Study Of Non-Standard Interaction Effects On The Solar Neutrino Day-Night Asymmetry

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Master Thesis

Study Of Non-Standard Interaction Effects On The Solar Neutrino Day-Night

Asymmetry

Cl´ ementine Vulliet

Theoretical Particle Physics, Department of Theoretical Physics, School of Engineering Sciences

Royal Institute of Technology, SE-106 91 Stockholm, Sweden

Stockholm, Sweden 2015

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Typeset in L

^A

TEX

TRITA-FYS-2015:12 ISSN 0280-316X

ISRN KTH/FYS/--15:12-SE Cl´ementine Vulliet, March 2015 c

Printed in Sweden by Universitetsservice US AB, Stockholm March 2015

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Abstract

Some years ago, SNO and KamLAND gave evidence for neutrino oscillations, which requires neutrinos to be massive. Neutrino oscillations occur both in vacuum and in matter. In the latter case may the effective potential affect the oscillation amplitude and increase the flavor transition probability. An interesting consequence of the matter effects is the day-night asymmetry of solar neutrinos, i.e., the difference in the neutrino rate measured during the night compared to the day. The Standard Model has to be extended in order to explain the generation of neutrino mass, and so-called non-standard interactions (NSI) naturally arise in many of these non- standard extension theories. In addition, some experimental neutrino anomalies are inconsistent with Standard Model expectations and could therefore be a hint for new physics. The aim of this work is to study the effects of NSI on the day-night asymmetry. We estimate the error in the asymmetry due to the uncertainty on the neutrino mixing parameters in the three flavor framework. Then, we build a two-flavor model which takes into account the dominant NSI in the solar neutrino sector, and we derive the expression for the day-night asymmetry in this model.

Finally, we study the impact of NSI on the asymmetry in a two step approach.

First, all the parameters are varied simultaneously in one set of computations, and secondly, each NSI parameter is varied individually when the others are set to zero. The results of the first step are mixed correlations of the NSI parameters with the asymmetry, but the results from the second step clarify the impact of each parameter. New limits can thus be set on the NSI parameters studied in the model, in order to satisfy the experimental measurements of the day-night asymmetry.

Key words : Solar Neutrinos, Neutrino Oscillations, Matter Effects, Day-Night Asymmetry, Non-Standard Interactions.

Sammanfattning

För n˚ agra ˚ ar sedan bevisade SNO och KamLAND att neutrinooscillationer, som indikerar att neutriner är massiva, existerar. Neutrinooscillationer sker i b˚ ade vaku- um och materia. I det senare fallet p˚ averkar den effektiva potentialen i materialet amplituden av oscillationerna och ökar sannolikheten för smakändringar. En annan intressant konsekvens av materiaeffekter är solneutrinernas dag-natt assymmetri, som ges av skillnaden i neutrinoflöde under natten jämfört med dagen. För att kunna förklara uppkomsten av neutrinomassa behöver Standardmodellen utvidgas.

I de flesta utvidgningar till Standardmodellen uppkommer s˚ a kallade ickestandardeffekter. Dessutom kan vissa experimentella resultat för neutriner inte förklaras av Standardmodellen, vilket kan vara en indikation p˚ a ny fysik. Syftet med det här arbetet är att studera ickestandardeffekter p˚ a dag-natt asymmetrin. Vi upp- skattar felet, som beror p˚ a osäkerheten i blandningsparametrarna, i assymmetrin i

iii

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iv Sammanfattning tresmaksmodellen. Därefter bygger vi en tv˚ asmaksmodell, som tar hänsyn till de do- minerande ickestandardeffekterna i solneutrinerssektorn. Vi härleder uttrycket för dag-natt asymmetrin i den här modellen. Slutligen studerar vi ickestandardeffekterna p˚ a assymmetrin i tv˚ a steg. Först varieras alla parametrar samtidigt i en mängd beräkningar. I det andra steget varieras varje parameter var för sig medan övriga parametrar satta till noll. I det första steget är resultaten att de olika parametrar- nas korrelationer med assymmetrin blandade. I det andra steget s˚ a p˚ avisas varje enskild parameters p˚ averkan. Slutligen kan begränsningar sättas p˚ a de parametrar som har studerats i modellen, fr˚ an experimentella data p˚ a dag-natt assymmetrin.

Nyckelord : Solneutriner, Neutrinooscillationer, Materiaeffekter, Dag-Natt Assym-

metri, Ickestandardeffekter.

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Preface

This Master Thesis is the result of six months of work between September 2014 and February 2015 in the Theoretical Particle Physics group at KTH. It was a great opportunity to devote a lot of time learning about the current challenges in particle physics, especially in the field of neutrino physics, which is very captivating in many aspects. Because they interact very rarely with matter, neutrinos are difficult to detect and several neutrino properties remain to be revealed. Solar neutrinos are of particular interest because they are produced in abundance in the Sun’s interior and most of them escape unattenuated, which make them good candidates for probing the solar internal structure. However, since the resolution of the solar neutrino problem, we know that neutrinos are not massless as predicted by the Standard Model (SM). Therefore, the SM is not the ultimate description of Nature and has to be improved in spite of its many successes, which opens possibilities for new physics.

This thesis is organized in five chapters. In Chapter 1, we present the relevant elements of neutrino physics for the remaining of the work, i.e., neutrino properties, solar neutrino production and the derivation of neutrino oscillation probabilities in the two and three flavor frameworks. In Chapter 2, the concept of the day-night asymmetry is introduced in the standard case. This asymmetry is a measure of the difference between the day rate of neutrinos, which are detected directly after propagation in vacuum from the Sun’s surface to the Earth, and the night rate of neutrinos, which have passed through Earth before detection. Expressions for the two and three flavor models are presented, and subsequently numerically estimated with MATLAB. The error induced by the uncertainty on the mixing parameters is also computed for the three flavor model in order to estimate the allowed range for a secondary non-standard transition process. In Chapter 3 the non-standard interactions, which are a natural consequence of neutrino mass generation, are introduced, and a model for the study of NSI together with solar neutrinos is built with motivations. The derivation of the oscillation probability with NSI is performed in analogy to the standard case of Chapter 1. In Chapter 4, we derive the day-night asymmetry expression when propagation NSI are taken into account, according to the model presented in Chapter 3. The asymmetry is then computed when the NSI parameters are varied to see the impact of each of them. The results

v

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vi Preface of these simulations are used to set new limits on the NSI parameter values. Finally, in Chapter 5, we summarize and discuss the results.

Acknowledgments

Now that my exchange in Sweden is almost over, I would like to thank all the

interesting and open-minded persons I met at KTH, and in particular my supervisor

Mattias Blennow for his patience, help and wise advice. I also wish to thank Stella,

Juan and Sushant who kindly shared an office with me for some months, all the

members of the Theoretical Particle Physics group, the teachers I had during my

first year of exchange, and all the other people of the department who share their

daily lunchbox, as I used to.

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Solar Neutrinos and

Elements of Neutrino Physics

Neutrino physics has emerged in the twentieth century and is nowadays a very rich field of particle physics regarding both experiments and literature. The first chapter of this thesis is intended to provide a simple introduction to neutrino physics and solar neutrinos with respect to history, experiments and theory.

1.1 Overview on neutrino discovery, experiments and status

1.1.1 Main stages in neutrino history

Neutrinos were postulated for the first time in 1930 by Wolfgang Pauli to explain the continuity of the β-decay spectrum. This decay reaction is characterized by the emission of an electron (the β

⁻

particle) from a nucleus:

A

Z

X →

^AZ+1

X

^′

+ e

⁻

+ ¯ ν

e

. (1.1) If the electron was the only particle produced, it would always carry away the same amount of energy and the spectrum obtained would be discrete. However, since the electron energy spectrum was continuous, another particle had to be involved, which Pauli named ”neutron” because it was electrically neutral. However, Chadwick discovered the neutron that we know today in 1932 through the energy measurement of the nuclei emitted in the target reaction:

4

2

He +

⁹

Be →

¹²

C + n, (1.2)

which required the emission of a particle of mass 1 u and null electric charge [1].

Afterwards, Enrico Fermi renamed Pauli’s particle the neutrino (small neutron) as

1

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2 Chapter 1. Solar Neutrinos and Elements of Neutrino Physics he integrated it into his theory of beta-decay [2]. The neutrino was assumed to be massless with spin-1/2. The concept of lepton number was introduced in 1953 by Konopinski and Mahmoud as a quantity which had to be conserved in reactions, but only six years later did Pontecorvo suggest that this number should depend on the flavor of the lepton involved in the reaction. As the beta-decay reactions predicted the neutrino cross-section to be extremely small, the direct measurement of this particle seemed difficult to achieve, although both Pontecorvo and Schwartz suggested future experiments [3, 4]. Neutrinos were measured experimentally for the first time in 1956 by Frederik Reines and Clyde Cowan close to the US nuclear reactor of Savannah River [5] (they actually measured electron anti-neutrinos), while the muon neutrino was first measured in 1962 at Brookhaven by Lederman, Schwarts and Steinberger [6], who received 1988 Nobel Prize ”for the neutrino beam method and the demonstration of the doublet structure of the leptons through the discovery of the muon neutrino”. Eventually, the tau neutrino, the last flavor to be discovered after the measurement of the tau lepton in 1975 at Stanford Linear Accelerator Center [7], was directly detected in 2000 at Fermi Lab in the DONUT experiment [8].

1.1.2 Current experiments

Now that all flavors of neutrinos have been found, different kinds of experiments continue to study those particles to discover more about their properties. We outline here the main experiment categories and name some important examples:

Reactor experiments use nuclear reactors as a neutrino source since a huge number of neutrinos are produced in nuclear decay and most of them can escape thanks to their very small interaction probability with matter. The Double CHOOZ experiment [9] (Ardennes, France) for example, consists in two detectors, one close (400 m) and one farther (1000 m) from the CHOOZ nuclear core and was started in April 2011. The aim of the two detectors is to reduce systematic errors and get a precise measurement of the third mixing angle θ

13

. One can also mention the Daya Bay experiment in China which consists of eight anti-neutrinos detectors located in three places within 1.9 km from six nuclear reactors and aims at the precise measurement of the mixing angle θ

13

[10].

Solar neutrino experiments study the intense flux of neutrinos produced in Sun’s thermonuclear reactions. The main experiments are the several phases of the Super-Kamiokande detector (Japan), a 50 kton pure water Cherenkov detector [11] which started taking data in April 1996, but was also involved in atmospheric neutrino measurements and the T2K accelerator experiment.

The SNO experiment (Sudbury Neutrino Observatory, Canada) also used a

Cherenkov detector filled with heavy water [12] and started recording data

in November 1999 to study solar neutrinos in particular. Finally, the Borex-

ino experiment, which is located in Gran Sasso, Italy, and started in 2007,

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1.1. Overview on neutrino discovery, experiments and status 3 uses a liquid scintillator detector to measure

⁷

Be solar neutrinos [13]. Con- trary to liquid scintillators, Cherenkov detectors are sensitive to the incoming neutrino direction, which enables a better separation of the signal from the background. On the other hand, scintillator detectors can achieve a better energy resolution and measure neutrinos of lower energy, which is the reason why Borexino studies

⁷

Be solar neutrinos instead of

⁸

B neutrinos which were the targets for SK and SNO.

Long and short baseline accelerator experiments have the advantage to set up and accurately control the production conditions of the neutrino beams and thus to design specific experiments. For instance, the OPERA experiment [14] in Gran Sasso, was designed to detect ν

τ

appearance in ν

µ

beams from CERN and was completed in 2008. Another major accelerator experiment is the T2K experiment [15], which consists of an accelerator (J-PARC) located in Tokai, of a near detector (at 280 m from the accelerator) and of the Super- Kamiokande detector, located 295 km away. T2K begun taking data in 2009 and aimed at detecting ν

µ

→ ν

^e

appearance. Finally, NOvA (NuMI Off-Axis Appearance) is another two detector experiment, consisting of a near detector located at Fermilab which started in 2013, and a far one located in northern Minnesota, at 810 km from the source, which started in 2014. NOvA studies the ν

µ

→ ν

^e

oscillations in order to determine the neutrino mass ordering (i.e., to determine if the mass ordering is normal: m

1

≤ m

²

≪ m

³

, or inverted:

m

3

≪ m

¹

≤ m

²

) and to measure the CP violation (i.e., the violation of the invariance of the laws of physics when a particle is replaced by its anti-particle and the space coordinates are inverted, which is believed to be linked to the matter-antimatter asymmetry of the Universe) [16].

Atmospheric neutrino experiments are committed to measure neutrinos produced in chain reactions resulting from cosmic ray interactions with nuclei in Earth’s atmosphere. Many unstable particles are produced in this processes together with muon neutrinos. The IceCUBE experiment is a 1 km

³

neutrino detector located under the Antarctic ice measuring atmospheric neutrinos as well as very high energy neutrinos due to violent events far from Earth and also involved in the search for dark matter [17]. The Super-Kamiokande detector mentioned previously is also involved in the study of the neutrino mixing in the atmospheric sector [18].

Other experiments study particular properties of neutrinos like KATRIN (Karl- sruhe, Germany) which is performing direct measurements of neutrino mass through the study of the spectrum of the β-decay of Tritium [19].

There are of course many other neutrino experiments which have not been

mentioned, but this thesis is focused on the solar neutrino day-night effect, and

thus only the Super-Kamiokande and SNO experiments will be studied further in

Chapter 2.

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4 Chapter 1. Solar Neutrinos and Elements of Neutrino Physics

Figure 1.1: Description of the particle content of the Standard Model [20].

1.1.3 Neutrinos in the Standard Model

Neutrinos are elementary particles of the Standard Model, which is a theoretical description of the particles responsible for the matter constitution of the Universe and of the four interactions of these particles, which are themselves represented by particles called gauge bosons (see Figure 1.1 for a schematic picture). The elementary particles can be classified into two types:

The fermions are described by Fermi-Dirac statistics, have an half-integer spin and are the constituents of matter,

The bosons are described by Bose-Einstein statistics, have an integer-spin and are the force carriers of the interactions between the fermions.

The four interactions of the Standard Model are the electromagnetic interaction, mediated by the photon and responsible for the electrostatic attraction between the electrons and the nucleus of atoms for example; the strong interaction, carried by gluons and responsible for the bond between protons and neutrons in a nucleus;

the weak interaction, mediated by W

^±

and Z

⁰

bosons for the charged-current and neutral-current respectively and responsible for the radioactive β-decay; and gravity, usually neglected in particle physics for strength scale reasons, but which is involved in the structure of the universe.

As depicted on Figure 1.1, there are two categories of fermions:

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1.1. Overview on neutrino discovery, experiments and status 5

The leptons are the electron, the muon, the tau and their corresponding neutrino ordered by increasing mass into three generations:



 ν

e

⁻



 ,



 ν

µ

⁻



 ,



 ν

τ

⁻





Contrary to the electron, the muon and the tau, neutrinos are electrically neutral and massless in the Standard Model, although neutrino oscillations require their mass to be non-zero (see Table 1.1).

The quarks are the constituents of the proton, neutron and all the other baryons and mesons. They are also arranged in three generations ordered with increasing mass:



 u d



 ,



 c s



 ,



 t b





The last piece to be added to the Standard Model is the Higgs boson, a spin-0 scalar particle of mass approximately equal to 125 GeV responsible for the so-called Higgs mechanism, i.e., the mass generation of the elementary particles except the neutrinos, the photons and the gluons. Postulated in 1964 by the 2013 Nobel prize winners Higgs [21, 22] and Englert [23], it was measured for the first time in 2012 by the CMS and ATLAS experiments at the LHC (Large Hadron Collider) [24, 25].

The discovery of this particle was the achievement of the Standard Model, however, there remains some shortcomings indicating that it has now to be extended, to manage in particular to:

- include gravity, - explain dark matter,

- find a satisfying mass mechanism for neutrinos.

The problem of neutrino mass comes from its currently unexplained smallness compared to the other fermions of the same generation. For instance, the third generation of fermions have the following masses:

m

t

= 1.7 · 10

²

GeV m

τ

= 1.8 GeV m

b

= 4.7 GeV m

3

≤ 2.3 · 10

⁻⁹

GeV

A mass ratio of 10

⁹

prevents the neutrino mass from being due to the Higgs mech-

anism. Other properties of neutrinos remain to be elucidated: their Dirac or Majo-

rana nature (i.e., whether the neutrino is its own anti-particle or not), the number

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6 Chapter 1. Solar Neutrinos and Elements of Neutrino Physics of mass eigenstates and the existence of sterile neutrinos (the number of flavor eigenstates lighter than 45 GeV has been measured to be 3 at the LEP [26] but other states of neutrinos are not excluded), whether CP is violated or not, and the sign of the mass hierarchy.

Particles Q m [GeV]

1

^rst

generation electron e

⁻

-1 0.0005 neutrino ν

e⁻

0 < 10

⁻⁹

2

^nd

generation muon µ

⁻

-1 0.106

neutrino ν

µ

0 < 10

⁻⁹

3

^rd

generation tau τ

⁻

-1 1.78

neutrino ν

τ

0 < 10

⁻⁹

Table 1.1: The charge and mass of the six fundamental leptons (data are retrieved from [27]).

1.2 Solar neutrinos and the solar neutrino problem

1.2.1 Solar neutrino production

The Sun is described by the so-called Standard Solar Model (SSM) [28]. It describes its internal structure and predict some of its properties, such as temperature, pres- sure and neutrino fluxes. As in any star, electron neutrinos are copiously produced in the Sun’s thermonuclear reactions. Because matter is almost transparent to neutrinos due to their very small cross-section, most of them escape the solar interior at a velocity close to the speed of light. This means that neutrinos can carry away information on the Sun’s internal structure and reach Earth within minutes. As they are the only particles known to do so, intense research has been performed on solar neutrinos.

The Sun produces energy through two main nuclear reaction cycles initiated by hydrogen fusion (see Figure 1.2). The pp chain reactions are dominant in lower mass stars as is the Sun, while the CNO cycle mainly takes place in higher metallicity stars since it requires some preexisting metals to act as catalysts. CNO reactions are a secondary process in the Sun, only occurring in the most central region of the core [28].

As depicted on Figure 1.2, neutrinos are produced at several stages of the pp-

chain. Each reaction produces neutrinos in a given energy range determined by

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1.2. Solar neutrinos and the solar neutrino problem 7

Figure 1.2: Two main energy production cycles in the Sun: pp chain reactions (left) and CNO cycle (right) (figure retrieved with permission from [28]).

the nuclei involved in the process. Solar neutrinos are named after their creation process, for instance

⁸

B neutrinos measured by SK and SNO are created in the pp-III reaction:

8

B →

⁸

Be

^∗

+ e

⁻

+ ν

e

, (1.3)

while

⁷

Be neutrinos measured by Borexino come from the pp-II reaction:

7

Be + e

⁻

→

⁷

Li + ν

e

. (1.4)

The expected global spectrum of solar neutrinos is reproduced from Ref. [29] in Figure 1.3.

Helioseismology covers the study of seismic waves propagating in the Sun, in- ferred from variations of the solar brightness. Its results include the SSM and consequently branching ratios for solar reactions, which in turn provide predictions for neutrino rates. These rates can also be measured experimentally to check the validity of the Sun’s description, and simultaneously to further study neutrino properties. When this was done with the first solar neutrino experiments, the measurements did not meet the predictions: the solar neutrino problem had emerged.

1.2.2 The solar neutrino problem

The first solar neutrino experiment is the famous chlorine experiment performed

by Davis in 1955 at Homestake (South Dakota) based on an inverse beta decay

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8 Chapter 1. Solar Neutrinos and Elements of Neutrino Physics

Figure 1.3: Solar neutrino spectrum: The CNO’s contributions are depicted in blue

while the pp’s are in black.

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1.2. Solar neutrinos and the solar neutrino problem 9 reaction stemming from the process suggested by Pontecorvo [30]:

37

Cl + ν

e

→ e

⁻

+

³⁷

Ar. (1.5)

The method used was to first let a 3900 liter tank of carbon tetrachloride buried underground (to shield it from cosmic rays) be irradiated by solar neutrinos, then to remove the

³⁷

Ar nuclei produced and eventually to count them, using their radioactivity, to deduce the solar neutrino rate. The result happened to be about one third of the expected rate [31], but as the solar neutrino fluxes were not known very accurately, the SSM was questioned and other experiments began to study solar neutrinos.

Kamioka, Super-Kamiokande’s predecessor, was originally designed to measure proton decay but eventually studied solar neutrinos too since its large dimensions were an asset in both cases. It was the first real-time neutrino detector and could establish their solar origin thanks to its direction sensitivity. Indeed the elastic scattering of the neutrinos on electrons, which reaction can be written:

ν

x

+ e

⁻

→ ν

^x

+ e

⁻

, (1.6)

results in highly energetic electrons, which emit Cherenkov light rings as they recoil.

The first upgraded version Kamiokande-II started data acquisition in january 1987 and was sensitive to all flavors of neutrinos although dominantly to the electron one by a factor of 6. In 1989, the collaboration could confirm the chlorine experiment results with a measured rate more than 50% smaller than the expected one [32].

The following upgrades of the detector enabled the lowering of the energy threshold, initially at 9.3 MeV, which reduced the target neutrino energy to a very small proportion of the solar spectrum, to 5 MeV for Super-Kamiokande.

Not all chemical elements allow neutrino absorption. Gallium was soon selected as particularly suited for solar neutrino detection since it has a low energy threshold (233 keV) for this reaction [28]. Two Gallium detector experiments started at the end of the 1980s: SAGE and GALLEX (and its follow-up GNO for Gallium Neutrino Observatory). Located in the Gran Sasso tunnel to take advantage of its natural background shield, GALLEX used an aqueous GaCl

3

solution detector containing 3 tonnes of Gallium to measure low energy neutrinos such as those from the pp-reaction using the process:

ν

e

+

⁷¹

Ga →

⁷¹

Ge + e

⁻

. (1.7)

GALLEX started recording data in 1991 and one year later published results that

confirmed the electron neutrino deficit claimed by the Homestake and Kamiokande

experiments, although for a different part of the solar neutrino spectrum. On

the other hand, the measured flux was consistent with matter enhanced flavor

oscillations in the Sun [33]. Due to the difficulty to extract a small number of

atoms from a multiple ton solution, which questioned the validity of the results,

GALLEX carried out a known source recording and obtained findings accurate

enough to establish the validity of the previous results [34].

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10 Chapter 1. Solar Neutrinos and Elements of Neutrino Physics

1.2.3 Solution to the solar neutrino problem

Some years earlier, in 1967, the idea of neutrino oscillations had been introduced by Pontecorvo, but as one single flavor was known, he could only imagine neutrino- antineutrino oscillations, by analogy to the neutral kaon oscillations. Nevertheless, he had predicted the solar neutrino deficit the year before it was measured by Davis. Oscillations between different flavors of neutrinos, in particular ν

e

and ν

µ

, were first discussed by Maki, Nagawa and Sakata in 1962 [35]. Some years later, the theory for matter enhanced neutrino oscillations was developped independently by Wolfenstein in 1978 [36] and Mikheyev and Smirnov in 1986 [37] and begun to be a viable alternative solution to the solar neutrino problem. However, other solutions were also considered, as resonant spin-flavor precession (i.e., spin-flavor conversion of Majorana neutrinos into non-electron anti-neutrinos induced by solar magnetic field interaction with neutrino spins), non-standard neutrino interactions, or violation of the equivalence principle (i.e., mixing between the flavor and gravity eigenstates) [38].

To put an end to the argument, an improved Cherenkov detector was started in 1999 in Canada: the Sudbury Neutrino Observatory (SNO), made of a 12 m- diameter acrylic sphere filled with ultra pure heavy water, i.e., water with

¹

H atoms replaced by deuterium,

²

H. The deuterium nuclei added two other reactions to the elastic scattering detection channel of Eq. (1.6) already present at the Kamiokande detector: the charged-current channel, only sensitive to electron neutrinos, and whose signal is the emitted electron’s Cherenkov light ring

ν

e

+ d → p + p + e

⁻

(1.8)

and the neutral-current channel, equally sensitive to every neutrino flavor, whose signal is the emitted neutron

ν

x

+ d → p + n + ν

^x

. (1.9)

Thanks to these reactions, SNO could measure the electron neutrino rate as well as the overall neutrino rate from the Sun independently. In 2001, the collaboration could provide strong evidence that the total solar neutrino flux was consistent with the SSM predictions, and that the electron neutrino deficit was due to flavor transitions [39]. Subsequent recordings from Super-Kamiokande [40] and SNO [41]

as well as the KamLAND reactor experiment [42] established these facts with more accuracy and could put constraints on the mixing parameter values.

1.3 Neutrino oscillations in vacuum

The flavor transitions observed by solar neutrino experiments are explained by

neutrino oscillations. While only flavor eigenstates of neutrinos, i.e., ν

e

, ν

µ

and ν

τ

,

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1.3. Neutrino oscillations in vacuum 11 can be produced or detected, the propagation eigenstates are the stationary states of the Hamiltonian, i.e., the solutions to the eigenvalue equation:

Hφ = i dφ

dt = Eφ (1.10)

where E = p

p

²

+ m

²

is the neutrino energy, which allows the solution:

|ν

ⁱ

(t) i = e

^−iEⁱ^t

|ν

ⁱ

(0) i. (1.11) The flavor states can be written as linear combinations of mass states to study their propagation. If the |ν

ⁱ

is have distinct masses, a phase difference arises during their propagation, changing the mass state composition of the initial flavor state. This phase difference is responsible for neutrino oscillations and explains how flavors different from those created can be detected.

The Hamiltonian in vacuum H

0

has the simplest expression in the mass eigenstate basis, indicated by the index m, which is, for n neutrino flavors:

H

0,m

= diag(E

1

, E

2

, ...E

n

) = p1 + 1 2E



 



m

²1

0 · · · 0 0 m

²2

· · · 0 .. . .. . . .. .. . 0 0 · · · m

²n



 



(1.12)

where the eigenstates are assumed to have the same momentum and, since m

i

is very small:

E

i

= q

p

²

+ m

²_i

≈ p + m

²_i

2E . (1.13)

In the remainder of this section, we will only consider the two and three flavor cases. Larger numbers of flavors would assume the existence of one or more sterile neutrinos, which will not be investigated in this work. The three neutrino framework provides a more accurate description of flavor transitions. However, since only electron neutrinos are produced in the Sun and θ

13

is small, the mixing with ν

3

remains small and the two-flavor basis (ν

e

,ν

µ

) is a good approximation regarding solar neutrino studies.

1.3.1 Two flavor case

In the two flavor framework, the flavor eigenstates are related to the mass eigenstates by a unitary matrix U , which only depends on the mixing angle in vacuum θ, such that:



 ν

e

ν

µ



 = U



 ν

1

ν

2



 =



 c s

−s c







 ν

1

ν

2



 , (1.14)

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12 Chapter 1. Solar Neutrinos and Elements of Neutrino Physics where c = cos θ and s = sin θ. The propagation of an electron neutrino after time t is thus given by:

|ν

^e

(t) i = ce

^−iE¹^t

|ν

¹

i + se

^−iE²^t

|ν

²

i (1.15) where E

1

and E

2

are the mass eigenstate energies and are given by Eq. (1.13). The amplitude of oscillations for ν

e

→ ν

^µ

is thus given by:

hν

^µ

|ν

^e

i = (−shν

¹

| + chν

²

|) · (ce

^−iE¹^t

|ν

¹

i + se

^−iE²^t

|ν

²

i)

= cs(e

^−iE²^t

− e

^−iE¹^t

)

= sin(2θ) exp

−i ∆m

²

t 4E

i sin ∆m

²

t 4E

, (1.16)

since

E

2

− E

¹

≃ p + m

²2

2E −

p + m

²1

2E

≡ ∆m

²

2E (1.17)

with ∆m

²

= m

²2

− m

²1

. This assumption, although not exact, simplifies the computation and a proper wave packet analysis of the coherent state yields the same result [27]. The transition probability is then given by:

P (ν

e

→ ν

^µ

, t) = |hν

^µ

|ν

^e

(t) i|

²

(1.18)

= sin

²

(2θ) sin

²

∆m

²

t 4E

. (1.19)

The term sin

²

(2θ) is the amplitude of oscillations and only depends on the mixing angle in vacuum. When θ is close to 0 or π, the mass eigenstates are almost aligned with the flavor states so the flavor transitions are not very probable. The other sine varies with time or equivalently, for relativistic neutrinos, with the traveled distance. The oscillation length is given by:

l

osc

= 4πE

∆m

²

(1.20)

and represents the distance between two consecutive maxima or minima of the transition probability. When the traveled distance is long, or when the source or the detector is of finite size, the second factor is averaged out and the probability becomes:

P (ν

e

→ ν

^µ

, L) = 1

2 sin

²

(2θ). (1.21)

For unitarity reasons, it immediately follows from Eq. (1.19) that the survival probability, i.e., the probability for a neutrino to be measured in the same flavor state as it was produced, is:

P (ν

e

→ ν

^e

, L) = 1 − P (ν

^e

→ ν

^µ

, L) = 1 − sin

²

(2θ) sin

²

∆m

²

L 4E

(1.22)

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1.3. Neutrino oscillations in vacuum 13

1.3.2 Three flavor case

In the three flavor framework, the so-called PMNS mixing matrix (for Pontecorvo- Maki-Nakagawa-Sakata) is parameterized by three mixing angles θ

12

, θ

13

and θ

23

and an additional CP-violating phase δ and is written as [43]:

U

PMNS

=



 

 

1 0 0

0 c

23

s

23

0 −s

²³

c

23



 

 



 

 

c

13

0 s

13

e

^−iδ

0 1 0

−s

¹³

e

^iδ

0 c

13



 

 



 

 

c

12

s

12

0 −s

¹²

c

12

0 0 0 1



 

 

(1.23)

=



 

 

c

12

c

13

s

12

s

13

s

13

e

^−iδ

−s

¹²

c

23

− c

¹²

s

23

s

13

e

^iδ

c

12

c

23

− s

¹²

s

23

s

13

e

^iδ

s

23

c

13

s

12

s

23

− c

¹²

c

23

s

13

e

^iδ

−c

¹²

s

23

− s

¹²

c

23

s

13

e

^iδ

c

23

c

13



 

 

(1.24)

where c

ij

= cos θ

ij

and s

ij

= sin θ

ij

. If the neutrinos were Majorana particles, the PMNS matrix would include two additional phases, but as neutrino oscillation experiments cannot separate between neutrino Dirac or Majorana character [44], the Majorana phases are neglected for the sake of simplicity. The expression for the transition probability is not as simple as in the two-flavor case. Formally, since

|ν

^e

i = X

3 i=1

U

ei

|ν

ⁱ

i, (1.25)

with i denoting the mass states, the propagation is given by:

|ν

^e

(t) i = X

3 i=1

U

ei

e

^−iEⁱ^t

|ν

ⁱ

i. (1.26)

The transition probability to another flavor state x is then:

P (ν

e

→ ν

^x

, L) = |hν

^x

|ν

^e

(L) i|

²

=

X

3 i=1

U

_xi^∗

U

ei

e

^−iEⁱ^L

2

. (1.27)

However, in the case of solar neutrinos, the condition

∆m

²₁₃

L

2E ≃ ∆m

²₂₃

L

2E ≫ 1, (1.28)

where ∆m

²_ij

= |m

²i

− m

²j

|, is fulfilled and the electron survival probability reduces to:

P (ν

e

→ ν

^e

, L) = s

⁴13

+ c

⁴13

P (1.29) where P is the two-flavor survival probability of Eq. (1.22) with θ → θ

¹²

and

∆m

²

→ ∆m

²12

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14 Chapter 1. Solar Neutrinos and Elements of Neutrino Physics

1.4 Neutrino oscillations in matter

Oscillations can be affected when neutrinos travel through matter. In particular, neutrinos can undergo coherent forward scatterings which alter their oscillation probabilities. The so-called MSW effect (for Mikheyev-Smirnov-Wolfenstein) describes the resonant enhancement of the flavor transition probability. The scatterer involved is a matter constituent, i.e., an electron, proton or neutron. Charged- current scatterings on electrons, mediated by W

^±

bosons, only concern electron neutrinos, while neutral-current scatterings on electrons, protons or neutrons, mediated by Z

⁰

bosons, equally affect all flavors of neutrinos. Therefore, the neutral- current adds the same term to each diagonal element of the Hamiltonian and ul- timately the same phase factor to each flavor, which will be canceled out in the transition probability expressions. One can thus neglect the neutral-current potential and only consider the charged-current one, which is equal to V

cc

= √

2G

F

N

e

where G

F

is the Fermi constant and N

e

the electron number density.

1.4.1 Two flavor oscillations in matter

In the case of matter effects, the Hamiltonian is more easily expressed in the flavor basis and the evolution equation becomes:

i d dt



 ν

e

ν

µ



 =



U



 E

1

0 0 E

2



 U

⁻¹

+



 V

cc

0 0 0











 ν

e

ν

µ





=







 c

²

E

1

+ s

²

E

2

cs(E

2

− E

¹

) cs(E

2

− E

¹

) s

²

E

1

+ c

²

E

2



 +



 V

cc

0 0 0











 ν

e

ν

µ



(1.30)

with E

i

the energy of the vacuum eigenstates. Performing the transformations c

²

E

1

+ s

²

E

2

=

c

²

E

1

− 1 2 E

1

+ 1

2 E

1

+

s

²

E

2

− 1 2 E

2

+ 1

2 E

2

= − 1

2 cos(2θ)(E

2

− E

¹

) + 1

2 (E

1

+ E

2

) (1.31) s

²

E

1

+ c

²

E

2

= 1

2 cos(2θ)(E

2

− E

¹

) + 1

2 (E

1

+ E

2

) (1.32) to simplify the diagonal terms and using the fact that E

2

− E

¹

=

^∆m_2E²

yields the Hamiltonian:

H

Mat,f

=



 −

^∆m4E²

cos(2θ) +

¹₂

(E

1

+ E

2

)

^∆m_4E²

sin(2θ)

∆m²

4E

sin(2θ)

^∆m_4E²

cos(2θ) +

¹₂

(E

1

+ E

2

)



+



 V

cc

0 0 0





(1.33)

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1.4. Neutrino oscillations in matter 15 Now, as the term

¹₂

(E

1

+ E

2

) appears in the two diagonal terms, it can be removed using the same argument as for the neutral-current potential. The Hamiltonian in matter can thus be written:

H

Mat,f

=



 −

^∆m4E²

cos(2θ) + √

2G

F

N

e ∆m² 4E

sin(2θ)

∆m²

4E

sin(2θ)

^∆m_4E²

cos(2θ)



 (1.34)

Then, the study of the propagation depends whether N

e

can be considered as constant or varies along the neutrinos’ trajectory.

Constant matter density case

In the case of constant matter density, N

e

does not vary and neither does the Hamiltonian. The oscillation probability can thus be computed in the same manner as in the vacuum propagation case, except that the vacuum mixing angle needs to be replaced by a matter mixing angle which permits the diagonalization of the Hamiltonian:

H

Mat,f

= ˆ U



 E ˆ

1

0 0 E ˆ

2



 ˆ U

⁻¹

with ˆ U =



 cos ˆ θ sin ˆ θ

− sin ˆθ cos ˆθ



 , (1.35)

where ˆ E

1

and ˆ E

2

are the energy states in matter and ˆ U is the mixing matrix in matter. The developed expression of the Hamiltonian is:

H

Mat,f

=



 −

¹2

( ˆ E

2

− ˆ E

1

) cos(2ˆ θ) +

¹₂

( ˆ E

1

+ ˆ E

2

)

¹₂

( ˆ E

2

− ˆ E

1

) sin(2ˆ θ)

1

2

( ˆ E

2

− ˆ E

1

) sin(2ˆ θ)

¹₂

( ˆ E

2

− ˆ E

1

) cos(2ˆ θ) +

¹₂

( ˆ E

1

+ ˆ E

2

)



 . (1.36) The equalization of Eqs. (1.34) and (1.36) yields:

tan(2ˆ θ) = 1

2 ( ˆ E

2

− ˆ E

1

) sin(2ˆ θ) 1

2 ( ˆ E

2

− ˆ E

1

) cos(2ˆ θ)

=

∆m

²

2E sin(2θ)

∆m

²

2E cos(2θ) − V

^cc

, (1.37)

since:

(E

1

+ E

2

) = tr H = V

cc

. (1.38) By analogy to the vacuum case, the propagation of |ν

^e

i in matter can be written:

|ν

^e

(L) i = ˆ U

e1

e

^{−i ˆ}^E¹^L

|ˆν

¹

i + ˆ U

e2

e

^{−i ˆ}^E²^L

|ˆν

²

i (1.39) and the oscillation probability becomes:

P (ν

e

→ ν

^µ

, L) = sin

²

(2ˆ θ) sin

²

| ˆ E

2

− ˆ E

1

|

2 L

!

. (1.40)

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16 Chapter 1. Solar Neutrinos and Elements of Neutrino Physics The oscillation amplitude is derived from Eq. (1.37):

sin

²

(2ˆ θ) =

∆m

²

2E sin(2θ)

!

²

∆m

²

2E sin(2θ)

!

²

+ ∆m

²

2E cos(2θ) − V

^cc

!

²

. (1.41)

Writing the energy difference as:

| ˆ E

2

− ˆ E

1

| = 1

2 ( ˆ E

2

− ˆ E

1

) sin(2ˆ θ) 1

2 sin(2ˆ θ)

(1.42)

and reinserting the numerator from the Hamiltonian in Eq. (1.34) and the denominator from Eq. (1.41), the energy difference is found to be:

| ˆ E

2

− ˆ E

1

| =

s ∆m

²

2E sin(2θ)

2

+ ∆m

²

2E cos(2θ) − V

^cc

2

. (1.43)

The oscillation length in matter is thus given by:

l

osc,Mat

= 2π

| ˆ E

2

− ˆ E

1

| = 2π q

∆m²

2E

sin(2θ)

²

+

^∆m_2E²

cos(2θ) − V

^cc

²

. (1.44) The shape of the oscillation amplitude given in Eq. (1.41) obviously allows a resonance, which occurs when the so-called MSW resonance condition is fulfilled:

V

cc

= ∆m

²

2E cos(2θ). (1.45)

Note that this condition requires the mass squared difference to be positive. This can be achieved if |ν

¹

i has the larger |ν

^e

i contribution. In the resonance, the amplitude is equal to one, and the flavor transition probability can also be equal to one for given distances. The mixing in matter is then maximal, even if the mixing in vacuum is very small, i.e., the oscillation probability in matter can be large even if it is very small in vacuum.

Varying matter density case

The more realistic case of varying matter density will be treated in the concrete

example of the propagation in the Sun in Chapter 4.

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1.4. Neutrino oscillations in matter 17

1.4.2 Three flavor oscillations in matter

In the three flavor framework, the Hamiltonian becomes:

H

Mat,f

= 1 2E U



 

 

m

²1

0 0 0 m

²2

0 0 0 m

²3



 

  U

⁻¹

+



 

 

V

cc

0 0

0 0 0



 

 

(1.46)

The resulting propagation equation is not easy to solve in general. However, in the case of solar neutrinos, the survival probability expression of Eq. (1.29) is still valid, when the two-flavor survival probability is recomputed from the evolution equation with the replacements θ → θ

¹²

, ∆m

²

→ ∆m

²12

and when the charged-current potential V

cc

is replaced by the effective potential V

eff

= c

²₁₃

V

cc

.

To conclude, this chapter has presented the main stages of neutrino history, in

particular concerning the solar neutrino problem. It has also introduced neutrino

oscillations and derived the transition probabilities in vacuum and in matter for

the two flavor case.

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18

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Chapter 2

The Day-Night Asymmetry

The day-night asymmetry is a natural consequence of matter effects on solar neutrinos. In this chapter, we will first introduce this notion, then derive its formula step-by-step for the two flavor case before extending it to the three flavor case. The error on the asymmetry due to the uncertainty on the mixing parameters will finally be computed to estimate the allowed range for sub-leading non-standard processes.

2.1 Definition and experiments

Matter effects on neutrino oscillations could theoretically induce a difference between day and night rates. Indeed, only electron neutrinos are produced in the Sun, but they may undergo flavor transition as they propagate. While neutrinos measured during daytime have only traveled through the Sun, those detected during nighttime have also crossed Earth’s matter. Thus, Earth matter effects can be seen directly, whereas the solar ones can only be deduced from the predicted production rates. This difference is quantified by the so-called day-night asymmetry, which is usually defined as the difference between the night (N ) and the day (D) rates, divided by their average rate:

A

dn

= 2 N − D

N + D . (2.1)

The experimental measurement of this quantity requires a real-time detector, sensitive to the incoming neutrinos’ direction so as to keep exclusively the signals coming from the Sun, and measuring neutrinos in the matter dominated region for oscillations. As a consequence, the detectors of interest are the Super-Kamiokande (SK) and the Sudbury Neutrino Observatory (SNO). In fact, SNO records were first consistent with zero asymmetry [45] because the statistical and systematic uncertainties were of the same order as the measured asymmetry, but it eventually found an asymmetry of A

dn

= 7.0% ± 4.9%(stat.)

^+1.3−1.2

%(syst.) [46]. Thanks

19

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20 Chapter 2. The Day-Night Asymmetry to its larger dimensions and larger detection number, SK combined four phases of the experiment showed an evidence for a non-zero asymmetry with a smaller uncertainty: A

dn

= 3.2% ± 1.1%(stat.) ± 0.5%(syst.), which deviates from zero at the 2.7σ level [47]. This means that the Earth matter effects regenerate electron neutrino flavor in the night rate.

The expression of the day-night asymmetry in the two flavor case will be derived in detail in the next section since it will be reused in Chapter 4. The expression in the three flavor case will only be given with the main arguments.

2.2 Formulae derivation in the two flavor case

A key point for the computation of electron neutrino survival probabilities is to assume that the flux which reaches Earth is incoherent. This means that interfer- ence between |ν

¹

i and |ν

²

i is lost during their propagation between the Sun and the Earth. There are several reasons motivating this assumption in the two flavor framework, among which the averaging over the neutrino production region in the Sun and the separation of |ν

¹

i and |ν

²

i wave packets during their travel to Earth [48] are some of the stronger.

2.2.1 Production and propagation in Sun and day survival probability

Under the assumption of incoherent propagation, the day survival probability, i.e., the probability for electron neutrinos to reach Earth during daytime, is given by:

P

S

= k

1

|hν

^e

|ν

¹

i|

²

+ k

2

|hν

^e

|ν

²

i|

²

(2.2) where |hν

^e

|ν

ⁱ

i|

²

= |U

^ei

|

²

is the probability for a mass eigenstate |ν

ⁱ

i to be measured as |ν

^e

i at Earth. k

ⁱ

is the fraction of the mass eigenstate |ν

ⁱ

i which exits the Sun, or equivalently which reaches Earth since it propagates as a mass eigenstate in the vacuum, and is equal to:

k

i

= Z

R⊙

0

drf (r)[cos

²

θ(r)P ˆ

_1i^s

+ sin

²

θ(r)P ˆ

_2i^s

], (2.3)

where ˆ θ(r) is the mixing angle in the Sun at radius r and cos

²

θ(r) (respectively ˆ sin

²

θ(r)) is the probability for a produced ˆ |ν

^e

i at radius r to be in the matter eigenstate |ν

¹

i (respectively |ν

²

i). P

ij^s

is the probability for a matter eigenstate

|ˆν

^j

i produced at radius r to exit the Sun as the mass eigenstate |ν

ⁱ

i, and f(r) is

(29)

2.2. Formulae derivation in the two flavor case 21 the neutrino production distribution function in the Sun. Obviously, under the assumption of no sterile neutrinos:

k

1

+ k

2

= 1. (2.4)

Furthermore, from the unitarity conditions:

P

ij

+ P

ii

= 1, (2.5)

P

ii

= P

jj

, (2.6)

P

ij

= P

ji

, (2.7)

for i 6= j and with some trigonometry manipulations, we obtain:

k

1

= 1

2 (1 + D

2ν

) and k

2

= 1

2 (1 − D

^2ν

) (2.8)

with

D

2ν

= Z

R⊙

0

drf (r) cos 2ˆ θ(r)(1 − 2P

^hop

) (2.9) where P

hop

= P

12

= P

21

is the hopping probability, i.e., the probability to jump from one eigenstate to the other. In the adiabatic case, one gets P

hop

= 0 because the eigenstates ”adapt” themselves to the medium and thus propagate independently. The adiabatic approximation is valid when the adiabaticity parameter at resonance satisfies the condition [49]:

γ

^r

= |∆E|

2 ˙ˆθ ≫ 1 (2.10)

where the time derivative ˙ˆθ is equivalent to the distance derivative

^dˆ_dr^θ

since the neutrinos are relativistic and t ≃ L = r for the propagation in the Sun. This parameter can be computed, using Eq. (1.43) estimated at the resonance, the numerator is given by:

| ˆ E

2

− ˆ E

1

| = ∆m

²

2E sin(2θ) (2.11)

and taking the derivative of − cos 2ˆθ(r) gives the denominator. From the relation:

Study Of Non-Standard Interaction Effects On The Solar Neutrino Day-Night Asymmetry

Master Thesis