A Study of Exotic ¯Νe → ¯Ν/E Oscillations in Double Chooz

A Study of Exotic ν¯e ν¯ → /e Oscillations in Double Chooz

Javier M. G. Duarte

Submitted to the Department of Physics

in partial fulﬁllment

of the requirements for the degree

of Bachelor of Science

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 2010 c 2010

Javier M. G. Duarte

Submitted to the Department of Physics in partial fulﬁllment of the requirements for the degree of

Bachelor of Science at the

Massachusetts Institute of Technology June 2010

c 2010 Javier M. G. Duarte

The author hereby grants to MIT permission to reproduce and distribute publicly paper and electronic copies of this thesis document in whole or in part.

Signature of Author Department of Physics May 7, 2010

Certiﬁed by Professor Janet M. Conrad Thesis Supervisor, Department of Physics

Accepted by Professor David E. Pritchard Senior Thesis Coordinator, Department of Physics

A Study of Exotic ν¯ ν¯ Oscillations in Double Chooz e → /e by Javier M. G. Duarte

Submitted to the Department of Physics on May 7, 2010, in partial fulﬁllment of the requirements for the degree of Bachelor of Science

Abstract

In this thesis, we estimate the sensitivity of Double Chooz, a reactorν ¯e ν¯e experiment, to detect “early” neutrino oscillations based on a three-active plus→ one-sterile, or (3 + 1), neutrino mixing model by implementing a least-squares ﬁt to the simu- lated electron antineutrino reactor spectrum1. By comparing the expected spectra from the null hypothesis and the (3 + 1) oscillation hypothesis at the Double Chooz near detector L = 200 m, we expect a modest sterile neutrino discovery potential, limited by the 2% reactor ﬂux uncertainty. This potential may be expanded by employing a Double∼ Chooz-like detector at a very short baseline L = 6 m from the reactor. At both baselines, the p-value for the null hypothesis was extremely small p 0.004, which is compelling evidence for rejecting this hypothesis. Very short baseline∼ antineutrino oscillation reactor experiments may help to resolve the current global incompatabilities between neutrino and antineutrino data sets.

Thesis Supervisor: Janet M. Conrad Title: Professor, Department of Physics

1The analysis was carried out in ROOT and RooFit with MINUIT

Acknowledgments

First and Foremost, I thank Professor Janet Conrad for being a wonderful adviser and mentor over the last four years. I also thank Dr. Lindley Winslow for guiding me throughout this analysis and the preparation of this manuscript.

I also thank my colleagues in the MIT Dark Matter and Neutrino group, Claire Thomas, Steve Jaditz, Sean Henderson, Kazu Terao, and Chris Jones, for their helpful discussions and company while working in 26-670.

I am very grateful to my fellow members of the class of 2010 in the physics and math departments, including Sara Campbell, Ian Rousseau, Eric Quintero, and Ye- lena Bagdasarova, Cinjon Resnik, and many others, without whom I certainly would not have survived Course 8 or 18. I especially thank my Junior Lab partner Sara Campbell; Our partnership and friendship have made me a better physicist.

My advisers and colleagues during my summer at CERN in 2009 were invaluable in my development as an “international” scientiﬁc collaborator (on the ATLAS experiment): Professor Homer Neal, Dr. Natalia Panikasvilli, Alexander Tuna, Jessica Muir, J.J. Stankowicz, Jeremy Ticey, Alex Palmer, Phil Hebda, and many, many more.

My Junior Lab (8.14) instructor Professor Nergis Mavalvala is responsible for most of my current knowledge of experimental methods and data/error analysis and for that, I thank her.

My summers spent at Fermilab working on MiniBooNE and SciBooNE in 2007 and 2008 resulted in a great increase in my knowledge of experimental particle physics and many fond memories. Thanks to Professor Mike Shaevitz, Professor Morgan Wascko, Professor Jocelyn Monroe, Matt Toups, Teppei Katori, Alexis Aguilar-Arevalo, Guilia Fanti, Dylan Nelson, Rhorry Gauld, Jamie Molaro, Danny Crow, and many others for teaching me physics.

My, freshman “Time” seminar adviser Professor Peter Fisher opened the door to particle physics for me, and I haven’t looked back since. I also thank Professor Seth Lloyd who taught me about time travel in quantum computing, and mentored me in my graduate shool decisions.

Other aspects of my life at MIT made the long days and nights working on physics problem sets seem much more bearable. My teammates on the MIT varsity soccer and varsity/club gymnastics teams, especially Tom Caldwell and Jake Shapiro, proved to me that true excellence has multiple dimensions, including athletics. My brothers in my fraternity ΦΣK, such as Joey Derosa, Matt Talpe, Gordon Wintrob, Antony Speranza, Dylan Rankin, Rich Whalley, Michael Melgar, and Steven Dabic, are some of my closest friends and continually inspire me with their work ethic as well as remind me to have fun every once in a while. My girlfriend, Catie Russell has been a beacon of sunshine, joy, and most importantly, laughter, during my ﬁnal stress-ﬁlled semesters at MIT.

Finally I dedicate this thesis to my mother, Bruny, and my sister, Rossy, for constantly supporting me throughout my undergraduate career. Contents

1 Introduction 1

1.1 History of Neutrino Experiments ...... 2

1.2 Motivation for a (3 + 1) Model ...... 8

2 Theory of Neutrino Oscillations 11

2.1 Sterile Neutrino Formalism ...... 14

3 Double Chooz 19

3.1 Detection Concept ...... 19

3.2 Multi-volume Detectors ...... 21

4 Nuclear Reactors 23

4.1 Reactor Neutrino Production ...... 24

4.2 Fission Rates and DRAGON ...... 25

5 The Likelihood Model 29

5.1 The Method of Least Squares ...... 30

5.2 Constructing the P.D.F. for dNν/dEe ...... 31

5.2.1 Null Hypothesis ...... 32

5.2.2 (3 + 1) Neutrino Oscillation Hypothesis ...... 33 ii CONTENTS

6 Sensitivity at Double Chooz 35

6.1 Generating Data ...... 36

6.2 Fitting with χ2 Minimization ...... 36

6.3 ∆χ2 Conﬁdence Levels ...... 38

7 Sensitivity at a Very Near Detector 41

8 Outlook and Conclusions 47 List of Figures

1-1 Summary of neutrino oscillation experiment results ...... 9

2 2 1-2ν ¯ global allowed ∆m14, sin (2θµµ) parameter space ...... 10

2-1 The Standard Model ...... 11

2-2 (3 + 1) neutrino mass hierarchy ...... 14

2-3 The survival probability P (¯νe ν¯e) for the standard theory...... 17 →

2-4 The survival probability P (¯νe ν¯e) for a (3 + 1) oscillation model. . 17 →

3-1 Inverse beta decay Feynman diagram ...... 20

3-2 The Double Chooz far detector...... 22

4-1 Relative ﬁssion rates of 238U, 235U, 239Pu, and 241Pu ...... 25

4-2 2D geometry of the nuclear reactor ...... 26

4-3ν ¯e Reactor Spectrum Shape and Ratio ...... 28

6-1 Expected spectra for null hypothesis and oscillation hypothesis at the Double Chooz near detector (L = 200 m)...... 37

6-2 Oscillation ﬁt at Double Chooz near detector ...... 38

2 2 6-3 Allowed regions in (sin (2θee), ∆m ) parameter space at the Double Chooz near detector (L = 200 m) ...... 40 iv LIST OF FIGURES

7-1 Expected spectra for null hypothesis and oscillation hypothesis at L = 6m...... 42

7-2 Oscillation ﬁt at a Double Chooz-like detector at 6 m ...... 44

2 2 7-3 Allowed regions in (sin (2θee), ∆m ) parameter space at 6 m . . . . . 45 Chapter 1

Introduction

Recent results [1, 2] have shown that despite tension between diﬀerent data sets, there exists a globally allowed region in parameter space for (3+1) neutrino mixing, a theory with one extra heavy neutrino, in short baseline (SBL) antineutrino experiments. In other words, all antineutrino experiments to date have not ruled out the possibility that there exists a heavy sterile neutrino. This possibility has deep implications for the Double Chooz reactor antineutrino disappearance experiment in France, which

2 1 aims to measure the neutrino mixing angle θ13 or limit it to sin (2θ13) < 0.025 [3] .

Double Chooz will utilize two identical detectors at 200 m and 1.05 km. Since the form of the neutrino survival probability is a sinusoid squared depending on

2 2 ∆m L/Eν, the far detector is placed at a local minimum in the survival probability . The purpose of the near detector is to precisely measure the full neutrino flux in order to constrain the flux prediction at the far detector (as well as reduce the systematic error on the θ13 measurement)[3]. However, the existence of large mass splittings of order ∆m2 1 eV result in “early oscillations” transforming the neutrinos before ∼ they have propagated to the near detector. Not only would the diminished flux at the near detector have ramifications for the Double Chooz analysis methodology, but it would give experimentalists a keen opportunity to observe new physics using an

1 Double Chooz seeks electron antineutrino disappearance, which we denote asν ¯e ν¯e 2 2 2 3→ 2 Assuming realistic values for the parameters, sin (2θ ) = 0.1, ∆m = 2.5 10− eV , and the 13 13 × average energy Eν = 2 MeV 2 CHAPTER 1. INTRODUCTION

existing experiment.

In this thesis, we present a study of a (3+1) neutrino mixing model near ∆m2 1 14 ' 2 2 2 2 eV and sin (2θee) = Ue4 (1 Ue4 ) 0.04 at the Double Chooz near detector. The | | −| | ' goal is to determine the sensitivity of the experiment to the parameters associated with the heavier sterile neutrino (which means it does not interact through the weak force). The method we use is to simulate the detected reactorν ¯e spectrum under the assumption of (3 + 1) neutrino mixing multiply it by the survival probability, include detector effects, and finally fit to the probability density function (p.d.f.) by χ2 minimization.

The rest of this chapter summarizes the history of neutrino experiments and provides the motivation for studying (3 + 1) sterile neutrino models given the current conﬂicted state of neutrino oscillation data.

Chapter 2 provides an overview of the theory of neutrino oscillations. Then, chapter 3 details the Double Chooz experimental design, with emphasis on the detection system. Chapter 4 outlines the basic physics of nuclear reactors.

Chapter 5 gives some statistics background and describes the least-squares method as it applies to this analysis, speciﬁcally, the construction of the p.d.f. for the number of detected electron antineutrinos and the χ2 function.

Chapters 6 explains the details of the analysis for the sensitivity to a (3 + 1) neutrino model at the Double Chooz near detector. Chapter 7 proposes a Double Chooz-like detector at a very short baseline of L = 6 m, and repeats the sensitivity analysis. Chapter 8 summarizes this work and discusses the signiﬁcance of this analysis for Double Chooz and future neutrino experiments.

1.1 History of Neutrino Experiments

Wolfgang Pauli ﬁrst proposed the existence of the neutrino to reconcile the principle of energy conservation with the process of beta decay in 1930 [4].

The data on beta decay had long since perplexed physicists. In 1914, Chadwick 1.1. HISTORY OF NEUTRINO EXPERIMENTS 3 measured the energy spectrum for beta particles coming from the apparent beta decay.

214Pb 214 Bi + β− (1.1) 82 →83

According to energy conservation, if the nuclei are at rest (before and after the decay) and no other particle is produced, then every beta particle should leave the decay with the same energy, E0, equal to the diﬀerence in rest mass energies between the parent and daughter nuclei. Chadwick’s result, however, was a continuous energy spectrum, showing that the beta particles were almost always ejected with energy Eβ less than

E0. The experiment demonstrated that there was some energy missing [5].

Pauli’s insight was that perhaps another particle (later dubbed the neutrino) was escaping the decay undetected, and taking with it, the missing energy E0 Eβ. Based − on this assumption, Fermi developed a very successful theory of beta decay, which predicted that the probability of neutrinos interacting was very small [6, 7]. After two decades without experimental confirmation, Reines and Cowan were able to use the Hanford nuclear reactor to measure the elusive particle [8]. The reactors of the 1950s were expected to produce 1019 antineutrinos per second [5]. Reines and Cowan hoped to use this large flux to overcome the small cross section barrier in order to detect the neutrino via the process of inverse beta decay (see Ch. 3). In 1953, they succeeded in providing the first tentative evidence for the existence of the neutrino, but it wasn’t until an improved experiment ran at the Savannah River reactor in 1958 that they obtained conclusive evidence of the existence of the neutrino[9].

In the following years, a new problem emerged: detecting neutrinos from the Sun. Solar neutrinos are produced in the core of the Sun, through the nuclear reactions that power the Sun [10]. In 1968, Ray Davis made the ﬁrst measurement of solar neutrinos deep underground in the Homestake mine (to reduce cosmic-ray backgrounds) using an enormous tank of chlorine [11]. The detection reaction involved a neutrino being absorbed by the chlorine to produce argon.

37 37 νe + Cl Ar + e (1.2) → 4 CHAPTER 1. INTRODUCTION

Davis accumulated argon atoms over a period of several months, but only measured 1/3 of the neutrinos that the standard solar model predicted [12]. Explaining this large deﬁcit became known as the solar neutrino problem [5].

Before the formulation of the solar neutrino problem, it was observed by E. P. Hincks and B. Pontecorvo in 1950 that the muon decays into an electron and two neutrinos [13]. µ e + ν + ν (1.3) → In 1957, B. Pontecorvo first suggested the possibility of neutrino-antineutrino ν ν¯ ↔ oscillations [14]. This suggestion had the necessary consequence that the neutrino had to be massive [15, 16]. Then in 1962, Z. Maki, M.Nakagawa and S. Sakata (MNS) suggested a two-neutrino mixing model, that is a model with two different types, or flavors of neutrinos [17]. Shortly thereafter, an experiment at Brookhaven produced conclusive evidence of a neutrino associated with the muon, called a muon neutrino

νµ, which was diﬀerent from the νe [18]. It wasn’t until 2000 that the DONUT collaboration conﬁrmed the existence of a third neutrino associated with the tau lepton, ντ [19].

The proposed solution put forth by MNS in [17] and Pontecorvo in [20] to the solar neutrino problem, was that electron neutrinos produced at the Sun were transforming into a diﬀerent ﬂavor, such as a muon neutrino, due to quantum mechanical mixing. The muon neutrino doesn’t interact via inverse beta decay and thus was not being detected.

Besides nuclear reactions, cosmic ray interactions in the Earth’s upper atmosphere also produce neutrinos. Pion decay followed by muon decay

π µ + νµ → (1.4) ↓ e + νe + νµ result in an expected ratio of two muon neutrinos for every electron neutrino coming from the atmosphere, known as atmospheric neutrinos. In addition, due to the 1.1. HISTORY OF NEUTRINO EXPERIMENTS 5

weakness of neutrino interactions, it was thought that equal numbers of atmospheric neutrinos should be detected from above as from below (from the other side of the Earth).

In 1989, the Kamiokande3 II experiment provided the first confirmation of the results of the Homestake experiment (the deficit of solar neutrinos) and measured the ratio of electron neutrinos to muon neutrinos from the atmosphere [21]. The Kamiokande experiment was intended to search for proton decay in the Kamioka mine in Japan. This experiment used a water detector to measure the particles’ Cerenkovˇ radiation. This form a radiation occurs when a particle traverses a medium at faster than the speed of light, causing lagging rings of light as the particle travels (like a Mach cone in a sonic boom). Solar and atmospheric neutrinos were detected by their elastic scattering off the water’s electrons

− − νx + e + νx + e (1.5) →

by observing the Cerenkovˇ rings of the recoil electron. This reaction is sensitive to

all three ﬂavors νe, νµ, and ντ , but with reduced sensitivity to νµ and ντ . Simulta- neously, another proton decay experiment, the IMB4 experiment, was running that was also sensitive to atmospheric neutrinos. Both Kamiokande and IMB measured a deviation from 2 in the ratio of muon-to-electron atmospheric neutrinos observed, indicating either an excess of electron-like events or a deﬁcit of muon-like events from the atmosphere [22].

The Super Kamiokande experiment succeeded the Kamiokande experiment with a water detector 25 times larger in the same mine. In 1998, Super Kamiokande found that the atmospheric deﬁcit was in muon neutrinos coming from the other side of the Earth [23]. Further, Super Kamiokande measured the deﬁcit of neutrinos from the Sun with the highest statistics of any solar experiment to date [24].

Finally, in 2002, the SNO5 collaboration presented compelling evidence that the

3Kamioka Nucleon Decay Experiment 4Irivine, Michigan, Brookhaven Experiment 5Sudbury Neutrino Observatory 6 CHAPTER 1. INTRODUCTION solar neutrino problem was due to neutrino flavor change. Using a 1 kiloton volume of heavy water, D2O, SNO measured solar neutrinos through the rates of the three interactions6: − νe + d p + p + e (CC) → νx + d p + n + νx (NC) (1.6) → − − νx + e νx + e (ES) → In the CC interaction, which is sensitive to only the electron flavor neutrino, SNO measured a deficit, whereas in the NC interaction, which is sensitive to all three flavors equally, SNO’s measurement is consistent with the Standard Solar Model [25, 12].

The ﬁnal source of neutrinos to be tapped was that of accelerators [26]. In principle, accelerator neutrino experiments can control both the neutrino beam energy and the distance to the detector, leading to more reﬁned experimental conditions. Neu- trino beams are created through the decay of pions and kaons and thus are mostly (−) (−) νµ orν ¯µ with a small νe background. The accessible experimental channels are νµ (−) (−) 7 disappearance, νe appearance, and ντ appearance . In 1996, the short baseline ex- 8 periment LSND measuredν ¯e appearance in aν ¯µ beam with 20 MeV 6 Eν 6 60

MeV and L = 30 m for an L/Eν 1. They observed an excess consistent with ∼ 2 2 2 2 0.2 eV 6 ∆m 6 10 eV over a range of 0.003 6 sin 2θ 6 0.03. The best ﬁt corresponded to ∆m2 = 1.2 eV2 and sin2 2θ = 0.003 [27].

Other accelerator experiments have probed a diﬀerent region of parameter space. Two long baseline experiments, MINOS and K2K, designed to be sensitive to the allowed atmospheric oscillation region, observed oscillation signals with ∆m2 3 23 ∼ × 3 2 2 10 eV and sin 223 1.0 despite having diﬀerent values of L and Eν (they had the ∼ same L/Eν) [28, 29].

In 2001, the MiniBooNE9 experiment, designed to conﬁrm or refute the LSND re-

6Charged Current (CC), Neutral Current (NC), and Elastic Scattering (ES) 7Oscillation searches can be categorized as appearance or disappearance. In an appearance analysis, one searches for an excess of a particular flavor of neutrino in a beam of a different flavor (such as νe in νµ), while in a disappearance analysis one searches for a reduced rate of interactions of a particular flavor with respect to the expectation (usually in a beam of the same flavor). 8Los Alamos Scintillator Neutrino Detector 9Mini Booster Neutrino Experiment 1.1. HISTORY OF NEUTRINO EXPERIMENTS 7

ˇ sult, began running at Fermilab. The MiniBooNE detector(500 tons of CH2 Cerenkov

spherical detector) was placed at L = 550 m from the neutrino beam (Eν 700 MeV) ∼ for an L/Eν 1. MiniBooNE’s initial search for νe appearance gave a null result, ∼ consistent with no oscillations [30]. Subsequent searches for νµ andν ¯µ disappearance 10 andν ¯e appearance also gave null results [31, 32, 33]

Coming full circle (from Reines and Cowan’s experiment), reactors are still a very attractive neutrino source due to their extremely high ﬂux. The CHOOZ experiment

set the best current limit on θ13 (see Ch.2) in 1998 with a single detector 1 km from the 2 Chooz nuclear reactor. CHOOZ provides the upper bound sin 2θ13 > 0.18 assuming ∆m2 = 2.0 10−3 eV2 (this limit is strongly correlated to the assumed value of atm × 2 ∆matm) [34, 35]. Multiple reactor experiments also ran at baselines ranging from 10 m to 1 km, but none saw a significant deficit [10]. Finally, KamLAND, the first long baseline reactor experiment, situated in the same Kamioka mine as Kamiokande, became the first reactor antineutrino experiment to detect a deficit of antineutrinos from a reactor [36]. Japan has 55 nuclear reactors along the coast while the Kamioka mine is in the center of the island. Due to this geometry, the 55 reactors can be thought of as one very large antineutrino source at L 200 km from KamLAND. This large ∼ distance requires that KamLAND have a very massive target: 1 kiloton of liquid scintillator (used to detect inverse beta decay). Interpreted as neutrino oscillations, the best fit to the KamLAND data gives a maximal mixing angle sin2 2θ = 1.0 and ∆m2 = 6.9 10−5 eV2 [37]. × The results of all neutrino oscillation experiments until 2008 are shown in figure 1-1. Almost all of the experimental evidence to date, ranging many orders of magni- tude in parameter space, favors neutrino oscillations to be the dominant mechanism causing neutrino flavor mutations during neutrino propagation. The three-neutrino PMNS11 matrix, defined in Ch.2, embodies the mixing in the leptonic sector and can

be parametrized in terms of 3 mixing angles (θ12, θ13, θ23) and a complex CP violat-

10 Searches forν ¯µ (¯νe) disappearance (appearance) were conducted in aν ¯µ beam. 11Pontecorvo-Maki-Nakagawa-Sakata 8 CHAPTER 1. INTRODUCTION ing phase (δ). Our knowledge of this matrix is far from complete–and we are most ignorant of the values of θ13 and δ. Double Chooz is a reactor antineutrino disappearance experiment, which is starting data collection in 2010 at the CHOOZ site, using two identical detectors, aims to minimize this ignorance by measuring θ13 or at 2 the least limit its value down to sin 2θ13 6 0.025. In order to do so, Double Chooz will need unprecedented precision, including a thorough understanding of the reactor antineutrino ﬂux at the near and far detectors. For this reason, an exploration of the possibility of a heavy sterile neutrino, which would distort the spectrum at the near detector, is necessary.

1.2 Motivation for a (3 + 1) Model

A global analysis of all short baseline (SBL) neutrino oscillation data sets reveals conflicting results. The most significant tension lies in reconciling the antineutrino and neutrino data. However, fits to antineutrino-only data sets, including appearance and disappearance experiments, are found to be significantly more compatible within a (3 + 1) sterile neutrino model. In figure 1-2, considering only antineutrino short baseline data sets (except LSND), the allowed 90%, 99%, and 3σ, confidence level

2 2 regions of sin (2θ) and ∆m14 parameter space are plotted [1].

2 2 The parameters sin 2θµe and sin 2θµµ in figure 1-2 represent “effective” mixing angles, that relate to the amplitudes of νµ νe and νµ νµ transitions. Likewise a → → 2 third parameter sin 2θee (not plotted) can be defined (see Eq. 2.12) which relates to the amplitude of νe νe survival. In the same global analysis of all SBL antineutrino → 2 experiments except LSND, the best fit values for a (3 + 1) model include sin 2θee = 2 2 0.043 and ∆m14 = 0.91 eV . In the rest of this thesis, we will assume the validity of this (3 + 1) model, in which the parameters take on these values, in order to analyze Double Chooz’s sensitivity to this model. 1.2. MOTIVATION FOR A (3 + 1) MODEL 9

CDHSW

CHORUS NOMAD NOMAD NOMAD

0 KARMEN2 CHORUS 10 MiniBo oNE Bugey LSND 90 /99 %

K2K SuperK 90 /99 % CHOOZ 10 –3 PaloVerde all solar 95% MINOS Cl 95%

] KamLAND 2 95%

[eV SNO 2 –6 95%

m 10 Super-K ∆ 95% Ga 95%

–9 10 νe↔ν X νµ↔ν τ νe↔ν τ νe↔ν µ All limits are at 90%CL unless otherwise noted 10 –12 10 –4 10 –2 10 0 10 2 tan 2θ http://hitoshi.berkeley.edu/neutrino Figure 1-1: Summary of the regions of ∆m2 and tan2 θ parameter space favored or excluded by various experiments. From reference [38]. 10 CHAPTER 1. INTRODUCTION

102 102 (3+1) (3+1)

10 10 ) ) 2 2

1 1 (eV (eV 2 2 41 41 m m ∆ ∆ -1 -1 10 ν SBL 90% CL 10 ν SBL 90% CL ν SBL 99% CL ν SBL 99% CL ν ν SBL 3σ CL MiniBooNE µ 90% CL 10-2 ν SBL without LSND 90% CL 10-2

-3 -2 -1 -3 -2 -1 10 10 10 1 10 10 10 1 2 θ 2 θ sin (2 µe) sin (2 µµ )

Figure 1-2: The allowed 90%, 99%, and 3σ CL regions from a combined analysis of FIG. 7:all The antineutrino allowed 90%, 99%, SBL and data3σ CL regionssets. from The a left combined plot analysis also shows of all antineutrino the 90% SBL CL data allowed sets. The re- left plot also shows thegion 90% obtained CL allowed from region a obtained combined from analysis a combined of anal allysis antineutrino of all antineutrino experiments experiments except except LSND (KARMEN, BNB-MB(¯ν), Bugey, and CHOOZ). The right plot also shows the 90% CL exclusion limit from [29]. The MiniBooNEν ¯ 13 µ disappearance(KARMEN, search excludes BNB-MB(¯ the parameterν) , Bugey, space to and the CHOOZ).right of the line The at right90% CL. plot See also text for shows more the details. 90% CL exclusion limit from [31]. The MiniBooNEν ¯µ disappearance search excludes the parameter space to the right of the line at 90% CL.

102 102 (3+1) (3+1)

10 ν SBL 90% CL 10 ν SBL 99% CL

) ν σ ) 2 SBL 3 CL 2

(eV 1 (eV 1 2 2 41 41 m m ∆ ∆

10-1 10-1 ν SBL 90% CL ν SBL 99% CL ν MiniBooNE µ 90% CL

-2 -2 10 -4 -3 -2 -1 10 -4 -3 -2 -1 10 10 10 10 1 10 10 10 10 1 2 θ 2 θ sin (2 µe) sin (2 µµ )

FIG. 8: The allowed 90%, 99%, and 3σ CL regions from a combined analysis of all neutrino SBL data sets. The right plot also shows the 90% CL exclusion limit from [29]. The MiniBooNE νµ disappearance search excludes the parameter space to the right of the line at 90% CL. See text for more details. are shown in Fig. 8. Current constraints from Mini- tineutrino data sets cannot be accommodated within a BooNE νµ disappearance are shown on the right panel (3+1) CPT-conserving sterile neutrino oscillation sce- of Fig. 8. Interestingly, fits to only neutrino SBL data nario. However, the constraining power of antineutrino 2 2 also yield a closed contour at 90% CL. The parame- SBL experiments alone on ∆m41 and sin 2θµe is remark- ter space, however, points to smaller mixing amplitudes able, and invites exploration of models that provide the relative to those preferred by the antineutrino-only fit. possibility of different oscillation patterns for neutrinos Neutrino-only fits and antineutrino-only fits are incom- versus antineutrinos. 5 patible, with a PG of 8.1 10− , as shown in Table IV. The large incompatibility× between antineutrino and neu- Figure 9 (left) shows a comparison of the BNB-MB(ν), trino SBL results suggests that the neutrino and an- BNB-MB(¯ν), and NUMI-MB event distributions for the neutrino-only best-fit parameters and antineutrino-only Chapter 2

Theory of Neutrino Oscillations

Neutrino oscillations are the natural phenomena by which a neutrino of one ﬂavor

(such as an electron neutrino νe) may transform into a neutrino of another ﬂavor

(such as a tau neutrino ντ ). Theoretically, this occurs if (1) there is a non-zero mass difference between at least two mass eigenstates and (2) the mass eigenstates and the weak, or flavor, eigenstates do not perfectly coincide, that is there is a nontrivial mixing matrix relating the two bases. Currently, the standard model (SM)1, represented in figure 2-1, does not incorporate massive neutrinos. 6

u c t γ up charm top photon

d s b g down strange bottom gluon QUARKS

Z νe ν ν μ τ Z boson electron muon tau neutrino neutrino neutrino Evidence of neutrino oscillation has e μ τ W accelerated the growing excitement in electron muon tau W boson neutrino physics. LEPTONS FORCE CARRIERS I II III

Much of what we know Figure about these 2-1: mysterious The three ﬂavors of elementary particles in the Standard Model. The particles has been learnedneutrinos very recently. are In in the the raised blocks. This image is taken from Ref. [39].

Standard Model of Particle Physics (ABOVE, RIGHT), the properties and interactions of known particles and forces are described in exquisite detail. But recent neutrino discoveries have exposed a crack in its K2K confirms atmospheric time-tested edifice – the Standard Model describes oscillations KamLAND confirms solar neutrinos (OUTLINED IN RED) as having no mass. As you oscillations * Nobel Prize for neutrino will read, the solution to the problem of Davis and astroparticle physics! SNO shows solar oscillation Bahcall’s missing neutrinos suggests otherwise – and to active flavor is casting a long shadow of doubt on this essential Super-K confirms solar deficit and “images” Sun scientific theory. Super-K sees evidence of atmospheric neutrino oscillations * Nobel Prize for ν discovery! LSND sees possible indication of oscillation signal * Nobel Prize for discovery of distinct flavors! Kamiokande II and IMB see supernova neutrinos Kamiokande II and IMB see atmospheric neutrino anomaly SAGE and Gallex see the solar deficit LEP shows 3 active flavors *Nobel Prize (ABOVE) awarded for neutrino discoveries Kamiokande II confirms solar deficit

1980 2005? 12 CHAPTER 2. THEORY OF NEUTRINO OSCILLATIONS

In the SM (with massless Dirac neutrinos), the neutrino mass eigenstates are degenerate and there is no difference between the mass and flavor eigenstates. Thus there is no mixing between mass and flavor eigenstates, electron number, muon number, and tau number are each strictly conserved in weak interactions, and there can be no neutrino oscillations [26, 40]. However, neutrino oscillations can treated phe- nomenologically, by assuming nonzero mass splittings nontrivial mixing. For a full theoretical treatment see [41, 42, 40].

In general, if there are N neutrino eigenstates of deﬁnite mass ν1, ν2, ν3, ... with

masses m1, m2, m3, ... and N neutrino eigenstates of deﬁnite ﬂavor νe, νµ, ντ , ..., then an N N unitary mixing matrix U, known as the Pontecorvo-Maki-Nakagawa-Sakata × matrix (PMNS matrix), characterizes how these states overlap. Assuming the mass

diﬀerence is too small to distinguish one mass eigenstate νj from another, each neu-

trino of deﬁnite ﬂavor να (where α = e, µ, τ, ...) produced by a source is the coherent superposition:

N X ∗ να = Uαj νj (2.1) | i j=1 | i

In quantum mechanics, free particle states Ψ(x, t) obey the Schr¨odingerequation, | i which dictates how a state evolves in time [43].

∂ Ψ(x, t) i | i = H Ψ(x, t) (2.2) ∂t | i

where H is the Hamiltonian. Suppose a neutrino να is born at position x = 0 and

time t = 0 in the lab frame, and the νj component has momentum pj and energy Ej. Then, after a time t, the neutrino is in the state2:

X ∗ ipj x −iEj t ν(x, t) = Uαj νj e e (2.3) | i j | i

1The standard model is a gauge theory of the strong (SU(3)) and electroweak (SU(2) U(1)) interactions with the gauge group SU(3) SU(2) U(1). It does not take gravitation into account.× × × 2using natural units ~ = c = 1 13

Since neutrinos are very light, they travel close to the speed of light. So, we may q 2 2 2 mj use the fact that mj pj to write Ej = pj + mj = pj + . To very good ∼ 2pj approximation, we also have pj ∼= Eν the beam energy. To zeroth order in the neutrino masses, we can approximate t ∼= x = L, the ﬁxed length from the source to the detector (in natural units) [44].

If we invert Eq. 2.1 to express νj as a combination of να ’s then we see | i | i 2 X X ∗ −im L/2Eν ν(L) = U e j Uα0j να0 (2.4) | i αj | i α0 j

We project the time evolved ket in Eq. 2.4 onto the detected neutrino bra to ﬁnd the

oscillation amplitude that a neutrino born as να will be detected as νβ.

Amp(να νβ) = νβ ν(L) → h | i X 2 ∗ −imj L/2Eν = Uαje Uβj (2.5) j

Finally, we square Eq. 2.5 to ﬁnd the oscillation probability. It is convenient to

express the distance traveled L in terms of meters, Eν in MeV, and the squared mass diﬀerences ∆m2 m2 m2 in eV2. Then the general probability that a neutrino ij ≡ j − i born as να will oscillate to νβ as a function of L is

P (να νβ) = δαβ → 2 X 1.27∆mijL 4 (U ∗ U U U ∗ ) sin2 αi βi αj βj E − i

where and denote the real and imaginary components, respectively [38]. R I 14 CHAPTER 2. THEORY OF NEUTRINO OSCILLATIONS

2.1 Sterile Neutrino Formalism

The standard theory of three neutrinos can be expanded by adding one or more sterile neutrinos, which do not couple to the weak vector bosons. We call a theory with three active and one sterile neutrino, a (3 +1) model. In this thesis, we will assume there is

a fourth ﬂavor state νs and a fourth mass state ν4, and that m4 m3, as portrayed in ﬁgure 2-2. 5 4 ∆ 2 m LSND

3 ∆ 2 m atm 2 ∆m2 1 solar

Figure 2-2: The mass hierarchy in our (3+1) neutrino mixing model. The approximate composite structure is represented by the colored ﬁll patterns. White represents the sterile neutrino νs, which comprises only a small fraction of the standard three neutrino mass eigenstates.

To determine the entries of the mixing matrix Uαj in Eq. 2.6, we must address how to parametrize the unitary mixing matrix U. To transform from the basis of ﬂavor eigenstates to the basis of mass eigenstates requires a four-dimensional rotation matrix involving complex phases to account for the possibility of CP violation. A general rotation in four dimensions can be realized with six two-dimensional rotations, Rij in the i j plane. For SBL experiments assuming a 3 + 1 mixing model, a convenient − parametrization is given in reference [45]. This form reduces to the standard three-

dimensional matrix by setting θi4 = 0.

(3+1) USBL = R14(θ14)R24(θ24)R34(θ34)R23(θ23, δ3)R13(θ13, δ2)R12(θ12, δ1) (2.7) 2.1. STERILE NEUTRINO FORMALISM 15

We notice that each term in the probability formula oscillates as a sine squared with

2 argument 1.27∆mijL/Eν. At the beam energy and short distance under consider-

ation at Double Chooz, L/Eν 10, we may ignore the standard mass diﬀerences ∼ 2 2 2 −3 ∆m12, ∆m13, ∆m23 < 10 compared to the extra mass splittings ∆m14 1. Ef- ∼ ∼ fectively, we are setting m1 = m2 = m3. Equivalently, we could say that the un-

detectable angles and phases (θ12, δ1), (θ23, δ2), (θ13, δ3) decouple from the oscillation formula. Therefore, we ignore this physically degenerate three-dimensional subspace (3+1) in the matrix USBL :

    Ue1 Ue2 Ue3 Ue4 c14 s14s24 s14c24s34 s14c24c34    − −      (3+1)  Uµ1 Uµ2 Uµ3 Uµ4   0 c24 s24s34 s24c34  U =    −  SBL   '    Uτ1 Uτ2 Uτ3 Uτ4   0 0 c34 s34      Us1 Us2 Us3 Us4 s14 c14s24 c14c24s34 c14s24c34 − − − (2.8) where snm = sin θnm and cnm = cos θnm. Furthermore, at Double Chooz we are looking forν ¯e disappearance. CP-violating phases can only be detected in appearance channels, so we may neglect the complex phases altogether. To convert ν oscillation probabilities toν ¯ oscillation probabilities, we assume CPT invariance holds [38].

P (¯να ν¯β) = P (νβ να) (2.9) → →

In this case, the oscillation formula in Eq. 2.6 forν ¯e disappearance becomes:

3 X 1.27∆m2 L P (¯ν ν¯ ) = 1 4 U 2 U 2 sin2 14 (2.10) e e ej e4 E → − j=1 | | | | ν

2 2 2 2 Using the fact that the mixing matrix is unitary so that Ue1 + Ue2 + Ue3 + Ue4 = | | | | | | | | 16 CHAPTER 2. THEORY OF NEUTRINO OSCILLATIONS

1, then substituting in the actual terms we get:

2 2 2 2 1.27∆m14L P (¯νe ν¯e) = 1 4 Ue4 (1 Ue4 ) sin → − | | − | | Eν 2 2 2 2 2 2 2 2 1.27∆m14L = 1 4s14c24c34(1 s14c24c34) sin − − Eν 2 2 2 1.27∆m14L 1 sin (2θee) sin (2.11) ≡ − Eν

where in the last equation we deﬁned a new mixing angle θee by the equation

sin(θee) sin(θ14) cos(θ24) cos(θ34) (2.12) ≡

To demonstrate the qualitative differences between the standard three-neutrino theory and our (3 + 1) model, refer to figures 2-3 and 2-4. In figure 2-3, theν ¯e survival probability is plotted as a function of distance from the reactor L, assuming

a monoenergetic Eν = 2 MeV (characteristic energy of antineutrinos from nuclear reactors). The blue vertical lines indicate the positions of the Double Chooz near detector L = 200 m and far detector L = 105 m. Thus, assuming these values for the parameters, it is clear why the Double Chooz near and far detector positions were chosen–the far site is at a maximum in the oscillation probability, whereas at the near site, the neutrinos haven’t had the chance to oscillate yet.

On the other hand, in ﬁgure 2-4, the result of additional mixing angles is a su- perimposed interference pattern. The “extra” oscillations have a small amplitude, but a much higher “frequency” (since it is inversely proportional to the very large ∆m2 1 eV2). In other words, within this (3 + 1), the neutrinos do have the chance 14 ∼ to oscillate away before they reach the near detector. We now turn to describing the details of the Double Chooz detector. 2.1. STERILE NEUTRINO FORMALISM 17

1 0.9 0.8 0.7 θ = 0.60 0.6 12 = 2 MeV ν θ = 0.16 0.5 13

) | E θ e 23 = 0.822

ν 0.4 2 -5 2 → ∆ m = 7.59 × 10 eV e 0.3 12 ν ∆ 2 × -3 2 P( m = 2.43 10 eV 0.2 23 Near Detector L = 200 m 0.1 Far Detector L = 1.05 km 0 1 10 102 103 104 105 L [m]

Figure 2-3: The survival probability P (¯νe ν¯e) for the standard theory. The parameter values chosen come from [38, 3]. →

1 θ 0.9 12 = 0.60 θ 0.8 13 = 0.16 θ 0.7 23 = 0.822 θ 0.6 14 = 0.115 = 2 MeV ν θ 0.5 24 = 0.08 ) | E

e θ 34 = 0.401 ν 0.4 -5 → ∆ 2 × 2 e 0.3 m12 = 7.59 10 eV ν ∆ 2 × -3 2 P( 0.2 m23 = 2.43 10 eV Near Detector L = 200 m 0.1 Far Detector L = 1.05 km 0 1 10 102 103 104 105 L [m]

Figure 2-4: The survival probability P (¯νe ν¯e) for a (3 + 1) oscillation model. The values chosen for the standard five parameters→ (three mixing angles and two squares mass differences) are from [38, 3], while the remaining four parameters (three mixing angles and one squared mass difference) were chosen to agree with the best-fit results within a (3 + 1) model from [1, 2]. 18 CHAPTER 2. THEORY OF NEUTRINO OSCILLATIONS Chapter 3

Double Chooz

Double Chooz is a reactor electron antineutrino disappearance experiment to run near the Chooz power plant in France. It aims to measure the neutrino mixing angle

θ13, a basic parameter in the standard theory of neutrino mixing. Over three years of 2 data collection, Double Chooz should be able to explore the range of sin 2θ13 from 0.2 to 0.03. To better model the ﬂux of reactor neutrinos and reduce the systematic errors, data collection will proceed in two stages: stage 1 will utilize a far detector at 1.05 km from the reactor (in the existing Chooz experiment hall) and stage 2 will employ an additional identical near detector at 200 m. In the standard theory of three neutrinos, the probability ofν ¯e oscillation P (¯νe ν¯x) is very small at 200 m, → while it becomes nearly maximal at 1 km (see ﬁgure 2-3). Therefore comparing the two datasets allows for cancellation of systematic errors producing a more precise measurement of θ13 [3].

3.1 Detection Concept

The detector design is optimized to detect reactorν ¯e with very high eﬃciency. The detector consists mostly of Gadolinium-loaded liquid scintillator (LS) surrounded by photomultiplier tubes (PMTs), which covert the scintillation light into an electrical signal. Inverse beta decay (IBD), in which aν ¯e interacts with a proton to produce a 20 CHAPTER 3. DOUBLE CHOOZ

positron and a neutron, is the main detection reaction[3].

+ ν¯e + p e + n (3.1) →

This process, represented by the Feynman diagram in ﬁgure 3-1, has an energy thresh-

old of Eν > 8.5 MeV. The positron scintillates until it annihilates with an electron ∼ to produce two 0.511 MeV γ’s. This is known as the prompt signal. The neutron thermalizes (i.e. loses kinetic energy through interactions with its environment) and captures on the Gadolinium (which has an exceptionally high thermal neutron cross section). This leads to an excited state of the Gadolinium, which then generates a cascade of γ-rays of total energy 8 MeV. On average, the neutron capture occurs ∼ 30 µs after the IBD event, so this is called the delayed signal. ∼

n + Gd Gd∗ → ↓ X Gd + γi i

The coincidence of the prompt and delayed signals is the signature of aν ¯e in Double Chooz [46].

νe e

u d d ¡ d u u Figure 3-1: Feynman diagram of inverse beta decay. The time axis points left-to-right. 3.2. MULTI-VOLUME DETECTORS 21

3.2 Multi-volume Detectors

The Double Chooz experiment set the current trend in θ13 experimental design: using several, multi-volume (layered), cylindrical, small detectors [47]. The Double Chooz far detector is displayed in ﬁgure 3-2. Target and γ-catcher

The far detector will consist of a target cylinder with a volume of 10.3 m3. This volume is filled with liquid scintillator loaded with gadolinium (Gd-LS) at a concen- tration of approximately 1 g/` . Surrounding this is a γ-catcher volume (21.5 m3), which also contains LS (but no Gd). This scintillating buffer is necessary to effectively measure the γ-rays which constitute both the prompt and delayed signals from

ν¯e events. Each of these volumes is contained in an acrylic shell, known as a vessel. The vessels will be transparent to photons with wavelengths above 400 nm, tightly contain the LS in both regions without leaks for 10 years, and be chemically compatible with the LS for at least 5 years. The last of these is the strongest constraint since no modiﬁcation of the liquid properties of the LS nor any degradation of the acrylic material can be tolerated during the data collection. Nonscintillating buﬀer and PMT support

A nonscintillating buffer encapsulates the double vessel (target and γ-catcher). This buffer, a 105 cm thick region of nonscintillating mineral oil (114.2 m3), serves to decrease the level of accidental background (mainly from PMT radioactivity). A final vessel of 3 mm thick stainless steel sheets and stiffeners surrounds this nonscintillating buffer. A total of 390 (10 inch) PMTs are mounted in a uniform array on the interior surface of the buffer vessel. Inner and outer veto systems

The inner veto is a 50 cm thick region filled with about 80 m3 of LS for both the near and far detectors. Another stainless steel vessel (thickness 10 mm) houses the inner veto region and supports an additional 78 PMTs. At the far detector, the outer veto will be an active proportional tube tracking system that will identify and locate “near-miss” cosmogenic muons. At the near detector, the outer veto system will 22 CHAPTER 3. DOUBLE CHOOZ consist of plastic scintillator strips with x/y measurement capabilities. The reason for using different outer veto systems lies in the fact that the far and near detector sites have different overburdens: the far site has about 300 meters-water-equivalent (m.w.e.), whereas the near site has about 114 m.w.e. Shielding at the far site will consist of 15 cm steel (weighing approximately 300 tons).

Figure 3-2: The Double Chooz far detector.

To summarize, the Double Chooz experiment is equipped to detect the unique signal ofν ¯e IBD events with very high eﬃciency ( 61% at the far detector and ∼ 44% at the near detector) using multi-volume detectors [3]. Additionally, two ∼ identical detectors will be deployed at diﬀerent baselines from the Chooz reactor (described in detail in the following chapter) to minimize the systematic error on the

ﬁnal measurement of θ13. Chapter 4

Nuclear Reactors

Nuclear reactors are constant, copious sources of electron antineutrinos up to energies of about 8 MeV. The high flux and low systematic uncertainty associated with the absolute flux (< 2%) make reactors optimal for precision oscillation tests [3]. The ∼ production ofν ¯e’s is actually the result of a chain of nuclear reactions. In general, reactors are fueled by a highly radiocative isotope, which undergoes fission. Fission is the process by which a heavy nucleus of an element is split into two lighter nuclei, often releasing free neutrons or photons. These lighter nuclei, called fission products, undergo other nuclear reactions during which electron antineutrinos are eventually produced. It is the beta decay of the daughters of these fission products that lead to electron antineutrinos.

− n p + e +ν ¯e (4.1) →

The Chooz nuclear power station consists of two twin pressurized-water (PWR) reactors of the most recent generation (N4). The core of each reactor is made up of 205 fuel assemblies. The vessel is ﬁlled with pressurized water (p = 155 bars) at temperature ranging from 280oC to 320oC. The water acts as a neutron moderator and cooling element [35]. 24 CHAPTER 4. NUCLEAR REACTORS

4.1 Reactor Neutrino Production

Electron antineutrinos are produced by the two nuclear reactors located at Chooz-B nuclear power station operated by the French company Electricitéde France (EdF) in partnership with the Belgian utilities Electrabel S.A./N.V. and SociétéPublique d’Electricitécores.

As mentioned, each N4 PWR has a thermal power of 4.27 GWth, a downtime of about one month per year, and a total of 205 fuel assemblies. Each fuel assembly is a square lattice structure of completed, pressurized fuel rods, which contain enriched uranium dioxide UO2 encased in Ziracaloy tubes about 4 m long [48]. The fuel is mainly composed of the nonfissile, abundant isotope 238U, which is enriched with 5% of the fissile isotope 235U. During the burn-up process the U isotopes breed ∼ 239Pu, and 241Pu. Three of these, 235U, 239Pu, and 241Pu, may undergo fission with just thermal neutrons, while 238U is only fissionable by fast neutrons. On average each fission releases 6ν ¯e’s, which, upon reaching the detector and being detected, typically have energies between 2 and 8 MeV1. Together, both reactors emit about

21 −1 10 ν¯es isotropically at full power [49].

Since each branch (full decay chain leading toν ¯e) of each isotope has a diﬀerent

ν¯e spectrum, the actual spectrum is a superposition of the diﬀerent possible spectra.

We use the “conversion” method to determine theν ¯e spectrum, which is based on the i fact that the antineutrino and the electron share the available energy E0 (also known as the endpoint energy for the ith branch) [50]. When possible, the β spectrum is determined experimentally. This is the case for 235U, 239Pu, and 241Pu, while Monte Carlo simulation is used for the β spectrum of 238U.

X i i ¯ Sβ(Eβ) = aiSβ(Eβ,E0, Z) (4.2) i

th i where ai is the amplitude of the i branch, and Sβ is the diﬀerential electron yield

1This includes the fact that in order to be detected the energy of the neutrino must be above the inverse beta decay energy threshold 4.2. FISSION RATES AND DRAGON 25

Figure 4-1: Relative ﬁssion contribution for the four dominant isotopes that produce ν¯e as a function of reactor operating time [40].

i per ﬁssion, which depends on the energy of the electron Eβ, the endpoint energy E0, ¯ and the average charge Z of the β-decaying nuclei. From this, theν ¯e spectrum can be constructed. X i i i ¯ Sν(Eν) = aiSβ(E0 Eβ,E0, Z) (4.3) i −

On the other hand, it has been found that we may parametrize Sν on an isotope- by-isotope basis. Thus, theν ¯e spectrum (diﬀerential neutrino yield per ﬁssion) from each isotope ` is " 2 # X ` k S`(Eν) = exp akEν (4.4) k=0

` where k is the power of Eν and the coeﬃcients ak can be found in reference [51].

4.2 Fission Rates and DRAGON

In order to proceed we must be able to calculate the fission rates f`(t), which are the number of fissions per second for the isotope `. In general, these fission rates vary as a function of time, since the quantity of each isotopes is changing in time. However, since these fission rates vary slowly, we may take f`(t) = f`(0) to first order. 26 CHAPTER 4. NUCLEAR REACTORS

The calculation of the fission rates can be done using a simulation of the reactor implemented in a lattice code called DRAGON, written by Ecole´ Polytechnique de Montréal. DRAGON simulates the neutron behavior of a unit cell or fuel assembly in a nuclear reactor [52]. In particular, DRAGON models the neutron transport equation, including neutron leakage, transport-transport, and transport-diffusion, in a user-defined geometry. We use the two-dimensional geometry, given in figure 4-2(a), in which each square cell represents a fuel assembly 16 16 structure of fuel rods. A × schematic of a fuel assembly, as represented in DRAGON, is shown in figure 4-2(b).

(a) The nuclear reactor core map. Each square cell (b) 16 16 fuel assembly in represents a 16 16 fuel assembly. DRAGON× × Figure 4-2: Two-dimensional geometry of the nuclear reactor.

After simulating the reactor in DRAGON, the ﬁssion rates f`(t) over a long pe-

riod of time can be calculated. Multiplying by the neutrinos per ﬁssion S`(Eν) and summing over the isotopes gives a double diﬀerential for the number of electron an-

tineutrinos produced Nν. 2 d Nν X = f`(t)S`(Eν) (4.5) dEνdt `

Noting as before that we can take f`(t) f` to be constant over a short period of ≡ time ∆t, we can calculate the shape of the expected reactor antineutrino spectrum at the detector by taking into account the probability that an antineutrino will interact 4.2. FISSION RATES AND DRAGON 27 at the detector,

Z ∆t dNν X = dt f`(t)S`(Eν)σ(Eν) dEν 0 ` X f`S`(Eν)σ(Eν) (4.6) ∝ ` where σ(Eν) is the cross section for inverse beta decay as a function of the energy of the electron antineutrino. In the low energy limit, the cross section for inverse beta decay can be written as a function of the energy Ee and the momentum pe of the outgoing positron [35]. 2π2~3 σ(Ee) = 5 7 peEe (4.7) mec fτn where the free neutron decay phase-space factor is f = 1.71465(15) [53] and the neutron lifetime is τn = 885.7(8) s [42]. We recall that the energy of the neutrino and the positron are related by

Eν = Ee + (Mn Mp) + (Eν/Mn) (4.8) − O where Mn and Mp are the masses of the neutron and proton respectively. Thus we may rewrite Eq. 4.6 in terms of the observable positron energy Ee as

dNν X f`S`(Eν)σ(Ee) (4.9) dEe ∝ ` where the Eν = Eν(Ee) is an implicit function of the positron energy Ee. Though this equation does not give us an accurate prediction for the total number of detected antineutrinos Nν, it does accurately predict the shape of the antineutrino spectrum at the detector. Figure 4-3 shows theν ¯e reactor neutrino spectrum of Eq. 4.9, normalized to one for two different times, t = 0 and t = 1015 days (in blue and red, respectively)2. The small difference in shape between these two spectra is justification for using f`(t) = f` constant fission rates to determine the neutrino spectrum shape for all time.

21000 days is characteristic of the longest continuous running time of a reactor 28 CHAPTER 4. NUCLEAR REACTORS

0.016

0.014 t = 0 days t = 1015 days 0.012

0.01

0.008

Fraction 0.006

0.004

0.002

0 2 4 6 8 Ee [MeV]

(a) Theν ¯e reactor spectrum at t = 0 days (in blue) calculated using Eq. 4.9 and DRAGON as a function of Ee and normalized to unity. The normalized reactor spectrum at t = 1015 days (in red) is plotted as well.

1.4

1.2

0.8

0.6

Normalized Ratio 0.4

0.2

0 2 4 6 8 Ee [MeV]

(b) The ratio ofν ¯e reactor spectra at t = 0 days and t = 1015 days. Horizontal line at 1 is plotted (black dotted line).

Figure 4-3: Theν ¯e reactor spectrum shape and ratio (t = 0 days)/(t = 1015 days) Chapter 5

The Likelihood Model

In general, the likelihood model tests the agreement between theory and experiment. Given a set of very large set of measurements (say N total) of a certain observable x it is often convenient to bin the values into a histogram. In other words, we can group the observations into frequencies of identical observations. If there are

m distinguishable values of x, denoted x1, x2, ..., xm, then one obtains a vector of

“binned” counts ~n = (n1, n2, .., nm). Each ni (also known as the bin content) is the

number of measurements that yielded x xi. So when one sums up all the bin ' contents, one gets N, the total number of measurements.

m X ni = N (5.1) i

N is also known as the normalization of the histogram of x. The standard deviation,

or the “error bar,” σi characterizes the uncertainty in the observed frequency ni. If the distributions of frequency measurements can be approximated as Poisson, then

we may use σi = √ni as the error [54]. The probability density function (p.d.f.) for an observable x is a nonnegative function which describes the probability of measuring a certain value (or range of values) ~ of the observable. It usually depends on unknown parameters θ = (θ1, θ2, ..., θp). The 30 CHAPTER 5. THE LIKELIHOOD MODEL

deﬁning property of the p.d.f f(x θ~) is how it connects to the physical probability. | Z b ~ ~ P (a 6 x 6 b θ) = f(x θ)dx (5.2) | a | where P (a x b θ~) is the probability of measuring x [a, b] given the parameters 6 6 | ∈ θ~.

Note that Eq. 5.2 implies that the p.d.f must be normalized to unity since the observable x must have a deﬁnite real value (that is, with probability 1).

Z ∞ f(x θ~)dx = 1 (5.3) −∞ |

An extended p.d.f F (x, θ~) is one that has an intrinsic expected normalization Nˆ (as part of the model parameters θ~). We may write the expectation values for the number ~ ˆ ~ of events in each bin, based on the model parameters, asn ˆi = F (xi, θ) = Nf(xi θ). |

5.1 The Method of Least Squares

We may deﬁne a function χ2(θ~) of the parameters, which quantiﬁes the level agreement between the extended p.d.f and the data.

m 2 X ni nˆi χ2(θ~) = − (5.4) σ i=1 i m ˆ ~ !2 X ni Nf(xi θ) = − | (5.5) σ i=1 i

The method of least squares proceeds by varying the set of parameters θ~ to achieve ~ a minimum. The least squares (LS) estimators for the parameters, denoted by θˆ, are determined by the minimum χ2.

∂χ2 = 0 , k = 1, 2, .., p (5.6) ∂θk 5.2. CONSTRUCTING THE P.D.F. FOR DNν/DEE 31

For good agreement between data and theory, the average spread between the data and the model should be about equal to the expected spread (the average error on a data point). In fact, the expectation value for the χ2 is

χ2 = m p ν (5.7) h i − ≡ which is the number of degrees of freedom (d.o.f.). In other words for a good ﬁt the normalized χ2/d.o.f. should be about equal to one.

The χ2, being a measured term itself, has a probability density distribution for ν degrees of freedom given by

1 2 f (χ2; ν) = χ2(ν−2)/2 e−χ /2 (5.8) χ 2ν/2Γ(ν/2)

The p-value, deﬁned as the probability of observing a value of χ2 that is larger than the particular value for the random sample of N observations with ν = m p degrees − of freedom is given by the following integral.

Z ∞ 2 pχ(χ ; ν) = fχ(z; ν)dz (5.9) χ2

For a more thorough treatment of maximum likelihood and least squares methods see Refs. [55] [54].

5.2 Constructing the P.D.F. for dNν/dEe

In order to determine the p.d.f. for the observable positron energy Ee, we must ﬁrst make an assumption about the parent distribution. We will proceed by positing two separate hypotheses and constructing the associated p.d.f.’s. The null hypothesis makes only the most general assumptions about the shape and normalization of the model. Speciﬁcally, the null hypothesis assumes the data (binned dNν/dEe counts) are the result of neutrinos propagating to the detector from a reactor at a distance 32 CHAPTER 5. THE LIKELIHOOD MODEL

Measured Terms Description Nν Measured number ofν ¯e signal events Oscillation Hypothesis Parameter θ~ (osc) Nˆν Expected number of events 2 sin\(2θee) ν¯e ν¯e survival mixing parameter 2 → ∆dm14 Squared mass diﬀerence between mass eigenstates Null Hypothesis Parameter θ~ (null) Nˆν Expected number of events

L with absolutely no oscillations. Conversely, the (3 + 1) neutrino oscillation hypothesis, or simply the oscillation hypothesis, assumes that the data are the result of neutrinos which may oscillate to other undetectable ﬂavors, predominantly due to

2 the large ∆m14 mass splitting, before interacting in the detector. That is, the oscil- 2 2 lation hypothesis assumes that parameters sin (2θee) and ∆m14, from Eq. 2.11, are non-vanishing.

Table 5.2 delineates the measured terms and the diﬀerent sets of parameters under the two hypotheses.

5.2.1 Null Hypothesis

In order to determine the extended p.d.f. for Ee we ﬁrst determine the reactor neutrino

spectrum shape from Eq. 4.9 as a function of positron energy Ee and normalize it to 1.

P (null) ` f`S`(Ee)σ(Ee) f (Ee) = R ∞ P (5.10) 0 dEe ` f`S`(Ee)σ(Ee) Then we simply multiply the p.d.f. in Eq. 5.10 by the proper expected number of (null) events, Nν to get the extended p.d.f.

(null) (null) (null) (null) F (Ee N ) = N f (Ee) (5.11) | ν ν 5.2. CONSTRUCTING THE P.D.F. FOR DNν/DEE 33

5.2.2 (3 + 1) Neutrino Oscillation Hypothesis

The theoretical model outlined in Ch. 2 is the basis for constructing the p.d.f for the number ofν ¯e events observed at a detector from a reactor.

We can construct the oscillation hypothesis p.d.f. by multiplying the null hypothe-

2 2 sis p.d.f. in Eq. 5.10 by theν ¯e survival probability P (¯νe ν¯e) P (Eν, sin (2θee), ∆m ,L) → ≡ 14 in Eq. 2.11 and normalizing,

P (E , sin2(2θ ), ∆m2 ,L) P f (t)S (E )σ(E ) f (osc)(E sin2(2θ ), ∆m2 ) = ν ee 14 ` ` ` ν e e ee 14 R ∞ 2 2 P | 0 dEeP (Eν, sin (2θee), ∆m14,L) ` f`(t)S`(Eν)σ(Ee) (5.12) (osc) Next, we scale by the proper expected normalization Nν , to get the extended p.d.f

(osc) (osc) 2 2 (osc) (osc) 2 2 F (Ee N , sin (2θee), ∆m ) = N f (Ee sin (2θee), ∆m ) (5.13) | ν 14 ν | 14

Further, given the expected number events in the absence of (3 + 1) oscilla- (null) tions, Nν , we can “oscillate away” events according to the survival probability 2 (osc) P (Eν, sin (2θee) to calculate Nν . In other words, the fraction

(osc) Z ∞ Nν 2 2 (null) (null) = dEeP (Eν, sin (2θee), ∆m14,L)f (Ee) < 1 (5.14) Nν 0 can be determined and it is always less than 1. This makes perfect physical sense since we expect to see a deﬁcit ofν ¯e events in oscillation hypothesis.

Note that L, the distance from the reactor source to the detector, has the ability to change the shape of the oscillation spectrum (in Eq. 5.12) and the total normalization (in Eq. 5.14). 34 CHAPTER 5. THE LIKELIHOOD MODEL Chapter 6

Sensitivity at Double Chooz

The Double Chooz near detector, located at 200 m from the reactor neutrino source may be sensitive to the signal from the heavy fourth neutrino (embodied in the

oscillation formula P (¯νe ν¯e)). This signal may be manifest in two distinct ways: → (1) changing the total number ofν ¯e events and (2) changing the shape of theν ¯e energy spectrum.

For effect (1) to be definite, the deficit ofν ¯e events must be greater than the uncertainty on the reactor flux, which is 2%. In other words, The deficit 1 ∼ − (osc) (null) Nν /Nν must be greater than 0.02. Effect (2) is a more promising signal, because no matter how much the overall normalization floats it will never fit a drastically different shape.

As an added detector eﬀect, each p.d.f is convoluted with a Gaussian of standard

1 deviation σE = 0.1 MeV, to simulate the detector energy resolution .

 −E2  Z exp( 2σ2 ) (α) ~ (α) ~ E (Ee θ(α)) = F (Ee E θ(α))   dE (6.1) F | − | · σE√2π where (α) = (null) or (osc). The effect of this “Gaussian smearing” is to smooth out some of the subtle shape differences, which would be undetectable due to the finite

1This is a worst case energy resolution, as Double Chooz expects to resolve ﬁner energies than this [3] 36 CHAPTER 6. SENSITIVITY AT DOUBLE CHOOZ energy resolution of a real detector.

6.1 Generating Data

To generate realistic data based on the oscillation hypothesis we first need to set the “truth” values of the oscillation parameters. Reference [1] performs a global fit over short baseline antineutrino experiments in order to estimate various parameters of (3 + 1) and (3 + 2) oscillation models. Specifically, we take the best-fit values of

2 2 2 sin (2θee) = 0.043 and ∆m14 = 0.91 eV , extracted from a ﬁt of antineutrino short ∗ baseline experiments assuming a unitary mixing matrix Uαj for a (3 + 1) model.

(null) Next, we can ﬁnd the expected value of the parameter Nν at L = 200 m. The

Double Chooz proposal expects the integrated rate ofν ¯e events at the near detector to be 161,260 y−1, including detector eﬃciency, dead time, and reactor oﬀ periods averaged over a year[3].

Given the expected values of all the parameters, we may generate the correct (null) number of events corresponding to a full Double Chooz run of 5 years: Nν = (osc) 806300 and Nν = 788929. This data generation was done using a Poisson process in RooFit. Figure 6-1 displays the null hypothesis p.d.f. and the oscillation hypothesis p.d.f. as well as the data generated from the oscillation hypothesis.

6.2 Fitting with χ2 Minimization

The χ2 function in Eq. 5.5 was minimized with respect to the parameters over 25

2 bins of Ee using Poisson errors for each bin . In the oscillation hypothesis fit there were three adjustable parameter: the overall normalization, the mixing amplitude, ~ (osc) 2 2 and the squared mass difference, θ(osc) = (Nν , sin (2θee), ∆m14). In the null (no oscillation) hypothesis fit, only one parameter was considered adjustable: the overall ~ (null) normalization θ(null) = Nν

2We used ROOT 5.26/00 [56] and RooFit v2-00-05 [57] with MINIUT to construct the p.d.f.’s and χ2 as well as minimize the χ2 6.2. FITTING WITH χ2 MINIMIZATION 37

60000

50000

40000

30000 / (0.268 MeV) ν

N 20000

10000

0 2 4 6 8 Ee [MeV]

Figure 6-1: The number of events per bin width for 25 bins of of positron energy Ee expected (before ﬁtting) for the null hypothesis (blue) and the oscillation hypothesis (red). The data generated from the oscillation hypothesis is also shown (black data points).

Figure 6-2 and table 6.1 show the fit results for both the null hypothesis and the oscillation hypothesis. The null hypothesis p.d.f. has floated down to better fit the generated data. However, due to the slight shape differences that survive the Gaussian smearing, the χ2/d.o.f. for the null hypothesis is still 1.8 times greater than the χ2/d.o.f. for the oscillation hypothesis. A particularly significant measure is the

2(osc) p-value for each ﬁt. The oscillation hypothesis has a minimum of χ min = 23.5 and (osc) 2(null) a p-value of p = 0.37, while the the null hypothesis has a minimum of χ min = 46.1 and a p(null) = 0.0043. The small p-value of the null hypothesis is considered statistically signiﬁcant and suggests that the null hypothesis is incompatible with the generated dataset [54]. 38 CHAPTER 6. SENSITIVITY AT DOUBLE CHOOZ

2 θ ± sin (2 ee) = 0.037 0.007 60000 ∆ 2 ± 2 m14 = 0.909 0.003 eV 50000 L = 200 m

40000

30000 / (0.268 MeV) ν

N 20000

10000

0 2 4 6 8 Ee [MeV]

Figure 6-2: The black points represent the generated data (based on the oscillation hypothesis). The red histogram represents the Gaussian-smeared prediction based on oscillation hypothesis. The blue histogram represents the Gaussian- smeared prediction based on null hypothesis. The ﬁtted parameters have values of 2 \2 2 2(osc) sin (2θee) = 0.037 0.007 and ∆dm14 = 0.909 0.003 eV , resulting in a χ min = 23.5 ±(osc) ± 2(null) and a p-value of p = 0.37. The null hypothesis ﬁt results in χ min = 46.1 and a p(null) = 0.0043

6.3 ∆χ2 Conﬁdence Levels

To find the confidence levels for our multiparameter fit, we minimize the χ2 func-

2 2 (osc) tion, keeping sin (2θee) and ∆m14 ﬁxed at particular values, but allowing Nν to vary during the minimization. Thus we are plotting the variation in ∆χ2 from the

2 2 2 minimum χmin. As shown in [54], the variation in ∆χ obeys the χ probability distribution for 1 degree of freedom. Recalling that the χ2 probability density distribution in Eq. 5.8 and setting the degrees of freedom ν = 1, we may calculate the p-value using Eq. 5.8. Putting this all together, the probability that a certain region of ∆χ2 covers the “true” parameter values is given by 1 p. − For example, since χ2/(ν = 1) 1 corresponds to 31.7% of the probability, our ≥ 6.3. ∆χ2 CONFIDENCE LEVELS 39

Parameter Truth Value Nν 788927 2 sin (2θee) 0.043 2 2 ∆m14 0.91eV Oscillation Fit Parameter Fitted Value Initial Value (osc) Nˆν 788066 892 788929 2 ± sin\(2θee) 0.037 0.007 0.043 2 ± 2 2 ∆dm14 0.909 0.003 eV 0.91 eV χ2/d.o.f = 1.068 ± pχ = 0.37 No Oscillation Fit Parameter Fitted Value Initial Value (null) Nˆν 787674 887 806300 χ2/d.o.f = 1.922 ± pχ = 0.0043 Table 6.1: This table summarizes the pre-fit and post-fit parameter values as well as the goodness of fit χ2 and p-values for L = 200 m.

∆χ2 < 1 corresponds to 68.3%. This ∆χ2 = 1 defines the 1σ confidence level contour, which represents a 68.3% probability of enclosing the the true parameter values. Similarly the confidence level contour for 2σ corresponds to the ∆χ2 = 4 contour (95.5% probability of enclosing the the true parameter values) and 3σ corresponds to ∆χ2 = 9 contour (99.7% probability of enclosing the the true parameter values). Figure 6-3 displays the 1σ, 2σ, and 3σ confidence level contours in the parameter

2 2 space of (sin (2θee), ∆m14) for the oscillation hypothesis. As one can see, the 2σ contour readily covers the true value of the parameters. 40 CHAPTER 6. SENSITIVITY AT DOUBLE CHOOZ

0.93

0.925

0.92 ] 2 0.915

[eV 0.91 2 14 m

∆ 0.905

0.9

0.895

0.89 0 0.02 0.04 0.06 2 θ sin (2 ee)

Figure 6-3: The allowed 1σ, 2σ, and 3σ confidence levels regions for the χ2 fit in 2 2 2 figure 6-2. The black point represents the best fit χ minimum (sin\(2θee), ∆dm14) = 2 2 2 (0.037, 0.909 eV ). The star represents the truth (initial) value (sin (2θee), ∆m14) = (0.043, 0.91 eV2). Chapter 7

Sensitivity at a Very Near Detector

The results of the previous chapter indicate that the fitted number of events for the null hypothesis N (null) dropped by 2.3% (with respect to the expected number of events for the null hypothesis) in order to fit the generated data based on the (3 + 1) oscillation hypothesis. However, this parameter is still within the accepted range given the 2% uncertainty on the flux. ∼

However, it can be seen from figure 2-4, that the first minimum of theν ¯e survival probability for a monoenergetic neutrino source with Eν = 2 MeV in a (3 + 1) model occurs near L 3 m. In fact with the reactor antineutrino energy spectrum shown in ∼ figure 4-3, the minimum occurs at approximately L = 6 m from the reactor. Thus if we place a Double Chooz-like detector at the “optimal” distance L = 6 m, and repeat the analysis of Ch. 6, we will see an even greater discrepancy between the oscillation hypothesis and null hypothesis reactor neutrino spectra.

There are many inherent diﬃculties in placing a detector very close to a nuclear reactor with such a high event rate. However, one experiment at the San Onofre Nuclear Generating Station (SONGS) shows that a detector at L = 25 m is possible [58]. The antineutrino ﬂux at the SONGS detector was 1017 m−2 s−1. To reduce the extremely high muon-induced background rate, a muon veto time of tµ = 100 µs was used to rejectν ¯e IBD candidate events that occur within a 100 µs window of a muon hit. The detector trigger rate above a 1 MeV threshold and the trigger rate for 42 CHAPTER 7. SENSITIVITY AT A VERY NEAR DETECTOR

the muon veto system were both 500 Hz. Using a similar muon veto time at our ∼ proposed detector at L = 6 m may achieve similar results in reducing background events.

Since the expected rate of events scales as 1/L2, the flux is much larger at this short distance. Therefore, we may acquire data over a much shorter period to achieve the same high statistics expected at the Double Chooz near detector for a 5 year (null) (osc) data set. Suppose we still expect Nν = 806300 and Nν = 776476. Then, figure 7-1 shows the pre-fit expected distributions for the number of events per energy bin width for 25 bins of Ee in the null hypothesis and the oscillation hypothesis at 6 m.

60000

50000

40000

30000 / (0.268 MeV) ν

N 20000

10000

0 2 4 6 8 Ee [MeV]

Figure 7-1: The number of events per bin width for 25 bins of of positron energy Ee expected (before ﬁtting) for the null hypothesis (blue) and the oscillation hypothesis (red). The data generated from the oscillation hypothesis is also shown (black data points).

Figure 7-2 displays the distributions after minimizing the χ2 and table 7.1 summarizes the results of the ﬁt, including extracted parameter values. Finally ﬁgure 7-3(a)

2 2 displays the allowed 1σ, 2σ, and 3σ conﬁdence level regions in the (sin (2θee), ∆m14) parameter space. 43

Parameter Truth Value Nν 776474 2 sin (2θee) 0.043 2 2 ∆m14 0.91eV Oscillation Fit Parameter Fitted Value Initial Value ˆ (osc) Nν 775269 880 776476 2 ± sin\(2θee) 0.039 0.008 0.043 2 ± 2 2 ∆dm14 0.839 0.061 eV 0.91 eV χ2/d.o.f = 0.782 ± pχ = 0.75 No Oscillation Fit Parameter Fitted Value Initial Value ˆ (null) Nν 775240 880 806300 χ2/d.o.f = 1.938 ± pχ = 0.0038 Table 7.1: This table summarizes the pre-fit and post-fit parameter values as well as the goodness of fit χ2 and p-values for L = 6 m.

After minimization, the fitted normalization for the null oscillation hypothesis drops by a full 3.9% with respect to the expected normalization. Furthermore, the null oscillation fit yields a χ2/d.o.f. of 2.5 times the χ2/d.o.f. from the oscillation hypothesis fit. Under these conditions, there is an even greater difference in the p-

2(osc) values for each ﬁt. The oscillation hypothesis has a minimum of χ min = 23.5 and a (osc) 2(null) p-value of p = 0.75, while the The null hypothesis has a minimum of χ min = 46.1 and a p(null) = 0.0038.

This wide discrepancy implies that under the right circumstances, our very near detector would be able to identify the (3 + 1) model as more compatible with reality than the null hypothesis. As this example shows, the (3+1) model is strongly favored when the difference between the number of expected events in the null hypothesis and the number of measured events differ by much more than 2% (the uncertainty in the flux) 44 CHAPTER 7. SENSITIVITY AT A VERY NEAR DETECTOR

2 θ ± sin (2 ee) = 0.039 0.008 60000 ∆ 2 ± 2 m14 = 0.839 0.061 eV

50000 L = 6 m

40000

30000 / (0.268 MeV) ν

N 20000

10000

0 2 4 6 8 Ee [MeV]

Figure 7-2: The black points represent the generated data (based on the oscillation hypothesis). The red histogram represents the Gaussian-smeared prediction based on oscillation hypothesis. The blue histogram represents the Gaussian- smeared prediction based on null hypothesis. The ﬁtted parameters have values of 2 2 2 sin\(2θee) = 0.039 0.008 and ∆dm = 0.839 0.061 eV . ± 14 ± 45

0.9

0.8 ] 2 0.7 [eV 2 14 0.6 m ∆ 0.5

0.4

0.3 0 0.05 0.1 2 θ sin (2 ee) (a) The allowed 1σ, 2σ, and 3σ confidence level regions for the χ2 fit in figure 7-2. The black point represents the best fit χ2 minimum 2 \2 2 (sin (2θee), ∆dm14) = (0.039, 0.839) eV ). The star represents the 2 2 2 truth (initial) value (sin (2θee), ∆m14) = (0.043, 0.91 eV ).

1 0.9 0.8 0.7 ] 2 0.6

[eV 0.5 2 14

m 0.4 ∆ 0.3 0.2 0.1 0 0 0.5 1 2 θ sin (2 ee) (b) A wider view of the allowed parameter space in ﬁgure 7-3(a). Note that the 3σ contour extends to the end of the physical region.

2 2 Figure 7-3: Allowed 1σ, 2σ, 3σ conﬁdence levels regions in (sin (2θee), ∆m ) parameter space at 6 m. 46 CHAPTER 7. SENSITIVITY AT A VERY NEAR DETECTOR Chapter 8

Outlook and Conclusions

In this thesis, we examined the consequences of a plausible (3 + 1) oscillation model at Double Chooz as well as a hypothetical very short baseline detector. We take into account much of the existingν ¯ data (but not the ν data) at short baselines in choosing

2 the truth values of theν ¯e oscillation survival amplitude, sin (2θee) = 0.043, and the 2 2 relevant squared mass difference, ∆m14 = 0.91 eV . Using this oscillation hypothesis we generated 5 years of data at the Double Chooz near detector distance of 200 m. Then using the method of least-squares to minimize a χ2 function, we compared the fit results of the oscillation hypothesis and the null (or no oscillation) hypothesis. We concluded that the oscillation hypothesis fit both the rate and shape significantly better. Further, the p-value, or the probability of obtaining a χ2 at least as extreme as the one that was observed, for the null hypothesis is extremely small at 0.0043, which strongly suggests rejecting the null hypothesis. However, the normalization of the (null) null hypothesis Nν floated down by only 2.3% in order to fit the data. This is just within the bounds of the known error on the reactor flux ( 2%), so these conditions ∼ may not give us a conclusive signal of a heavy fourth neutrino at the Double Chooz near detector.

(null) We repeated this analysis at the more optimal distance of 6 m. This time, Nν dropped 3.9% in order to fit the data. This is nearly twice the value of the flux uncertainty. This effect coupled with the even smaller null hypothesis p-value of 48 CHAPTER 8. OUTLOOK AND CONCLUSIONS

0.0038, may be enough to conclusively determine the existence of a heavy fourth neutrino. The added bonus of using a detector at such a short baseline is the extremely short period of data collecting necessary to reach this conﬁdence level.

Though, to first approximation, these results are accurate and demonstrate that Double Chooz shows some promise in its sensitivity to “early oscillations,” several effects may limit Double Chooz’s ultimate sensitivity (and that of the very near detector). Firstly, at the very short baseline detector, the extremely high background rates would have to be tempered with corresponding selection cuts (such as the muon time veto discussed in Ch. 7) on theν ¯e interaction candidate sample. Luckily, though, 18 2 −1 the extremely high flux coming from the reactor at this distance, > 10 ν¯e m s , ∼ ensures that we would not be statistics-limited.

One approximation that may be rectiﬁed in future analysis is accounting for the

burn-up aﬀect in the reactor, embodied in the time-varying ﬁssion rates f`(t). Another condition currently neglected (at both L = 200 and L = 6 m) is that of the detector’s dimensions. To correct for this, we would need to average the oscillation probability

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