estimated uncertainties foreach error source. To produce this figure, the absolute errors of thefive groups (statistical, detector response, neutrino interaction, neutrino oscillation, and regularization), are individually calculated, by
repeating the above toy MC method but without inclusion of theother four groups. It can be seen that currently, the neutrino crosssection errors are the dominant source of systematic uncertainty.
5.5
5.5.1

results and discussions
Measured flux spectra

Figure 42 shows the obtained νe and νµ energy spectra, using all SK-I
to SK-IV data. The energy binning, average bin energy (x-axis values),
and measured value and error (y-axis values and errors), are also
written in Tab. 13.
4 Also note that in this case, the index j does not refer to the data bins, but is simply
the second index of the covariance matrix.

101measurement of the energy spectra of the neutrino flux

1
0.8
0.6
0.4
0.2

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

0
-0.2
-0.4
-0.6
-0.8
-1

Figure 40: Normalized error correlation matrix Ĉij of the observed flux values Φi . The matrix contains both νe and νµ , with νe ranging overbins 1 → 11 and νµ over bins 12 → 23. Strong correlations
are seen between νe and νµ as many systematics apply to both
flavours in similar ways, although some anticorrelated systematics also exist.

∆Φ/ Φ [%]

102

35

νµ

νe

30

total
statisticalcross section
detector
oscillation and regularization

25
20
15
10
5
0
0

2

4

6

8

10

12

14

16

18

20

22
i binsFigure 41: Breakdown of the estimated uncertainties in the measured flux
values depending on each error group (statistical, detector response, neutrino cross section, neutrino oscillation, and regularization). The x-axis represents the output spectrum bins as defined in Tab. 13. The “total” histogram represents the error when
all error sources are considered simultaneously.

E2Φ [GeV/cm2/sec/sr]

5.5 results and discussions

10-1
10-2

νµ

10-3
10-4

νe

Ratio

Ratio

10-5

1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2

νe

1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
-1

νµ

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Log (E / GeV)
10

Figure 42: The measured flux energy spectra, using all SK-I to SK-IV data,for νe (blue) and νµ (red). The flux values are shown multiplied by E2 , following a common convention. Error bars include all statistical and systematic uncertainties. The curves show
the HKKM11 [19] model’s prediction with (solid) and without(dashed) neutrino oscillation (same color scheme). The lower partof the figure shows the ratio of data / HKKM11 prediction.

103

measurement of the energy spectra of the neutrino flux

The measured energy spectrum agrees with oscillated HKKM11
flux, within the estimated uncertainties. The unoscillated flux is alsoplotted, where the deficit of νµ flux due to neutrino oscillation becomes apparent below 100 GeV.

Ratio

5.5.2

Ratio

104

Comparison to flux models

1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2

νe

1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
-1

νµ

HKKM11

BartolHKKM07

Fluka

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Log (E / GeV)
10

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

The observed fluxes are compared graphically to several flux models, including the HKKM11 [19], HKKM07 [18], Fluka [20], and Bartol [21] models in Figure 43.
To perform a quantitative comparison, we perform a χ2 test whichtakes into account the error correlation matrix, defined as
N N

χ2 = ∑ ∑(Φi − ΦMC,i )T Cij−1 (Φ j − ΦMC,j )
i

(62)

j

where Φi is the observed flux and ΦMC,i is the expectation of the flux
model being tested, at the i-th energy bin, and Cij is the error covariance matrix (calculated as described in Sec. 5.4 and shown in Fig. 40).
Both statistical and systematic uncertainties are thus taken into account in the χ2 calculation, and the results are shown in Table 14
for three cases: νe and νµ , νe only, and νµ only. Considering the combined νe + νµ test, the χ2 values do not differ significantly between
the flux models, however HKKM11 has the best fit to the data. As
our measurement prefers HKKM11 above HKKM07, we can identify

5.5 results and discussions

energy range
log10 ( Ei [GeV])

average energylog10 Ei [GeV]

measured flux

error

[GeV cm−2 sec−1 sr−1 ][%]1.21 × 10−2

±17

νe

−0.8 → −0.6

−0.6 → −0.4

−0.4 → −0.2
−0.2 → 0.0
0.0 → 0.2

−0.71

−0.51

1.46 × 10−2

−0.09

1.37 × 10−2

0.10

0.2 → 0.4

0.30

0.6 → 0.8

0.70

0.4 → 0.6
0.8 → 1.0
1.0 → 1.5

1.72

−0.6 → −0.4

-0.51

−0.2 → 0.0
0.0 → 0.2

0.2 → 0.4

0.4 → 0.6

0.6 → 0.8

0.8 → 1.0

1.0 → 1.5

1.5 → 2.0

2.0 → 3.0

3.0 → 4.0

-0.32
-0.09
0.10
0.30
0.51
0.71
0.90
1.21
1.73
2.40
3.39

±21

9.42 × 10−4

νµ

±15

6.09 × 10−3
2.32 × 10−3

1.22

±15

±17

3.73 × 10−3

0.90

±16

1.16 × 10−2

8.55 × 10−3

0.50

1.5 → 2.0

−0.4 → −0.2

1.50 × 10−2

−0.27

2.03 × 10−4

±19
±22
±20

±15
±18

1.58 × 10−2±23

1.86 × 10−2

±16

1.77 × 10−2
1.68 × 10−2

1.38 × 10−2

9.59 × 10−3

6.68 × 10−3

4.79 × 10−3

2.62 × 10−3

1.20 × 10−3

2.49 × 10−4

1.46 × 10−5

±18
±18

±20

±23

±23

±19

±14

±14

±16

±19

Table 13: Neutrino flux binning NiCC , and measurement results, using SK-I
to SK-IV data. The measured flux values Φi are given multiplied2

105

by Ei , following a common convention. The error includes both
statistical and systematic uncertainties.

106

measurement of the energy spectra of the neutrino fluxχ2
ν e (11 dof)ν µ (12 dof)

ν e and ν µ (23 dof)

hkkm11

8.5

19.5

21.8

hkkm07

15.4

21.7

22.2

bartol

10.9

28.8

30.7

fluka

9.0

18.7

25.6

Table 14: The results of a χ2 test of several flux models against the measured flux, according to Eq. (62). The number of degrees of freedom (DOF) in each test are also shown.

the updates to the hadron simulation [19] (which cause changes inthe energy region below 1 GeV) as bringing slightly better agreementwith the data. Looking at each flavour individually, the agreement forthe predicted νe spectra for each model is good, however the νµ testshows a slight inconsistency at the 2.0 σ level for the HKKM07 model,
and at the 3.0 σ level for the Bartol model.5
We also evaluate the agreement of the data with each model’s normalization γ and spectral index α, by testing modified energy spectra
0ΦMC,ibased on each model ΦMC,i , which were defined previously in
Eq. (59). The χ2 is then calculated by Eq. (62), and the best-fit valuesof the ∆α and ∆γ parameters are obtained by minimizing the χ2 , andare shown in Fig. 44. The normalization and spectral index agrees
within the 1 σ error for every model, except from the fitted spectral
index of the Fluka νµ spectrum which deviates by 2.4 σ.
In summary, while none of the current flux models are generally
inconsistent with our data measurement, there are some tensions inthe νµ spectrum for the HKKM07 and Bartol models, and in the spectral index of the Fluka model. The HKKM11 model gives the best fit
to the current data.
5.5.3

Cross-check by SK period

Considering the final unfolded spectra in Fig. 43, some systematic
shift in shape is clearly seen for both the νe and νµ unfolded fluxes. In
order to verify the consistency of this tendency across each SK period,
the energy spectrum unfolding of each period is calculated and compared in Fig. 45, where only statistical errors are shown. While theerror bars are relatively large, the spectral distortion seems consistentacross SK periods, and not attributable to statistical error. The origin
5 Due to the error correlation, the χ2 statistic for the νe + νµ test is not simply a combination of the χ2 statistic of the individual tests. In principle we should also definea trials factor for the individual tests, since we are applying a related test to a subsetof the data, but such a factor is assumed to be small as the tests are not completely
correlated.

Normalization ( ∆ α)

5.5 results and discussions

0.2
0.1
0
-0.1

Spectral index (∆ γ )

-0.2

νµ

νe

0.06
0.04
0.02
0
-0.02
-0.04
-0.06

HKKM11 HKKM07

Bartol

Fluka

Ratio

Ratio

Figure 44: The best-fit and 1 σ error of the normalization and spectra index
parameters ∆α and ∆γ, obtained by χ2 calculation against eachmodel with the parameters artificially shifted.

1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2

νe

1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
-1

νµ

-0.5

SK1

SK3

SK2

SK4

0

0.5

1

SK1-4

1.5

2

2.5

3

3.5

4

Log (E / GeV)
10

Figure 45: Flux ratio to the HKKM11 model for each SK period. Error bars
include only statistical uncertainty in this figure.

107

measurement of the energy spectra of the neutrino fluxof the spectral distortion is however within the range of the estimatedsystematic effects, in particular the neutrino interaction uncertainties,
and so at present we cannot identify whether or not this indicates adirect disagreement with the HKKM11 model.
5.5.4

E2 Φ [GeV cm-2 sec-1 sr-1]

108

Comparison to other measurements

10-1
10-2
10-3
10-4
10-5

Super-Kamiokande I-IV ν eSuper-Kamiokande I-IV ν µ

10

-6

10

-7

Frejus ν eFrejus ν µ
IceCube/DeepCore 2013 ν e

10-8
10-9
-1

IceCube 2014 ν e
IceCube ν µ unfolding
IceCube ν µ forward folding
AMANDA-II ν µ unfolding
AMANDA-II ν µ forward folding

0

1

2

3

4

5

Log (E/GeV)
10

Figure 46: The measured energy spectra of atmospheric νe and νµ at SK,
shown in comparison to measurements by Frejus [28], AMANDAII [29, 30], and IceCube [31, 33, 32]. The HKKM11 flux curves
are also shown in solid (with oscillation) and dashed (without
oscillation) lines.

Figure 46 shows our flux measurement compared with the resultsfrom other experiments. Our measured data provides significantly
improved precision up to 100 GeV, and a direct measurement is madefor the first time below 320 MeV. At higher energies, our measurement overlaps with the AMANDA-II and IceCube νµ measurements.
In the future, this overlap should allow a better constraint to be placedon the flux normalization at the energies beyond 100 GeV, which will
be helpful to the understanding of the astrophysical neutrino flux.

5.6 measurement of neutrino and antineutrino fluxes

5.6
5.6.1

measurement of neutrino and antineutrino fluxes
Data selection and binning

In this section we make use of the neutrino and antineutrino samples, as defined in Sec. 4.2.3. As a reminder, we cannot separate on
an event by event basis, but have some ability to statistically “enrich” a sample’s ν or ν̄ content by a certain percentage, based on the
kinematic differences. In principle, we could then measure the ν/ν̄
ratio directly if we completely understand and account for the crosssection and kinematic differences between any two data samples, although in practice this is quite difficult.6 In the samples defined inSec. 4.2.3 we thus additionaly made use of the theoretical ν/ν̄ ratios
in order to estimate the ν and ν̄ content in each sample – therefore
we should keep in mind that any measurement with these samples
is model-dependent. Nonetheless, the errors on the theoretical prediction of the ν/ν̄ ratios are of the order 1%, and are not calculated
directly from the normalizations, which have errors of the order 10%.
A measurement of these samples can thus provide new informationon the ν and ν̄ normalizations due to both kinematic or flux normalization differences.
The purity and statistics of the various ν/ν̄ separated samples, as
was shown in Tab. 6 on page 72, is quite variable. We wish to perform a sub-selection of these samples to define νe -like, ν̄e -like, νµ -like,and ν̄µ -like samples, which are as high-purity7 and high statistics as
possible. With regards to the purity and statistics, increasing one generally comes at a cost to the other, so the criteria for choosing the“best” samples is somewhat arbitrary. We choose a guiding principlethat a sample should be at least 70% pure to be included in the ν-likesamples, and ∼ 30% (∼ 40%) pure to be included in a ν̄e -like (ν̄µ like) samples. The requirements for the ν̄ samples are of course lower,
due to the predominance of ν over ν̄ in the atmospheric flux before
separation.
Firstly, both of the π 0 -like samples and the “multi-ring other” sample are discarded, due to their high NC and ντ backgrounds. The
PC and UPMU samples, while not shown in Tab. 6, generally haveνµ and ν̄µ purities of around 60% and 30% respectively, and these
6 For example, a measurement of the νµ /ν̄µ ratio was attempted by considering thedifference in the µ-like single-ring nd = 0 and nd = 1 samples (i.e. without orwith a Michel electron), whereby about 20% of µ− are captured on oxygen, thus
enriching the νµ content of the nd = 0 sample by a known amount. However, evenin this simple case, many second-order kinematic effects were seen contribute, andthe study was not completed on time to be included here.
7 While the estimated wrong flavour, wrong sign, and NC backgrounds will be subtracted during the unfolding process, and the uncertainty of the background subtraction is accounted for in the systematic error estimations, a high-purity sample
minimizes the bias and also protects against spurious results in the (pessimistic)
case where some systematics were not considered or were underestimated.

109

110

measurement of the energy spectra of the neutrino flux

samples are also discarded. For the remaining FC samples, a study
is performed examining the purity as a function of the reconstructed
momentum Erec , in case it may be possible to fulfill our stated purity
requirements by choosing binning that uses only energy sub-rangesof some samples. This is found to be possible only for the “single-ring
sub-GeV µ-like nd = 1” sample, where we can raise the ν̄µ purity to∼ 40% by using only events with 1000 < Erec < 1330 MeV.
The remaining samples and energy-ranges that satisfy our purity
requirements are then selected, with the exception of the “single-ring
sub-GeV e-like nd = 0” sample – while the νe purity is high in thissample, the ν̄e contamination is somewhat higher than the other νe like samples, and by discarding this sample we have the ability toperform an extremely good separation of νe and ν̄e in our νe -like sample. The final data samples and binning are shown in Tab. 15. Compared to the ν + ν̄ study, the low statistics present more of a problem
than the energy resolution, and the binning is generally chosen to
keep statistical errors below 20% for each bin (even for the smaller
SK-II dataset). Also, the maximum Erec limit in some samples is also
reduced, due to the much lower statistics at high energies. In summary, we obtain the following (relatively speaking):• νe -like sample: high purity, low statistics• ν̄e -like sample: low purity, medium statistics (covers Erec > 1330 MeV
only)• νµ -like sample: high purity, high statistics• ν̄µ -like sample: medium purity, medium statistics ( covers Erec >1000 MeV only).
For the output binning, shown later in Tab. 16, we again consider thepredicted resolution and energy coverage of Eν compared with Erec ,and modify the Eν bins to minimize the statistical error in each bin.
The first estimation of the detector response matrix by MC is shownin Fig. 47.
5.6.2 Systematic errors
5.6.2.1 New systematic errors
The accuracy of the division of samples into ν and ν̄-like have associated systematic errors that must be estimated, primarily comingfrom the accuracy of the neutron and Michel-electron tagging procedures, while other systematics were shown in [84] to be relatively
smaller. These errors control migration between νe and ν̄e , and between νµ and ν̄µ ; we assume that the uncertainty on the backgroundof wrong-flavour and NC interactions, which was already estimatedSpectrum binning (ν e and ν µ)

5.6 measurement of neutrino and antineutrino fluxes

9
8
100
7
80

6
5

60
4
40

3
2

20
1

Spectrum binning (ν e and ν µ)

0
0

2

4
6
8
10
12
Data binning (ν e-like and ν µ-like)

0

6
60
5
50
4
40
3

30

2

20

1

10

0
0

1

2
3
4
5
6
7
Data binning (ν e-like and ν µ-like)0

1
Figure 47: The initial estimation of the detector response matrix A−
ji for theν/ν̄ separated unfolding, showing ν (top) and ν̄ (bottom) for SK-I.

111

112

measurement of the energy spectra of the neutrino flux

bin edges
sub-sample

bins
log10 ( Erec [GeV])ν e -likesingle-ring sub-GeV e-like n d = 12-1.0, 0.3, 0.2

single-ring multi-GeV ν e -like

1

0.0, 1.7

single-ring multi-GeV ν̄ e -like

30.0, 0.25, 0.5, 2.2

multi-ring ν̄ e -like

1

0.0, 2.2

single-ring sub-GeV µ-like n d = 05-0.8, -0.6, -0.4, -0.2, 0.0, 0.2

single-ring sub-GeV µ-like n d = 22-0.8, -0.2, 0.2

multi-ring ν µ -like

3-0.8, 0.1, 0.3, 1.2

single-ring sub-GeV µ-like n d = 1

10.0, 0.2

single-ring multi-GeV ν̄ µ -like

20.0, 0.4, 1.0ν̄ e -likeν µ -likeν̄ µ -like

Table 15: Energy binning definitions for the data bins for the ν/ν̄ separated
unfolding.

in the νe + ν̄e and νµ + ν̄µ samples, is similar in the ν/ν̄ separated samples. The decay electron tagging efficiency uncertainty is modeled in
the SK systematic error database, causing normalization changes ataround the 1∼10% level depending on the sub-sample and SK period,
and was included on the samples separated by nd .
The uncertainty on the neutron tagging efficiency was estimatedin [84] using a Americium/Beryllium neutron source, producing an
accurately known intensity of low energy neutrons, placed in thetank; the results from this measurement were compared with the neutron tagging MC, and the difference between the two of ∼ 10% was
taken as the systematic error. For the MC itself, two simulations wererun using different software packages (FLUKA and GEANT3), however
the difference between them was negligible. The uncertainty on the
decay electron tagging efficiency is included in the SK database, similarly being calculated by a comparison between MC and cosmic-ray
muon decay electron data, and is at the 1% level. Finally, the systematics on the hadronic differences in the interactions were also estimatedin [84] by comparing various theoretical (e.g. NEUT) and experimental data (e.g. the CHORUS νµ beam experiment), and found at the2 ∼ 10% level, depending on the sample and energy range.
The final reconstruction systematic errors on the ν/ν̄ separation forthe subsamples used in this analysis were calculated as follows: approximately 5% for single-ring multi-GeV νe -like, 1% for single-ring
multi-GeV ν̄e -like, 1% for multi-ring ν̄e -like, 1.5% for multi-ring νµ -

Unfolded / true

5.6 measurement of neutrino and antineutrino fluxes

1.5
1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5
-1

νe
∆ α = +0.1, ∆ γ = -0.025∆ α = -0.1, ∆ γ = +0.025∆ α = +0.2, ∆ γ = -0.05∆ α = -0.2, ∆ γ = +0.05

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Unfolded / true

Log10(Eν [GeV])
1.5
1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5
-1

νe
∆ α = +0.1, ∆ γ = -0.025∆ α = -0.1, ∆ γ = +0.025∆ α = +0.2, ∆ γ = -0.05∆ α = -0.2, ∆ γ = +0.05

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Unfolded / true

Log10(Eν [GeV])
1.5
1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5
-1

νµ
∆ α = -0.1, ∆ γ = +0.025∆ α = +0.1, ∆ γ = -0.025∆ α = -0.2, ∆ γ = +0.05∆ α = +0.2, ∆ γ = -0.05

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Unfolded / true

Log10(Eν [GeV])
1.5
1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5
-1

νµ
∆ α = -0.1, ∆ γ = +0.025∆ α = +0.1, ∆ γ = -0.025∆ α = -0.2, ∆ γ = +0.05∆ α = +0.2, ∆ γ = -0.05

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Log10(Eν [GeV])

Figure 48: The regularization error estimated on the four ν/ν̄ separated samples.

like, and 2% for single-ring multi-GeV ν̄µ -like, where the percentages
are given by averaging the systematic error over the included energy
ranges.8
5.6.2.2

Recalculation of previously defined errors

The regularization error, as was estimated in Sec. 5.4.4, was re-estimatedusing the same method, with the result shown in Fig. 48. For νe andνµ , generally the uncertainties are at the 1% level, rising to the 3%
level for bins on the edge of the distributions; by keeping the number of output bins small, we see that we were able to retain a similar
8 Although the errors consider migration between e.g. νe and ν̄e , the percentages may
be larger on one sample than the other, due to the relative size of the samples.

113

114

measurement of the energy spectra of the neutrino flux

accuracy to the ν + ν̄ study, which had < 1% in the central bins andsimilarly 3% for the edge bins. For the antineutrino measurement
however, the errors are much larger at the 5 ∼ 15% level. This is dueto the fact that number of input bins is small, and they are crowded
together a relatively narrow range of Erec , which leads to a poorer
resolution in true neutrino energy Eν . Unfortunately, it is difficult toimprove the choice of binning without a significant loss of purity.
The oscillation errors, SK systematic errors, and statistical errors
were also re-calculated for the new input and output binnings. The
oscillation errors are similar to the previous study, at the 2 ∼ 3%
level for most output bins (in the previous study a small numberof the νe + ν̄e bins had an error of < 1%, which was possible dueto their narrow energy range). The systematic errors are also similar,
with an additional 1 ∼ 3% error coming from the ν/ν̄ separation
uncertainties. The statistical errors on the other hand are somewhat
higher in the new study, at the 8 ∼ 10% level for all νe , ν̄e , and ν̄µbins, and slightly lower at 5% for νµ (whereas in the ν + ν̄ study, theerrors were generally at the ∼ 3% level, although somewhat moreenergy-dependent and rising to 8 ∼ 10% in the very highest energybins).
5.6.3 Results and discussions
The unfolded fluxes are shown graphically in Fig. 49, numericallyin Tab. 16, and compared with various flux models in Fig. 50. The
statistical error components are also plotted separately, to illustrate
their relatively larger size compared to the previous study. The results mostly agree with the flux models within statistical and systematic error, except for the νe flux in the region below 1 GeV, which ismeasured higher than predicted, and the νµ flux in the 2 to 3 GeV
region, which is slightly lower than predicted.
For the νe flux, considering the strong correlation of the systematic
errors between energy bins, and that all νe data bins are higher than
predicted, the νe measurement suggests that the flux is systematicallyhigher than the default prediction. As the ν̄e flux is slightly lower,some contribution of this increase may be due to the νe /ν̄e anticorrelated systematics; but as they are only at the few-percent level, there
may also be some contribution due to cross-section differences (whichhave the largest systematic uncertainty, and affect the ν and ν̄ fluxes
differently). For the data point below 1 GeV, we unfortunately do nothave any ν̄e data to compare with, as we were not able to make aclear ν̄e -enhanced measurement in that region. However if the νe fluxis already systematically high, the upwards fluctuation of this bin (already significant at less than 2 σ) could also be reasonably explained
as a statistical fluctuation. In any case, as the flux models agree well in5.6 measurement of neutrino and antineutrino fluxes

this region, this datapoint does not have strong discriminating power
between the flux models.
For the νµ flux, we see the same general systematic shape as wasseen in the νµ + ν̄µ unfolded flux. However, the dip at around the2 ∼ 3 GeV region is more pronounced in the unfolded νµ flux, andless so in the unfolded ν̄µ flux. In this case the disagreement is hard to
explain within the estimated νµ /ν̄µ anticorrelated systematics, again
at the few-percent level. Therefore, we again conclude that the default
cross-sections may be more accurately modeled for ν̄µ than νµ at thisenergy range.
Approaching the TeV scale, the errors on the predicted ν/ν̄ ratios
increase substantially (from the SK database, a 16% error at 1 TeV
is assumed for the HKKM11 model). As can be extrapolated from
Fig. 50, the νµ /ν̄µ ratio of the Bartol flux strongly disagrees with theother models. We thus see that if they could be extended to a higher
energy scale, this kind of measurement could be particularly usefulin discriminating between the flux models.
In general, we presented here the first separated measurement ofthe ν and ν̄ atmospheric neutrino fluxes. Despite being a modeldependant measurement, it may be interpreted as providing more
information on the relative normalizations of the ν and ν̄ fluxes whencompared to the ν + ν̄ measurement. Our results generally agree withthe flux models, but do hint at some interesting ranges for further examination. In particular, the planned addition of gadolinium to SK
should allow an excellent separation of neutrinos and antineutrinos,
in turn allowing a more accurate investigation of the systematic differences in the event numbers, and a direct measurement of the ν/ν̄
ratios.

115

116

measurement of the energy spectra of the neutrino flux

energy range
log10 ( Ei [GeV])

average energylog10 Ei [GeV]

measured flux

error

[GeV cm−2 sec−1 sr−1 ][%]1.09 × 10−2±20

2.15 × 10−3±20νe-0.6→0.0
0.0→0.3-0.3

0.3→1.5

0.82

0.1→0.4

0.25

0.4→0.7

ν̄e

1.17-0.6→-0.3-0.45-0.1→0.1

0.01

0.1→0.3
0.3→0.5νµ0.0→0.2

0.01

0.2→0.4
0.4→1.5

±23

ν̄µ

±18

1.00 × 10−2±17

2.23 × 10−3

0.88

±18

±18
±23
±22

9.10 × 10−3

±23

2.57 × 10−3

±28

6.76 × 10−3

0.29

±21

8.27 × 10−3

5.08 × 10−3

0.40
0.95

5.84 × 10−4

8.41 × 10−3

0.20

0.5→1.5±24

1.00 × 10−2

-0.20

±21

5.06 × 10−3

2.94 × 10−3

0.54

0.7→2.0-0.3→-0.1

6.77 × 10−3

0.16

±21

Table 16: Neutrino flux binning, and measurement results, for the ν/ν̄ separated unfolding using SK-I to SK-IV data. The measured flux val2

ues Φi are given multiplied by Ei , following a common convention.
The error includes all statistical and systematic uncertainties.

E2 Φν [GeV cm-2 sec-1 sr-1]

5.6 measurement of neutrino and antineutrino fluxes

10-2
10-3
10-4
10-5

νµ
νe

10-6

HKKM11 νµHKKM11 νe

10-7
10-8
-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

E2 Φν [GeV cm-2 sec-1 sr-1]

Log10(Eν[GeV])
10-2
10-3
10-4
10-5

νµ
νe

10-6

HKKM11 νµHKKM11 νe

-7

10

10-8
-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Log10(Eν[GeV])
Figure 49: The measured ν (top) and ν̄ (bottom) fluxes, compared with theoscillated (solid line) and unoscillated (dashed line) HKKM predictions.

117ν e flux / HKKM11

measurement of the energy spectra of the neutrino flux

2
1.8
1.6

νe

1.4
1.2
1
0.8
0.6

HKKM07

0.4

Fluka

0.2

Bartol

0
-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Log (E [GeV])
ν e flux / HKKM11

10

ν

2
1.8
1.6

νe

1.4
1.2
1
0.8
0.6

HKKM07

0.4

Fluka

0.2

Bartol

0
-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Log (E [GeV])
ν µ flux / HKKM11

10

ν

2
1.8
1.6

νµ

1.4
1.2
1
0.8
0.6

HKKM07

0.4

Fluka

0.2

Bartol

0
-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Log (E [GeV])
10

ν µ flux / HKKM11

118

ν

2
1.8
1.6

νµ

1.4
1.2
1
0.8
0.6

HKKM07

0.4

Fluka

0.2

Bartol

0
-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Log (E [GeV])
10ν

Figure 50: The measured flux ratios between the data and HKKM11 model,
for each neutrino type, with other models overlaid. Colored error
bars represent statistical and systematic uncertainties combined,
while black error bars represent statistical uncertainties only.

MEASUREMENT OF THE AZIMUTHAL SPECTRA OF
THE NEUTRINO FLUX

6.1

introduction

Neutrino experiments have often focussed on the zenith angle distributions of the neutrino flux, as neutrino path-length is a strong
function of the zenith angle, and the oscillation effects may be investigated. However, the azimuthal distributions also contain some interesting physics. The geomagnetic rigidity cut-off of primary cosmic
rays at the Earth’s surface is an anisotropic function, as discussed inSec. 2.2.2, where Fig. 7 showed the rigidity cut-off at the SK site in detail as a function of zenith and azimuthal angles. The strongest effectof this anisotropy is the well-known suppression of west-going primary cosmic rays (illustrated clearly in Fig. 6), causing an east-west
anisotropy in the secondary cosmic ray flux. This effect is of course
expected to appear in the neutrino flux, where it has been previously
discovered for νe , but indicated only with low significance (< 2 σ) forνµ , in a previous Super-Kamiokande measurement [135].
Recent atmospheric flux models have also progressed from onedimensional to three-dimensional calculations, including modelingthe bending of cosmic ray secondary particles in the geomagnetic
field, which has led to significant modifications of the predictions of
the azimuthal distributions [19]. While the dominant effect remains
the primary west-going suppression, the asymmetry is enhanced byan east-west dipole effect caused by muon bending, with a strengththat depends strongly on the energy and zenith angle. Measuring indetail the azimuthal distributions can thus be used as a cross-checkto the flux simulations, in particular to test the accuracy of their implementations of geomagnetic effects; due to the increased statistics
since the previous measurement, we may also have the power to discover the expected azimuthal anisotropy in the νµ flux.
6.2
6.2.1

event classification and analysis method
Selection and binning

The azimuthal analysis uses only the FC sample, as described inSec. 4.2, and selects all FC e-like and µ-like single-ring events. Thesesamples have a very high neutrino flavour purity, estimated as 94%
νe for the e-like sample, and 98% νµ for the µ-like sample.

119

6

120

measurement of the azimuthal spectra of the neutrino flux

Each neutrino event is binned by the azimuthal angle φ, zenith
angle θ, and reconstructed energy of the produced lepton Erec . The
azimuthal angle φ is defined as the clockwise angle between true
south and the lepton’s reconstructed forward-going direction, and isbinned into 12 evenly-sized azimuthal bins from 0→360 degrees. The
zenith angle θ is binned as 5 evenly-sized cos(zenith) bins from -1 to+1, where +1 represents an up-going lepton. The reconstructed energy
Erec is binned into 4 energy bins starting at 0.1 (0.2 for µ-like), 0.4,
1.33, and 3.0 GeV, where the last bin is unbounded from above. Theenergy resolution of the lepton per event, even in the lowest energybin, is estimated as better than 7% (3%) for e-like (µ-like) events. Thisbinning was chosen following the MC prediction, in order to show
zenith and energy regions in which the predicted east-west effects
have various different strengths, while keeping statistics high in eachbin.
All data from the SK-I to SK-IV periods are summed together, fora total of 13,061 (e-like) and 12,711 (µ-like) events. The MC events are
generated as described in Sec. 4.5, and binned in a similar way using
their reconstructed variables.
6.2.2

Asymmetry parameter

The azimuthal distributions will be examined by plotting them foreach energy bin while summing over zenith angle, and for each zenith
bin while summing over energy. To quantify the east-west dipole
asymmetry in each plot, we define the parameterA=

neast − nwest
neast + nwest(63)

where neast (nwest ) represents the number of events with azimuth angle between 0→180 (180→360) degrees.
To calculate the significance of a nonzero east-west effect, we test a
reduced sample containing the middle two energy bins and middle
three zenith bins, which is to say all events in the 0.4→3.0 GeV energy
range with |cos(θ )| < 0.6. These criteria were optimized by MC toselect the event sample giving the largest predicted value of A/∆A,
which is defined as the significance in units of σ.
6.2.3

Zenith dependency of asymmetry shape

The primary shape of the the rigidity cutoff effect is approximated as
an east-west dipole shape, as can be clearly seen in Fig. 7 on page 22.
However, away from the horizontal, it can be seen that the rigidity
cutoff takes on a slightly more complicated shape. While the A parameter measures the strength of the east-west dipole asymmetry depending on energy and zenith, we also wish to test if the shape of

6.2 event classification and analysis method

this asymmetry is affected by this zenith dependency of the rigidity
cutoff shape.
In principle we could measure the relative flux of the neutrino as
a complicated function of zenith and azimuth and compare with therigidity cutoff map, but in practice the statistics are not high enoughto accurately measure such differences. We can however define acoarser parameter, to attempt to measure some zenith dependencyof the neutrino flux’s azimuthal distribution shape. According to the
MC prediction, it should be possible to measure these second-ordereffects by the phase-shift parameter B in the fitting
k1 sin(φ + B) + k2

(64)

for each zenith plot (with k1 , k2 free parameters). This parameter essentially fits the azimuthal alignment of the asymmetry dipole depending on the zenith angle, and is shifted away from zero for a noneast-west alignment of the dipole. By toy MC, we see on average a
2.0 σ significance of a zenith-dependent B parameter, when combining both e-like and µ-like samples.
The actual fitting method of the data to the MC prediction uses
the B parameter as a function of the five zenith bins (shown later in
Fig. 60 on page 132), plotted separately for e-like and µ-like events.
We define our hypothesis H for the bin contents as the shape predicted by MC, but also allow a free-floating overall normalization α1and y-axis scaling α2 of the MC shape separately for e-like and µ-likeµ µevents, such that we have four free parameters H = H (α1e , α2e , α1 , α2 ).
We then define, for data counts N, the statistic∆χ2 ≡ 2 ln

LM

LM (N|H)µN | H (α2e = 0, α2 = 0)(65)

where L M is a Gaussian maximum-likelihood estimator, which finds
the best-fit α parameters.1 In the denominator however, we restrict
the α2 parameters to be zero, which is equivalent to fitting a constantfunction for the e-like and µ-like plots – i.e. no zenith dependencyof B. Since the difference in parameter space between the hypotheses in the numerator and denominator is two, by Wilks’ theorem (asexplained in Appendix B) the statistic ∆χ2 should be distributed asa χ2 (k = 2) distribution (where k is the number of degrees of freedom). We may thus extract the significance of rejecting the no-zenithdependency model as P χ2 (k = 2) ≥ ∆χ2 .

1 We use the notation ∆χ2 , as for Gaussian L M this test is equivalent to the differencein the χ2 statistic of the two hypotheses.

121

122

measurement of the azimuthal spectra of the neutrino flux

East-going events

0.018

0.018

West-going events

0.016

0.016

0.014

0.014

0.012

0.012
0.01

0.01

0.008

0.008

0.006

0.006

0.004

0.004
0.002

0.002
0
-10

-8

-6

-4

-2

0

2

4

6

8

0
-10

10

PID Likelihood (e-like ↔ µ-like)

0.018

East-going events

0.016

0.016

0.014

0.014

0.012

0.012

0.01

0.01

0.008

0.008

0.006

0.006

0.004

0.004

-6

-4

-2

0

2

4

6

8

10

PID Likelihood (e-like ↔ µ-like)

West-going events

0.002

0.002
0
-10

-8

-8

-6

-4

-2

0

2

4

6

8

10

Ring Counting Likelihood (single ↔ multi)

0
-10

-8

-6

-4

-2

0

2

4

6

8

10

Ring Counting Likelihood (single ↔ multi)

Figure 51: The ring counting and PID likelihood variables for the SK IV SubGeV data, separated into east and west-going events, shown with
best-fit spline function based on the MC distribution.

6.3
6.3.1

systematic uncertainties
Azimuthally-dependent detector systematics

The SK detector is considered to be highly azimuthally symmetric,
and no azimuthally dependent systematic reconstruction errors are
implemented in the SK systematic error database (as was describedin Sec. 5.4 and summarized in Appendix A). The possible existenceof such errors was nonetheless investigated.2
6.3.1.1 PID and ring-counting
Possible particle identification and ring-counting biases were checkedusing the distribution of their likelihood variables, for each detector
period. The nominal likelihood shapes from the MC events were fitted
to two sets of data, one for east-going events and one for west-going
events, for sub and multi-GeV samples separately. Figure 51 shows
an example of such fittings for the SK-IV sub-GeV sample, where the2 A possible cause of such errors could be, for example, a residual magnetic field
inside of the SK tank. While the geomagnetic field is in principle compensated by
Helmholtz coils, a previous measurement of the remaining field throughout the tankshowed that while the mean field in the vertical direction was less than 0.1 mG asmall horizontal mean field at the 30 mG level remained, which was expected toeffect PMT collection efficiency at the 1∼2% level [79].

Ee [MeV]

6.3 systematic uncertainties

50
2.5
40
2
30
1.5
20
1

10

0
-1

0.5

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Cos(θµ→e)

Figure 52: The predicted decay rates for the Michel decay µ− → e− + ν̄e + νµ(µ+ → e+ + ν̄µ + νe ). Here Ee is the electron (positron) energy,and θµ→e is the inner angle between the muon spin and electron(positron) momentum. Column normalized.

shape was fitted allowing a shift or a re-scaling along the x-axis. The
fitted shapes of the distributions were consistent between the east
and west-going events, and the error in either of these parameters
was considered negligible.
6.3.1.2 Reconstructed energy
The variance of reconstructed momentum scale by azimuth was checkedusing electrons (and positrons) coming from the decay of cosmic-ray
muons which come to a complete stop in the detector. The energy
spectrum of such electrons is the well-known Michel spectrum. Sucha sample can be selected by requiring a fitted muon track that comesto rest in the detector, and a single contained electron-like event that
follows, in addition to strict cuts on the decay vertex position, decay
time, and muon and electron fitter goodness.
However, there are several factors relating to the muon polarizationthat must be corrected for. Figure 52 shows the muon decay rate as afunction of the the angle between the muon spin and the electron momentum, and the electron’s energy. It can be seen that in the forward
direction of polarization, the electron tends to be given more kinetic
energy. Thus, if the muon flux at SK has an azimuthally-dependent
polarization, we should expect a real energy bias in the Michel electrons that must be corrected for. In fact, even an azimuthally-independent
average polarization will cause an azimuthal bias, as the azimuthal
decay distribution will be weighted by the muon flux intensity, which
differs depending on the azimuth angle due to the shape of the mountain above SK.
At least some muon polarization should be expected consideringthe Lorentz boost between the parent frame of the atmospheric π

123

124

measurement of the azimuthal spectra of the neutrino flux

Figure 53: The mean energy of decay-electrons depending on the azimuth
angle, with the average energy normalized to one.

or K and the lab frame. Previous measurements by the Kamiokande
detector [136] (which take into account capture and depolarization ofµ− in water) suggest an average forward muon polarization of 0.26,
which is implemented into the stopping-muon MC. In order to reduce
this polarization effect, we implement a cut requiring the stopping
muon direction to be within 20° of down-going, which is the tightest
cut possible while keeping reasonable statistics.
The final result of the azimuthal energy scale check is shown in
Fig. 53 for MC and data. While the MC shows no clear systematic bias,
the data shows a small and roughly sinusoidal azimuthal bias of theorder 0.6%. Although this may be plausibly due to a poor modeling of
the muon polarization, it may be due to a genuine bias in the detector.
An azimuthal energy calibration systematic is therefore implemented
as a sinusoidal effect, with a magnitude of 0.6% for a 1 σ value of theerror strength. The final effect of this systematic on our data sample
is shown later in Tab. 17.
6.3.2

Second-order systematics due to azimuthal flux asymmetries

Considering the neutrino flux at SK, the energy spectrum, averagepath length, and ν/ν̄ ratio, are expected to change slightly dependingon the azimuth angle [68] due to the geomagnetic effects.3 Any detector systematic that depends on these factors will thus be azimuthally
dependent, in second-order, and must be considered.
To give a clear example of what we mean by a “second-order” dependence, consider for example the MC true energy spectra of µ-likeevents shown in Fig. 54. We see that east-going sub-GeV events haveon average a slightly lower energy than west-going events; since some

3 These changes are expected to be too small to be directly measurable, which is why
this study considers only on the flux normalization as a function of azimuth.

6.3 systematic uncertainties

east-going

0.05

west-going
0.04

0.03

0.02

0.01

0
200

400

600

800

1000

1200

Eν [MeV]Figure 54: The energy spectra for the µ-like single-ring sub-GeV events, estimated by MC, separated into east-going and west-going events.
Plots are normalized to unity.

detector reconstruction uncertainties are larger at those lower energies, a larger systematic error is applied to the east-going events.
Many systematics in the SK systematics database, described in Sec. 4.5.4
and Appendix A, are a function of neutrino energy. The systematics
software was modified to process events as a function of azimuthal
angle (in addition to zenith angle and neutrino energy), and we calculate their effects at 1 σ in the azimuthal binning, including crosssection errors (15), detector reconstruction errors (53), neutrino oscillation model errors (6), and flux model errors (21). The detector-related
errors are numerous as most of them are modeled separately for each
SK period, however since the selection of single-ring events is simpler
than the multiple sub-samples used in the previous energy spectrum
analysis, comparatively less reconstruction-related errors are applicable.
6.3.3

Implementation of the systematics

For the parameter A in Eq. (63), any systematic shift in the total event
number that applies equally to east-going and west-going events willcancel out, and give no contribution to the final error estimation. Conversely, any systematic that effects east-going and west-going events
differently will give some contribution to the final error.
Similarly, when we plot the full azimuthal distributions, we are interested in testing their shape as a function of azimuth, rather than
measuring the absolute normalization of the neutrino flux. Thus if we
consider a systematic effect that applies equally to all azimuthal bins,
it should not contribute to the systematic error estimation. For eachsystematic error, we define it’s “normalization” component as the average effect on all azimuthal bins at, and it’s “azimuthally-dependant”

125

126

measurement of the azimuthal spectra of the neutrino flux

component as the effect on each bin relative to the average effect,
at the 1 σ level. The normalization component is discarded, and theazimuthally-dependant components of all systematics are combinedin quadrature to calculate the final systematic error in each azimuthal
bin.4
The systematics with the strongest effects on the azimuthal analyses are listed in Tab. 17. Cross-section errors are still dominant, andthere is some contribution from relative flux normalizations, eventhough these errors are only azimuthally dependent in second order(as described in the previous section). Some detector reconstruction
uncertainties for SK-IV and SK-I are also noticeable, due to their highlivetimes compared with other SK periods.
6.3.4

Azimuthal angle reconstruction

Finally, it is important to note that the dominant factor in mis-reconstructing
the neutrino direction does not come from detector inadequacies, butfrom the fact that in CCQE interactions at lower energies the neutrino direction and the produced lepton direction are poorly correlated. This is not modeled as a systematic error, but the effects are
generally accounted for by the high statistics of the SK MC, whenused in forward-fitting procedures. The correlation for reconstructed
single-ring events is plotted by energy in Fig. 55, which shows that forinteractions at less than 400 MeV there is only a faint ability to evendiscriminate the neutrino’s forward direction from its backwards one.
Events in the range 400 < Eν < 1330 are generally correlated atleast within 90°, which allows a good separation between east andwest-going events. Finally, multi-GeV events are very well correlated
within ∼ 15°.
6.4
6.4.1

results and discussions
Azimuthal distributions

The azimuthal distributions, using the lepton’s reconstructed energy
and direction, are shown in Fig. 56 for data and MC events. The values of the dipole anisotropy parameter A corresponding to each plot
are shown in Fig. 57.
6.4.1.1 Monte Carlo distributions
To understand the origin of these distributions from the true flux
shape, we also show the same MC events rebinned by true neutrino
direction in Fig. 60, and first discuss only the MC predictions.
4 For a meaningful comparison, the MC must also normalized to the data in this case;the required normalization change is less than 2%.

6.4 results and discussions

systematic error

size of effect [%]Neut axial mass parameter

0.59

MC statistical error

0.49

CCQE cross-section ratio

0.46

Axial mass parameter in QE and single π

0.35

Flux relative normalization < 1 GeV

0.34

Single meson cross-section

0.34

Flux relative normalization > 1 GeV

0.25

Neutral / charged pion ratio

0.22

Coherent pion cross-section

0.17

Deep inelastic scattering Q2 at low W

0.10

Relative normalization for sub/multi-GeV FC

0.10

Single pion ν/ν̄ ratio

0.08

CCQE ν/ν̄ ratio

0.08

Deep inelastic scattering model differences

0.08∆m23 error (from T2K)

0.07

Azimuthal energy calibration (SK IV)

0.06

Fiducial volume (SK IV)

0.06

ν/ν̄ ratio 1 < Eν < 10 GeV

0.05

Overall energy calibration (SK IV)

0.05

Azimuthal energy calibration (SK I)

0.05

Others0.88

Table 17: Each systematic error and the total effect of their azimuthal component, given as a percentage shift away from the central MC values
for a 1 σ shift in the error, when binned in the azimuthal analysis
binning. "Others" represents the sum of 74 systematics with individual effects < 0.05%.

127

128

measurement of the azimuthal spectra of the neutrino flux

0.1

νe

0.08

Eν < 400
400 < Eν < 700
700 < Eν < 1000
1000 < Eν < 1330
1330 < Eν

0.06

0.04

0.02

0

0

20

40

60

80

100

60

80

100

120

140

160

180

120

140

160

180

0.1

νµ

0.08

0.06

0.04

0.02

0

0

20

40

ν → l inner angle [deg]Figure 55: The neutrino → lepton true direction correlation for the singlering events used in the azimuthal analysis, showing νe (top) andνµ (bottom). Neutrino energies Eν are in MeV.

6.4 results and discussions

The MC truth plots show that at Eν < 0.4 GeV the true anisotropy ishigh, however the MC reconstructed plots show a diluted anisotropy,
as was expected due to the poor correlation between the incoming
neutrino and outgoing lepton directions. The anisotropy is strongest
in the MC reconstructed plots at around ∼ 1 GeV, and it also forevents around cos(θ ) = 0.
The cos(θ ) > 0.2 bins the MC truth plots show a more complicatedshape, which is due to the rigidity cutoff having a suppressive effectin both east and west directions for up-going neutrinos. As this effectis strongest for sub-GeV events within ∼ 10° of up-going, it is also
washed out in the MC reconstructed plots.
6.4.1.2

Data distributions

Considering now the observed data events, we see that the current
data give excellent agreement with the MC expectations (based on
the HKKM11 flux model). The χ2 value (based on a weighted vs.
unweighted events χ2 test [137]) for the energy (zenith) distributionsin Fig. 56 is 87.6 for 96 bins (106.6 for 120 bins), which for the MC
prediction as the null hypothesis gives a p-value of 0.69 (0.79). The
flux simulation is thus strongly consistent with the current data.
The final azimuthal distributions, using the reduced sample optimized for discovery of the asymmetry effects, are shown in Fig. 59.
The final A parameters are found to be Ae = 0.153 ± 0.015(stat) ±0.004(syst) for e-like events, and Aµ = 0.108 ± 0.014(stat) ± 0.004(syst)
for µ-like events. The east-west effect is thus seen at a significancelevel greater than 6σ (8σ) for the µ-like (e-like) samples. This is thefirst time it has been seen convincingly in the νµ flux.
6.4.2

Zenith dependency of asymmetry shape

Figure 60 shows the MC and data results for the B parameter, were
the data matches well with the MC, and shows a 2.2 σ indication
for the existence of a zenith dependence of the azimuthal asymmetry
shape (using the analysis method as explained in Sec. 6.2.3, combining the significance from the e-like and µ-like samples together). Thisis the first indication that the geomagnetic effect produces an asymmetry which is more complicated than the “east-west” effect that is
usually assumed.
In future, detectors such as Hyper-Kamiokande should be able toperform an improved test of the B parameter. By assuming similarsystematic errors, but increasing statistics by a factor of 10 compared
to SK (corresponding to a few years of observation by HK), we findthe estimated sensitivity to observe a non-zero B parameter as 5.9 σ.

129

130

measurement of the azimuthal spectra of the neutrino flux

520

240

0.1 < Erec < 0.4 [GeV]

500

220

480

200

460

180

440

-1.0 < cos( zenith ) < -0.6

160

420
140

400
260

380
550
500

-0.6 < cos( zenith ) < -0.2

240

0.4 < Erec < 1.33 [GeV]

220
200
180

450

160

400

140
320

350

300

-0.2 < cos( zenith ) < 0.2

280

300

260
240

160

220

1.33 < Erec < 3.0 [GeV]

200
180
160

140

280

120

0.2 < cos(zenith ) < 0.6

260
240

100

220
200

80

180

110

160

3.0 < Erec [GeV]

100
260

90

240

80

220

70

200

0.6 < cos(zenith ) < 1.0

180

60
0

60

120

180

240

300

360

160
0

60

120

180

240

Azimuth [deg]
420
400

300

360

Azimuth [deg]320

0.1 < Erec < 0.4 [GeV]

300

380

280

360

260

340

240

-1.0 < cos( zenith ) < -0.6

220

320

200

300
280

300

-0.6 < cos( zenith ) < -0.2

280

650

260

0.4 < Erec < 1.33 [GeV]

240

600

220
200

550

180

500

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450

260

-0.2 < cos( zenith ) < 0.2

240

400

220
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180

180

1.33 < Erec < 3.0 [GeV]

160
140

160
240

140

0.2 < cos(zenith ) < 0.6

220
200

120

180

100
55
50

160
140

3.0 < Erec [GeV]

120
220

45

0.6 < cos(zenith ) < 1.0

200

40
35

180

30

160

25

140

20
0

60

120

180

240

300

Azimuth [deg]

360

120
0

60

120

180

240

300

Azimuth [deg]Figure 56: Single-ring e-like (top) and µ-like (bottom) events for the SK-I to
SK-IV data (error-bars) and MC (error-boxes), by reconstructedazimuthal angle. The left plots show reconstructed energy ranges
summing over all zenith angles θ, and right plots show zenith
angle ranges summing over all energies Erec . The plots are zerosuppressed to show the data/MC shape comparison clearly.

360

parameter A

6.4 results and discussions

e

0.14

parameter A

e

0.14

0.12

0.12

0.1

0.1

0.08

0.08

0.06

0.06

0.04

0.04

0.02

0.02

0

0

-0.02

-0.02

-0.04

-0.04

µ

0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
-0.02
-0.04
0.5

1

1.5

2

2.5

3

3.5

4

µ

0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
-0.02
-0.04
1

4.5
5
Erec [GeV]

0.6

0.2

-0.2

-0.6

-1
cos θ

Figure 57: The parameter A depending on zenith angle or energy, for the
e-like (top) and µ-like (bottom) events, using the SK-I to SK-IV
data (error bars) and MC (error-boxes). The highest energy bin isunbounded from above, but is plotted only to 5 GeV.

700
600

350

0.1 < Erec < 0.4 [GeV]

300

500

200

400

150
100

300

50

200

0
450

100

400

0
800
700

0.6 > cos θ > 0.2

350
300

0.4 < Erec < 1.33 [GeV]

250
200

600

150
100

500

50

400
300

0
500

200

400

100
0
250

νμ
νe

1.0 > cos θ > 0.6

2500.2 > cos θ > -0.2

300
200

1.33 < Erec < 3.0 [GeV]

100

200

0

150

400
350

100

-0.2 > cos θ > -0.6

300
250

50

200
150

0
100

100
50

3.0 < Erec [GeV]

0

80

300

-0.6 > cos θ > -1.0

250

60

200

40

150
100

20

50

0

0

60

120

180

240

300

360

Azimuth φ [deg]

0

0

60

120

180

240

300

360

Azimuth φ [deg]Figure 58: Reconstructed e-like (blue) and µ-like (red) MC events, as inFig. 56, but re-binned using the MC true neutrino direction (forboth zenith and azimuth angles) instead of the reconstructed lepton direction.

131

measurement of the azimuthal spectra of the neutrino flux

700

850
800

e

650
600

µ

750
700

550
500

650
600
550

450
400
350
0

60

500
0

120 180 240 300 360
Azimuth φ [deg]60

120 180 240 300 360
Azimuth φ [deg]Figure 59: e-like (left) and µ-like (right) events for the SK I-IV data (errorbars) and MC (error-boxes), optimized for calculation of the
final A parameters, selected by 0.4 < Erec < 3.0 GeV and|cos(zenith)| < 0.6.

parameter B [deg]

132

80

e

80

60

60

40

40

20

20

0

0

-20

-20

-40

-40

-60

-60

-80
1

µ

-80
0.6

0.2

-0.2

-0.6

-1

cos θ

1

0.6

0.2

-0.2

-0.6

-1

cos θ

Figure 60: The parameter B depending on zenith angle, for the e-like (left)
and µ-like (right) single-ring events using the SK-I to SK-IV data(error bars) and MC (error-boxes).

6.4 results and discussions

6.4.3

Flux model comparisonsAzimuthal flux distributions were also published by the Bartol groupin 2003 [21]. Although this is older than the current HKKM group’smodel, the major relevant improvements (a three dimensional calculation with bending of secondary cosmic rays in the atmosphere) were
included. The Bartol data contains 18 azimuthal bins, but only a single zenith bin from −0.5 < cos(θ ) < 0.5. By reweighting the MC fromthe default HKKM11 azimuthal spectra to the Bartol azimuthal spectra (for events covered by the above range only), we created a new setof plots similar to Fig. 56, and obtained a χ2 statistic for the Bartol
model. The largest differences (up to 25%) in the azimuthal spectraof the flux models occur at < 400 MeV, but due to the poor neutrinolepton directional correlation, the observable differences in that range
are small. The HKKM11 model has a χ2 value lower by 1.0 units, but
considering the number of bins such a difference seems insignificant,and we cannot draw any strong conclusions on flux model preference
from the azimuthal distributions.

133

M E A S U R E M E N T O F T H E C O R R E L AT I O N B E T W E E N
THE SOLAR CYCLE AND THE NEUTRINO FLUX

7.1

introduction

The solar magnetic activity cycle is an oscillatory change in the solar density and structure, which causes a change in the level of solar
plasma emissions, with an average period of approximately 11 years.
The cosmic ray flux at earth is well known to be dependent on thesolar activity [138]. This is because the plasma flux, or “solar wind”,
acts as a barrier by scattering cosmic rays entering the solar system
and approaching the Earth; therefore during solar activity maxima
the cosmic ray flux is reduced. Consequently the atmospheric neutrino flux is predicted to reduce – although this has not yet beenmeasured.1
Historically, the solar activity was measured by its correlation withthe appearance of sunspots, which are regions of intense surface magnetic activity causing a drop in temperature in that region, often easily visible from Earth. Since 1948, the use of neutron monitors (NMs)provides a method to accurately and constantly track the neutron
flux at the Earth’s surface resulting from cosmic ray impacts [139].
As the primary cosmic ray flux is measured by devices mounted on
balloons or spacecraft (which have limited flight times), it is not itself constantly measured, but the NM counts are generally believed
to be well-correlated with the primary cosmic ray activity. Some corrections may however be necessary depending on the measurement
site; firstly the relative atmospheric pressure, which alters the mean
free path of particles in the atmosphere, must be corrected for. The
down-going rigidity cutoff (an effect discussed in Sec. 6.1) at the sites
must also be considered, as the plasma wind affects the low-energy
primary flux more strongly, and areas with a low rigidity cutoff havea relatively higher flux at lower energies.
We thus plan to test for a correlation of the atmospheric neutrino
flux with the solar cycle, by searching for a correlation between theneutrino flux at SK and the neutron detection rates at various NMs
operated by other institutes. This method is able to test on very shorttimescales of O(1 h), as both neutrino and neutron observations takeplace on Earth, and the propagation of the solar wind within the solar
system (with speed of the order 100 km s−1 ) need not be considered.
1 As was mentioned in Sec. 2.3.2, a modulation of the neutrino flux is also expected
due to seasonal atmospheric variations, however at the SK site these variations areat the  1% level [68] and assumed to be negligible in this analysis.

1357

correlation between the solar cycle and the neutrino flux

cos θ

136

-1

1

-0.8

0.98

-0.6

0.96

-0.4

0.94

-0.2

0.92

0

0.9

0.2

0.88

0.4

0.86

0.6

0.84

0.8
1

10-1

0.82
1

10

102

3

10

104

5

10

Eν [GeV]Figure 61: The reweighting function for the νe flux taken from the HKKM
model, for the highest modeled solar activity corresponding to a
Climax NM count of 3500 counts hr−1 ×0.01. Here θ is the zenith
angle of arrival at SK.

We assume that at a given time the neighborhood of the Earth can betreated as a heat bath with constant temperature, and we may expect
a good correlation between the neutrino flux at the SK site and theneutron flux monitored at the NMs in various locations around theEarth.
7.1.1

HKKM flux model predictions

Data was obtained from the HKKM flux group [74] that predicts therelative normalization change of the νe , ν̄e , νµ , and ν̄µ fluxes depending on the solar cycle. This data uses the count rate of a specific NM,
the Climax NM [140], as the parameter corresponding to the degreeof solar activity. Such a single parameter may be used as the model
assumes that after corrections for the local air pressure at the monitor, all NMs are linearly correlated with the solar flux modulation –even though if two monitors are located at different rigidity cutoffs,
the gradient of the correlation would be expected to be different. The
HKKM model then calculates the effect at the SK site, extrapolating
the effect of the solar modulations in all directions, where rigidity
cutoff is a function of direction.
As an example, Fig. 61 shows the expected reweighting required forthe νe flux at the SK site during the highest solar activity covered by
the model. This corresponds to a Climax NM count of 3500 counts hr−1
×0.01 (where the factor of 0.01 in the count rate is the conventional
notation for the NM data), which was the average count rate overa period roughly corresponding to the 22nd solar maximum.2 Such
2 Here “22nd” refers to the cycles measured since sunspot counting began on Earth;the 22nd cycle ran from 1986 to 1996.

7.2 event classification and analysis method

137

reweighting functions are provided at intervals of 50 up to 4150 counts hr−1
×0.01, which is not the absolute maximum value recorded by the Climax NM, but rather a solar activity below which the change in ν flux
is calculated to be negligible and the reweighting function approaches
unity.
From Fig. 61, we see that only the neutrino flux below 1 GeV isstrongly affected, so we may focus our analysis on the SK “sub-GeV”
event samples as described in Sec. 4.2.2, which have reconstructed
lepton energies Erec < 1330 MeV. It is also seen that there is a strong
zenith-dependence of the flux suppression. Although the model assumes no directional asymmetry in the solar wind effects, a directional asymmetry occurs at the SK site due to the fact that the rigidity
cutoff causes a relatively higher low-energy neutrino flux around theEarth’s geomagnetic polar regions compared to the equatorial regions.
Since the solar wind has a stronger suppressive effect on low-energy
particles, the biggest relative flux reductions are in the polar regions.
After considering the geometric effect of translating from global coordinates to zenith coordinates at the SK site, the predicted differencein the suppression effect appears as an up-down asymmetry.
7.2
7.2.1

event classification and analysis method
Super-K data selection

We select the sub-GeV samples only for this study, since the multiGeV samples show a negligible solar modulation effect. Table 18 shows
all sub-GeV samples, from which we further select only the singlering e-like and µ-like samples. The other sub-GeV samples have either
low purity, or high systematic normalization errors; as we fit the νeand νµ data according to separate models, and the solar modulation
effect is at most a ~10% effect, these samples are essentially not useful. Studies were done to check if using an energy sub-range of the
sub-GeV data could increase the statistical power, however no strong
benefits were seen.
The data we use in this analysis are thus simply categorised aseither e-like or µ-like, up-going or down-going (based on the fitted
lepton direction), and by SK period – resulting in a total of 2 × 2 × 4 =16 samples. For this analysis we use the very latest SK dataset, which
covers up to April 2015; compared to the analyses in Sec. 5 and Sec. 6,
this gives an additional six months of data.
7.2.2 Neutron monitor data
We search for NM data [141] from monitors that have been active andwell-maintained throughout the entire SK experimental period. While
the Climax NM does not fall into this category (having shut down in4400

Climax NM [counts hr -1 × 0.01]

Climax NM [counts hr -1 × 0.01]

correlation between the solar cycle and the neutrino flux4400

p = 390.90
p = 0.649
1
p = 1.24×10-4

4200
4000

p = -29540
p = 2.26
1
p = -1.49×10-4

4200
4000

2

3800

2

3800

3600

3600

3400

3400

3200

3200

3000

3000

2800

2800

2600

2600
2400

2600

2800

3000

3200

3400

3600

3000

3500

4000

Newark NM [counts hr × 0.01]

Climax NM [counts hr -1 × 0.01]

4400

4400

p = 6.0250
p = 0.375
1
p = 5.43×10-6

4200
4000

4000

2

5000

p = -313.40
p = 0.528
1
p = 4.52×10-6

4200

3800

4500

Thule NM [counts hr -1 × 0.01]-1

Climax NM [counts hr -1 × 0.01]

138

2

3800

3600

3600

3400

3400

3200

3200

3000

3000

2800

2800

2600

2600
6500

7000

7500

8000

8500

9000

9500

10000

10500

11000

McMurdo NM [counts hr × 0.01]
-1

5500

6000

6500

7000

7500

8000

8500

Kerguelen NM [counts hr-1 × 0.01]

Figure 62: Fitted conversion functions between the pressure-corrected datafrom several NMs to the Climax neutron monitor. Each data point
represents the neutron count recorded at each monitor during agiven hour.

the 90s), we find four such NMs – the Thule, McMurdo, Kerguelen,and Newark monitors [129, 142]. These stations all monitor downgoing neutrons with relatively low rigidity cutoffs (0.3, 0.3, 1.14, and
2.4 GV respectively), and thus have good sensitivity to changes inthe solar cycle, which as stated affects most strongly the low-energy
cosmic ray flux. In particular, the Thule and McMurdo monitors in thenorth and south polar regions are very sensitive to the solar activity.
To obtain an “equivalent Climax NM count” for each of these NMs,
we compare the counts of the monitors during times when they wereboth operational, as shown in Fig. 62. Although the correlation isalmost as well fitted by a linear fit (as is assumed by the HKKM
model), a second-order polynomial is used to account for a small
non-linearity.
The χ2 -statistic per degrees-of-freedom, considering the statisticalerrors only, is however unsatisfactorily large on these plots, indicating some systematic differences in the NM monitors. To minimise
systematic error, we define a “Climax NM parameter”, which is theaverage of the four NM values after each was converted to the equivalent Climax NM count using the fitted polynomials. The systematic
error on this final parameter at 1 σ is estimated, by taking the average
RMS of the four counts, as 15.8 counts hr−1 ×0.01.
The variance of this Climax NM parameter over the SK operating
period is shown in Fig. 63, showing that with recent data included, almost two solar maxima are covered by SK data (despite unfortunately
some downtime between SK-I and SK-II).

Climax NM parameter [counts hr -1 × 0.01]

7.2 event classification and analysis method

SK-I

4400

SK-II
SK-III
SK-IV

4200

4000

3800

3600

3400
1996

1998

2000

2002

2004

2006

2008

2010

2012

2014

2016

Year

Figure 63: The values of the “Climax NM parameter” plotted over the operational history of the SK detector, where each point represents
a neutrino event. Two solar maxima can be clearly seen, as cansome areas where SK was not operational (such as the maintenance works between SK periods). Some sharp downward peaks
can be seen, which will be analysed separately as described inSec. 7.2.7.

7.2.3

Parameterization of the effectUsing the HKKM model, the SK MC events are reweighted based on
their true neutrino energy and zenith angle, using reweighting functions such as those in Fig. 61. The fractional decrease in the numberof events in each reconstructed sample, calculated using the SK-I→IV
MCs weighted appropriately by livetime, is shown in Fig. 64. It canbe seen that the suppression effect is not linear in the NM count parameter, and only has a significant effect compared to the relative
normalization systematic errors of ∼ 3% at NM count values below
around 3700. Comparing with Fig. 63, We see that only the data obtained during the solar maxima at the end of SK-I and beginning of
SK-II, and some the SK-IV data covering the most recent solar maxima, will be sensitive to the solar modulation. Of course, data in the
low-solar activity periods is also essential, in confirming the expected
event rates against which to compare any relative decrease.
The four functions in Fig. 64 give the prediction of the suppression effect in a more realistic way than a simple linear fitting would
achieve for each of the four data samples, however it is possible thatthe effect is stronger or weaker than the prediction in reality. To test
for this possibility when opening the data, we define a function f s (α)
for each sample s which takes its shape from the predicted functions,but allows them to be rescaled by the single continuous parameter α.
We define α = 0 to represent no solar activity and α = 1 to represent the default prediction of the HKKM model, but higher or lower

139

Surviving fraction of events

correlation between the solar cycle and the neutrino flux

1
0.99
0.98
0.97
0.96
0.95

µ -like down-going

0.94µ -like up-going

0.93

e-like down-going
0.92

e-like up-going

0.91
0.9
3500

3600

3700

3800

3900

4000

4100

Climax NM parameter [counts hour -1 × 0.01]

Figure 64: The effect of reweighting by solar activity on the SK-I→IV reconstructed data samples, calculated from the SK MC according to
the HKKM model.

Surviving fraction of events

140

1.4
1.2
1
0.8

α = -1

0.6

α=0
0.4

α=1

0.2

α=2

0
3200

3400

3600

3800

4000

4200

4400

Climax NM parameter [counts hour -1 × 0.01]

Figure 65: The function f s (α) showing an example of the parameterizationof the solar activity effect by a single parameter α, in the casewhere the sample s is the e-like up-going sample.

7.2 event classification and analysis method

sampleevents141

error [%]

purity [%]
sk-i

sk-ii

sk-iiisk-iv

Single-ring e-like

10,892

95.0

2.80

2.82

3.00

2.35

Single-ring µ-like

10,763

98.2

2.53

2.87

3.13

2.48

Single-ring π 0 -like

545

45.5

2.2

2.5

2.2

2.1

Multi-ring µ-like

823

93.6

5.2

8.9

7.5

4.2

PC

374

99.1

4.6

6.3

6.6

5.8

Table 18: The SK-I→IV data, counting sub-GeV events only. The purity isdefined as the fraction of correct flavour interactions, estimated by
MC (for the single-ring π 0 -like sample, the “correct flavour” is assumed to be νe ). The error represents the relative normalization
error depending on SK period, calculated by combining all applicable systematic errors in quadrature, as described in Sec. 7.2.4.

values are also possible. As an example, functions for the SK-I e-likesample as a function of α are shown in Fig. 65. We thus have a familyof functions that can simultaneously be fitted to the 4 data samples
in each of the 4 SK periods, in order to measure the overall strengthof the solar modulation effect by the single parameter α.
7.2.4

Systematic errors

As the solar modulation effect is a relative normalization shift as afunction of the solar activity, any systematic error that affects the overall normalization of all SK data may be ignored. However, since thedatasets of the SK-I to SK-IV periods are combined, we must consider errors that are dependent on detector period, and may causea systematic shift in the relative normalization of each period. Sucherrors can arise from detector changes such as the differences in the
PMT properties and distribution in the tank, or the replacement of theelectronics. Table 18 shows the overall calculated systematic errors forall samples by detector period (although only single ring e-like andµ-like are used in the analysis).
The breakdown of the individual effect of each systematic is shownin Table 19. The dominant detector-dependent errors (>1% level) are
the fiducial volume cut and the ring separation uncertainty; for the
e-like sample the single ring π 0 fitter uncertainty also contributes,
and for the µ-like sample the decay electron tagging uncertainty contributes (although this error is much reduced with the SK-IV electronics). While these errors are considered as completely independent foreach detector period, in reality there may be some correlation forcertain errors, which would mean that the error in the relative normalization would be somewhat conservative.

correlation between the solar cycle and the neutrino flux
142

Azimuthal energy calibration

Up / down energy calibration

Energy calibration

Single ring PID

Ring separation

Non-ν µ backgroundNon-ν e background

FC / PC separation

FC reduction

1.12

0.01

< 0.01

0.12

0.23

1.49

-

0.50

-

0.20

sk-i

< 0. 01

1.12

0.01

< 0.01

0.18

0.66

1.47

-

0.20

-

0.20

sk-ii

2.00

0.01

0.83

0.01

0.01

0.30

0.26

1.88

-

0.10

-

0.80

sk-iii2.00< 0.011.00

0.01< 0.01

0.26

0.28

0.51

-

0.10

-

0.30

sk-iv

2.00

1.29

-

0.03

0.03

0.40

0.18

0.70

0.21

-

0.01

0.2

sk-i

2.00

1.35

-

0.01

0.03

0.61

0.50

1.33

0.11

-

<0. 01

0.20

sk-ii

2.00

1.32

-

0.02

0.06

0.99

0.19

1.56

0.11

-

0.01

0.80

sk-iii

2.00

0.78

-

0.02

0.01

0.85

0.22

0.84

0.12

-

< 0.01

0.30

sk-iv

effect on µ-like [%]Single ring π 0 fitter< 0.012.00

effect on e-like [%]Decay electron tagging

2.00

error

Fiducial volume cut

Table 19: Breakdown of SK period-dependent systematic error effects in the single ring e-like and µ-like samples.

Total observation time [day]7.2 event classification and analysis method

500

400

300

200

100

0

3400

3600

3800

4000

4200

4400

Climax NM parameter [counts hr -1 × 0.01]

Figure 66: Total observation time tobs
at a particular NM count bin i (SKi
I→IV inclusive).

As these errors are not dependant at all on the Climax NM parameter, we simply combine them in quadrature, and assign a total relative
normalization error for each SK period, for the e-like and µ-like samples separately. There are thus a total of 8 systematic errors.
7.2.5

Fitting method

The histogram in Fig. 66 shows the total observation time in days tobs
ifor SK I-IV inclusive, binned by the Climax NM parameter averaged
over 1-hour periods (with bins i = 1 → 25). The minimum valueon the Climax NM parameter in these plots is 3300 counts hr−1 ×0.01;
there is a small amount of observation time below this value, butthese data will be treated separately, as discussed later in Sec. 146.
In the case of no solar modulation effect, we can simply scale thishistogram by the expected daily event rate for each sample rs takenfrom SK MC, to calculate a prediction for data binned in the sameway. In the case of a solar modulation effect, we must also apply thereweighting functions f s (α) (as shown in Fig. 65). Explicitly, we define
our hypothesis Hs,i as the HKKM solar modulation prediction, given
for each sample s in each bin i, by
Hs,i (α) = tobs
× rs × f s,i (α)i(66)

where f s,i (α) is simply f s (α) evaluated at the center of bin i. For example, Fig. 67 shows the number expected of events for the e-likeup-going sample, for both the Hs,i (α = 0) and Hs,i (α = 1) hypotheses.3
3 On this plot, we should consider the uncertainty in the NM parameter, which may
cause events to be binned incorrectly. This may affect the fitting due to the limited
number of events at high solar activity. Even though such migration should be small

143

correlation between the solar cycle and the neutrino flux

Expected events (e-like up-going sample)

144

3

10

102

10

3400

3600

3800

4000

4200

4400

Climax NM parameter [counts hour -1 × 0.01]

Figure 67: The predictions for the e-like up-going sample (one of four samples) in the case of no solar activity correlation (Hs,i (α = 0),
black line), and of a correlation as predicted by the HKKM model(Hs,i (α = 1), blue points with statistical error).

Our 8 systematic errors (as discussed in Sec. 7.2.4) are taken into
account by varying our hypothesis as follows. We define e as thevector of systematic “error pulls”, which holds the strength of eachsystematic in units of σ, and define a new hypothesis as0Hs,i(α, e, k s ) = Hs,i (α) × Ss,i (e) × k s(67)

where Ss,i is the systematic modification, which has a strength depending on e and a shape depending on the relative fraction of each
SK period in each bin i, and k s represents four free-floating overall
normalization parameters, allowing a change in the the total νe andνµ flux normalizations (consistently across the SK-I to SK-IV periods).
0 ( α ) and H 0 ( α =The comparison between the two hypotheses Hs,is,i0) and the data is mathematically performed as follows. Considering Fig. 65, we expect data counts in many bins low enough that theGaussian approximation for statistical errors will be poor. In such acase we should consider directly the Poisson distribution for expected
events, and use the likelihood ratio method to compare our MC anddata. A brief mathematical description of this method is given in Ap-

since the error on the NM parameter (estimated as 15.8 counts hr−1 ×0.01 in Sec. 137)
is much less than the bin size, it was estimated by unsmearing the distribution witha Gaussian function. This showed that any bias is negligible in the sensitive region,and should not contribute to the final result. In the areas where the observation timeis quickly-varying as a function of the NM parameter, there may be some migration
between bins at the ∼ 10% level. However from e.g. Fig. 67, we can see that thishappens only in the low solar activity region, and would have no effect on the fitted
α-function which is flat in that region.

7.2 event classification and analysis method

pendix B, which for data counts Ni leads us to consider the likelihood
ratio


0 ( α, e, k )L0M Ni | Hs,isΛ ≡ 2 ln(68)
0 ( α = 0, e, k )L0M Ni | Hs,iswhere L0M is defined as
L0M ≡ L M + e2

(69)

where L M is itself the maximum likelihood estimator described in Appendix B, which finds the best-fit for the parameters α, e, and k s of H 0(the extra penalty term e2 is included here to constrain the systematic
pulls, to avoid unrealistically high systematic effects). Importantly, inthe denominator of Eq. (68), α is restricted to be zero, which means
there is one less free parameter than in the numerator; by Wilkes’
theorem, we can then expect that Λ is distributed as a χ2 (k = 1) distribution4 and the significance√ to5 reject the no-solar-correlation modelHs,i (α = 0) can be taken as Λ.
7.2.6

Toy MCToy datasets were generated for each of the 16 samples, for both α = 0
and α = 1 hypotheses, in order to test the sensitivity to observe along-term solar activity correlation. Each dataset was created by generating a random set of systematics e according to a Gaussian distribution, scaling the expected number of events by the systematics,
then generating events in each bin based on a Poisson distribution.
An example of a toy dataset, showing the best-fit α solar correlation
model, is shown in Figure 68. A large number of toys are generated
and passed through the fitting procedure, and several tests are performed on the results to ensure that the fitting procedure functions
as expected.
For example, for the α = 1 sample, first the distribution of obtained
α parameters is fit with a Gaussian function. The resulting parameters
are mean µ = 1.03 ± 0.02 and width σ = 0.57, which confirms that the
fit is unbiased. Similarly, the systematic error pull parameters fit witha mean consistent to zero, and the overall sample normalizations fit
with a mean consistent to the nominal data rate used in the toy MC,
confirming that the fitting of the systematic errors is also not biased.
Next, the observed distribution of the likelihood ratio is fit with a χ2function, giving degrees-of-freedom k = 96.0 ± 0.4, consistent with
4 In principle, the minimization over systematics can mean that Wilkes’ theorem does
not hold, and our test will not return a result distributed exactly as χ2 (1). However,tests using our toy MC (Sec. 7.2.6) showed that the impact of including the systematics seems to be negligible in the region of interest. p
5 Which is somewhat analogous to the common case of χ2 .

145

2

e-like up-going

1.8
1.6
1.4
1.2
1

Neutrino events [count day -1]

Neutrino events [count day -1]

correlation between the solar cycle and the neutrino flux

2

1.6
1.4
1.2
1
0.8

0.6

0.6

0.4

0.4

0

e-like down-going

1.8

0.8

0.2

0.2
3400

3600

3800

4000

4200

4400

0

4600

3400

3600

3800

2

µ -like up-going

1.8
1.6
1.4
1.2
1

2

1.2
1

0.6

0.4

0.4

TOY MC

0.2
3600

3800

4000

4200

4400

4600

4600

1.4

0.6

3400

4400

1.6

0.8

0

4200

µ -like down-going

1.8

0.8

0.2

4000

Climax NM parameter [count hr -1 × 0.01]

Neutrino events [count day -1]

Climax NM parameter [count hr -1 × 0.01]

Neutrino events [count day -1]

146

0

3400

Climax NM parameter [count hr -1 × 0.01]

3600

3800

4000

4200

4400

4600

Climax NM parameter [count hr -1 × 0.01]

Figure 68: Example of a toy MC dataset, showing the four event type samples integrated over SK I-IV. To aid visualization (though contrary
to the fitting procedure), here the best-fit solar correlation modelis shown without being systematically shifted, and the data aresystematically shifted and shown with Poisson error bars (using“Pearson’s-χ2 intervals” as defined in [143]).

96 free parameters√ (100 bins minus 4 sample normalizations). Finally,
the significance Λ is plotted, and is distributed consistent with aχ2 (k = 1) as expected; by taking the mean of this distribution, we
find that we expect on average a 1.75 σ sensitivity to observe a nonzero solar activity correlation.
7.2.7

Search for effects during Forbush decreases

The search for a long-term correlation between the neutrino flux andthe solar activity, as described above, used events down to a Climax
NM parameter of 3300 counts hr−1 ×0.01. In fact, the effective parameter sometimes drops significantly below this value, down to a minimum of ∼2800 counts hr−1 ×0.01, although the statistics are quite low
and spread over a range of values. Such events are listed in Tab. 20;
the cause of these events is believed to be large magnetically-charged
coronal mass ejections, causing relatively short and high-intensity increases in the solar plasma flux [144]. On Earth these are associatedwith particularly large decreases in the cosmic ray flux and are often
termed “Forbush decreases”. While there is some ambiguity in theterm, a common definition is any event that causes a > 10% decreasein the intensity of cosmic rays on the Earth’s surface.
The reason for not using these time periods in the above analysis was the low statistics, and also that we suspect the correlation

7.3 results and discussions

start time (utc)

end time (utc)

observation time [hr]2000/07/15, 18:00

2000/07/17, 21:00

50

2001/04/11, 23:00

2001/04/13, 14:00

38

2003/10/29, 11:00

2003/11/01, 00:00

61

2003/11/01, 00:00

2003/11/04, 13:00

67

2005/01/19, 00:00

2005/01/19, 13:00

13

total:

229

Table 20: The periods for which the Climax NM parameter drops below
3300, corresponding to strong Forbush decrease events.

between NMs at various rigidities and locations may not hold during Forbush decrease events. For these reasons, the HKKM simulation of the neutrino flux decrease is only modeled only down to3500 counts hr−1 ×0.01, and although we extend the correlation down
to 3300 counts hr−1 ×0.01 by a polynomial fitting, we believe extending it any further may be unrealistic.
Thus, while no theoretical prediction of the fractional neutrino flux
decrease during these events is available, these may be the most sensitive times to measure a solar effect on the atmospheric neutrino flux.
We thus make a simple test against the null hypothesis, by counting
the number of events observed during all such periods over all four
samples (in order to gain sufficient statistics), and comparing withthe expected nominal event rate.
7.3

results and discussions

7.3.1

Continuous correlation search

The data for the continuous correlation search, summing over SK
I-IV and dividing into the four sample types (e-like or µ-like, andup-going or down-going) are shown in Fig. 69. The test statistic depending on α is shown in Fig. 70. The best-fit value of alpha is α =+(0.57, 1.12, 1.66)
0.62−(0.58, 1.18, 1.80) , with errors given at (1 σ, 2 σ, 3 σ). The rejection
power of the null hypothesis α = 0 is 1.06 σ. The data in Fig. 69 seemto be in good agreement with the model, and the four plots together
have a χ2 statistic of 88.0 for 100 DOF, which is clearly reasonable.6
The significance of rejecting the null hypothesis is lower than themean value predicted by toy MC with α = 1, but still a reasonably
likely result according to that MC, at the p = 0.26 level. We show alsoin Fig. 71 the measured event rates binned per year, which is not used

6 The α-function based on the HKKM model however gives a similar goodness-of-fit
to a simple linear fit, with χ2 = 87.9.

147

2

e-like up-going

1.8
1.6
1.4
1.2
1

Neutrino events [count day -1]

Neutrino events [count day -1]

correlation between the solar cycle and the neutrino flux

2

1.6
1.4
1.2
1
0.8

0.6

0.6

0.4

0.4

0

e-like down-going

1.8

0.8

0.2

0.2
3400

3600

3800

4000

4200

4400

0

4600

3400

3600

3800

2

µ-like up-going

1.8
1.6
1.4
1.2
1

2

4400

4600

1.6
1.4
1.2
1
0.8

0.6

0.6

0.4

0.4

0

4200

µ-like down-going

1.8

0.8

0.2

4000

Climax NM parameter [count hr -1 × 0.01]

Neutrino events [count day -1]

Climax NM parameter [count hr -1 × 0.01]

Neutrino events [count day -1]

0.2
3400

3600

3800

4000

4200

4400

0

4600

3400

3600

-1

3800

4000

4200

4400

4600

Climax NM parameter [count hr -1 × 0.01]

Climax NM parameter [count hr × 0.01]

Figure 69: The test for a solar modulation correlation using SK-I to SK-IV
data, showing the best-fit α-function (for α = 0.62) across four
data samples.

Λ

148

20
18
16
14
12

3σ

10
8
6

2σ

4

1σ

2
0
-2

-1

0

1

2

3
α

Figure 70: The test statistic, defined as ∆χ2 , as a function of α. The significance levels are drawn according to Wilks’ theorem.

5
4.5

4400

4
4200

Events day -1

Climax NM parameter [counts hr -1 × 0.01]

7.3 results and discussions

3.5
3

4000
2.5
2

3800

1.5
Fit (α = 0)

3600

1

Fit (α = 1)
Data

0.5

3400
1996

1998

2000

2002

2004

2006

2008

2010

2012

2014

0

2016

Year

Figure 71: The measured daily event rate summing over the four selected
Super-K data samples, compared against a linear function (α =0) and the predicted solar correlation function (α = 1), where
both functions are scaled to their best-fit normalizations. The NM
count parameter, as in Fig. 63, is shown in the background forreference. Error bars represent statistical error. This plot is forillustrative purposes and is not used in the final analysis.

in the analysis, but gives an easily understandable demonstration of
the sensitivity of our data.
We also perform the continuous correlation search on sub-samplesof the data: first we re-run the analysis for each SK period I to IV
separately, then we re-run the analysis for all SK periods but fitting
using each sub-sample individually. The results are shown in Fig. 72.
Data from the SK-III period alone cannot give any result, as it has
no observation time above the minimum solar activity required tocause any effect according to the HKKM model. Although SK-II and
SK-IV prefer a low value of α, the statistical power is lower and notinconsistent with the overall result.7 Somewhat interestingly, the elike samples prefer no correlation, while the µ-like samples prefermore the expected α = 1 correlation; the significance is however nothigh enough to draw any strong conclusions.
Since the current HKKM model is used to apply solar activity
weights in the SK MC, the strength of the correlation should be tested
again by further measurements if possible, to ensure that the MC solar weighting does not bias the interpretation of the measurement ofother neutrino parameters. For the Hyper-Kamiokande case, an estimation of the expected sensitivity to a long-term correlation depends
strongly on the intensity and number of solar maxima observed in
the future, which cannot be accurately predicted. However, if HK had
observed the same time periods as SK, we calculate that a 5 σ (3 σ)
7 Indeed much of the fitting power comes from the combination of the data, and theoverall result is thus not given by the average of these individual results.

149

correlation between the solar cycle and the neutrino flux

α

150

3
2
1
0
-1
-2
-3

SK I

SK II

SK III

SK IV

e-like up

e-like downµ-like up µ-like down

Sub-sample

Figure 72: The best-fit α parameter obtained by performing the analysis on
sub-samples of the data, showing SK periods on the left, and datasub-sample types on the right. The red band shows the combined
result. The SK-III data point is absent, due to a lack of any datain the sensitive solar-active region, and thus a complete inability
to perform the fit.

discovery sensitivity would have been expected for the true case of
α = 1 (α = 0.62).
7.3.2

Event rate during Forbush decreases

During the coronal mass ejection periods described in Tab. 20, theSK detector was operational for a total of 7.21 days. Using the fitted normalization constants of the α-function in the previous analysis, which corresponds to the expected daily event rate with no solar
modulation effect, we expect 31.80 ± 0.17 sub-GeV single-ring eventsin this time. The actual number of events recorded by the detector
was n F = 20, which by consulting the Poisson distribution gives
P(n F ≤ 20) = 0.017, corresponding to rejection of the null hypothesis of no solar activity correlation at the 98.3% (2.38 σ) significancelevel.
This significance level is higher than the long-term correlation search,
and while not high enough to claim evidence of a solar correlation, is
nonetheless an indication that it may be possible to measure such aneffect with high accuracy in the next generation neutrino detectors. In
the Hyper-K case, based simply on the observed SK event counts, wewould expect a 3 σ sensitivity from even a single Forbush decrease
event.

8

CONCLUSIONS

Measurements of the atmospheric neutrino flux were performed using the water-Cherenkov detector Super-Kamiokande, which has theworld’s largest dataset of atmospheric neutrino interactions with an
exposure of approximately 315 kton yr. The atmospheric νe + ν̄e andνµ + ν̄µ fluxes were measured and analysed as a function of energy,
azimuthal direction, and time, with a detailed analysis of systematic
errors.
the energy spectra were measured in the range 100 MeV up
to 10 TeV. Our measured data provided significantly improved precision up to 100 GeV, and the first data below 320 MeV. While noneof the current generation flux models are strongly inconsistent with
our data measurement, there was some preference for the HKKM11
model as the best-fit to our data. Energy spectra measurements werealso obtained for the neutrino and antineutrino fluxes separately, using enriched neutrino and antineutrino samples; these were the firstsuch direct measurements obtained for the atmospheric neutrino flux,
and also showed agreement with the current flux models.
the azimuthal spectra showed an east-to-west dipole asymmetry in the neutrino flux, caused by the geomagnetic field, for bothneutrino flavours at > 5 σ. This represents the discovery of the effect for the νµ flux. The strength of this east-west effect was shown todepend on neutrino energy and zenith as predicted by the HKKM11
model. Furthermore, an indication of a predicted shift of the dipole
asymmetry angle depending on the zenith angle was seen at the 2.2 σlevel, which is the first measurement that explores the geomagnetic
field effects beyond a simple east-west asymmetry. These measurements give confidence that the neutrino flux simulations correctlymodel the complicated effects of the geomagnetic field.
a study of the time correlation between the atmospheric
neutrino flux and the solar magnetic activity cycle was performed.
The type of correlation predicted by the HKKM group was calculated
to have a relatively minor effect on most of the Super-Kamiokande
data, but by searching for a long-term correlation using approximately
14 years of data, a slight preference for such a correlation was seenat the 1.1 σ level. By examining several much shorter periods (not included in the long-term analysis) corresponding to especially strongsolar activity, from across the SK operational period for a total expo-

151

152

conclusions

sure of 7.1 days, an indication for some decrease in the atmospheric
neutrino flux was seen at the 2.4 σ level.
Summary and Future
These direct measurements of the atmospheric neutrino flux tested
the theoretical models with improved precision, and searched for several new physical effects not previously measured. While these effects
were seen at a range of significance levels, even those with relatively
low significance are interesting indications, that may be further studied by the next generation of neutrino detectors. Our measurements
are in general consistent with the current generation flux models,
which gives confidence in our understanding and modeling of theatmospheric neutrino flux.
In future, combining our measurements with those from other current generation neutrino detectors, which are sensitive at distinct
but overlapping energy regions, even more accurate data may be obtained. For example, constraints given by the improved precision of
our energy spectra measurement combined with other measurements
at higher energies can help to accurately determine the astrophysical neutrino spectra. In general, further improvements to the atmospheric flux measurements could also provide feedback to the flux
simulations, and further a better systematic understanding of the atmospheric neutrino flux as both a background and a signal source.

Part IV
APPENDIXAS K S Y S T E M AT I C E R R O R D ATA B A S E

Here we list all systematic errors in the SK systematic error database,excluding some non-standard systematics described in the text (suchas the azimuthal energy scale systematic, or the ν/ν̄ separation uncertainties).
a.1

flux

The flux-related errors are based in the estimated uncertainties on
the HKKM11 [19] atmospheric neutrino flux simulation. If numbersin brackets are given, they show the estimated errors at 1 σ.
• Absolute normalization at Eν < 1 GeV• Absolute normalization at Eν > 1 GeV• ν/ν̄ ratio at Eν < 1 GeV (1%)• ν/ν̄ ratio at 1 < Eν < 10 GeV (1.5%)• ν/ν̄ ratio at Eν > 10 GeV (2.5% until 30 GeV, then increases as
0.081 × log10 ( Eν [GeV]) − 0.093)• νe /ν̄e ratio at Eν < 1 GeV (2.5%)• νe /ν̄e ratio at 1 < Eν < 10 GeV (2.5%)• νe /ν̄e ratio at Eν > 10 GeV (4% until 100 GeV, then increases as
0.067 × log10 ( Eν [GeV]) − 0.103)• νµ /ν̄µ ratio at Eν < 1 GeV (1%)• νµ /ν̄µ ratio at 1 < Eν < 10 GeV (3%)• νµ /ν̄µ ratio at Eν > 10 GeV (3% until 50 GeV, then increases as
0.105 × log10 ( Eν [GeV]) − 0.123)• Relative normalization of the FC multi-GeV sample (as depending on the flux uncertainties, not the detector systematics)• Relative normalization of the PC-stopping and UPMU-stopping
samples (as above)• Up / down ratio• Horizontal / vertical ratio

155

156

sk systematic error database• K/π ratio (where K and π are the parent particles of the neutrinos)• ν path length (zenith-angle dependant)• Solar activity (compared to a fitted constant in each SK-I to SKIV period)a.2

cross section

The cross-section related errors come from uncertainties in the NEUT
predictions, and are explained in Sec. 4.5.1.
• Axial mass
• Axial mass for QE and single-pion interactions• CCQE cross-section (energy dependant, by comparison to othermodels)• CCQE ν/ν̄ cross-section ratio (energy dependant)• CCQE νe /νµ cross-section ratio (energy dependant)• Single meson cross-section• DIS cross-section• DIS model differences• Coherent π cross-section• NC / CC ratio• CCQE cross-section for ντ and ν̄τ• π 0 /π ± ratio• Single pion cross-section ratio for ν/ν̄
• DIS q2 at high W• DIS q2 at low W
a.3

detector

The detector errors come from uncertainties in the reconstruction abilities of the detector. The reduction and reconstruction errors are explained in Sec. 4. Generally, each of these errors is applied seperately
for SK-I to SK-IV.
• Ring separation• Single-ring PID

A.3 detector• Multi-ring PID• Energy calibration (the overall energy scale of the entire detector)• Up / down relative energy calibration• FC reduction efficiency• PC reduction efficiency• UPMU reduction efficiency• Fiducial volume cut efficiency• FC / PC separation• Non-νe background in e-like samples• Non-νe background in multi-GeV single-ring e-like samples• Non-νe background in multi-GeV multi-ring e-like samples• Non-νµ background in µ-like samples• Hadron simulation (in SKDetsim; responsible for NC contamination into single-ring events)• UPMU reduction energy cut• UPMU reduction path length cut• UPMU background reduction of non-showering events• UPMU background reduction of showering events• UPMU background reduction of stopping events• UPMU stopping / through-going separation• UPMU showering / non-showering separation• PC stopping / through-going separation (top cap)
• PC stopping / through-going separation (barrel)• PC stopping / through-going separation (bottom cap)
• Two-ring π 0 sample normalization• Polfit single-ring π 0 (moves samples between single-ring π 0 andsingle-ring e-like)• Decay electron tagging efficiency• Decay electrons from pions (the number of decay electrons able
to be tagged in various hadronic events)

157

158

sk systematic error database• Multi-GeV multi-ring other separation (where the “other” sample contains events not considered well-reconstructed enoughto be placed into the other multi-ring samples)• Multi-GeV multi-ring e-like νe /ν̄e separation (from an older likelihood; not used in this thesis)
a.4

oscillation

Systematic errors related to uncertainties on the neutrino oscillation
parameters (not used in the oscillation analysis, or in the atmospheric
neutrino flux energy spectra measurements in this thesis).
• θ12
• θ13
• θ23 (error size taken from the T2K experiment)• ∆m212• ∆m223 (error size taken from the T2K experiment)• Matter effects

B

POISSON LIKELIHOOD METHOD

b.1

poisson likelihood ratio

The Poisson distribution is given byhn e−h(70)n!
where h is the number of events predicted by a hypothesis H, and n
is the number of observed events. Summing over all data bins i, thelog-likelihood of one particular given set of data N is!
hini e−hiln L( N | H ) = ∑ ln
(71)
ni !
i
L(n|h) =and using this likelihood, the ratioL( N | H )λ H = −2 ln
L( N | N )(72)

can be shown [145] to be approximately distributed as χ2 ( NDF ) in
the limit n → ∞. If the hypothesis is a function of j free parameters,then NDF ≈ i − j − 1.
b.2wilks’ theorem

The maximum likelihood estimator L M ( N | H ), for the hypothesis H,
is defined as the case where L( N | H ) is maximised by modifying theparameters of H. By Wilks’ theorem [146], if we consider an alternate
hypothesis F that is a subset of H with a reduction in parameter spaceof k parameters, then the ratio
Λ = 2 ln

LM (N|H)
L M ( N | F)(73)

is asymptotically distributed as χ2 (k ) in the case that F is true. Thisis an important result that allows us to perform a significance test
between nested hypotheses, without the need for a toy MC study tounderstand the result.
Using the notation from Eq. (159), we can write
Λ = λF − λH(74)

and for the Poisson case, since λ ∼ χ2 ( NDF ), we can make an analogy of Λ to the commonly-used gaussian ∆χ2 statistic. In particle
physics, the statistic Λ is often written as ∆χ2 , even when a poisson
likelihood method is being used.

159

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We gratefully acknowledge the cooperation of the Kamioka Mining
and Smelting Company. The Super-Kamiokande experiment has been
built and operated from funding by the Japanese Ministry of Education, Culture, Sports, Science and Technology, the U.S. Department ofEnergy, and the U.S. National Science Foundation. Some of us havebeen supported by funds from the Research Foundation of Korea(BK21 and KNRC), the Korean Ministry of Science and Technology,
the National Research Foundation of Korea (NRF- 20110024009), theEuropean Union H2020 RISE-GA641540-SKPLUS), the Japan Society
for the Promotion of Science, the National Natural Science Foundation of China under Grants No. 11235006, the National Science andEngineering Research Council (NSERC) of Canada, and the Scinet
and Westgrid consortia of Compute Canada.
external data used
We acknowledge the NMDB database [141], founded under the European Union’s FP7 programme (contract no. 213007) for providing
data. The Newark, Thule, and McMurdo neutron monitors of the Bartol Research Institute are supported by the National Science Foundation. Kerguelen neutron monitor data were kindly provided by the
French Polar Institute (IPEV, Brest) and by Paris Observatory.
We are particularly grateful for the use of the atmospheric neutrino
simulation data released by the HKKM group, available at [147], andfor additional data provided by M. Honda by private communication.
colophon
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