Short Baseline Neutrino Oscillations with the Soli∂ Experiment a Likelihood Method to Discern Signal from Background

Short baseline neutrino oscillations with the SoLi∂ experiment A likelihood method to discern signal from background

Simon De Kock

Promotor: Prof. Dr. Dirk Ryckbosch Supervisor: MSc. Ianthe Michiels

A thesis submitted in partial fulfillment of the requirements for the degree of Master in Physics and Astronomy

Academic year: 2018 - 2019

Acknowledgements

First of all, I would like to thank Prof. Dr. Dirk Ryckbosch for the opportunity to be part of the SoLid experiment. Thanks to him I was lucky enough to experience an internship last summer at WIPAC in Madison where I learned a lot about different aspects of scientific research. Most importantly, the internship taught me how to question my own work in a consequent way; namely by examining all the steps of a process in detail to minimize the chance of errors. The knowledge and skills obtained had a big impact on how I approached the problems encountered in this thesis.

Thanks are owed to my supervisor, Ianthe Michiels, for her help throughout the year. The many ’sanity checks’ to avoid and explain problems were highly appreciated. The help with the structured presentation of my research and proofreading this work is also valued.

In addition, I would also like to thank the other member of the SoLid team at Ghent University, Giel Vandierendonck, for proofreading this thesis and his suggestions on the rate calculations.

Many thanks are owed to my mother, Hilde Ureel, for proofreading this thesis.

I would also like to extend thanks to my fellow thesis students for the conversa- tions on and off topic. The many answered questions, suggestions, discussions and Disney music are appreciated.

iii iv

Lastly, I wish to thank some people that had an indirect influence on my thesis. I would like to thank my parents and brother for tolerating my need to communicate on my passion for physics and science. To my friends, a big thanks for making the last year of my studies full of highlights. The last thanks are owed to the members of the VVN for keeping me engaged in science communication. The meetings, talks, and trip were a nice shift of focus. I would lastly like to thank them for letting me use their platform to publish my science communication video on neutrinos. Contents

Acknowledgements iii

List of Figures vii

List of Tables x

Wetenschapscommunicatie xiii

Inleiding xv

Introduction xvii

1 Neutrino physics 1 1.1 The standard model of particle physics ...... 1 1.2 Neutrinos in the Standard Model ...... 4 1.2.1 A brief history of neutrinos ...... 4 1.2.2 Solar neutrinos ...... 10 1.3 Neutrinos beyond the Standard Model ...... 13 1.3.1 Neutrino oscillations ...... 13 1.3.2 Neutrino mass hierarchy ...... 17 1.4 Neutrino anomalies ...... 19 1.4.1 Accelerator anomaly ...... 20 1.4.2 Gallium anomaly ...... 20 1.4.3 Reactor anomaly ...... 22 1.4.4 Possible solutions ...... 22 1.5 Sterile neutrinos ...... 24

v vi Contents

2 The SoLi∂ experiment 27 2.1 The BR2 reactor ...... 28 2.2 Detector principles ...... 29 2.2.1 Detecting positrons ...... 31 2.2.2 Neutron detection ...... 31 2.2.3 Signal collection ...... 33 2.3 The SoLid detector ...... 33 2.3.1 Detector layout ...... 34 2.3.2 Background reduction ...... 34 2.3.3 Detector prospects ...... 36 2.4 Detector challenges ...... 38

3 Detector data 39 3.1 Reconstruction ...... 39 3.2 Real and Fake IBD signal ...... 41 3.2.1 Accidental background ...... 41 3.2.2 Correlated background ...... 42 3.3 Event properties ...... 43 3.4 Initial cuts ...... 45 3.5 Correlations ...... 46

4 Likelihood method 49 4.1 Possible methods ...... 50 4.1.1 Rectangular cuts ...... 50 4.1.2 Likelihood method ...... 50 4.1.3 Machine learning ...... 51 4.2 Likelihood method ...... 51 4.2.1 Uncorrelated and correlated background ...... 52 4.2.2 Chosen variables ...... 54 4.2.3 Likelihood and global-likelihood functions ...... 54 4.3 Correlated data ...... 59 4.3.1 Using an n-dimensional Gaussian ...... 59 4.3.2 Transform to set of less correlated variables ...... 60 4.3.3 Shift the power balance of input variables ...... 60 Contents vii

5 Improving the likelihood method 63 5.1 Correlated data ...... 64 5.2 ROC curve evaluation ...... 64 5.2.1 Background selection ...... 68 5.2.2 Uncertainty in efficiency calculations ...... 70 5.3 Introducing extra cuts ...... 71 5.4 Results for reactor-off data ...... 74 5.4.1 Two background ROC analysis ...... 74 5.4.2 One background ROC analysis ...... 75 5.4.3 Additional cut(s) ...... 76 5.4.4 Final selection of cuts ...... 77 5.4.5 Results from rectangular cuts ...... 78

6 IBD analysis 79 6.1 Selection of data ...... 79 6.2 Reactor-on Analysis ...... 81 6.2.1 The procedure ...... 81 6.2.2 Results ...... 83 6.2.3 Comparison to theoretical predictions ...... 85

7 Conclusion & Outlook 89

A Index of likelihood functions 93

B Poisson and Exponential distributions 97

List of Figures

1.1 The standard Model of particle physics ...... 5 1.2 Energy spectrum of β emitter ...... 6

1.3 Z boson cross-section and the relation to Nν ...... 9 1.4 The main fusion channels according to SSM ...... 10 1.5 Energy spectrum of solar neutrinos ...... 11 1.6 Oscillation probabilities for an initial electron neutrino ...... 15 1.7 Representation of the possible neutrino mass hierarchies ...... 19 1.8 Results of a combined study of GALLEX and SAGE ...... 21 1.9 Measurements of reactor neutrino experiments at short baselines ... 23 1.10The case for a fourth low mass neutrino ...... 25

2.1 Schematic of the BR2 reactor building ...... 29 2.2 A PVT cube and the IBD signal ...... 30 2.3 Difference in positron and neutron signals ...... 32 2.4 Different stages of the assembly of the SoLid detector ...... 35 2.5 Background reduction for the SoLid detector ...... 36 2.6 Exclusion sensitivity of the SoLid experiment ...... 37

3.1 Uranium decay chain ...... 44

4.1 Distribution of time difference for reactor-off data ...... 53 4.2 Probability distributions of event types and properties ...... 55 4.3 Example of a global likelihood distribution ...... 57 4.4 Cumulative distribution of a global likelihood...... 58

ix x List of Figures

5.1 Example of 10 ROC curves ...... 65 5.2 ROC curve with both quality measures ...... 66 5.3 Best ROC curves using both backgrounds ...... 67 5.4 Best ROC curves using one background ...... 69 5.5 Best ROC curves for ∆r ...... 72 5.6 Best ROC curves for ∆t ...... 73

6.1 Environmental change in pressure in the detector hall ...... 80 6.2 Ratio plots of the final results for all five parameters ...... 84 6.3 Distributions of time between occurrences of successive events .... 87 List of Tables

1.1 The fundamental forces in physics ...... 3 1.2 Latest values of all neutrino oscillation parameters ...... 17

3.1 Initial cuts for future analysis...... 45 3.2 Correlation matrices for the different event types ...... 47

5.1 Best cuts using both backgrounds ...... 68 5.2 Best cuts using one background ...... 69 5.3 Best cuts for ∆r ...... 73 5.4 Best cuts for ∆t ...... 74 5.5 Initial number of events ...... 74 5.6 Cuts on λ using both backgrounds, with results ...... 75 5.7 Cuts on λ using one background, with results ...... 76 5.8 Cuts on λ and ∆r with results ...... 76 5.9 Cuts on λ and ∆t with results ...... 77 5.10Cuts on λ, ∆r and ∆t with results ...... 77 5.11Final cut values and result ...... 78 5.12Rectangular cut values and result ...... 78

6.1 Total detection time for the two types of data ...... 81 6.2 Counts for the individual event types and their errors ...... 83

A.1 Likelihood functions and their associated index...... 95

Wetenschapscommunicatie

Voor het luik wetenschapscommunicatie van de thesis werd gekozen om een filmpje te maken over neutrino’s. Er is geopteerd om dit in het Nederlands te doen aangezien het Engelstalige aanbod reeds groot is.

Het filmpje bespreekt in 7 minuten enkele van de belangrijkste ontdekkingen en de vooruitgang binnen de neutrino-fysica. Om de video zo toegankelijk mogelijk te maken werd besloten de nodige wetenschappelijke voorkennis tot een minimum te beperken.

Het werd voor een breed publiek beschikbaar gemaakt via YouTube onder een Cre- ative Commons-licentie met naamsvermelding en kan bekeken worden via volgende link: https://youtu.be/02mtBvu0Ybo

xiii

Inleiding

Neutrino’s behoren tot de meest populaire en intrigerende deeltjes binnen de ele- mentaire deeltjesfysica. Hoewel hun detectie alles behalve makkelijk is, lijken ze de meest geschikte kandidaat om (in)direct ongekende fysica waar te nemen.

Het pad van de neutrino fysica is gelegd door heel wat theoretisch en experimenteel werk. Iets meer dan 100 jaar na de eerste tekenen van neutrino’s staat de kennis heel wat verder dankzij wetenschappers als Pauli, Fermi, Pontecorvo, Reines, Perl en vele anderen. De hedendaagse beschrijving van deze deeltjes, welke steunt op drie smaken (soorten) van neutrino’s die hun smaak aanpassen tijdens het doorkruisen van tijd en ruimte, is sinds het begin van deze eeuw experimenteel bevestigd door de metingen van SNO en Super-KameokaNDE experimenten.

Om deze oscillaties beter te meten en te begrijpen werden nieuwe experimenten opgestart, deze bleken tekorten of excessen in vergelijking met de theoretische kennis te meten. De metingen in allerlei soorten experimenten lijken de standaard drie neutrino oscillatie hypothese in vraag te stellen. Deze van het model afwijkende metingen worden anomalieën genoemd.

Een nieuwe generatie aan experimenten loopt om ondermeer deze anomalieën verder te onderzoeken. Indien deze anomalieën aanwezig blijven kan het goed zijn dat neutrino’s de eerste deeltjes zijn, in lange tijd, die fysica vertonen buiten de huidige theoretische modellen. Zo is een mogelijke verklaring voor al deze anoma- lieën de introductie van een vierde speciaal neutrino genaamd steriele neutrino’s. Dit steriele neutrino is enkel onderhevig aan de zwaartekracht.

xv xvi Inleiding

In hoofdstuk 1 worden de theoretische en experimentele geschiedenis van neutrino’s in kaart gebracht. Het hoofdstuk start bij de theoretische voorspelling van neutrino’s en eindigdt bij de mogelijke introductie van steriele neutrino’s.

Één van deze nieuwe generatie experimenten, waar men op zoek gaat naar steriele neutrino’s, is het SoLid experiment. De antineutrino’s van een nucleaire reactor worden gebruikt om neutrino oscillaties waar te nemen en om mogelijke tekenen van steriele neutrino’s te observeren. In hoofdstuk 2 worden zowel de reactor als de detector van het SoLid experiment besproken.

Hoofdstuk 3 beschrijft hoe de metingen in het experiment worden omgezet in bruik- bare data. Deze data worden dan gebruikt om verder te zoeken naar steriele neutrino’s. Er zijn echter veel processen die de resultaten van de detector beïnvloeden. Om het aandeel van deze achtergrondprocessen zo veel mogelijk te verkleinen en zo veel mogelijk signalen te vinden is verdere analyse nodig.

In hoofdstuk 4 worden enkele methoden kort besproken waarna er één wordt gekozen om verder op te focussen: de likelihood methode. In hoofdstuk 5 wordt deze methode verder verfijnd om optimale resultaten te bekomen. De resultaten van deze methode worden in hoofdstuk 6 besproken. Introduction

Neutrinos are among the most popular and intriguing particles in elementary particle physics. Although their detection is anything but easy, they seem to be the most suitable candidate for (in)directly observing new physics.

The path of neutrino physics has been laid by a great deal of theoretical and experimental work. Just over 100 years after the first signs of neutrinos, science is much further along thanks to scientists such as Pauli, Fermi, Pontecorvo, Reines, Perl and many others. The contemporary image of three neutrino flavors oscillat- ing into each other has been experimentally confirmed since the beginning of this century by the measurements of SNO and Super-KameokaNDE experiments.

In order to better measure and understand these oscillations, new experiments were started. These measured a lack or excesses of neutrinos compared to theoretical knowledge. The measurements in all kinds of experiments seem to question the standard three-neutrino oscillation hypothesis. The measurements that deviate from the model are called anomalies.

A new generation of experiments is underway to further investigate these anomalies. If these anomalies remain present, it may well be that neutrinos are the first particles (in a long time) that show physics outside the current theoretical models. A possible explanation for all these anomalies is the introduction of a fourth special neutrino called sterile neutrinos. This sterile neutrino is only subject to gravity.

xvii xviii Introduction

In chapter 1 the theoretical and experimental history of neutrinos is mapped. The chapter starts with the theoretical prediction of neutrinos and ends with the possible introduction of sterile neutrinos.

One of this new generation of experiments, where people are looking for sterile neutrinos, is the SoLid experiment. The antineutrinos from a nuclear reactor are used to observe neutrino oscillations and to observe possible signs of sterile neutrinos. In chapter 2 both the reactor and the detector of the SoLid experiment are discussed.

Chapter 3 describes how the measurements in the experiment are converted into usable data. These data are then used to continue the search for sterile neutrinos. However, there are many processes that influence the results of the detector. To reduce this background and find as much signal as possible, further analysis is needed.

In chapter 4 a few methods are briefly discussed, after which one is chosen to focus on further: the likelihood method. In chapter 5 this method is further refined to obtain optimal results. The results of this method are discussed in chapter 6. Chapter 1

Neutrino physics

Nature has always looked like a horrible mess, but as we go along we see patterns and put theories together; a certain clarity comes and things get simpler.

– Richard P. Feynman, QED - The Strange Theory of Light and Matter

This chapter will introduce the Standard Model of particle physics in section 1.1, in section 1.2 the history of neutrinos in the Standard Model is discussed. Section 1.3 deals with the unusual properties of neutrinos that are not incorporated in the Standard Model, this is further highlighted in section 1.4 where neutrino anomalies are introduced. A possible solution for these anomalies is discussed in section 1.5.

1.1 The standard model of particle physics

The goal of physics is to explain and model observable phenomena through a math- ematical description, making humanity able to accurately predict these phenomena and use them in our advantage. Many physicists are and have been focused on a bigger goal, bundling many observations together in a single theory. By looking

1 2 1.1. The standard model of particle physics for common mechanisms they try to unite different theories into a single unified theory.

One of these unified theories is called the Standard Model (SM), this theoretical framework describes all known subatomic particles and the forces that act between them. In physics, there are four fundamental forces; the electromagnetic force, gravity, the weak nuclear force, and strong nuclear force. All of these forces are incorporated in the SM, with the exception of gravity. It turns out gravity has some fundamental differences compared to the other forces. The hope is to one day bring gravity together with the other known forces; this unification is generally known as a Theory of Everything. It turns out one can compare the strength of these four forces, this comparison can be found in table 1.1. It is clear that gravity is much weaker than the other forces and it is thus not needed to accurately describe the subatomic world.

The combination of theoretical models and observations, that confined these models, made it possible to put the SM in its final form by the middle of the 1970s. The model uses the language of Quantum Field Theory (QFT), a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. The SM consists of a unification of the electro-weak theory (which itself is a unification of the electromagnetic and weak nuclear force) and the theory of quantum chro- modynamics (QCD) which describes the strong nuclear force. In this model, the interactions between particles are mediated by gauge bosons, which are integer spin particles. Only particles that have a charge associated with a certain boson will be able to interact through that force1. All gauge bosons and associated charges are mentioned in table 1.1. All other particles in the SM are half-integer spin particles named fermions, they are the basic building blocks of all matter seen around us.

A further subdivision can be made between bosons; massless and massive bosons. The Photon and the Gluon are massless, the W , Z0 and Higgs bosons do have a mass of respectively 80.38GeV, 91.19GeV and 125.18GeV.

1Except for the Higgs boson which does not mediate a force. Chapter 1. Neutrino physics 3

Force Relative strength 2 Gauge boson(s) Charge

Strong nuclear 1 8 Gluons Colour Electromagnetic 10−3 Photon Electric Weak nuclear 10−8 W , Z0 Weak isospin Gravity 10−37 Graviton3 Mass

Table 1.1: The fundamental forces in physics. [1]

Fermions too can be divided into smaller categories, which are the three generations of fermions. The first generation gives us the building blocks for the matter we observe in our daily lives, these contain the lightest most stable particles. The second and third generation can be toughed of as heavier versions of the first, but unstable due to their higher mass.

A second way of separating fermions in groups is through the forces they interact with. A clear distinction can be made between those that hold colour charge (named quarks) and those that do not (named leptons). As the absence of charge results in no interaction with the associated force, leptons cannot interact via the strong nuclear force. This presence/absence of colour charge has big implications, as the strong nuclear force allows only colourless particles to exist freely. This results in two or three quarks combining together to make a composite particle with no colour, called Hadrons. Composite particles made of two quarks are called Mesons, those with three quarks Baryons. There are two quarks in each generation and three generations, this makes for a lot of possible combinations which leads to a lot of Hadrons. The situation was particularly confusing in the late 1960s when the quark was not yet discovered but hundreds of Hadrons types had been observed. The most common Hadrons (Baryons) are the neutron (udd) and proton (uud), which are the building blocks of the atomic nucleus.

Leptons, on the other hand, are not restricted in this way and can exist freely. There are also three generations observed for leptons, each with two leptons; an electrically charged lepton (named electron (e), muon (µ) or tau (τ)) and a neutral

2Compared to the the strong nuclear force. 3The Graviton is the hypothetical quantum of gravity, it has not been measured nor is it used in established theories. 4 1.2. Neutrinos in the Standard Model lepton (named neutrino (ν)). For each charged lepton there is a corresponding neutrino, they are arranged in a doublet structure of weak isospin. This weak isospin doublet structure can also be found in quarks within a generation. Each element of the doublet can transform into its counterpart by emission or absorption of a W boson. Right-handed fermions do however not couple to the W boson in the SM (due to maximal parity violation), they will only interact weakly with the Z boson. As the SM assumes neutrinos to be massless4, this means the model only assumes left-handed neutrinos.

The SM also predicts that for every particle there is an antiparticle. Antiparticles have the same mass but opposite charges. This means a total of 24 fermions and 13 bosons5 are expected in the SM. A detailed description of all the particles in the SM can be found in figure 1.1.

1.2 Neutrinos in the Standard Model

Neutrinos are the odd ones out in the SM: only upper limits on their masses are known, which are much lower than the masses of the other particles in the SM. Fur- thermore, they are the only fermions with no charge and no right-handed partners. For these reasons and more they are interesting particles to study.

1.2.1 A brief history of neutrinos

Theoretical evidence

In 1914 the first signs of neutrinos were observed by J. Chadwick when studying the spectrum of a β emitter [5]. β emission is the process of radioactive decay of an atomic nucleus to a lighter one by the emission of an electron or a positron.

4Back then it appeared neutrinos where massless. 5Boson antiparticles of have already been accounted for. Chapter 1. Neutrino physics 5

gravitational force (mass) 2 1/2 eV 32 − outside 10 × 6 ... sterile neutrino graviton ? < standard model

weak nuclear force (weak isospin) 0 GeV

18 electromagnetic force (charge) . bosons H Higgs 125 Goldstone 1 strong nuclear force (color) GeV 1 1 19 . color Z 91 13 bosons 1 1 g γ gluon photon force carriers GeV 38 charge colors mass spin . W 80 1 1/3 − 2/3 1/2 1/2 1/2 1/2 −

R/G/B R/G/B generation MeV τ rd GeV 2 GeV . 0 GeV . neutrino 18 2 80 . . 3 t b τ ν top bottom tau τ 173 4 1 < → 1 1/3 − 2/3 1/2 1/2 1/2 1/2 −

R/G/B R/G/B nd and 2 unstable matter MeV µ keV MeV 66 GeV . 0 neutrino 190 . 3 . c s µ ν charm strange muon µ 1 95 105 < 12 fermions 1 1/3 increasing mass − 2/3 1/2 1/2 1/2 1/2 − (+12 anti-fermions) R/G/B R/G/B st e MeV 1 eV MeV MeV 2 neutrino 2 7 511 . . . u d e ν up down electron e 2 4 0 < standard matter

6 quarks 6 leptons (+6 anti-quarks) (+6 anti-leptons)

Figure 1.1: The standard model of particle physics. [2][3][4] 6 1.2. Neutrinos in the Standard Model

Until then the consensus was that a β emission followed the following process:

n → p + e (1.1)

Using momentum and energy conservation for a two body decay, in 1927 C. D. Ellis and W. A. Wooster calculated that for a quantized nucleus both particles are expected to have a fixed energy [6]. As the proton is confined in the core it will (to a good approximation) stay at rest, the smaller electron will thus escape with all of the energy which results in a fixed expected energy spectrum for a given source. However, Chadwick measured a continuous electron spectrum, this can be found in figure 1.2.

Figure 1.2: Measured electron energy spectrum of a β emitter in black and the expected spectrum (according to reaction 1.1) in red. [7]

In 1930 W. Pauli postulated a theoretical explanation for this result, which was further developed by E. Fermi in 1933. By introducing a new chargeless, low mass fermion they were able to explain the continuous β spectrum; this new particle would be hard to detect and gives fewer constraints on the kinematics of the electron in this interaction. The new fermion had to be chargeless as it would otherwise already have been detected and charge conservation cannot be violated. The study of the kinematics of the interaction and the associated observed spectrum hints towards a mass of the particle that is much smaller than the mass of the electron. Fermi coined the name ’neutrino’ (Italian for ”little neutral one”) to distinguish it from the then recently discovered neutron [8]. This then new theoretical explana- Chapter 1. Neutrino physics 7 tion introduced a new way of describing β emitters:

n → p + e + ν

Shortly after Fermi published his theory of beta decay, H. Bethe and R. Peierls calculated the cross-section (σ) of this interaction. Their first calculations showed that σ < 1044cm2 [9], a cross-section that was much smaller than any other process known at that time. It was the first sign of a new type of interaction that was unknown at the time.

Experimental evidence

Detecting neutrinos was (and still is) a complicated problem since most particle detectors use the electric charge or the heavy mass of particles to detect them. Neither of these options is possible for neutrinos. Detecting the effects of neutrinos interacting with matter makes it possible to detect them. The interaction creates secondary particles that do have charge and thus can be detected. The big disadvantage with this method is the extremely small cross-section of neutrinos, large fluxes of neutrinos are needed to detect them in a reasonable time.

The first neutrino detection experiment was done by C. Cowan and F. Reines, they published their results of the first direct measurement of neutrinos in 1965 [10]. Using W. Ganchangs proposal to experimentally detect neutrinos, they placed a detector near a nuclear reactor. This reactor should, according to predictions have a high (anti)neutrino flux (≈ 5×1013s−1) . The process used for detection is called inverse beta decay (IBD).

ν + p → n + e+

In an IBD one can convert ν into e+, these then quickly annihilate with electrons. The annihilation leaves a clear signal of two resulting γ-rays which are rather easy to detect. They decided to use the Hanford site for their preliminary experiment and later moved the experiment to the Savannah River plant. In 1995 the Nobel prize in physics was given to the first detection of neutrinos to Reines only, as Cowan died in 1974. 8 1.2. Neutrinos in the Standard Model

Since the muon was already known at that time, it was wondered if the neutrinos associated with them were different from those associated with electrons. In 1962 L. M. Lederman, M. Schwartz and J. Steinberger tried to answer this question by performing an experiment at the Brookhaven National Laboratory. Using the 1GeV Alternating Gradient Synchrotron (AGS) beam they created pions [11], by hitting protons on a beryllium target, which decay in flight. The pion decays dominantly via a muon:

π+ → µ+ + ν

If different generations of neutrinos would exist, there would be a neutrino associated with the electron (νe) and one with the muon (νµ). If the neutrinos created in this experiment were able to create both electrons and muons further down the line, no second generation exists. The experiment found that only muons were created by these neutrinos. The second generation of neutral leptons was discovered, for this discovery Lederman, Schwartz and Steinberger were rewarded with the Nobel Prize in 1988.

Between 1974 and 1977 a series of experiments was done at SLAC by M. L. Perl and collaborators. They detected anomalous events that showed signs an unknown particle. Further investigation revealed a new charged lepton [12], for this discovery Perl shared the Nobel prize with Reines in 1995. The discovery of this third lepton, the τ particle, was immediately followed by the prediction of a third neutrino flavour. This lead some to suggest the existence of the third generation of quarks. The first quark of this third generations was indeed discovered in 1977 [13] which resulted in a further belief that a third neutral lepton would be discovered.

In 1995 the last quark of the third generation had been detected [14], leaving only one fermion to be discovered according to figure 1.1. In 2001 the third flavour of neutrinos was finally discovered at DONUT, this flavour was produced through the decay of charmed mesons [15]. Chapter 1. Neutrino physics 9

Before 1989 no limits had been set on the number of generations of neutrinos or other fermions. This lead to a series of precision measurements that were done between 1989 and 1995 at LEP, one of which was measuring the cross-section of the Z-boson. In the SM this is predicted to be partially related to the number of low mass neutrino flavours:

ΓZ = 3Γ (Z → l + l) + Γ (Z → hadrons) + Nν Γ(Z → νl + νl)

At LEP they measured the cross-section to an incredible precision and obtained

Nν = 2.92 0.05 [16]. The result of this detection can be seen in figure 1.3. Again this was generalised to all fermions for symmetry reasons.

Figure 1.3: Measurements of the hadron production cross-section around the Z resonance for several experiments. The curves indicate the predicted cross-section for two, three and four neutrino generations. [16]

The results of all these experiments confirmed the SM, only the Higgs boson which had long been predicted by theory was left to be discovered. Eventually, the discovery of the Higgs boson was announced in 2012 [17]. With this last detection, it appears the SM is complete and able to explain all possible observations. 10 1.2. Neutrinos in the Standard Model

1.2.2 Solar neutrinos

As the understanding of particle physics increased some physicist tried applying this knowledge to different phenomena. They tried applying it to other fields with unsolved problems, such as astronomy. One of the big unanswered problems in astronomy was the inner working of the Sun, they were interested in the composition and the reaction mechanisms that explained their observations. One of the main ingredients in tackling this problem was the increasing knowledge in nuclear fission and fusion. By the middle of the 1960s, the first calibrated models were created, called Standard Solar Models (SSM). These models consisted of a large number of fusion cycles that follow up one another. The current version of the SSM can be found in figure 1.4.

pp pep + + 2 + 99.77% 0.23% + − + 2 p + p → H + e + νe p + e + p → H + νe

−5 2H 10 % 84.92% 2 + 3 3 + 4 + H + p → He + γ He + p → He + e + νe 15.08% hep 0.1% 3He +4 He →7 Be + γ 7Be 99.9% 7 − 7 7 + 8 Be + e → Li + νe Be + p → B + γ ppI 8B 3 3 4 + 7 + 4 4 8 8 ∗ + He + He → He + 2p Li + p → He + He B → Be + e + νe ppII 8 ∗ →4 4 Be He + He ppIII

Figure 1.4: The main fusion channels according to SSM, the pp-channel which is most prominent is marked in yellow. The different processes which produce neutrinos are also highlighted in red. The SSM only predicts production of electron neutrinos. [1][18]

To check the validity of the SSM an experiment was proposed that would detect the neutrinos associated with these cycles, called solar neutrinos. In order to plan and build an experiment, more information about these solar neutrinos is needed as most detectors have a threshold of the neutrino energy. In figure 1.5 the energy spectrum of neutrinos from the different fusion channels can be found. Chapter 1. Neutrino physics 11

Figure 1.5: Energy spectrum of solar neutrinos and the sensitivity of different detectors. According to SSM, the total integrated solar neutrino flux is approximately 38 −1 2 ×10 νes .[19]

In order to detect as many neutrinos as possible, a detector with the lowest possible energy threshold should be constructed. As seen in figure 1.5, the best candidates to detect neutrinos are Gallium and Chlorine.

By the late 1960s, the SSM was being tested by R. Davis and J. N. Bahcall with a detector using chlorine as the active volume, deep in the Homestake mine in South Dakota. This detection mechanism relies on the following reaction:

37 37 − νe + Cl → Ar + e (Eν ≥ 0.814MeV)

The concentration of Argon in the detector will thus be proportional to the flux of neutrinos over the period of exposure. Other processes induced by cosmic rays can can mimic neutrino signals in this detector. Therefore, these types of experiments are located deep underground to shield from cosmic rays as much as possible.

Despite the fact that neutrino fluxes are huge, the observed rate is small due to the weak interacting nature of the particle. In order to deal with these small units a new notation was introduced, the Solar Neutrino Unit (SNU = 10-36s−1) which is a measure for the number of interactions per second. Bahcall calculated the 12 1.2. Neutrinos in the Standard Model theoretically predicted flux, assuming the SSM are correct:

R (Cl, SSM) = 8.5 0.9 SNU

The final results found a surprising discrepancy between the experiment and the prediction [20].

R (Cl, exp) = 2.56 0.16(stat) 0.16(sys) SNU

Only one third of the total predicted flux was measured.

To check if no errors were made in the preliminary results and to rule out any problems that were detector related, three new experiments (GNO, GALLEX, and SAGE) were constructed using a different detection mechanism. Each of these new experiments used the same Gallium based method:

71 71 + νe + Ga → Ge + e (Eν ≥ 0.233MeV)

As their detection method was different, they would detect solar neutrinos at a different rate. The theoretical prediction for these types of detectors were:

+8.1 R (Ga, SSM) = 127.9−8.2 SNU

Each of the three experiments found similar results that agreed very well with each other. The final results of these Gallium based detectors gave about half of the expected flux [21].

+7.1 R (Ga, GALLEX) = 73.4−7.3 SNU

+5.5 R (Ga, GALLEX + GNO) = 69.3−5.5 SNU

+6.5 R (Ga, SAGE) = 70.8−6.1 SNU

Although the detectors relied on different detection mechanisms they all observed the same deficit in their measurement of solar neutrinos. Soon physicists stated wondering whether the SSM might be wrong, however other types of astronomical observation agreed very well with the SSM.

It became clear that the SM did not have all the answers to explain this strange observation of solar neutrinos. Chapter 1. Neutrino physics 13

1.3 Neutrinos beyond the Standard Model

By the end of the 20th century these Chlorine and Gallium experiments had come to a conclusion and the results were clear: some part of the description of neutrinos was missing in the SM. As most physicists were unwilling to give up the SSM, new solutions were looked for.

1.3.1 Neutrino oscillations

The most prominent solution was actually already introduced back in 1962; Maki, Nakagawa and Sakata assumed the two flavour eigenstates (the state in which particle interactions are described: νe,µ) are a mixture of two neutrino mass eigenstates (the state relevant for propagation of particles in vacuum: ν1,2)[22]. In their calculations they assumed the mass eigenstates propagate as plane waves in the form:

i(⃗p1.⃗x−E1t) −iϕ1 |ν1 (t)⟩ = |ν1⟩ e = |ν1⟩ e

i(⃗p2.⃗x−E2t) −iϕ2 |ν2 (t)⟩ = |ν2⟩ e = |ν2⟩ e

At the time they used a two flavour description of the flavour (or weak) eigenstates. These flavour eigenstates are related to the mass eigenstates through a 2×2 unitary matrix, most easily represented as:      

νe cos θ sin θ ν1   =     νµ − sin θ cos θ ν2 where θ is called the mixing angle.

From calculations based on the previous assumptions, it turns out that neutrinos are allowed to oscillate: after some time T and distance L the probability of an electron neutrino becoming a muon neutrino is non-zero [1]. These oscillations are, however, only possible if the different flavours of neutrinos have different masses.  [ ]  2 2 ∆m12 eV 2 2   P (νe → νµ) = sin (2θ) sin 1.27 L [km] (1.2) Eν [GeV] 14 1.3. Neutrinos beyond the Standard Model

The work was translated in a three flavour basis by B. Pontecorvo and V. Gribov in 1969 [23]. They first introduced what is now known as the Pontecorvo-Maki- Nakagawa-Sakata (PMNS) matrix:

  Ue1 Ue2 Ue3      UPMNS = Uµ1 Uµ2 Uµ3

Uτ1 Uτ2 Uτ3

The PMNS matrix is most commonly parameterized by three mixing angles (θ12, θ23 6 and θ13) and a single complex phase called δ that allows for CP violation .

      −iδ 1 0 0   cos θ13 0 sin θ13e   cos θ12 sin θ12 0         ×   × −  UPMNS = 0 cos θ23 sin θ23   0 1 0   sin θ12 cos θ12 0 iδ 0 − sin θ23 cos θ23 − sin θ13e 0 cos θ13 0 0 1

After some calculation, one finally finds the three flavour equivalent of equation 1.2 [1][24]:

CP conserving z }|  [ ] { 2 2 ∑ ( ) ∆mij eV P (ν → ν ) = δ − 4 Re U ∗ U U ∗ U ∗ sin2 1.27 L [km] α β αβ αi βi αj βj E [GeV] i>j ν  [ ]  2 2 ∑ ( ) ∆mij eV + 2 Im U ∗ U U ∗ U ∗ sin 2.54 L [km] (1.3) αi βi αj βj E [GeV] i>j ν | {z } CP violating

2 − 2 where ∆mij = mi mj . As the only imaginary parts in the PMNS matrix are due to the complex phases, it is clear from equation 1.3 that the CP violating parameter in the matrix can have an effect on the oscillation probability. In figure 1.6 the oscillation probability of an electron neutrino that ends as every possible neutrino flavour is found.

The theory of neutrino oscillations described above is explicitly done in a vacuum and is therefore called vacuum neutrino oscillations.

6If neutrinos turn out to be Majorana particles, a fourth matrix will have to be added that introduces two additional complex phases: α1 and α2. Chapter 1. Neutrino physics 15

Figure 1.6: Probability of finding νe , νµ and ντ as a function of the distance per en- 2 2 2 ergy using equation 1.3. Assuming sin (θ13) = 0.0256, sin (θ23) = 0.413, sin (θ12) = 2 ≈ 2 × −3 2 2 × −5 2 0.314, ∆m32 ∆m13= 2.32 10 eV and ∆m12 = 7.59 10 eV , normal mass hierarchy and δ = 0.[25]

A different effect that also leads to neutrino oscillation is an effect of interaction with matter. This effect is called the Mikheyev-Smirnov-Wolfenstein (MSW) effect and appears when neutrinos travel through very high-density objects. As solar neutrinos originate in the center of the sun, these conditions are indeed met. The MSW effect enhances the effects of vacuum oscillations of the dense region travelled, depending on the density.

In order to find out which of the previous explanations is correct, a new series of solar neutrino experiments started. This time the detector had to be sensitive to each flavour of neutrino as the sum of the three should result in the total predicted flux. Two big experiments looked into the problem of these vanishing neutrinos: the Sudbury Neutrino Observatory (SNO) and Super-Kamioka Neutrino Detection Experiment (Super KamiokaNDE). Both used Cherenkov detection7 to find signs of

7When particles traverse a medium with a speed greater than the speed of light in that medium, they radiate. This radiation is sent out in a cone like shape in the direction of propagation. The detection of this cone makes it possible to detect the particle and reconstruct its movements. 16 1.3. Neutrinos beyond the Standard Model the presence of different flavours of neutrinos. As one can see in figure 1.5 the threshold energy for these types of detectors is much higher, thus a lower flux is expected. One is however better at measuring neutrinos that originate from the Sun, as Cherenkov detectors allow path reconstruction.

In 2002 the SNO collaboration released their results [26]:

SNO +1.57 +0.55 ϕNC = 6.42−1.57(stat)−0.58(sys)

SSM +1.01 ϕNC = 5.05−0.81 where NC stand for the neutral current, a detection channel that is sensitive to all three neutrino flavours. These results are in good agreement with theory compared to the Homestake, GNO, GALLEX or SAGE results. This measurement was able to confirm neutrino oscillation but unable to differentiate between vacuum oscillations and the MSW effect. The Super KamiokaNDE, however, measured neutrino oscillation in atmospheric neutrinos8 [27], as the atmosphere can hardly be considered dense these neutrinos have definitely not been affected by the MSW effect. One can calculate the mixing angles from these results, when compared there is some difference between the SNO data and current results. It turns out the SNO detector did measure the MSW effect [28]. In 2015 A. B. McDonald (director of SNO) and T. Kajita (a lead scientist at Super-KamiokaNDE) received the Nobel prize in physics ”for the discovery of neutrino oscillations, which shows that neutrinos have mass”9.

A completely different solution to the problem was proposed by B. Pontecorvo in 1967, right when the first measurements of Homestake were released [29]. At that point, it had become clear the electron and muon neutrino were different particles and the data somehow suggest CP violation. Pontecorvo suggested there could be a process where, when a neutrino becomes an anti-neutrino (or the other way around), a neutrino is obtained that does not interact via the weak force. He called this a sterile neutrino. The sterile neutrinos Pontecorvo described did not agree with observations and were thus forgotten.

8Neutrinos produced as decay products in hadronic showers resulting from collisions of cosmic rays with nuclei in the upper atmosphere. 9This description is misleading as both detectors measured a different effect. Chapter 1. Neutrino physics 17

2 Finding all neutrino oscillation parameters (∆mij, θij and δ) is done at many detectors across the world, the latest values for these parameters can be found in table 1.2.

Table 1.2: Latest values of all neutrino oscillation parameters [30]. Figure 1.6 uses very similar results (except for δ) and gives quite an accurate description of the 2 ≈ 2 real process. One can indeed assume ∆m13 ∆m32 as they are both orders of 2 magnitude bigger than ∆m21.

Parameter Normal hierarchy Inverted hierarchy [ ] 2 −5 2 10 ∆m21 10 eV 7.53 0.187 [ ] 2 −3 2 ∆m32 10 eV 2.444 0.034 −2.53 0.05

2 +0.013 10 sin (θ12) 0.307−0.012

2 +0.019 +0.023 sin (θ23) 0.512−0.022 0.536−0.028 [ ] 2 −2 10 sin (θ13) 10 2.18 0.07

+0.18 10 δ 1.37−0.16

1.3.2 Neutrino mass hierarchy

Thanks to SNO and Super-KamiolaNDE there is no question neutrino oscillations do take place, this has some big implications for the SM. The model had thus far assumed neutrinos to be massless, but equation 1.3 clearly shows there must be a mass difference between the different types of neutrinos.

Some neutrino masses can be measured through observation of nuclear decay, others need neutrino oscillations studies to obtain results. As neutrino masses are very small, it is hard to find definitive answers for the value of the mass but one can, however, find upper limits for them. Future experiments are needed to pinpoint the exact value of the mass.

10 The values of these do not depend on the type of hierarchy. The exact measurements (with signs) of the other parameters will determine the hierarchy. 18 1.3. Neutrinos beyond the Standard Model

The upper limit for the mass of the electron neutrino in figure 1.1 was found by studying the energy distribution of the electron in the β decay of tritium (3H). Hav- ing a spectrum similar to figure 1.5, the tail end of the spectrum is observed. The maximal energy of the electron gives information about the minimum energy needed to produce a neutrino, this can be translated in an upper limit for the mass of electron neutrinos. A new experiment is build to give more precision to current results, the Karlsruhe Tritium Neutrino (KATRIN) experiment is designed to reach a mass sensitivity of 0.2eV (90% C.L.) [31].

2 The other neutrino masses can be found through analysis of the different ∆mij mass differences. A big problem arises, as the masses are squared one cannot identify the hierarchy of the masses. There are two possible solutions: the normal hierarchy and the inverted hierarchy of neutrino masses, these can both be found in figure 1.7.

In table 1.2 latest values of the neutrino oscillation parameters can be found. To measure the real value of a square-mass difference and the mixing angles, one needs multiple sources that have different densities between source and detector. The results will vary dependent on the density due to the MSW effect. This difference makes it possible to calculate the actual square-mass difference instead of 2 the absolute value. This has already been done for the ∆m21 where the Sun and an artificial source were used. The Earth will be used as the dense region to measure the other parameters in the same way, but since the density is much lower than the Sun the MSW effect is much smaller. Better precision measurements of these parameters are thus needed to find the actual values, this will for once and for all determine the mass hierarchy of the neutrinos.

Further information about the mass of neutrinos can be obtained from cosmological observations and models. As the abundance and mass of neutrinos have an effect on the formation of the early universe, constraints can be found on the mass of these neutrinos. These constraints will vary dependent on the used model and the latest observations. When loose criteria are used, one finds [33]: ∑ mν < 0.118eV Chapter 1. Neutrino physics 19

Figure 1.7: Representation of the possible neutrino mass hierarchies and the flavour ( ) composition of each mass eigenstate. Note that ∆m2 is equivalent to ∆m2 and ( ) atm 32 2 2 ∆m sol is equivalent to ∆m21.[32]

This value is quite close to the minimum total mass required for inverted hierarchy (≃ 0.1eV), in the next decade several experiments will attempt to determine the neutrino mass hierarchy [34].

1.4 Neutrino anomalies

The previous section gives a clear picture of the shortcomings of neutrinos within the SM, however, there are experiments that seem to detect deviations of this neutrino oscillation theory. This section will discuss the different anomalies, compared to the standard theory of neutrino oscillation, detected in different detectors and experiments. 20 1.4. Neutrino anomalies

1.4.1 Accelerator anomaly

In 1996 the Liquid Scintillator Neutrino Detector experiment (LSND) was investi- gating νµ → νe osculations with high sensitivity and neutrino cross-sections. These

νµ were produced in the same way as Lederman, Schwartz and Steinberger did to detect muon neutrinos. After the analysis was done, a 3.5 σ excess of νe was found over the background [35]. This was the first result that did not fit in the three-generation oscillation model.

During the same years, a different experiment ran with a similar setup, called the

KARMEN experiment. Although it was set up to measure ∆me,µ, the measurements could also be used to confirm the anomaly found at the LSND experiment [36]. No such anomaly was found, but since KARMEN used a different baseline than LSND it could not fully exclude the possibility of an anomaly.

In order to know the origin of the anomaly, a new experiment started in 2007, called the MiniBooNE experiment. It investigated both νµ → νe and νµ → νe oscillations and found a 3.0 σ excess of electron neutrinos over the background [37]. The latest results of the MiniBooNE were released in October 2018 and they still confirm a significant excess of electron-like events. This latest report claims the results are consistent in energy and magnitude with the excess of events reported by the LSND, and the significance of the combined LSND and MiniBooNE excesses is 6.0σ [38].

1.4.2 Gallium anomaly

Around the same time as the LSND experiment, two other experiments were run- ning; GALLEX and SAGE. Both experiments used Gallium (Ga) as a detection material which would transform in Germanium (Ge) through neutrino capture, as explained in section 1.2. To test these detectors researchers places well known radioactive materials in the detector that decay as follows:

51 − 51 37 − 37 Cr + e → V + νe Ar + e → Cl + νe Chapter 1. Neutrino physics 21

As almost every molecule of Ge found in the detector originates from a neutrino that comes from one of these sources 11, one can find the total neutrinos captured in the experiment. Theoretical calculations can also be done to find this rate of capture. It turns out that theory and experiment do not align, there is a 2.7 σ difference between theory and experiment for the neutrino capture rate. Even taking into account the large uncertainties of the detector, still, a 1.8 σ scarcity of experimental observation is found when compared to theory [39][40]. In figure 1.8 all individual results and the result of a combined analysis are presented.

Figure 1.8: Results of a combined study of GALLEX12 and SAGE neutrino capture using known test sources. A combined result of observed over predicted rate is 0.870.05. [41]

11The flux of neutrinos from these sources must be sufficiently high compared to the background. 12Two separate test were done using 51Cr. 22 1.4. Neutrino anomalies

1.4.3 Reactor anomaly

A third anomaly is measured when comparing theoretical predictions to the measurement of neutrino fluxes in nuclear reactor experiments. Many experiments (eg. Bugey, ILL, Goesgen, ROVNO, ...) are set up around nuclear reactors to measure neutrino oscillations13. Focusing on experiments with a short baseline, located between 10m and 100m from the reactor, an anomaly is present. The expected rate of νe is lower than theoretically predicted, the ratio of observations over theoretical calculations gave a small deficit: 0.9800.024. Later, the theoretical calculations were improved and the expected flux of the reactors was now expected to be 3.5% higher [42]. This means the ratio of neutrino detections over theory had to be recalculated and now a 3.1σ difference is found between theory and experiment [43]. In figure 1.9 the combined results are found, after improved calculations of the expected neutrino flux.

1.4.4 Possible solutions

All previous anomalies are consistent, when measuring at distances between 10m and 100m from the source the experiments report one of two things:

• When νe (and νe) are produced at the source, a deficit of them is observed when compared to the three flavour neutrino oscillation theory.

• When starting from other types of neutrinos, an excess of electron(anti)neutrinos is seen.

These anomalies must have a sound physical explanation to solve the problem.

One of the explanations is that the physics inside the reactor might not be fully understood. The problem might lie in particular with the reactor flux calculations, these can be done with different methods that give somewhat different results. It is not unlikely that in some of these methods something is missing, after all, one has to take into account the different decay chains of the reaction products with

13Nuclear reactors make a good environment for these experiments as huge neutrino fluxes are present originating from a very confined source. Chapter 1. Neutrino physics 23 their branching fractions and decay rates. Although this explanation works well for the reactor anomaly, it is less convincing for the other two anomalies.

Figure 1.9: The weighted average14 of 19 measurements of reactor neutrino experiments operating at short baselines. [44]

A different explanation states it might be possible that correlated effects between measurements have been overlooked. This solution is not very likely since many different types of reactors and detectors were used for the reactor anomaly. The other two anomalies should be largely uncorrelated with the results from the reactor anomaly from a design perspective.

14Including correlations. 24 1.5. Sterile neutrinos

A last and very exciting explanation is signs of new physics. When the three flavour picture of neutrino oscillations is questioned the measurements might make sense, this requires us to introduce an additional neutrino that is unlike any other discussed so far. These new neutrinos are called sterile neutrinos, why they are called sterile will be explained in the following section.

1.5 Sterile neutrinos

In section 1.3 the suggestion made by Pontecorvo was mentioned briefly, the current explanation of sterile neutrinos goes far beyond his initial ideas.

As is already discussed above there are only 3 generations of light weakly interacting neutrinos (see figure 1.3). In order for other light neutrinos to exist they cannot interact with the weak force, this makes them only susceptible to gravity. To differentiate between these two types of neutrinos a new name is introduced; neutrinos that interact through the weak force are called ”active” neutrinos, while those that do not are called ”sterile” neutrinos, thus all neutrinos mentioned above are active neutrinos15.

In theory one can propose as many sterile neutrinos as one sees fit, but for now, only one generation of sterile neutrinos will be introduced; this is called the 3 + 1 model. If this additional sterile neutrino is added to the discussion16, a lot of the aforementioned anomalies can be explained.

Sterile neutrinos can be introduced in QFT in the form of Majorana particles. If one would introduce the Majorana neutrinos in the same way as the SeeSaw-mechanism 17, but with an extra term for non-active neutrinos (νRand νL ), a general Lagrangian can be constructed18 [44]:

15Below, when talking about neutrinos, it will be explicitly stated when talking about sterile neutrinos. All other mentions of neutrinos have to be considered as ’active’ neutrinos. 16with a mass at the eV scale. 17This mechanism can explain the fact that neutrinos have extremely low masses compared to other fermions. 18 The only difference with the Majorana equation being MR ≠ 0. Chapter 1. Neutrino physics 25

    ( ) R 1 mL mD νC  L = − νL νL + h.c. 2 C R mD MR ν

Where mL, mD and MR are respectively the Majorana mass for the active neutrino, the Dirac mass term for massive particles and the mass term for the sterile neutrinos. The dimensions of the Majorana mass matrix (mL) is 3 × 3 (3 generations).

The dimensions of the sterile neutrino mass matrix (MR) is nR × nR with nR the L R number of sterile neutrino generations, now assuming nR =1. The ν and ν are L R Weyl spinors that represent the left- and right-handed neutrino and νC and νC their conjugates. By diagonalizing the mass matrix one can determine the physical neutrino mass states as the eigenstates of the mass matrix. This leads to determining the mass of any sterile neutrino. If mD ≠ 0 there is mixing between the active and sterile neutrinos, this is indeed required to explain the anomalies.

Assuming the introduction of one more generation of neutrinos (be it sterile), equation 1.3 can still be used given some additions to UPMNS. Using the recorded data, 19 2 one can find which values for the matrix UPMNS+ and ∆msterile give us the best fit to the data. Using the same notation as the PMNS matrix but with one extra generation, one finds the best fit through the data; this can be seen in figure 1.10.

Figure 1.10: A comparison of the fit of the SM neutrino mixing solution (red) 2 (sin (θ13) ≈ 0.039) to one where one sterile neutrino is introduced on top of the

2 2 ≈ SM (blue). A least mean square fit finds ∆msterile = 0.5 eV and sin (θ14) 0.036. [44]

19 Notation used after the addition of the θi4 elements and possible extra CP violating terms. 26 1.5. Sterile neutrinos

Comparing this to the recorded data, the addition of a sterile neutrino is able to explain the reactor anomalies. At this time, the three flavour oscillation analysis is excluded at 99.8%, corresponding to roughly 3 σ [44].

There are a few ways to detect these sterile neutrinos. One could do more investigation into the anomalies and find indirect evidence or, one could look at the direct effect of introducing an extra sterile neutrino in the SM.

This last approach can be done in two ways [45]:

• Sterile neutrinos will leave a clear signal in the oscillation probability of electron (anti)neutrinos close to their source, this can be seen in the blue line of figure 1.9. Studying this region of oscillation has not been done until now, the results could confirm or deny the best fit introduced in figure 1.9.

• The only force that might have an effect on sterile neutrinos is gravity. Hence, the existence of sterile neutrinos can have observable effects in astrophysical environments and in cosmology. Looking for extreme events might give hints about the existence of these sterile neutrinos.

To prove the existence of sterile neutrinos in terrestrial experiments, worldwide efforts towards the search for light sterile neutrinos have been steadily growing since 2011. Several new neutrino reactor experiments will report on the short- baseline νe disappearance in the coming years. They are making measurements of the energy spectrum at different distances in order to obtain information on neutrino oscillations that are independent of the neutrino flux calculations. These much more localised and precise measurements will at least be able to constrain better limits on the 3 + 1 model and might possibly prove the existence of this hypothetical particle. Chapter 2

The SoLi∂ experiment

The hardest problems of pure and applied science can only be solved by the open collaboration of the worldwide scientific community.

– Kenneth G. Wilson, Acceptance Speech Nobel Prize in Physics 1982

To evaluate the 3 + 1 neutrino model several experiments have been suggested to prove or reject the existence of a sterile neutrino in a given parameter range. One of the many experiments taking on this task in the SoLid collaboration, it is an acronym for Search of Oscillations with a 6Li Detector. The experiment is located at the BR2 nuclear research reactor of SCK•CEN1 in Mol, Belgium. Originally proposed by Oxford University, it has grown to a collaboration involving 12 institutes from 4 countries (the United Kingdom, the United States of America, France and Belgium). Four Belgian institutes are involved with the project: SCK•CEN and the universities of Brussels, Antwerp and Ghent.

This chapter will discuss the unique reactor and why it was chosen in section 2.1. In section 2.2 the physics of particle detections are addressed, the knowledge of which is combined in section 2.3 to discuss the full detector. The chapter ends with section 2.4, where the observed problems of the detector or its design are discussed.

1Belgian research centre dealing with peaceful applications of radioactivity.

27 28 2.1. The BR2 reactor

2.1 The BR2 reactor

When searching for sterile neutrinos on the eV scale, there are a lot of constraints on the detector design. As the SoLid detector is studying the neutrino reactor anomaly, figure 1.10 displays the relevant range to study electron (anti)neutrinos: 1 to 10 meters from the source. Predictions in this range give an important difference between an extra sterile neutrino and the classical three-neutrino oscillatory model. Just like the detector constraints, there too are constraints on the reactor that can be used. Below some of the most important criteria are listed:

• The detector needs to be placed close to the reactor core. Very few nuclear reactors have accessibility that close to the core.

• A reactor with a very small core is needed, if the core is too big the results will be smeared as the exact origin of the neutrino cannot be determined.

• Given the first two conditions one also needs small outer walls between the reactor core and the place where the detector is set up.

• The neutrino flux from the reactor needs to be calculated to very good precision. A reactor with a high purity of fuel will have fewer isotopes, this makes the calculations of the flux more precise.

For these reasons the BR2 reactor was chosen, it is a medical and research reactor with a very small core. The very intense fuel rods are all located in a space smaller 3 than 1m (deffective ≈ 50cm), this gives an almost point-source-like signal. In Figure 2.1 one can see the outer walls of the reactor are quite small, the outermost part is only about 5m from the center of the reactor core. The reactor also uses highly enriched Uranium (235U) with over 93% purity [46] and has a variable power output between 45 and 100 MW [47]. These properties make the BR2 reactor a good candidate for the location of the SoLid experiment.

The reactor has some other welcome properties that are not explicitly needed. For one the reactor is only active about 150 days in a year in evenly spread one-month cycles, this makes it possible to do necessary background measurements between different reactor runs. Chapter 2. The SoLi∂ experiment 29

Figure 2.1: Schematic of the BR2 reactor building, the reactor core is coloured green. The location of the SoLid experiment is also marked. [46]

As the SoLid experiments require a very low background for optimal results, no other experiments are allowed on the same floor. As all other beam ports on that level are closed the background from the reactor is significantly reduced.

From the flux calculations of the reactor, it is expected that the SoLid detector will see about 1800 neutrinos per day.

2.2 Detector principles

Neutrinos are very hard to detect, as most particle detectors use the charge of a particle for detection. As mentioned before, neutrinos are detected by the secondary particles that originate from a neutrino interacting with a proton or neutron. When optimising the neutrino detection rate one looks for those interactions 30 2.2. Detector principles whose cross-section is as high as possible2 which is, in this case, inverse beta decay (IBD):

+ νe + p → n + e (Eν ≥ 1.81MeV)

The SoLid detector uses an innovative organic scintillator technology called polyvinyl- toluene (PVT), which enables the detection of electron antineutrinos from the reactor core. The antineutrinos will interact with the protons located in the PVT and produce two secondary particles; a neutron and a positron. To obtain high spatial resolution a lot of small PVT cubes (5×5×5 cm3) are used, a depiction of which can be found in figure 2.2.

(a) (b)

Figure 2.2: (a) A polyvinyl-toluene (PVT) cube and (b) the interaction of a neutrino in PVT cubes. [48]

2The cross-section increases with the energy of the neutrino, however the energy distribution for neutrinos from the BR2 reactor drops for higher energies. Somewhere between these two effects, one finds the neutrino energy that has a maximal probability. This energy is dependent on the detection process and the used reactor. Chapter 2. The SoLi∂ experiment 31

2.2.1 Detecting positrons

When charged particles traverse a medium, the charge ionizes the atoms around them. These excited atomic states then decay back to their ground state, their excitation energy will be lost in the form of light. This process is called scintillation, it stops the moment a charged particle has lost to much energy.

Some materials are better at producing scintillation than others, one of the main reasons PVT was chosen is for its scintillating capabilities. What makes it even better is that PVT is optically transparent and produces a high light yield, two qualities that are important for the effectiveness of the detector.

These PVT cubes will radiate light when a positron is created within, simulations show the positron starts radiating within the first centimeter after creation [49]. This allows us to very accurately determine the location of the positron as well as the timing; one expects a very short very bright light signal when a positron is created.

2.2.2 Neutron detection

As neutrons are not charged, one cannot make use of the scintillating properties of the PVT cube. A very common process to detect neutrons uses the 6Li (n, α) 3H neutron reaction:

6 →3 4 3Li + n 1 H +2 α + 4.78MeV (2.1)

In order to capture neutrons 6Li needs to be added to the PVT cubes, every cube gets two layers of LiF of approximately 250µm thickness on opposite sides of the cube. The LiF is enriched with 6Li to optimise the reaction described above as 6Li is known to have a large cross-section for neutrons.

The neutrons produced from IBD are too energetic to take part in the 6Li reaction, they first have to be slowed down. Neutrons can only slow down when they can deposit part of their energy in an atom through collision, this is done easily for atoms with a low amount of protons (low Z) but much harder for those atoms with 32 2.2. Detector principles high Z. The best element for slowing down neutrons is thus H, in PVT a lot of these atoms are present.

Neutrons produced from IBD first scatter on (low Z) atoms, performing a random walk (approximately 15 cm according to simulations [49]). Once they have lost enough energy they can interact with the LiF layers where energetic α particles are formed. To detect these particles the LiF layers are doped with ZnS(Ag) (phosphorus grains), α particles will ionize the grains and these will slowly decay to their ground state producing a lot of light in the process. As these α particles have high energy, a large number of states are exited both in the scintillator and the grains. This will result in multiple recombinations at different rates, giving us a very different signal compared to positron detection.

A clear difference in the light signal of the positron and neutron is thus expected, as the neutron has to thermalize one expect this signal to be later in time. This can be seen in figure 2.3 and enables us to identify particles base on the pulse shape (called pulse shape discrimination).

Figure 2.3: Signal of positron in red and neutron in blue, the multiple recombination are clearly visible for the latter. [49] Chapter 2. The SoLi∂ experiment 33

2.2.3 Signal collection

Figure 2.3 shows a clear difference in light signal between the two types of particles, a useful property although this light has to first be collected somewhere. One uses wavelength shifting fibers3 (3×3 mm2) which are placed in the groves of the PVT cubes (see figure 2.2a). Multiple fibers are used, in each cube, they cross perpendicular enabling good spatial resolution. To obtain this resolution the light of one cube cannot leak into another one, for this a reflective polymer wrapping called Tyvek is used.

The signal from these fibers is transported to the silicon photomultipliers (SiPM) that too have a surface area of 3×3 mm2. The sensors used in the SiPMs are so- called Multi-Pixel Photon Counters (MPPC), they contain an array of 3600 pixels. Each of these pixels is sensitive to a single photon as the sensor is operated in Geiger mode4, the total output of the SiPMs is measured in the number of pixel avalanches.

2.3 The SoLid detector

The detector is a collection of PVT cubes that together form a larger whole, this collection of PVT cubes results in a very high spatial resolution. As the detector is next to a nuclear reactor and above ground, one also needs to account for background sources that give the same signals as IBDs. Therefore extra shielding and veto triggers are installed.

3The scintillation light from PVT and ZnS(Ag) is approximately 420nm, this is considered blue light. In order to trap and this light and guide it to the detector, a material with high internal reflection is used. The wavelength shifting (from blue to green) part is done so the detector will work optimal as well as to avoid light leakage into neighbouring cells. 4The bias voltage is set above the breakdown voltage, once a single electron-hole pair is created (photon detection) the avalanche will sustain itself. 34 2.3. The SoLid detector

2.3.1 Detector layout

The individually wrapped PVT cubes are stacked on top of each other in two dimensions, resulting in a detection plane of 16×16 cubes. This plane is held together with polyethylene (PE) sealing bars for extra structure. The cubes are stacked in such a way the groves (see figure 2.2) align, making it possible to insert the wavelength shifting fibers. Since every cube has four groves (two times as many as shown in previous figures) located on opposite faces, a unique combination of signals from two fibers results in the localisation of an event in a specific cell. The redundancy of this second pair results in excellent localisation op the scintillation signals.

Each detection plane is surrounded by an aluminium frame where the SiPM are connected to the wavelength shifting fibers on one end, on the opposite end of the fibers mirrors are installed to get even better detection efficiency. The whole frame is now covered by two additional sheets of Tyvek to make sure light from this frame does not leak into neighbouring ones.

Ten of these frames are then combined to make a module, the Phase I SoLid detector consists of 5 modules resulting in a total of 12,800 PVT cubes which makes about two tonnes of detector material. The different parts of the detector can be seen in figure 2.4.

The good pulse shape discrimination makes it possible to identify particles, thus needing only one readout system. On top of this, the detector has extremely high segmentation which allows for 3D reconstruction. The expected energy resolution of this detector is modest at 1MeV; σE/E ∼ 14% [45].

2.3.2 Background reduction

Cosmic rays give rise to large backgrounds in neutrino detectors, therefore most neutrino detectors are located deep underground. As no reactors are found here this is not a feasible option to reduce the background of the SoLid detector.

There are two large backgrounds to be considered and reduced; the cosmic rays (primarily muons) and the background due to the reactor. Chapter 2. The SoLi∂ experiment 35

Figure 2.4: Different stages of the assembly of a SoLid detector module. [50][51]

To lose most of the muon background a veto system is installed on the detector5; scintillator plates are placed on the top, bottom and sides of the detector, these are connected to photo-multiplier tubes6 (PMT) that detect the scintillation light. If the veto system is triggered (charged particle passes through the scintillators) this information is saved to make muon identification easier.

The reactor itself also produces a large background, the high radioactivity of the detector and the concrete result in extra measures that need to be taken. To stop environmental neutrons and gamma rays from the detector, extra shielding on detector modules are installed. The whole detector is also covered with thick slabs of high-density polyethylene (HDPE), these plastic bricks (0.5×0.75×0.83 m3) are filled with water as this is a good neutron moderator. The top of the detector is covered with PE slabs for additional background reduction.

What makes the reduction on background more difficult is the constraints of the size on the shielding, if it becomes to thick the baseline of the detector increases; this would cancel out the major benefits of the combination of this reactor and detector.

5As most muons can travel through a lot of material before they are stopped, a veto system is considered a standard way of dealing with them compared to trying to physically stop them. 6A photon detection device that converts photons into electrons at the photo-cathode, the converted electrons are multiplied using a series of dynodes. 36 2.3. The SoLid detector

Figure 2.5: Plastic brick walls filled with water and the polyethylene are placed around and on top of the SoLid detector. Their respective total masses can be found on the lower right side of the figure. [52]

2.3.3 Detector prospects

At the time of writing the detector takes data as part of Phase I of the SoLid experiment, it is planned for 450 days of measurements while the reactor is on; this stage will take approximately three years. There are plans for a Phase II where an additional module (CHANDLER) will be added with an expected energy resolution at 1MeV of σE/E ∼ 6% [53], the potential upgrade will have total sensitive detector mass up to 3t [54].

The experimental sensitivity towards resolving the anomalies and/or detecting evidence of a sterile neutrino is shown in figure 2.6. The star in the center of the 2 figure represents the current best fit for the 3 + 1 neutrino model in terms of ∆m41 2 and sin 2θ14 [55].

Detection of sterile neutrinos will have big consequences for particle physics, it would be the first discovered particle beyond the SM. It can even have consequences on other branches of physics, mainly for cosmology7.

7Sterile neutrinos at the eV scale can make up a small component of dark matter. Chapter 2. The SoLi∂ experiment 37

Figure 2.6: Exclusion sensitivity of the SoLid experiment in the 3+1 sterile neutrino oscillation parameter space. [54]

2 If the particle cannot be found the experiment will find new upper limits for ∆m41 2 and sin 2θ14, this would weaken the case for light sterile neutrinos (as the best fit would be excluded) and might even reject the 3 + 1 model altogether.

In any outcome, the experiment will have an impact on the current study of reactor anomalies. It will help to improve the antineutrino flux calculations of reactors thanks to the special properties of the BR2 reactor and the SoLid experiment. These calculations will be useful for future reactor experiments using electron antineutrinos. The experiment is also useful for its use of new technologies (cubes), it can teach us what improvements can be made for future experiments to obtain an even better spatial resolution and/or background rejection in future experiments. 38 2.4. Detector challenges

2.4 Detector challenges

During the construction, calibration, and run of Phase I of the experiment a few problems arose:

• Each PVT cube has its own sensitivity, for this reason, all cubes were individually labeled and their masses recorded. To correctly calibrate the whole detector one uses a known external source where the individual cube mass is taken into account.

• During the construction of the detector the masses of the LiF : ZnS(Ag) changed [56]. The increase in mass came from an additional backing, as the thickness of the screens is stable one can conclude the density of 6Li in the screens is decreasing. These plates also appeared to have more impurities which introduce more background. The cubes with the thinnest LiF : ZnS(Ag) were added on the outside of the detector. They are known to have a better signal to noise ratio and can be used to further test background-rejection methods on the detector and see the effects on the two types of plates. Chapter 3

Detector data

It is a capital mistake to theorize before one has data. Insensibly one begins to twist facts to suit theories, instead of theories to suit facts.

– Arthur C. Doyle, A Scandal in Bohemia

This chapter will addresses the reconstruction of signals described in chapter 2 to usable information in section3.1. In section 3.2 the origins will be discussed for events that look like IBD but are not. Section 3.3 deals with some of the relevant properties saved from an IBD-like event, this information is utilized in the following chapters. Studying these properties is used in section 3.4 to determine initial cuts for the first selection of data. Lastly, section 3.5 reviews with the correlations present in this selection.

3.1 Reconstruction

Light pulses of cubes go through a unique combination of perpendicular wires within a limited time frame. These pulses are recorded by the SiPMs that translate the pulse into an electronic signal (waveform). A program called Saffron is then used to translate collections of waveforms into usable data for further analysis. Although

39 40 3.1. Reconstruction the program translates a great amount of information, below only the neutron- positron coincidence reconstruction is discussed.

The reconstruction starts with matching and labeling waveforms in a limited time, this is called clustering. There are 254 cubes in a frame and the reconstruction links specific signals to specific cubes, each cube has a horizontal (x) and vertical (y) position in the frame. The position between the different frames is saved in its ’stacked position’ (z).

Once this first stage of reconstruction is completed, the clusters (or ”light events”) are labeled using pulse shape discrimination. This is followed by looking for IBD like signals. Starting by looking for matches within a frame, if a neutron-like and positron-like signal are found close enough to one another in timing they are considered a match. If these matches are not found within a frame, one can look between frames but has to take into account only a limited time window between both signals. There are very little constraints on these windows; to find the actual IBD signals a more detailed analysis is needed.

Additional analysis is done to check if none of the events have other known causes such as cosmic muons. This additional filter tries to removes as many of these events as possible. The resulting events are saved in ROOT1 files, where many of the features found through reconstruction are saved. One can, for instance, determine the difference in a certain property (L) between the neutron and positron

√(∆L). Using this information one can construct even more properties such as ∆r = (∆x)2 + (∆y)2 + (∆z)2.

Properties like the energy deposition of the positron, can be found by studying the estimated energy deposition in the detector. The waveforms of these PVT signals are typically short in time with a pulse length less than 300ns and the amplitude is proportional to the energy deposited in each cube [57]. The knowledge of the positron energy gives a good estimation for the energy of the original electron antineutrino (Eνe ).

1ROOT is an object-oriented program and library developed by CERN. It is designed for particle physics data analysis and contains several features specific to this field. Chapter 3. Detector data 41

3.2 Real and Fake IBD signal

Some of the reconstructed signals do not have their origin as IBD events, it is important to understand the different backgrounds in order to optimise the reconstruction and the IBD analysis. To study these backgrounds the SoLid detector is recording, even if the BR2 reactor is not in operation. Studying the recorded events in the reactor-off period can reveal a great deal about the specific characteristics of the background which in term might hint at how to minimize the background acceptance of the detector. For the SoLid detector, background events can be divided into two categories: accidental and correlated backgrounds.

3.2.1 Accidental background

When two individual unrelated particles interact in the detector, within a small time window of one another, they can (accidentally) be seen as an IBD signal. Any two or more particles that can mimic the signal of a positron and neutron signal in the detector can be considered part of this background. As these two interactions in the detector are completely uncorrelated, thus sometimes called uncorrelated background, one expects the rate of these interactions to be constant.

One of the best-known examples of this type of background is the time coincidence between a positron (also called prompt) signal and a neutron signal faked by-products of the fission processes inside the reactor. The production of gamma rays is common in this environment and the signal of Compton scattered electrons looks very similar to a prompt signal, meanwhile, the neutron signal might come from an environmental neutron or from the fact that the reactor hall has many activated materials that emit neutrons. Due to the high rate of gamma rays from the reactor, this coincidence occurs rather frequently. It is, therefore, one of the most dominant forms of background for the experiment. 42 3.2. Real and Fake IBD signal

3.2.2 Correlated background

When the origin of the detection of an IBD like event is a single particle which puts in motion a process that mimics real events, these backgrounds are called correlated as the detection of the event has a single particle at its cause.

Correlated backgrounds can be further divided into two groups, dependent on the origin:

• Highly penetrating particles: When these particles enter the detector they can cause activation2 or spalla- tion3 in the detector medium. Both muons and environmental neutrons can give rise to this type of background. The muons can kick a neutron out of the core of an element, the muon itself can be seen as a prompt signal and the kicked neutron will be correctly identified. Environmental neutrons on the other hand can hit hydrogen atoms, these appear to the detector as prompt signals while the neutron is again correctly identified.

• Nuclear decay of radioactive elements: Nuclear decay in the detector can be due to two main reasons: the detector material can be activated (due to processes described above) and the natural radioactivity from the environment. One of the main reasons for this natural radioactivity can be found in the Uranium decay chain, specifically when Radon is produced. Radon is a naturally occurring gas but its concentration will increase close to a nuclear reactor. In figure 3.1 one can see one specific decay that has small enough half-life time4 to mimic an IBD event: the Bismuth-Polonium-Lead decay. Once Bismuth decays, Polonium soon follows resulting in a β− and α particle within less than 200µs; the electron will mimic the signal of the positron while the alpha particle will interact directly with the ZnS(Ag) grains and skip reaction 2.1.

2Creation of a nuclear unstable atom that will decay. 3The break-up of a bombarded nucleus into several parts. 4A measure for the lifetime of a particle before it decays. Chapter 3. Detector data 43

Other radioactive decays also contribute, these are mainly attributed to the contaminated5 materials inside the detector which originate either from activation or lax quality control at the time of construction.

3.3 Event properties

Each event has many properties, in this section all the relevant properties, that are used later, are listed and their meaning is briefly explained.

• ∆x is the horizontal distance measured in cubes between the cube associated with the neutron and the one associated with the positron.

• ∆y is the vertical distance measured in cubes between the cube associated with the neutron and that of the positron.

• ∆z is the number of frames between the cube associated with the neutron and that of the positron.

• ∆r is the radial distance between the neutron associated cube and the positron one.

• ∆t is the timing difference between the detection of the neutron and positron in nanoseconds.

• Eprompt is the energy associated with the positron, measured in MeV.

• tNearest Neutron gives the time between the current reconstructed neutron and the previous or following one in nanoseconds. If this difference is too small, it could be the case that a highly energetic particle interacted with ZnS(Ag) grains of different cubes, mimicking the signal of multiple neutrons.

• V ol is a measure of the amount of light detected for the total IBD-like event. This measures all light associated with the event to obtain a total volume of the event.

5Materials are never made from completely raw materials containing only specific elements, one can specify the purity of the material if needed. 44 3.3. Event properties

Figure 3.1: The decay chain of Uranium can account for part of the correlated background in the SoLid detector. [58] Chapter 3. Detector data 45

3.4 Initial cuts

By studying different event properties and their probability distributions, it becomes clear that not all events are equally interesting. Using Monte Carlo samples one can study the effect of pure IBD events (A more detailed discussion is given in chapter 4), this information can be used for a first selection of the data.

The first selection of events is made to remove possible reconstruction errors and less interesting events. In table 3.1 all initial cuts on data, used in a further stage, can be found.

Table 3.1: Initial cuts for future analysis.

Properties Cut range

∆x, ∆y, ∆z [−10, 10]

∆r [0, 17.33]

∆t [−500 000, −40 000] and [0, 460 000]

Eprompt [2, 10]

tNearest Neutron [200 000, ∞[

Vol [1, ∞[

Some of the choices in table 3.1 need to be explained a bit further. The cut on ∆r comes straight from its definition, plugging in the maximal values for the three spatial parameters. The choice for the cut on ∆t is more complicated, one sees an almost flat distribution for negative timed events except for events very close to 0. Here a large rise is seen in the distribution, most probably from small errors in reconstruction. To lose this anomaly in the distribution, the selected timings in negative timing differences are shifted but the size of the window is the same for positive and negative timed events. As at least one cube must have detected something, the cut on the volume is explained. To improve the identification of real neutron signals, the cut on the nearest neutron timing assures the detected particle was truly a single neutron detection. 46 3.5. Correlations

The cut on the energy of the positron is a careful balance between cutting out some of the noise and having sufficient data to use for further analyses. During the first stage of analysis, (see chapter 4 figure 4.2) the minimum energy was changed multiple times until the figure had the required result: smooth distributions with small error bars.

3.5 Correlations

Once the cuts are applied, a first-order measure of the correlation between properties can be calculated using the correlation factor; a single value between -1 and 1 that gives a measure to what extent two properties are correlated. When rep- resenting these properties as an n-dimensional collection ⃗x = (∆x, ∆y, ∆z, ∆r, ∆t), a correlation matrix can be introduced.

Because many of these properties are linked in some way, one cannot use already implemented functions in the ROOT framework to obtain the correlation. The correlation coefficient between two properties should be calculated given the other properties do not change, resulting in the following evaluation for the correlation matrix C:

Cov (xi, xj) Cij = ρxi,xj = σxi σxj ∃k ≠ i, j for which xk=Cte In order to obtain these results Monte Carlo sampling of the data are done: the values of xk are fixed through a random number generator after which one loops over the data, the events that pass the requirements are then used to calculate the correlation coefficient. As requiring all xk to be strictly the same results in very few matches the selection criteria are relaxed; all values of xk that do not differ to much from the generated fixed ones are accepted for further calculation. Another problem arises since the spatial parameters are closely linked; if for instance one wants to calculate ρ∆x∆r, one cannot fix both ∆y and ∆z as this would fix ∆r given a certain ∆x. This constraint determines which xk have to be fixed. This procedure is done many times in order to remove the effects of fixing xk at specific values.

As one uses Monte Carlo sampling the results will be less accurate as doing the full analysis, however such a high dimensional problem takes an incredible amount of computing time compared to the sampling. The correlation matrices and their Chapter 3. Detector data 47 accuracies for the different types of data can be found in table 3.2. They are split in three types: accidental events (∆t <0), uncorrelated and correlated events (∆t >0) and the simulated data. Section 4.2 will further discuss the need for this separation of the data.

Table 3.2: Correlation matrices for the tree event types, all factors with an absolute value larger than 0.1 have been tagged in red. Since the correlation matrices are symmetric, only the upper half is displayed.

∆x ∆y ∆z ∆r ∆t

∆x 1.0 0.023 0.102 −0.084 0.114 0.197 0.149 0.003 0.0302 ∆y 1.0 0.048 0.150 0.310 0.155 −0.037 0.075 ∆z 1.0 0.258 0.247 0.007 0.065 ∆r 1.0 0.057 0.058 ∆t 1.0 (a) Accidental events

∆x ∆y ∆z ∆r ∆t

∆x 1.0 0.021 0.135 0.029 0.069 0.209 0.173 −0.012 0.064 ∆y 1.0 0.026 0.036 0.114 0.169 0.046 0.049 ∆z 1.0 0.291 0.114 −0.002 0.009 ∆r 1.0 0.055 0.095 ∆t 1.0 (b) Uncorrelated and correlated events

∆x ∆y ∆z ∆r ∆t

∆x 1.0 0.013 0.113 0.036 0.170 0.177 0.183 −0.015 0.016 ∆y 1.0 −0.007 0.086 0.329 0.164 −0.015 0.091 ∆z 1.0 0.145 0.181 −0.016 0.095 ∆r 1.0 0.013 0.063 ∆t 1.0 (c) Simulated events

Despite the large errors in table 3.2, one can still see some trends between the different types of events. They seem to correlate most between the spatial parameters and ∆r as can be expected, form the definition of ∆r. Furthermore, there seems to be little to no correlation between each of the spatial parameters and ∆t. In conclusion, all correlations appear small when compared to the spatial correlations.

Chapter 4

Likelihood method

It is common sense to take a method and try it. If it fails, admit it frankly and try another. But above all, try something.

– Franklin D. Roosevelt, Looking Forward

In chapter 3 the many possibilities that give rise to an IBD-like signal are mentioned. One of the main problems of analysing the detector data is to discern real IBD signals from fake ones. Many possible methods exist to discriminate between these types of events, this thesis focuses on discerning the different types of signals.

Section 4.1 introduces some of the methods to discriminate between these types of events, section 4.2 goes into detail on one of them: the likelihood method. Lastly, the effect of correlated data on the likelihood method is discussed in section 4.3.

The aim is to find real IBD signals in the haystack of background events using the likelihood method. In this chapter the likelihood method is discussed in detail and in chapter 5 the optimisation of this procedure to the problem at hand is covered.

49 50 4.1. Possible methods

4.1 Possible methods

Knowing the different event properties (section 3.3) allows us to discern different types of events. As has been mentioned in section 3.2, three types of events can be considered: uncorrelated background, correlated background, and signal. One can use many possible techniques to find these IBD signals, below three types will be mentioned: rectangular cuts, the likelihood method, and machine learning. Each of these methods will use data that pass the initial cuts as mentioned in section 3.4.

4.1.1 Rectangular cuts

By introducing more stringent cuts than the initial selection cuts on the event properties, one is able to select a subset of the data. The content of the subset entirely depends on the values of the cuts. Through the use of Monte Carlo generated data sets and reactor-off data one can find which event properties maximally distinguish between the different types of events. Further improving cut values can yield a clear difference between signal and background.

The disadvantage of this method is that it is rather time demanding to find good event properties and accompanying cut values.

4.1.2 Likelihood method

Unlike rectangular cuts, the use of a likelihood method is more clearly defined. The likelihood method offers a multivariate approach for background rejection of the IBD analysis on an event-by-event basis. After the choice of relevant parameters, one uses probability distributions for the different types of events. Through evaluation of each event in these probability distributions, one can create a likelihood function for a given type of data. Combinations of these likelihood functions result in a single new parameter that can be used in further cuts. Using reactor-off Chapter 4. Likelihood method 51 data and simulated events one is able to find optimal likelihood functions and cut parameters.

The advantage of this method compared to the previous one is the clearly defined recipe on how to obtain a cut value, through use of different data sets one can then optimise this cut value by using different likelihood functions.

4.1.3 Machine learning

Once relevant event properties have been identified the used data events need to be tagged as signal or background. This tagged data (called training data) is fed into a model that will optimise itself to identify events with the highest possible accuracy, this is mainly dependent on the choice of model. These models have been constructed so that they will make predictions and decisions without being explicitly programmed to perform that task. Once the model has trained to a sufficient degree it will be fed untagged data, in which the model will then tag all events.

The advantage of this method is the possibility to let many models (possibly with different event properties) train at the same time. The challenge of machine learning is identifying the most relevant event properties and choosing the correct models.

4.2 Likelihood method

Early on the choice was made to focus on the likelihood method for this thesis. There are two reasons for this choice: it was previously demonstrated to be more effective than the rectangular cuts [59] and from the previous discussion of machine learning it appears more like a ”black box”, a trained model will tag events but the user has little control on the selection once the model is chosen. The likelihood method gives more control to the user in all stages of the process. It is thus a good intermediate between rectangular cuts and machine learning, which makes it useful to compare results from the different methods discussed above. 52 4.2. Likelihood method

4.2.1 Uncorrelated and correlated background

Section 3.2 and 4.1 discussed the use of three types of events. In section 3.2 the origin of these three types is described, however, it has not been mentioned how these different types of events are labeled. As mentioned before, one does not need to worry about the signal; this is simulated without any background, these events are thus clearly marked.

It is, however, more tricky to discern uncorrelated and correlated background. Through the understanding of the physical processes discussed in section 3.2 one can identify the uncorrelated background quite easily. Since these events consist of accidental detection of a neutron and a positron like signal with no causal link, a constant background of these signals is expected over the entire detection time window. A good part of these events will thus have a time difference (∆t) that is negative, meaning a neutron detection followed by a positron detection. Selecting all events in reactor-off data with a negative time difference (∆t <0) results in a good selection of uncorrelated events.

The selection of correlated events is quite a bit harder since reactor-off data consists of both types of background. It is thus impossible to definitively say that any event with a positive time difference (∆t >0) is correlated or uncorrelated. However, it is possible to distinguish the probability distributions of the two types of background. Since all positive time differences consist of one of these two backgrounds, one can use the probability distributions of uncorrelated background to find the correlated background probability distributions. Assuming the time window for a positive and negative time difference is the same, one can subtract the probability distributions of any property A of the events with a negative time difference from those with a positive time difference.

Correlated A Reactor Off A − Reactor Off A P ( ) = P∆t>0 ( ) P∆t<0 ( )

This yields a new probability distribution that can be labeled as correlated background.

In principle, this procedure can be used to obtain a probability distribution for any type of event property A. However, some properties have no distinct features in Chapter 4. Likelihood method 53 their probability distributions; one such property is the time difference for the uncorrelated background. No variation in the uncorrelated background is expected, but due to statistical fluctuations in the detection, some variation observed. In- stead of subtracting the two probability distributions like before, one fits a uniform distribution through the uncorrelated time difference distribution. Subtracting this fit from the positive time distribution results in a correlated background distribution for the time difference. Due to some features of the detector this ∆t <0 distribution is not flat which can be observed in figure 4.1, this needs to be taken into account when searching for the time difference distribution of correlated events.

t) 5000 ∆ P (

4000

3000

2000

1000

× 3 0 10 −400 −200 0 200 400 ∆ t (ns) Figure 4.1: Distribution of time difference for reactor-off data. One can clearly see two different plateaus for negative time difference.

It appears two plateaus are present, this can be explained by the fact that the readout window before and after a neutron detection is not 0µs. This window makes the detector more sensitive in one part of the time difference distribution, namely in the window [−200, 500] µs [60]. A uniform distribution is fitted to the uncorrelated time difference distribution for time differences in the window [−200, 0[ µs, this fit will then be subtracted from the positive time-difference distribution to obtain a ∆t distribution for correlated events. 54 4.2. Likelihood method

4.2.2 Chosen variables

The different event types can be compared through the probability distributions of different properties. In order to make a good comparison one should look at the shape of the distribution and not the absolute number of counts, by normalising the distribution one obtains the same effect. Comparing these distributions, the resemblances and differences are clear between the three types of events. By carefully studying many of the event properties, one can find those that give the biggest differences between the event types in normalised probability distributions. The final selection contains six relevant event properties. The distribution of prompt energy had to be left out of the discussion as it might distort later analysis of the energy spectrum. This leaves us with the following five relevant event properties: ∆x, ∆y, ∆z, ∆r and ∆t. The probability distributions can be found in figure 4.2.

4.2.3 Likelihood and global-likelihood functions

For this method one starts by labeling event by event, as has been explained in section 4.2.1 this introduces some problems: type of background cannot be directly related to a specific ∆t event. To solve this problem, a new set of labels is introduced: signal, uncorrelated background (∆t <0) and correlated + uncorrelated background (∆t >0).

Using complementary data sets other than section 4.2.1 (that pass the cuts of section 3.4) and the probability distributions from figure 4.2, the likelihood method can be implemented.

The first step of this method consists of evaluating event properties in the probability distributions of figure 4.2. Each event can thus be evaluated for three probability distributions: signal, uncorrelated background and correlated background. The evaluation of event properties is done by looking at the function value of the probability density at the specific value of that event property. Notice the result of this evaluation is always between 0 and 1 as the probability distributions have been normalised. The following notation will be used for the evaluation of an event Chapter 4. Likelihood method 55 P (∆ x) P (∆ y) 0.5 0.5 x) y) ∆ ∆ 0.45 P 0.45 P P ( signal P ( signal

0.4 Puncorr 0.4 Puncorr P P 0.35 corr 0.35 corr

0.3 0.3

0.25 0.25

0.2 0.2

0.15 0.15

0.1 0.1

0.05 0.05

0 0 −10 −5 0 5 10 −10 −5 0 5 10 (a) PP(∆ (∆x z)) ∆ x (cubes) (b) PP(∆ (∆y r)) ∆ y (cubes) 0.5 r) z) ∆ ∆ 0.45

P P ( P P ( signal signal 0.25 0.4 Puncorr Puncorr P P 0.35 corr corr 0.2 0.3

0.25 0.15

0.2 0.1 0.15

0.1 0.05 0.05

0 0 −10 −5 0 5 10 0 2 4 6 8 10 12 14 16 18 (c) P (∆z) ∆ z (cubes) P (∆ t) (d) P (∆r) ∆ r (cubes) 0.2 t) ∆

P ( 0.18 Psignal

0.16 Puncorr P 0.14 corr

0.12

0.1

0.08

0.06

0.04

0.02 × 3 0 10 0 50 100 150 200 250 300 350 400 450 (e) P (∆t) ∆ t (ns)

Figure 4.2: Probability distributions of different event types (different colors) for the event properties of interest (different figures). In figure 4.2a and 4.2b a small deficit in the center of the uncorrelated distributions can be seen which is unexpected. This can be attributed as an effect of the used reconstruction algorithm [61]. Negative √ ∆t are shifted to positive ∆t in figure 4.2e and ∆r = ∆x2 + ∆y2 + ∆z2. 56 4.2. Likelihood method

(i), with the focus on event property (A), for one event-type probability distribution (type):

type PA (i) ∈ R | 0 ≤ P ≤ 1 (4.1)

Using this notation one can make combinations of evaluations of different properties for the same event. In general, these combinations are multiplied and thus receive the name likelihood. There is however no reason to specifically use multiplication, in theory, one can also add or subtract these evaluations. Since this technically no longer qualifies as a likelihood, the term ’extended likelihood’ is introduced to include all possible combinations of evaluation of different properties; no longer guaranteeing a combined number between 0 and 1. Even though it is in general allowed to subtract these evaluations, only manipulations that result in positive real numbers are considered. This choice is made to guarantee values between 0 and 1 in the final step of the method.

The following notation is used for the extended likelihood of an event i, for an event-type probability distribution type: [ ] type type type type type type ∈ R L (i) = f P∆x (i) ,P∆y (i) ,P∆z (i) ,P∆r (i) ,P∆t (i) (4.2)

As mentioned before the likelihoods for the different event-type probability distributions are now combined into a single function, called the global likelihood function. In general, any combination of extended likelihood functions is allowed but again constraints are introduced, requiring the combination of likelihood functions to be normalized at all times: all used likelihoods are added to the denominator of the global likelihood to guaranty all global likelihood values are between 0 and 1. The following notation is used for the global likelihood: [ ] GL (i) = g LSignal (i) ,LCorrelated (i) ,LUncorrelated (i) ∈ R | 0 ≤ GL ≤ 1 (4.3)

Applying this method to each tagged event results in three global likelihood distributions, one for each label discussed at the start of this section:

GLSimulated GLUncorrelated GLUncorrelated + Correlated Chapter 4. Likelihood method 57

All one has to do now is find good function f (equation 4.2) and g (equation 4.3) to obtain maximal difference between signal and background. Below an example is provided of this method in practice.

For the following example, the most straight forward functions are chosen for the extended and global likelihood, keeping in mind the extra constraints. The following extended likelihood function will be used:

type type × type × type × type L (i) = P∆x (i) P∆y (i) P∆z (i) P∆t (i) (4.4) as ∆r is a combination of ∆x, ∆y and ∆z it was left out. The chosen global likelihood function is the most trivial normalisation possible to expect good signal to background discrimination:

LSignal (i) GL (i) = (4.5) LSignal (i) + LCorrelated (i) + LUncorrelated (i)

Once chosen, these equations need to be used for every tagged event. Using the aforementioned functions, the global likelihood distributions can be constructed and can be found in figure 4.3.

Signal Corr + Uncorr Uncorr Normalized Count

10−1

10−2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 GL value

Figure 4.3: Global likelihood distribution from the three types of tagged data using equation 4.4 and 4.5 for extended and global likelihood. 58 4.2. Likelihood method

From figure 4.3 it is hard to obtain a clear cut value as one cannot see what amount of data from each of the event types are left over once a cut is applied. Using cumulative distributions of these global likelihoods does reveal this information, this can be found in figure 4.4.

1 Normalized Count

10−1

signal 10−2 corr + uncorr uncorr

− 10 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 GL value

Figure 4.4: Cumulative distribution of figure 4.3.

Figure 4.4 gives a very clear picture of the result of the used method, a substantial difference is noticed in the global likelihood distribution of the signal compared to the two backgrounds. For instance, using a cut value (λ) from 0.5 to 1 will contain mostly signal and some correlated + uncorrelated background, however, nearly all uncorrelated background is lost. Chapter 5 discusses how to optimise the procedure to find ideal cut values, but figures such as 4.4 are the first sign of a good choice of functions. Only those extended and global likelihood functions are used that give clear differences, seen on cumulative distributions, between signal and background. Chapter 4. Likelihood method 59

4.3 Correlated data

Until now ideal conditions have been assumed, where no correlations between the different event properties are taken into account. However, this is known not to be the case as at least one of the variables (∆r) is explicitly correlated with the other spatial measurements through their relation (see section 3.1).

Below three different methods are discussed that can remove these correlated effects to a large extent. Most of these methods make some less than perfect assumptions, but it is better than leaving the correlation untreated. When two or more event properties are highly correlated, their mutual information will appear twice or more in the extended likelihood. If these properties are bad discriminators, the likelihood gets biased by this mutual information; this will lead to sub-optimal performance of the method.

4.3.1 Using an n-dimensional Gaussian

For this method, the collection of all event properties is used as a n-dimensional vector: ⃗x = (∆x, ∆y, ∆z, ∆r, ∆t). In order to account for the correlations in ⃗x, an n-dimensional Gaussian distribution is introduced, centered at origin, which is dependant on an n × n covariance matrix C [62]: [ ] 1 ⃗yT C−1⃗y G (⃗y) = √ exp − (4.6) (2π)n | C | 2 Since ⃗x are generally not Gaussian distributed, a parameter transformation of ⃗x to ⃗y is introduced. Assuming the probability density function of ⃗y is given by equation 4.6, the new probability distributions P ′ is obtained with the following relation:

′ type n n P⃗x d ⃗x = G (⃗y) d ⃗y

Rewriting this and using the definition of the Jacobian one finds:

n ′ ∏ P type type xi type P⃗x = G (⃗y) type = c (⃗x) P⃗x i=1 gyi P type here xi are the one dimensional probability densities from figure 4.2. To be clear, a probability distribution of ⃗x is just a classic likelihood function. Written out, the 60 4.3. Correlated data improved probability distributions/likelihood function is: [ ] ⃗yT (C−1−I)⃗y exp − n ′ ′ 2 ∏ P type = Ltype = √ P type ⃗x | | xi C i=1

These new likelihood functions can be used to construct a global likelihood function where the effects of correlations are much smaller: ( ) ′ ′ ′ GL′ = h LSignal ,LCorrelated ,LUncorrelated here h is a function that satisfies all the criteria of section 4.2.3.

The main advantage of this method is that it is very efficient in dealing with the correlated data and reducing its effects on the likelihood technique. The main disadvantage is it only considers classic likelihoods and cannot be extended, unlike section 4.2.3 suggests.

4.3.2 Transform to set of less correlated variables

Just like the previous method, the goal is to construct a new set of variables that have no linear correlations, thus the correlation matrix becomes diagonal. Through introduction of the covariance matrix (Cij = cov (Xi,Xj)) the linear correlation of the n-dimensional collection ⃗x can be removed, using the following transformation: ⃗y = C−1/2⃗x [63].

This method is much more straightforward than the previous method, the main disadvantage is that different event types have different correlations. This makes it hard to get a clear expression for the covariance matrix C.

4.3.3 Shift the power balance of input variables

The standard likelihood method is constructed of correlated event properties and uncorrelated event properties, each of these is manipulated with the same weight effectively giving the correlation higher weights. One easy way to counter this is to introduce unequal weights (Wi). Taking the power of each evaluated probability Chapter 4. Likelihood method 61 and its specific weight, one can remove some of the effects of correlation on the likelihood method.

Even if all event properties are correlated this method can be used in some cases. Let’s assume all properties are correlated with ∆r. One can, in that case, suppress the effect of ∆r through the use of W∆r. An extra constraint is introduced in this ∑ N method, the sum of the weights cannot change ( i=1 Wi = N)[63]. Thus the following expression is found for the extended likelihood, this time, less affected by correlation:

{[ ] [ ] [ ] [ ] [ ] } ′ W∆x W∆y W∆z W∆r W∆t type type type type type type L (i) = f P∆x (i) , P∆y (i) , P∆z (i) , P∆r (i) , P∆t (i)

The main advantage is the quick implementation of this method, the main disadvantage is one has to choose one weight for all event types: not all correlations will be lost. Unlike the last two options, this one is a bit more trial-and-error, but it works well for small correlations.

Chapter 5

Improving the likelihood method

My success, part of it certainly, is that I have focused in on a few things.

– Bill Gates, Interview with Peter Jenning

In chapter 4 the likelihood method is discussed in detail, in this chapter its practical implementation is described and the criteria used for a good selection of functions and cut values are discussed. Section 5.1 deals with the correlations present in reactor data and simulation to make a choice between the methods described in section 4.3. Section 5.2 introduces the measure that selects the best (extended and global) likelihood functions. In section 5.3 attention is given to the addition of further cuts and its effect on the end result. The chapter concludes in section 5.4 which lists the most optimal functions, additional cuts, cut values and the results obtained from them.

63 64 5.1. Correlated data

5.1 Correlated data

To improve the likelihood method one tries to account for the correlations discussed in section 3.5. The correlations in table 3.2 clearly differ based on the event type. As these correlations appear small and most have the same sign, the third method described in section 4.3 will be used: shifting the power balance of input variables. Contrary to intuition, the weight for correlated events has to be increased and decreased for uncorrelated events because all evaluated probabilities lie between 0 and 1 and thus scale differently.

5.2 ROC curve evaluation

This section discusses the selection of extended and global likelihood functions, all functions will be labeled with Li and GLi. One can find the function associated with the index i in appendix A.

In section 4.2 a first selection is described for a good likelihood function, however, it is insufficient for further use. A more quantitative measure is needed than cumulative distributions that are distinctive between the different event types. ROC curves are introduced for a more quantitative measure of the likelihood function.

A ROC curve (Receiver Operating Characteristic curve) displays the results from the likelihood method in a two-dimensional plane. The ordinate represents the signal and the abscissa the background, both ranging from 0 to 1. For each cut of the global likelihood value λ, a certain amount of signal and background will pass. The fraction of the data that passes divided by the total amount is the true measure used in a ROC curve. Plotting this measure for both signal and background results in a single point on the plane, by varying the cut parameter λ one obtains a curve on the plane as can be seen in figure 5.1. This curve can be obtained for many combinations of extended and global likelihood functions. The most optimal point on the curve is (0, 1), meaning only signal is present. All curves that come close to this point are useful for further analysis, a next step is thus comparing different curves (different functions) and selecting the best-performing ones. A Chapter 5. Improving the likelihood method 65 decent quantitative measure for the performance of a combination of functions is calculating the area under the curve (A), the results of which can also be found in figure 5.1. This measure is a good measure but all curves with a similar area ( 5%) will need a second test to findBest the ROC optimal CurvesY1 cut value.

1 Signal

GL0 L8 A =0.895

GL L A =0.640 0.8 1 6 GL L A =0.858 2 4

GL3 L7 A =0.842

GL L A =0.857 0.6 5 4 GL L A =0.796 7 5

GL8 L5 A =0.796

GL L A =0.586 0.4 9 4 GL L A =0.848 12 1 GL L A =0.890 14 1 0.2

0 0 0.2 0.4 0.6 0.8 1 Background

Figure 5.1: Comparison of different functions, see appendix A to find the function associated with the index. Some functions clearly perform much better than others as can be seen with their areas. Here the background is the sum of both types of backgrounds.

In general, the choice of the cut value λ is completely left to the user. If one wants a very pure result with little background it is possible to obtain this with the downside of having very little events to work with. On the other hand one could want a result with no loss of signal, this too can be obtained with the downside of having a lot of background to deal with. In general, one is looking for a good middle ground. The most used measure of the optimal cut value is Youden’s J statistic (J)[64].

Youden’s J statistic is a quite trivial measure to find the optimal cut value. By looking from the maximal deviation of the ROC curve from the equilibrium of signal and background (dashed black line in figure 5.2), the optimal cut value is found. 66 Best ROC CurvesY05.2. ROC curve evaluation

1 dO Signal

0.8

0.6

0.4

0.2

0 0 0.2 0.4 0.6 0.8 1 Background Figure 5.2: ROC curve of figure 4.4 where λ was changed in steps of 0.01 from 0 to 1. This plot contains two graphical representations of measures for the best cut value: Youden’s J statistic (J) and the distance from optimal (dO).

The measure should always be positive for likelihood functions that perform well on the area test, resulting in the following expression for the J statistic:

J = S − B here S represents the passed fraction of the signal and B that of the background, maximising J will result in the optimal cut value.

As this J statistic is very dependent on the smoothness of the ROC curve it does not appear to be the most optimal measure, we thus introduced our own measure.

The distance to optimal (dO) makes use of the knowledge of the most optimal point (0, 1) and measures the radial distance between this point and any on the curve. This measure seems more intuitive as it is less dependent on the quality of the ROC Chapter 5. Improving the likelihood method 67 curve. Through the use of the Pythagorean theorem one finds: √ 2 2 dO = B + (1 − S) here S and B are the same as above, minimising dO results in an optimal cut value.

It is clear these two methods are different and thus are not expected to give exactly the same results. However, it is expected they identify the same functions as optimal, with a (small) difference in the optimal cut value λ.

After selection of the most optimal cut values for the most optimal functions, it is useful to represent the top contenders and see the difference between the different measures as can be seen in figure 5.3. The optimal cuts associated with these ROC curves can be found in table 5.1Best. ROC CurvesY0

1 Signal

GL3 L9 A =0.894

0.8 GL4 L9 A =0.894

GL4 L9 A =0.894

GL3 L9 A =0.894

0.6 GL4 L14 A =0.894

GL0 L15 A =0.894

GL14 L15 A =0.894

0.4 GL3 L19 A =0.894

GL4 L19 A =0.894

GL3 L15 A =0.894 0.2

0 0 0.2 0.4 0.6 0.8 1 Background

Figure 5.3: The best five ROC curves using the J statistic measure (first 5) and the five best for distance to optimal (d0) measure (last 5). It is clear both measures work complementary and produce nearly the same curves. The results of both measures can be found in table 5.1. This figure is obtained by including both background types in the analysis. 68 5.2. ROC curve evaluation

Figure 5.3 clearly shows both measures do not give any fundamental differences, in the future the distance to optimal measure will be used as it appears more intuitive for calculating the optimal cut value. It turns out this measure gives a lower background acceptance, this second reason will be discussed in detail in section 5.4.

Table 5.1: The Youden’s J statistic and distance from optimal of the best performing functions from figure 5.3 ranked from best to worst and the λ value associated with the most optimal point.

Ranking Functions λ range J value Ranking Functions λ range d0 value

1 GL3L9 [0.39, 1] 0.661068 1 GL0L15 [0.48, 1] 0.253655

2 GL4L9 [0, 0.61] 0.661068 2 GL14L15 [0, 0.52] 0.253655

3 GL4L9 [0, 0.62] 0.660567 3 GL3L19 [0.49, 1] 0.253745

4 GL3L9 [0.38, 1] 0.660567 4 GL4L19 [0, 0.51] 0.253745

5 GL4L14 [0, 0.6] 0.660549 5 GL3L15 [0.49, 1] 0.253764

(a) Youden’s J statistic (b) distance to optimal (dO)

5.2.1 Background selection

The previous treatment of background is summing both types and looking at the resulting ROC curve. From chapter 4, more specifically figure 4.4, it is clear both backgrounds behave very differently when a λ cut is applied. It turns out that the uncorrelated events are present when the lambda cut is very loose. More stringent cuts like the ones above result in almost no correlated background.

By applying the same analysis as before on a single background (correlated) the ROC curve is expected to cover less area as the total number of background events has dropped compared to the previous discussion. The exact same techniques as before are used but now leaving one of the backgrounds out of the discussion. The effect on the acceptance of uncorrelated background is further examined in section 5.4. The result of this one background analysis can be found in figure 5.4 and the accompanying measures in table 5.2. Chapter 5. Improving the likelihood method 69 Best ROC CurvesY0

1 Signal

GL14 L15 A =0.860

0.8 GL0 L15 A =0.860

GL14 L3 A =0.860

GL0 L3 A =0.860

0.6 GL4 L11 A =0.860

GL14 L11 A =0.860

GL0 L11 A =0.860

0.4 GL4 L8 A =0.860

GL3 L8 A =0.860

GL14 L13 A =0.860 0.2

0 0 0.2 0.4 0.6 0.8 1 Background

Figure 5.4: The best five ROC curves using the J statistic measure (first 5) and the five best for distance to optimal (d0) measure (last 5). The results of both measures can be found in table 5.2. This figure is obtained by including only correlated background.

Table 5.2: The Youden’s J statistic and distance from optimal of the best performing functions from figure 5.4 ranked from best to worst and the λ value associated with the most optimal point.

Ranking Functions λ range J value Ranking Functions λ range d0 value

1 GL14L15 [0, 0.55] 0.580625 1 GL14L11 [0, 0.46] 0.306959

2 GL0L15 [0.45, 1] 0.580625 2 GL0L11 [0.54, 1] 0.306959

3 GL14L3 [0, 0.54] 0.580573 3 GL4L8 [0, 0.46] 0.306978

4 GL0L3 [0.46, 1] 0.580573 4 GL3L8 [0.54, 1] 0.306978

5 GL4L11 [0, 0.57] 0.580540 5 GL14L13 [0, 0.47] 0.307031

(a) Youden’s J statistic (b) distance to optimal (dO) 70 5.2. ROC curve evaluation

5.2.2 Uncertainty in efficiency calculations

The introduction of a new cut parameter comes with an uncertainty on the result of the cut. All errors will be calculated for S and B, thus they too lie between 0 and 1. Two new variables are introduced: N is the total number of events before the cut and k is the number of events that pass the cut. One expects the best estimate for the unknown true efficiency to be ϵˆ = k/N.

Interested in the efficiency given the number of total events and those that pass, Bayesian theorem will be used: P (k; ϵ, N) P (ϵ; N) P (ϵ; k, N) = (5.1) C here C is a normalisation constant and P (ϵ, N) is the probability assigned to the true efficiency before considering data. As ϵ is only an accepted value between 0 and 1, it is reasonable to assume:   1 if 0 ≤ ϵ ≤ 1 P (ϵ; N) = (5.2)  0 else

Using normalisation of equation 5.1 in combination with definition 5.2 one finds: ∫ ∫ ∫ ∞ ∞ P (k; ϵ, N) P (ϵ; N) 1 1 P (ϵ; k, N) dϵ = dϵ = P (k; ϵ, N) dϵ = 1 (5.3) −∞ −∞ C C 0 The term on the right hand under the integral of equation 5.3 is nothing but the probability of k events passing with a fixed efficiency (ϵ) and total number of events (N). This is given by a binomial distribution:  

N − P (k; ϵ, N) =   ϵk (1 − ϵ)N k (5.4) k

The following condition is thus obtained:

  ∫ ∫ N 1 1   k − N−k N! k − N−k C = ϵ (1 ϵ) dϵ = − ϵ (1 ϵ) dϵ (5.5) k 0 k!(N k)! 0

The right-hand side of equation 5.5 is a Beta function: B (k + 1,N − k + 1). Using its definition in terms of Gamma functions one finds: N! 1 C = = (5.6) (N + 1)! N + 1 Chapter 5. Improving the likelihood method 71

Substitution of equations 5.4 and 5.6 in equation 5.1 and assuming 0≤ ϵ ≤1 results in:

(N + 1)! − P (ϵ; k, N) = ϵk (1 − ϵ)N k k!(N − k)!

One now needs to check if this distribution does indeed correspond with the expec- tation: ϵˆ = k/N. In order to find the most probable value of this distribution the mode (ϵ) is studies, by solving dP (ϵ; k, N)/dϵ =0 one finds:

k mode (ϵ) = N

In this case the mode is a good estimator for the real efficiency since it is consistent, unbiased and efficient.Lastly, the variance on the efficiency is calculated(V (ϵ))[65] [66]:

V (ϵ) = ϵ2 − ϵ2 ∫ 1 = ϵ2P (ϵ; k, N) dϵ − [ϵP (ϵ; k, N) dϵ]2 0 (k + 1) (k + 2) (k + 1)2 = − (5.7) (N + 2) (N + 3) (N + 2)2 here again the definition of the Beta function was used to simplify much of the work. Taking the square root of equation 5.7 one finds the uncertainty on the efficiency. In all of the ROC curves above, these uncertainties are plotted; for large k and N the errors are expected to be very small which is indeed the case for these curves.

5.3 Introducing extra cuts

In section 4.2 the probability distributions for the different types of events are studied. These are useful for the likelihood method but can have even more use, figure 4.2 shows a clear difference in signal and background for P (∆r) and P (∆t).

Using this knowledge, one can impose further cuts on the results obtained from the already discussed likelihood method. Just like before a ROC curve can be constructed by cutting at different values. A first parameter that can be used for additional cuts is ∆r, the result can be found in figure 5.5 and the accompanying 72 5.3. Introducing extra cuts measures can be found in table 5.3. It is clear from figure 5.5 that this cut is worse at separating signal from background but it still has some effect. Using the distance from optimal measure determines the optimal cut on ∆r, these too can be found in table 5.3.

Since almost all uncorrelated background has been filtered out through the likelihood method, one can now just sum both backgrounds (unlike before) as it makes almost no difference in finding theBest optimal ROC CurvesY1 cut value.

1 Signal

GL14 L15 GL L 0.8 0 15 GL14 L3

GL0 L3 GL L 0.6 4 11 GL14 L11

GL0 L11 GL L 0.4 4 8 GL3 L8

GL14 L13 0.2

0 0 0.2 0.4 0.6 0.8 1 Background

Figure 5.5: ROC curve for the ∆r cut for the ten best likelihood functions. Starting from the data that passes the likelihood cut, ROC curve is obtained.

By using table 5.3 one can now fix the cut on ∆r, this further filters the data. These results can be used again as the starting point of another cut. Further cuts will, however, have a similar effect as a cut on ∆r, they will reduce the signal much more than the background.

A different cut can be obtained completely independent of the ∆r one, the most obvious one is ∆t. Again the backgrounds are treated as a sum of both backgrounds, the ROC curve for a cut in ∆t can be found in figure 5.6 and the accompanying measures can be found in table 5.4. In section 5.4 the usefulness of these cuts will be determined. Chapter 5. Improving the likelihood method 73

Table 5.3: Best ∆r cuts for figure 5.5 determined by distance from optimal.

Ranking Functions λ range ∆r range d0 value

1 GL4L11 [0, 0.57] [0, 2.0] 0.650894

2 GL4L8 [0, 0.46] [0, 1.9] 0.657333

3 GL3L8 [0.54, 1] [0, 1.9] 0.657333

4 GL14L13 [0, 0.47] [0, 1.9] 0.66303

5 GL14L11 [0, 0.46] [0, 1.9] 0.663175

6 GL0L11 [0.54, 1] [0, 1.9] 0.663175

7 GL14L15 [0, 0.55] [0, 2.0] 0.664082

8 GL0L15 [0.45, 1] [0, 2.0] 0.664082

9 GL14L3 [0, 0.54] [0, 1.9] 0.666879

10 GL0L3 [0.46, 1] [0, 1.9] 0.666879

Best ROC CurvesY1

1 Signal

GL14 L15

0.8 GL0 L15

GL14 L3

GL0 L3

0.6 GL4 L11

GL14 L11

GL0 L11

0.4 GL4 L8

GL3 L8

GL14 L13 0.2

0 0 0.2 0.4 0.6 0.8 1 Background

Figure 5.6: ROC curve for the ∆t cut for the ten best likelihood functions.

Additional cuts on top of the ones discussed here will not be studied in detail but are completely analogous to the method described above. 74 5.4. Results for reactor-off data

Table 5.4: Best ∆t cuts for figure 5.6 determined by distance from optimal.

Ranking Functions λ range ∆t range d0 value

1 GL4L15 [0, 0.55] [0, 44000] 0.602370

2 GL0L15 [0.45, 1] [0, 44000] 0.602370

3 GL14L3 [0, 0.54] [0, 44000] 0.602433

4 GL0L3 [0.46, 1] [0, 44000] 0.602433

5 GL4L11 [0, 0.57] [0, 44000] 0.606231

6 GL14L11 [0, 0.46] [0, 40000] 0.626292

7 GL0L11 [0.54, 1] [0, 40000] 0.626292

8 GL4L8 [0, 0.46] [0, 40000] 0.628342

9 GL3L8 [0.54, 1] [0, 10000] 0.628342

10 GL14L13 [0, 0.47] [0, 40000] 0.624777

5.4 Results for reactor-off data

Below, the effect of the different cuts described above is examined. Monte Carlo simulations are used to test how well these cuts work on the signal part of reactor- on data, reactor-off data are used to study the effect of the cuts on the background signal. Since all tests start from the same events that pass initial cuts (section 3.4), the amount of data before cuts are applied is reported in table 5.5. All results will be reported as the percentage of initial events that survive the cuts.

Table 5.5: Number of events per event type that pass initial cuts. All further results will be reported as the percentage of these events that survive the cut(s).

Simulation (Sim) Correlated + uncorrelated (Corr + Uncorr) Uncorrelated (Uncorr) 32690 82515 29252

5.4.1 Two background ROC analysis

A first analysis of the likelihood method was done by considering both types of background together. Using the cuts described in table 5.1, the amount of data that survives the cut is studied. In section 5.2, the distance from optimal as the measure of the final result was chosen. So far, both Youden’s J statistic and distance Chapter 5. Improving the likelihood method 75 from optimal have been reported as it is useful to compare the results at least once. For this cut both shall be discussed, all further results will only report on the distance from optimal measure.

In table 5.6 the results of the cuts can be found, some of the cuts give exactly the same results. This effect is explained in detail in appendix A. All future tables will not report these double results.

It is clear that the distance from optimal measures results in a smaller fraction of both signal and background. However both drop almost equal percentages when compared to the J statistic measure, this gives rise to a higher fraction of the data being signal (signal to background/noise ratio). This is the biggest reason to select the distance from optimal over the J statistic measure.

Table 5.6: Percentage of data that survives the reported cut, ranked from best to worst. The first five results are obtained using Youden’s J statistic while the second five use the distance from optimal measure.

Ranking Functions λ range Simu (%) Corr + Uncorr (%) Uncorr (%)

1 GL3L9 [0.39, 1] 91.69 0.15 33.82 0.16 2.33 0.09

2 GL4L9 [0, 0.61] 91.69 0.15 33.82 0.16 2.33 0.09

3 GL4L9 [0, 0.62] 92.01 0.15 34.29 0.17 2.44 0.09

4 GL3L9 [0.38, 1] 92.01 0.15 34.29 0.17 2.44 0.09

5 GL4L14 [0, 0.60] 91.78 0.15 34.02 0.16 2.34 0.09

1 GL0L15 [0.48, 1] 86.34 0.19 28.43 0.16 1.47 0.07

2 GL14L15 [0, 0.52] 86.34 0.19 28.43 0.16 1.47 0.07

3 GL3L19 [0.49, 1] 85.73 0.19 27.89 0.16 1.49 0.07

4 GL4L19 [0, 0.51] 85.73 0.19 27.89 0.16 1.49 0.07

5 GL3L15 [0.49, 1] 86.25 0.19 28.35 0.16 1.52 0.07

5.4.2 One background ROC analysis

For the one background ROC analysis, only the correlated+uncorrelated background is used as a measure of the total background. This did not guarantee it would lead to low amounts of uncorrelated background except for suggestions from figures like 4.4 that showed a clear difference between the global likelihood functions. 76 5.4. Results for reactor-off data

Table 5.2 introduces different cuts and different optimal functions, the results of the cuts can be found in table 5.7.

Table 5.7: Percentage of data that survives the reported cut, ranked from best to worst only accounting for the correlated+uncorrelated background.

Ranking Functions λ range Sim (%) Corr + Uncorr (%) Uncorr (%)

1 GL0L11 [0.54, 1] 81.83 0.21 24.74 0.15 1.17 0.09

2 GL4L8 [0, 0.46] 82.68 0.21 25.35 0.15 1.30 0.07

3 GL14L13 [0, 0.47] 82.35 0.21 25.12 0.15 1.21 0.06

From the results in table 5.7 it is clear the single background ROC analysis gives more stringent cuts which result in fewer events passing, but a higher fraction of the total passed data are simulated. These three results will be used on which further cuts can be placed.

The fraction of signal in the total data that passes a cut are a good measure to check if a cut is more optimal, however, it is not the only measure that needs to be taken into account. Another measure is the total number of simulated events, these have to be sufficiently high as one otherwise gets low statistics so that errors can reduce the significance of the results.

5.4.3 Additional cut(s)

Above, possible additional cuts are discussed such as ∆r and ∆t which are imposed on top of the one background cuts. The cut values for ∆r cuts can be found in table 5.3. In table 5.8 one finds the results of these additional cuts on the data.

Table 5.8: Percentage of data that survives the reported cuts λ and ∆r, ranked from best to worst accounting for a single background1.

Ranking Functions λ range ∆r range Sim (%) Corr + Uncorr (%) Uncorr (%)

1 GL4L8 [0, 0.46] [0, 1.9] 47.08 0.21 12.74 0.15 0.23 0.07

2 GL14L13 [0, 0.47] [0, 1.9] 47.24 0.21 12.89 0.15 0.24 0.06

3 GL0L11 [0.54, 1] [0, 1.9] 47.01 0.21 12.71 0.15 0.23 0.09

1Additional cuts such as ∆r can be done without introducing additional errors, the only error on the total cut is due to the likelihood part. This is discussed in detail in section 5.2 Chapter 5. Improving the likelihood method 77

The effect of this cut on simulation and background could already have been predicted from figure 5.5. This cut can still be used, as a decent portion of the signal is still present and almost all the background is lost.

One can do a very similar analysis on ∆t as has been discussed in section 5.3, the results of which can be found in table 5.9.

Table 5.9: Percentage of data that survives the reported cuts λ and ∆t, ranked from best to worst accounting for a single background1.

Ranking Functions λ range ∆t range Sim (%) Corr + Uncorr (%) Uncorr (%)

1 GL0L11 [0.54, 1] [0, 40 000] 49.15 0.21 11.96 0.15 0.51 0.0.09

2 GL4L8 [0, 0.46] [0, 40 000] 49.44 0.21 12.25 0.15 0.60 0.07

3 GL14L13 [0, 0.47] [0, 40 000] 49.24 0.21 12.03 0.15 0.53 0.06

One can also combine the cut on ∆r and ∆t to obtain new results, these can be found in table 5.10.

Table 5.10: Percentage of data that survives the reported cuts λ, ∆r and ∆t ranked from best to worst accounting for a single background1.

Ranking Functions λ range ∆r range ∆t range Sim (%) Corr + Uncorr (%) Uncorr (%)

1 GL0L11 [0.54, 1] [0, 1.9] [0, 40 000] 30.64 0.21 6.04 0.15 0.10 0.09

2 GL4L8 [0, 0.46] [0, 1.9] [0, 40 000] 30.70 0.21 6.07 0.15 0.10 0.07

3 GL14L13 [0, 0.47] [0, 1.9] [0, 40 000] 30.70 0.21 6.07 0.15 0.10 0.06

5.4.4 Final selection of cuts

As discussed before a first selection needs to be made based on the signal to background ratio, thus using the single background analysis. To choose which additional cuts will study the differences between tables 5.8, 5.9 and 5.10.

Table 5.10 has by far the best signal to noise ratio of all the tables but has the disadvantage of losing a lot of signal. As one needs high statistics for the further implementation of this method, a combined cut on both ∆r and ∆t is unwanted. 78 5.4. Results for reactor-off data

On first sight table 5.9 looks to have better results when compared to table 5.8. The signal to noise ratio is better and the amount of signal is better too. The one disadvantage table 5.9 has, is the higher uncorrelated background. As will be seen in chapter 6, it is more optimal to reduce one background as much as possible since it reduces the error on the final calculation of the neutrino flux from the reactor. Thus, the cut on ∆r shall be used for the analysis in chapter 6, using the best performing function from table 5.8:

Table 5.11: Final cuts that will be used in the following part of this thesis1.

Ranking Functions λ range ∆r range Sim (%) Corr + Uncorr (%) Uncorr (%)

1 GL4L8 [0, 0.46] [0, 1.9] 47.08 0.21 12.74 0.15 0.23 0.07

5.4.5 Results from rectangular cuts

As discussed in chapter 4 there are many ways to filter signal from background. To compare the global likelihood method with at least one other method, some results of these rectangular cuts are presented below. It is expected these results will be of lesser quality as fewer parameters can be cut. The lower quality is either because of lower signal to noise ratio or due to the loss of too much signal. The best results are discussed in table 5.12, where one looks for results in the percentage of signal that corresponds well to previous optimal results and observes the effect on the resulting background.

Table 5.12: Rectangular cuts that come close to the one background global likelihood cuts in terms of signal1.

∆r range ∆t range Sim (%) Corr + Uncorr (%) Uncorr (%)

[0, 10] [0, 100 000] 81.52 35.06 12.27 [0, 2] [0, 100 000] 42.23 12.51 0.15 [0, 2] [0, 50 000] 30.99 7.37 0.10

It is clear from table 5.12 that rectangular cuts decrease the signal to noise ratio compared to the likelihood method. Chapter 6

IBD analysis

By no process of sound reasoning can a conclusion drawn from limited data have more than a limited application.

– J. W. Mellor, Higher Mathematics for Students of Chemistry and Physics

In this chapter, the final analysis of reactor-on data is discussed. It reveals how the likelihood method can be used to find real IBD events in the haystack of similar looking background events. Section 6.1 deals with the selection of the used data for the analysis. In section 6.2 the final analysis is discussed and the results are mentioned.

6.1 Selection of data

Section 6.2 explains how the IBD signal is calculated, however, before this analysis can be done one has to select the data. The analysis needs both reactor-on and reactor-off data, giving extra constraints to the selected files. Two major criteria are used for this selection:

79 80 6.1. Selection of data

• Order of events As discussed in chapter 3, the reactor has some effect on the background experienced by the SoLid detector. To get the best possible representation of the background, one needs to use reactor-off results that follow shortly after a reactor-on period. This way the possible extra background due to the reactor is present in the reactor-off data too.

• Similar environment There are a few properties of the environment that can change the background. One of the most important ones is the atmospheric pressure, as it can have an effect on the amount of Argon residing in the detector (see section 3.2 for the effect of Argon). If the atmospheric pressure of the two sets of files is too different, sub-optimal results are expected as the backgrounds are not fully comparable. In figure 6.1 the pressure as a function of the days is represented, the days from which data will be used are displayed in coloured bands.

Figure 6.1: Pressure as a function of time in the detector hall in the summer of 2018. The reactor-on (green) and reactor-off(red) period are displayed at the bottom of the graph. The selected reactor-on date is shown in light green, the reactor-off date in gold. The grey bar represents a different data set used for the build up of the probability distributions (see figure 4.2).

The used detector data does have a difference in atmospheric pressure but the difference is rather small and does not appear to be a problem. From figure 6.1 it might seem that the detector measured for a full day but this is not the case. Chapter 6. IBD analysis 81

Furthermore, the detector measures longer when the reactor is offline. In table 6.1 the total measurement time is mentioned for the selected reactor-on and reactor-off data.

Table 6.1: Total detection time for the two types of data. The analysis of chapter 4 and 5 need a lot of events to work well. As neutrinos interact very rarely a large amount of detection time is needed.

Data type Date Detection time (s)

Reactor on 08/07/2018 62404.4 Reactor off 17/07/2018 78725.6

6.2 Reactor-on Analysis

The method to find real IDB on top of the large background is discussed first, followed by the results obtained from it. Lastly, a comparison is made with theoretical prediction.

6.2.1 The procedure

The analysis of the previous chapter resulted in a selection of cuts using the likelihood method, these cuts can be found in table 5.11. One can apply these cuts to different data types. Looking at the number of events that pass the cuts makes it possible to calculate the total amount of actual IBD events. For this final analysis, both reactor-on and reactor-off data are used.

Once again, the reactor-off data (Doff) can be split up in two parts: ∆t > 0 and ∆t < 0. The second one is a very good measure of the uncorrelated background Uncorr (Boff ) for the reactor-off data. Combining the two parts of the data, a result for the correlated background is found:

Corr − Uncorr Boff = Doff Boff (6.1)

The same thing can be done for reactor-on data:

Corr+Signal − Uncorr Don = Don Bon (6.2) 82 6.2. Reactor-on Analysis

As the data are selected so that the correlated background should stay approximately the same, one can now calculate the total amount of IBD signals. Keeping in mind the difference in total measuring time, a re-scale factor is introduced: T R = on (6.3) Toff Combining equations 6.1, 6.2, and 6.3 one finds:

Corr+Signal − × Corr NIBD = Don R Boff (6.4)

It can be useful for further analysis to look for these results as a function of some variable ∆i:(NIBD (∆i)). Plotting the distributions of events that pass the cuts as a function of some parameter (e.g. ∆t), one can study if in a certain region a clear excess is seen. If this is the case, the information can be used for a more focused search using the same likelihood method (by reducing the range of the initial cut values in table 3.1).

To find the statistical errors on these counts equation 6.4 needs to be written out completely: [ ] − Uncorr − × − Uncorr NIBD = Don Bon R Doff Boff (6.5)

The value of the standard deviation of radiation events (such as IBD count values) is related to the actual number of photons counted during the measurement. Theoretically, the value of the standard deviation is the square root of the mean of the measurements. Each of the counts (D and B) in equation 6.5 has an error associated with it (σ). The total error on the number of IBD events is obtained by summing all the individual errors in quadrature. For this the scaling factor needs to be kept in mind, all reactor-off errors need to be scaled by R resulting in an error on the correct scale: σR . √ ∑ ( ) R 2 σIBD = σi i

This study will only discuss statistical errors, systematic errors are present due to detector biases and reconstruction errors. At this point they are not fully understood, they require a lot of time to find and since this is a complete study in itself they were omitted. Thus they are left out of all future results but cannot be forgotten. Chapter 6. IBD analysis 83

6.2.2 Results

The results of the individual event types of equation6.5 can be found in table 6.2. Using the information in table 6.1 one can calculate the scale factor:

R = 0.792682

The combination of all these results gives the total number of IBD events found by using the likelihood method.

NIBD = 198.232 123.465 (6.6)

Although a clear excess is observed, the error is rather large. When assuming a no signal hypothesis we find a 1.60σ deviation. To claim real IBD events have been measured is not founded with this deviation. When compared to another analysis done on data from an earlier stage of the experiment (with only one module recording data), our result are compatible with the 1.94σ observed there [67].

Table 6.2: Counts for the individual event types and their errors. The total counts of background and reactor-on data are also mentioned but as they have not been properly scaled, no errors are reported for these counts.

Event type Counts Errors

Doff 10511102.523 Uncorr Boff 67 8.18535 Corr Boff 10444

Don 853792.3959 Uncorr Bon 60 7.74597 Corr+Signal Don 8477

As suggested before, one can also do this analysis without losing the underlying distributions as was done above. The result of this can be found in figure 6.2 where the two data types of equation 6.4 can be found as well as the difference of the two divided by the background. These figures provide additional information about the chosen cuts as well as revealing where one might look to find a better signal. 84 6.2. Reactor-on Analysis

4000

Corr+Signal Corr+Signal

Entries 3500 Don Entries Don 3500 Corr Corr 3000 R x Boff R x Boff 3000

2500 2500

2000 2000

1500 1500

1000 1000

500 500

0 0 ∆ off ∆ off x (cubes) 0.2 y (cubes) off off -B 0.1 -B B B on on 0.1 D D

0 0

−0.1 −0.1 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 ∆ x (cubes) ∆ y (cubes) (a) NIBD (∆x) (b) NIBD (∆y)

1000 4000 Corr+Signal Corr+Signal Entries Don Entries Don

3500 Corr Corr R x Boff R x Boff 800 3000

2500 600

2000

400 1500

1000 200 500

0 0 ∆ off ∆ off z (cubes) t (ns) off off -B -B

B 0 B on on

D D 1 −0.1 −0.2 0 −0.3 −1 ×103 −3 −2 −1 0 1 2 3 0 20 40 60 80 100 120 140 160 180 200 220 ∆ z (cubes) ∆ t (ns) (c) NIBD (∆z) (d) NIBD (∆t)

1200 Corr+Signal

Entries Don

1000 Corr R x Boff

800

600

400

200

0 ∆ off 0.6 r (cubes) off -B B on 0.4 D 0.2 0 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ∆ r (cubes) (e) NIBD (∆r)

Figure 6.2: Distribution of events that pass the cuts of table 5.11 as a function of the five selected parameters. Below the two distributions, the difference between both is given divided by the background. These ratio plots show the ranges where a clear excess of the data is seen. Chapter 6. IBD analysis 85

Even though no hard cut was introduced on the time difference, it appears the likelihood method did indeed have an effect on the selection of time differences. The ratio plots reveal that there is no clear signal in one specific range of any parameter except for maybe ∆r between 0.4 and 0.8. Lastly, it is interesting to note the asymmetry in the signal to background (∆i > 0 has a bigger excess in events) for the three spatial parameters, this could already be seen somewhat in figure 4.2c.

It has to be noted the likelihood cut was introduced to deal with the large background experienced by the detector. However, when all cuts are removed we observe a lot more events:

No Cuts NIBD = 2406.39 424.103 (6.7) which results in a higher significance. When assuming a no signal hypothesis we find a 5.67σ deviation, which is a significant difference from the no signal hypothesis, it is large enough to discard this hypothesis.

This last result is somewhat strange, it is expected from chapter 5 that the introduction of extra cuts will purify the signal which should thus lead to a higher significance. As the code has been checked extensively and on many different files including limited parts of files which all give the same result, it thus seems unlikely there is an error in the code. The only possible explanation we can think of at this time is that we have too few signal events with the additional cuts, thus decreasing signal but not the error. As the error does not scale linear with the number of IBD counts in reactor-on data, this might be a solution to the observed count. Further tests are needed to check if this explanation is indeed correct.

6.2.3 Comparison to theoretical predictions

As no timing information can be found in the simulation files, it is impossible to compare the theoretical number of IBD events directly with the ones calculated above. To be more general, in the future it is useful to talk about rates instead of the absolute count. A total of 1800 neutrino events are expected per day, this number is calculated for a period of 24 hours with no detector efficiencies accounted for. As 86 6.2. Reactor-on Analysis the detector has a trigger efficiency for IBD events of 78% [46] and an uptime of approximately 70% (see table 6.1) we find the following rate: 1800 × 0.78 × 0.7 RTheory = = (1.138 0.057) × 10−2 s−1 IBD 1 day As some small approximations are made and no error is given on the number on IBD events in a day, nor the efficiency, we assume an error of 5%. This rate is expected if our analysis would accept all IBD events, in table 5.11 it is found that only 47.08% of real IBD events survive the cut. 1800 × 0.78 × 0.7 × 0.4708 RExpected = = (5.356 0.268) × 10−3 s−1 (6.8) IBD 1 day Dividing result 6.6 by the total detector-on run-time, we find:

Measured × −3 −1 RIBD = (3.176 1.978) 10 s (6.9)

The results of 6.8 and 6.9 are found within a 1σ deviation from one another and thus very compatible.

As these results match, one can wonder about the previously discussed problem when no cuts were applied (result 6.7). In that case, we find the following results:

Measured × −3 −1 RIBD = (38.561 6.796) 10 s (6.10)

This result gives even more reason for further investigation of the problem, as the rate is not at all comparable with 6.8, the measured IBD events cannot all be real.

There is a second way of calculating the rates, completely independent of the results obtained in the previous section. As IBD events are inherently seen as counts, this process is best described by Poisson statistics. It can be proved that a given number of occurrences in a fixed period (Poisson) can transform into an exponential distribution with the time between occurrences of successive events as time flows by continuously, this is shown in appendix B. Plotting the time differences of successive events, one finds an exponential distribution. Thus an exponential is fitted to the distribution:

f(C, λ, t) = C exp (−λt) (6.11) Chapter 6. IBD analysis 87

where C is a scaling variable and λ is the rate associated with the type of data. As there are too few events for the negative time differences, no fits can be obtained. They can however be found for the two positive time differences, these can be found in figure 6.3.

0.07 0.07

0.06 0.06

0.05 0.05 Normalized count Normalized count

0.04 0.04

0.03 0.03

0.02 0.02

0.01 0.01

0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ∆ t (s) ∆ t (s) (a) Reactor-off data with C = (3.536 0.055) × 10−2 (b) Reactor-on data with C = (3.576 0.061) × 10−2 and λ = 3.455 0.045 and λ = 3.520 0.051

Figure 6.3: Normalised distributions of time between occurrences of successive events and the fitted exponential from equation 6.11.

The uncorrelated background has a very small rate (order 100 in a day as seen above). When treating this background as negligible we can find a direct measure for the IBD rate. Keeping in mind that the individual rates of figure 6.3 are not dependant on the total measuring time, one can subtract the two rates obtained from the fit and use the errors on the fit as the errors of the rate.

Fit − − −1 −1 RIBD = λOn λOff = (3.52049 3.45366) s = (0.067 0.068) s (6.12)

As the errors on the fits of the figures are too high, it is not possible to calculate a useful rate. Only an order of magnitude calculation is possible here. Using more data, it might be possible to obtain decent and independent results. This method could, in that case, be used as a check for the standard method to obtain the rate; division by the runtime.

Chapter 7

Conclusion & Outlook

Scientists have become the bearers of the torch of discovery in our quest for knowledge.

– Stephen Hawking, The Grand Design

The Standard Model works very well, but the neutrinos have made it difficult a number of times. Recent measurements have observed anomalies that can no longer be seen as fluctuations. A possible explanation for these abnormal measurements is the introduction of a sterile neutrino. The Solid experiment tries to detect these sterile neutrinos indirectly by taking short distance oscillation measurements. Modern detector technology is used to achieve high spatial resolution in the detector.

Since the end of 2017, the complete Phase I detector has been taking data, both when the nuclear reactor is on and off. This gives a good measure of the background present in this experiment. As the reactor runs in one-month cycles one always has recent background measurements that can be used.

After much work that has been done in recent years in design, construction, and reconstruction, the data sets can be used for analysis. A first step is to filter out as much background signal as possible from all IBD-like events. There are different

89 90 methods for this. In this thesis, it was decided to work with a likelihood method. The method was described in chapter 4 and further optimized in chapter 5.

We found the likelihood method had a higher efficiency for removing background (for a fixed number of true IBD events). Future results (mainly focused on machine learning) will have to be compared to the results described in chapter 6. Here, we do observe a neutrino rate that is compatible with the theoretical predictions (when an error is introduced for the last one). Different methods of analysing the rate give different results, this has to be studied in further detail as we see no reason these two measurements should be different. Furthermore, the rate calculations based on the fit of an exponential distribution should also be done for simulated events as soon as the simulations allow for it.

A different problem was found when applying no cuts on the data, both the true count and the significance of the number of IBD events increased when no cuts were applied. The resulting rate of this cut was too high compared to the expected theoretical rate. After many tests we only found one plausible solution; a lack of data might explain the good performance associated with the lack of a cut. To test this, the same method as described in chapter 5 and 6 must be followed, this time using much more data.

When dealing with the rectangular cuts in section 5 we noticed the cuts were less effective than the likelihood method. As long as the cuts are not too harsh, they can be used as a stronger initial cut. This, in turn, might optimise the likelihood method even more as fewer background events get trough. We recommend a ROC curve analysis to obtain optimal initial cut values. It is also recommended to take a further look in the cut on ∆r, from both sections 5.3 and 6.2 it is clear more stringent cuts can be applied. Figure 6.2 gives some indication it might be useful to introduce a cut that does not start at 0 (e.g [1, 3]).

Lastly, we have to stress the importance of the systematic errors which have been excluded from the thesis as mentioned earlier. The results of this will have strong implications for the capability to find a high signal to background ratio. Chapter 7. Conclusion & Outlook 91

To conclude, this thesis took steps in evaluating the possibility to use the likelihood method to filter background in further stages of the analysis. Although the results described above do not look very promising, the method may not be abandoned until our suggestions have been implemented and results do not improve.

Whether or not sterile neutrinos are discovered, the future of neutrino physics is assured.

Appendix A

Index of likelihood functions

In previous chapters, likelihood functions have been discussed. Since many vari- ations of these functions are used they are referred to with an index. The index associated with the global and extended likelihood functions can be found in table A.1, where the best performing functions have been highlighted in red. The atten- tive reader will notice that we deviated somewhat from the original constraints set in section 4.3, not all powers used to suppress correlations sum to an integer. A few functions where chosen that deviate to check the initial assumption and as it turns out, one of these deviations still scores very well (L11). In the end, the choice was made to keep with the original assumption, thus equations like L11 will are left out of the final analysis.

The equations were obtained in different stages, a first stage tested all the basic functions (L0 - L7), from this L3 performed the best by far. The second stage of functions relied on modifying L3 to treat the correlation effects, this is how L8 − L21 were obtained.

93 94

A few redundancies can be found in table A.1b, multiple equations lead to the exact same results, this can be seen in section 5.4. For instance GL0 and GL14 are closely related through: GL0 = 1 − GL14. This means one will obtain different optimal λ cuts that have this same relation, this can be seen in tables 5.1 and 5.2. These functions were left as a test for the method and the code; since the end result only requires one extended and one global likelihood function it is not possible to run into any problems by including them in the analysis. Appendix A. Index of likelihood functions 95

Table A.1: Likelihood functions and their associated index.

Index Extended likelihood function Index Global likelihood function 0 P∆x × P∆y × P∆z × P∆t

LSim 0 Sim Uncorr Corr 1 P∆x × P∆y × P∆z × P∆r × P∆t L +L +L

× LCorr 2 P∆r P∆t 1 LSim+LUncorr +LCorr ( ) P +P +P ∆x ∆y ∆z × × LUncorr 3 3 P∆r P∆t 2 ( ) LSim+LUncorr +LCorr P∆x+P∆y +P∆z × 4 3 P∆t LSim 3 LSim+LCorr

5 P∆x × P∆y × P∆z LCorr 4 LSim+LCorr 6 P∆x × P∆y × P∆z × P∆r ( ) LSim 5 LSim+LUncorr P∆x+P∆y +P∆z × 7 3 P∆r ( ) Uncorr 1.2 6 L P∆x+P∆y +P∆z × 0.8 × LSim+LUncorr 8 3 P∆r P∆t ( ) 0.8 Uncorr P∆x+P∆y +P∆z 1.2 L × × 7 Uncorr Corr 9 3 P∆r P∆t L +L ( )1.2 10 P∆x+P∆y +P∆z × P × P 0.8 LCorr 3 ∆r ∆t 8 LUncorr +LCorr ( )0.8 P∆x+P∆y +P∆z 1.2 11 × P × P LSim+LUncorr 3 ∆r ∆t 9 ( ) LSim+LUncorr +LCorr P∆x+P∆y +P∆z × 0.9 × 0.9 12 3 P∆r P∆t |LSim−LUncorr | ( ) 10 LSim+LUncorr +LCorr 13 P∆x+P∆y +P∆z × P 0.9 × P 1.1 3 ∆r ∆t Uncorr Sim ( ) |L −L | 11 LSim+LUncorr +LCorr P∆x+P∆y +P∆z × 1.1 × 0.9 14 3 P∆r P∆t ( ) |LCorr −LUncorr | 12 Sim Uncorr Corr P∆x+P∆y +P∆z × 1.1 × 1.1 L +L +L 15 3 P∆r P∆t ( )1.2 |LUncorr −LCorr | P +P +P ∆x ∆y ∆z × 0.9 × 0.9 13 LSim+LUncorr +LCorr 16 3 P∆r P∆t ( ) Corr Uncorr P∆x+P∆y +P∆z × 0.7 × 1.3 14 L +L 17 3 P∆r P∆t LSim+LUncorr +LCorr ( )1.1 P∆x+P∆y +P∆z 0.7 1.2 Sim Corr 18 × P∆r × P∆t L +L 3 15 LSim+LUncorr +LCorr ( )0.8 P∆x+P∆y +P∆z 1.1 1.1 19 × P × P | Sim− Corr | 3 ∆r ∆t 16 L L ( ) LSim+LUncorr +LCorr P∆x+P∆y +P∆z × 1.3 × 0.7 20 3 P∆r P∆t | Corr − Sim| 17 L L ( )0.9 LSim+LUncorr +LCorr P∆x+P∆y +P∆z 1.3 0.8 21 × P∆r × P∆t 3 (b) Global likelihood GL (a) Extended likelihood L

Appendix B

Poisson and Exponential distributions

In chapter 6 the relation between the Poisson distribution (for a fixed time window) and the exponential distribution (for a time between occurrences of successive events) is discussed. As this relation is not immediately clear to most, below an argument is made to make it more plausible.

For this discussion, the following notation is used: nT is the number of arrivals during a period T (Poisson), xT is the time it takes for one more particle to be detected assuming a particle was detected at time T . From these notations one can immediately conclude the following:

(xT > t) ≡ (nT = nT +t) (B.1)

Indeed, the left hand side of equation B.1 implies no particle has been detected in the time window [T,T + t]. This in turn means that the count of number of arrivals at time T + t is the same as at time T .

Using the complement rule:

P (xT ≤ t) = 1 − P (xT > t) (B.2)

97 98 which can be transformed by using equation B.1:

P (xT ≤ t) = 1 − P (nT +t − nT = 0)

This right hand side can be rewritten:

P (nT +t − nT = 0) = P (nt = 0)

Which, as discussed above is clearly Poisson distributed, where λ is the average number of detections per unit of time and t a quantity of time units:

(λx)0 P (n = 0) = exp (−λt) t 0!

Combining everything together one finds:

P (nT +t − nT = 0) = exp (−λt)

Or, when substituting in the original equation:

P (xT ≤ t) = 1 − exp (−λt) (B.3)

When plotting the time differences between events and normalizing to unity, one effectively has made a probability distribution P (xT > t). This distribution is thus expected to be exponentially declining with a rate (λ) that also happens to represent the average detected particles per unit of time. The time unit is determined by the scale of the plot. Bibliography

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