Beyond the Standard Model with neutrino physics

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Master of Science in Physics

Theoretical physics

Master thesis

Beyond the Standard Model with

neutrino physics

by

Robert-Jan Hagebout

6182178

60 ECT

September 2013 - November 2014

Supervisor:

dr. T.M. Nieuwenhuizen

Examiner:

Prof.dr. P.J.G. Mulders

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Abstract

Neutrinos physics has gained increased interest due to the discovery of the oscillating property of neutrinos. This proved that neutrinos are massive and provided clear evidence of physics beyond the Standard Model. In this thesis an overview of the theory and experimental data of neutrino oscillations, the search for neutrino masses and their corresponding Dirac or Majorana nature is given. Additionally the possibility that neutrinos might be relevant for Dark Matter is discussed. The theory of neutrino oscillations will be derived in detail for the often used two neutrino model, the currently accepted three neutrino model and expanded models for sterile neutrinos for oscillations in vacuum as well as the modifications due to the interactions with matter. Since the nature of neutrinos is unknown at this point, the implications will be discussed of how the Dirac and/or Majorana terms in the Lagrangian affect the eigenmasses of the neutrinos. An overview of experiments that lead to the current knowledge of neutrino oscillations is given and experimental anomalies will be discussed.

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1 Introduction

The neutrino was first theoretically postulated by W. Pauli in 1930 to explain unresolved problems with beta decay. The proposed new particle firstly solved the problem of how the observed continuous energy spectrum of electrons emitted in beta decay could conserve energy and secondly the conservation of angular momentum when the mother and daughter atom could have spin under the emission of an electron. This new fermionic particle would have no mass and no electric charge. [96]

C. Cowan and F. Reines detected the electron anti-neutrino in 1956 as a result of a reactor experiment where the reaction ¯νe+ p → n + e+ was used. This discovery would later be awarded

the 1995 Nobel Prize in Physics. In 1962, L. Lederman, M. Schwartz and J. Steinberger where the first to detect interactions of the muon neutrino, for which the 1988 Nobel Prize in physics was awarded. In 2000, the DONUT collaboration of FermiLab found evidence of the existence of the tau neutrino. [66]

In 1967, S. Weinberg and A. Salem formulated the modern form of the Standard Model (SM) based on an SU(2) × U(1) gauge model proposed by S. Glashow in 1961. In this model, neutrinos are massless and do not mix. In the late 1960s, the flux of solar neutrinos was measured. The first experiment to measure this was the Homestake experiment. This experiment measured a deficit, later confirmed by other experiments, in the solar neutrino flux. This deficit was known as the solar neutrino problem and was resolved in the late 1990s by the theory of neutrino oscillations. In 2002, R. Davis and M. Koshiba won part of the Nobel Prize in Physics for their contribution to the solar neutrino problem. [54]

The concept of neutrino oscillations was first proposed in 1957 by B. Pontecorvo motivated by the kaon oscillation phenomenon. The theory of neutrino oscillations was finally developed in 1975/76 by S. Eliezer and A. Swift , H. Fritzsch and P. Minkowski , S. Bilenky and B. Pontecorvo. In 1985, S. Mikheyev and A. Smirnov (expanding on 1978 work by L. Wolfenstein) noted that flavor oscillations can be modified when neutrinos propagate through matter. This is known as the MSW effect, which is important to understand solar neutrinos. The Super-Kamiokande experiment was the the first experiment to provide strong evidence of neutrino oscillations in 1998. Later the oscillations of flavor neutrinos where confirmed by reactor and accelerator experiments. In 2012, the reactor experiment Daya Bay provided evidence of a nonzero value of the last unknown mixing parameter. [54]

Over the past decade, neutrino physics has gained increasingly more interest, mainly because of the successes in solving the solar neutrino problem. In addition, neutrino physics has so far provided the only evidence of beyond SM physics. There are still unresolved questions about neutrinos, like the Dirac or Majorana origin. Additionally, there have been experiments that show interesting anomalies that may lead to new neutrino flavors or might hint for the still unknown dark matter.

The goal of this master project is to understand the oscillations between neutrino flavors, the Dirac or Majorana origin, the models for mass generation and how the theory of neutrino physics is used in experiments.

In section 2 the theory of neutrino oscillations are covered in detail. The section will start with the commonly used simplification to the mixing of two neutrinos. Then the established model of the mixing of three neutrinos will be covered. Because there are hints that there might be more then 3 flavor neutrinos, described in section 4, the oscillation model will be expanded with sterile neutrinos. The oscillation models will be covered in vacuum as well as in matter, which leads to the MSW effect. Useful limits that are used in neutrino experiments will be derived.

Section 3 will cover the nature of neutrinos. The differences between Dirac and Majorana neutrinos will be discussed. This section will show the Dirac and Majorana terms in the Lagrangian and cases where the Lagrangian will be a combination of these terms. This nature of the mass

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and the Majorana phases.

In section 4 an overview of important neutrino experiments will be given. The experiments that have contributed to the establishment of present day neutrino physics will be discussed. Experiments that show anomalies and recent analyses that might hint to the existence of sterile neutrinos will also be discussed. The necessary theory in order to understand the experiments that has not been discussed in earlier sections, such as the effective neutrino mass in double beta decay, will be given in this section.

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2. Neutrino mixing

One of the biggest discoveries of the last 15 years is that a neutrino flavors state can oscillate into an other neutrino flavor state. Only if flavor neutrinos are superpositions of mass eigenstates, a neutrino with initial flavor state να can oscillate into a final flavor state νβ during propagation.

This section will discuss the phenomenon of neutrino mixing. The neutrino oscillation probabilities will be derived for vacuum and in matter and the necessary oscillation probabilities for neutrino oscillations will be given. Additionally, models for sterile neutrinos will be discussed.

2.1 2 Neutrino mixing in vacuum

Consider only two neutrino flavors, electron and muon neutrinos, which are superpositions of two mass eigenstates as follows [37]

|νe> |νµ> = U|ν1> |ν2> = Ue1 Ue2 Uµ1 Uµ2 |ν1> |ν2> . (2.1)

Where U is called the mixing matrix and obeys U U†= I, Ue1 Ue2 Uµ1 Uµ2 U∗ e1 Uµ1∗ U_e2∗ U_µ2∗ =1 0 0 1 . thus the mixing elements should satisfy the relations

Ue1Ue1∗ + Ue2Ue2∗ = 1

Ue1Uµ1∗ + Ue2Uµ2∗ = 0

Uµ1Ue1∗ + Uµ2Ue2∗ = 0

Uµ1Uµ1∗ + Uµ2Uµ2∗ = 1

Therefore a solution of the mixing matrix is

U = cos θ sin θ − sin θ cos θ

. (2.2)

where θ is the mixing angle that defines how different the flavor states are from the mass states. This is similar to the lepton sector where the mixing angle θ is called the Cabibbo angle. Notice that this is not a unique solution, but since all the parameterizations are related, they give the same physical result.

The most general notation of the mixing between two neutrinos is therefore written as |νe> |νµ> = cos θ sin θ − sin θ cos θ |ν1> |ν2> . (2.3)

Sometimes it is more useful to write the mass eigenstates in terms of flavor neutrinos. |ν1> |ν2> =cos θ − sin θ sin θ cos θ |νe> |νµ> (2.4)

As a neutrino propagates through free space, like all fundamental particles, it obeys the evolution equation

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Where i=1,2. The solution is a plane wave solution

|νk(x, t)>= e−iΦk|νk(0, 0)> .

Where Φk= Ekt − pkx. Using this plane wave solution, one can write the evolution as

|ν1(x, t)> |ν2(x, t)> =e −iΦ1 ₀ 0 e−iΦ2 |ν1(0, 0)> |ν2(0, 0)> =e −iΦ1 ₀ 0 e−iΦ2 cos θ sin θ − sin θ cos θ |νe(0, 0)> |νµ(0, 0)> .

Since it is more useful to write an evolution of flavor neutrino states, one can write this as |νe(x, t)>

|νµ(x, t)>

=

cos2_θe−iΦ1_{+ sin}2_θe−iΦ2 _{− cos θ sin θe}−iΦ1_{+ sin θ cos θe}−iΦ2 − cos θ sin θe−iΦ1_{+ sin θ cos θe}−iΦ2 _sin2_θe−iΦ1_{+ cos}2_θe−iΦ2

|νe(0, 0)>

|νµ(0, 0)>

.

Because neutrino mixing is interesting to investigate, one would need to know the oscillation probabilities of producing a neutrino νeand measuring it at a later time as νµ.

The oscillation transition amplitude is

<νµ(x, t)|νe(0, 0)>= cos θ sin θ(eiΦ2− eiΦ1) <νe(0, 0)|νe(0, 0)>

+ (sin2θeiΦ1_{+ cos}2_θeiΦ2_{) <ν}

µ(0, 0)|νe(0, 0)> .

For a pure νethat is created at the source one has the following initial conditions:

<νe(0, 0)|νe(0, 0)> = 1

<νµ(0, 0)|νe(0, 0)> = 0

Therefore the oscillation probability of an electron neutrino oscillating into a muon neutrino is P (νe→ νµ) = | <νµ(x, t)|νe(0, 0)> |2

= sin2(2θ) sin2(Φ2− Φ1

2 ). (2.6)

For the oscillation phases, one can write: Φ2− Φ1 = (E2− E1)t − (| ~p2| − | ~p1|)x and because

neutrino masses are considered to be very small in comparison with other particles, one can safely assume that neutrinos are relativistic: t = x = L and | ~pi| =pEi2− m

2 i ≈ Ei(1 − m2_i 2E2 i ) Therefore Φ2− Φ1can be written as Φ2− Φ1= ( m2 2 2E2 − m 2 1 2E1 )L = ∆m 2 21 2E L. Where ∆m2

21≡ m22− m21 is the mass squared difference and the energies of the neutrinos during

propagation, either as a pure ν1 or as a pure ν2, is assumed to be equal, E1= E2= E.

Therefore the most general notation of an electron neutrino oscillating into a muon neutrino is P (νe→ νµ) = sin2(2θ) sin2(

∆m2 21L

4E ). (2.7)

Or when natural units are restored and ∆m2_{, L and E are expressed in eV/c}2_{, km and GeV}

respectively, the oscillation probability can be written as

P (νe→ νµ) = sin2(2θ) sin2(1.27∆m221

L

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2.2 2 Neutrino mixing in matter

If neutrinos propagate through matter, the total energy of the neutrino will increase as a result of an increased potential. If the potential is different for different neutrino flavors, this will lead to matter induced oscillations. [37]

Considering the time evolution of the neutrino mass eigenstates, the evolution equation in matrix form looks like

i~d dt ν1 ν2 = Hν1 ν2 . (2.9)

For flavor α and β this would be να νβ = cos θ sin θ − sin θ cos θ ν1 ν2 = Uν1 ν2 and therefore i~d dt ν1 ν2 = U†_i~d dt να νβ = Hν1 ν2 = HU†να νβ .

Since constants in the Hamiltonian are irrelevant when one calculates the oscillation probabil-ities, one can write the Hamiltonian in vacuum as

H = c 4 2E m2 1 0 0 m2 2 . (2.10)

Multiplying on the left with U gives i~_dtd ν_να β = U HU†να νβ , (2.11) where U HU† = c 4 4E(m 2 1+ m 2 2) 1 0 0 1 + c 4 4E∆m 2 21 − cos(2θ) sin(2θ) sin(2θ) cos(2θ) = H0+ c4 4E∆m 2 21 − cos(2θ) sin(2θ) sin(2θ) cos(2θ) ≡ Hf

is called the free Hamiltonian and H0= c

4 4E(m 2 1+ m22) 1 0 0 1 .

If neutrinos propagate in a nonzero potential V, then the total energy of the neutrino increases with V, so i~_dtd ν_να β = (Hf+ V ) να νβ . (2.12)

Because the potential can be different for different neutrino flavours, the potential is labeled as i~_dtd ν_να β = Hf να νβ +Vα 0 0 Vβ να νβ

Contrary to electrons, muons and tauons are unstable and will decay. Therefore, unless the medium produces muons or tauons, most matter will consist of electrons. Because of this fact, electron neutrinos can interact with electrons through the charged current interaction, fig.2.1a. At the same time e, µ and τ neutrinos can interact with matter through the neutral current interaction, fig.2.1b.

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¯ νe e− W− ¯ νe e− νe e− W+ e− νe

(a) The Feynman diagrams for charged current inter-actions νe, νµ, ντ e−, p, n Z0 νe, νµ, ντ e−, p, n

(b) The Feynman diagram for neutral cur-rent interactions

Figure 2.1: The interactions of e, µ and τ neutrinos in matter. Electron neutrinos can interact through the charged current and neutral current interactions, while muon and tau neutrinos can only interact with matter through the neutral current interaction.

all neutrino flavours. Therefore it appears as a constant in the equation and can be subtracted without affecting the result for the oscillation probability.

If the Hamiltonian is transformed back to the vacuum mass eigenstates and the matter potential is included, the mass state evolution will look like

i~_dtd ν_ν1 2 = 1 2E " c4m 2 1 0 0 m22 + 2EVcc

_cos2_θ _{cos θ sin θ}

cos θ sin θ sin2(θ)

#_ν 1 ν2 (2.13) =c 4 2E m2 1+ 2EVcc c4 cos 2_θ 2EVcc c4 cos θ sin θ 2EVcc c4 cos θ sin θ m 2 2+ 2EVcc c4 sin 2_θ ν1 ν2 . (2.14)

This matrix is no longer diagonal, therefore the mass eigenstates in vacuum are unequal to the mass eigenstates in matter. In order to find the matter induced mass eigenstates, one has to diagonalize the matrix to the form

Hm= c4 2E m2 1m 0 0 m2 2m . (2.15) Where m2

1mand m22m are the modified mass eigenvalues in matter.

One can do this by calculating the eigenvalues λ as follows: 0 = m21+2EVc4cccos 2_{θ − λ} 2EVcc c4 cos θ sin θ 2EVcc c4 cos θ sin θ m 2 2+ 2EVcc c4 sin 2_{θ − λ} = λ2− (m2 1+ m 2 2+ 2EVcc c4 )λ + m 2 1m 2 2+ 2EVcc c4 m 2 1sin 2_{θ +} 2EVcc c4 m 2 2cos 2_θ Therefore λ±= 1 2 " m21+ m22+ ∆V c4 ± r ∆m2 21cos(2θ) − ∆V c4 2 + (∆m2 21)2sin 2 2θ # (2.16) Where ∆V = 2EVcc= 2 √ 2GFNeE.

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Therefore the effective mass splitting in matter is ∆m2_21m= m2_2m− m21m= λ+− λ−= r ∆m2 21cos(2θ) − ∆V c4 2 + (∆m2 21)2sin 2_2θ = ∆m2₂₁ s cos(2θ) − ∆V ∆m2 21c4 2 + sin22θ (2.17) = ∆m2₂₁A. Where A = r cos(2θ) − _∆m∆V2 21c4 2

+ sin22θ. The behaviour of this factor is visualized in fig. 2.2.

Figure 2.2: The behaviour of the factor A = r

cos(2θ) − 2√2GFNeE/∆m2c4

2

+ sin22θ that describes how the neutrino parameters in matter differ from their value in a vacuum as a function of the neutrino energy multiplied with the electron density of the medium. The difference between the 12 and 13 sectors, which are the mixing of ν1 with ν2 and ν1 with ν3 respectively, have been

obtained by using the current best known values of the mixing parameters, given in section 4.9.[50]

2.2.1 MSW effect

The matter potential does not only affect the mass splitting, but also the mixing among the mass eigenstates. To write the mixing angle in terms of a mixing angle in matter, the fact is used that

UmHmUm† = Hf+ Vcc 0 0 0 , (2.18) where Um= cos θm sin θm − sin θm cos θm (2.19) is the mass modified mixing matrix and θmthe effective mixing angle in matter. This written out

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This gives the following relations regarding the effective mixing angle in matter θm sin 2θm= sin 2θ r cos(2θ) − _∆m∆V2 21c4 2 + sin22θ (2.21) cos 2θm= cos(2θ) − _∆m∆V2 21c4 q (cos(2θ) − ∆V ∆m2 21c4 )2_{+ sin}2_2θ (2.22) tan 2θm= sin 2θ cos(2θ) − _∆m∆V2 21c4 (2.23) = tan 2θh1 − ∆V ∆m2 21c4cos 2θ i−1 (2.24)

From these equations it can be seen that this leads to a resonance if ∆V = 2√2GFNeE =

∆m2

21c4cos(2θ). This corresponds to an electron density of

NeR=

∆m2

21c4cos(2θ)

2√2GFE

. (2.25)

The transition probability can now be written in the form of matter modified mixing angles/ mass splitting.

Pm(νe→ νµ) = sin2(2θm) sin2(1.27∆m221m

L

E) (2.26)

When the resonance occurs at ∆V /∆m2

21c4 = cos(2θ), then sin(2θm) = 1. Therefore the

probability takes the form of

Pm(νe→ νµ) = sin2(1.27∆m221m

L E). This means that if the resonance region is wide enough, L = _1.27∆mE 2

21m

, then Pm(νe→ νµ) = 1

and the probability is maximal and leads to total transitions between 2 flavours. This is called the Mikheyev-Smirnov-Wolfenstein (MSW) effect. Some examples of this resonance are given in figure 2.3.

Also notice that if ∆V = 0, the mass modified parameters reduce to the parameters in vacuum, ∆m2_21m= ∆m21 and sin2(2θm) = sin2(2θ), as expected. If the matter is very dense, ∆V → ∞,

then sin2(2θm) → 0 and ∆m221m→ ∞. So in very dense matter, oscillations cannot occur.

An other important effect is that if ∆m221 is replaced with −∆m221, sin 2

(2θm) will have a

different value. Therefore one can use matter effects to determine the sign of the mass squared difference i.e. the mass hierarchy. [37, 75, 54]

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(a) A plot of P(νe → νµ) for constant distance,

L=10000 km, in the 12 sector.

(b) A plot of P(¯νe → ¯νµ) for constant distance,

L=10000 km, in the 13 sector.

(c) A plot of P(¯νe→ ¯νµ) for constant electron

num-ber density, Ne= 1030m−3 in the 12 sector.

(d) A plot of P(¯νe→ ¯νµ) for constant electron

num-ber density, Ne= 1030m−3 in the 13 sector.

Figure 2.3: Examples of the MSW effect in the oscillation probabilities P(νe→ νµ) for a constant

oscillation distance, a and b, and for constant electron number density of the medium, c and d. The best known values for the mixing angles and mass squared differences have been used to visualize the difference between the 12 and 13 sector. [50]

2.3 3 Neutrino mixing

Previously, neutrino mixing for two flavors have been discussed. This principle can be very useful for some approximations. However, since there are three active neutrino flavors, νe, νµ and ντ,

this section will now discuss the mixing among three neutrino flavors.

2.3.1 Mixing matrix

For the mixing of 3 neutrinos, the flavor states are superpositions of 3 mass eigenstates. The mixing matrix for 3 neutrinos has the form

  νe νµ ντ  =   Ue1 Ue2 Ue3 Uµ1 Uµ2 Uµ3 Uτ 1 Uτ 2 Uτ 3     ν1 ν2 ν3   (2.27)

or when the mass eigenstates are expressed as superpositions of flavor states   ν1 ν2 ν3  =   U_e1∗ U_µ1∗ U_{τ 1}∗ U_e2∗ U_µ2∗ U_{τ 2}∗ U_e3∗ U_µ3∗ U_{τ 3}∗     νe νµ ντ  .

Because this matrix is unitary, U†U = 1, one can write  Ue1 Ue2 Ue3 U U U  U_e1∗ U_µ1∗ U_{τ 1}∗ U∗ U∗ U∗  =  1 0 0 0 1 0  .

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Which gives the relations Ue1Ue1∗ + Ue2Ue2∗ + Ue3Ue3∗ = 1 Uµ1Uµ1∗ + Uµ2Uµ2∗ + Uµ3Uµ3∗ = 1 Uτ 1Uτ 1∗ + Uτ 2Uτ 2∗ + Uτ 3Uτ 3∗ = 1 Ue1Uµ1∗ + Ue2Uµ2∗ + Ue3Uµ3∗ = 0 Ue1Uτ 1∗ + Ue2Uτ 2∗ + Ue3Uτ 3∗ = 0 Uµ1Uτ 1∗ + Uµ2Uτ 2∗ + Uµ3Uτ 3∗ = 0

Or when U U†= 1 is used, it also gives the following relations

Ue1Ue1∗ + Uµ1Uµ1∗ + Uτ 1Uτ 1∗ = 1

Ue2Ue2∗ + Uµ2Uµ2∗ + Uτ 2Uτ 2∗ = 1

Ue3Ue3∗ + Uµ3Uµ3∗ + Uτ 3Uτ 3∗ = 1

A 3x3 matrix has 9 independent parameters, of wich are 3 mixing angles and 6 phases. However 5 out of 6 phases can be chosen in such a way that only 1 phase remains physical. Therefore the mixing matrix for 3 neutrinos has 3 mixing angles and 1 physical phase. [54]

The mixing matrix can be written as a product of 3 neutrino sectors. U = U23U13U12

The choice can be made that in the 12 sector, the mixing angle θ12 mimics the Cabbibo angle in

the CKM matrix in the lepton sector. The other choice that is made is that the only physical phase is in the 13 sector. Then the matrices look like

U12=   cos θ12 sin θ12 0 − sin θ12 cos θ12 0 0 0 1   (2.28) U13=  

cos θ13 0 sin θ13e−iδ13

0 1 0

− sin θ13eiδ13 0 cos θ13

  (2.29) U23=   1 0 0 0 cos θ23 sin θ23 0 − sin θ23 cos θ23   (2.30)

With these choices made [37], the mixing matrix is called the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix and looks like

UP M N S =   c12c13 s12c13 s13e−iδ13 −s12c23− c12s23s13eiδ13 c12c23− s12s13s23eiδ13 c13s23 s12s23− c12s13c13eiδ13 −c12s23− s12s13c23eiδ13 c13c23  . (2.31)

Where cab= cos(θab) and sab= sin(θab). Additionally, δ13is sometimes called the Dirac phase or

the CP violating phase δCP. An overview of the current best values of the parameters is given in

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2.3.2 Oscillation probability

If a neutrino is created in a pure ναstate, it looks like

|να(0, 0)>= Uα1|ν1> +Uα2|ν2> +Uα3|ν3> .

The neutrino propagates as a plane wave.

|να(x, t)>= Uα1e−iΦ1|ν1> +Uα2e−iΦ2|ν2> +Uα3e−iΦ3|ν3> .

Where Φi= Eit − ~pi· ~x

After travelling a distance L and treating neutrinos as relativistic Φi= Ei− |pi|L ≈ (Ei− |pi|)L ≈ (|pi| + m2 i 2Ei − |pi|)L = m 2 i 2Ei L.

Expressing the neutrino state in terms of flavour states and arranging the terms gives |να(L)>= X β=e,µ,τ X k=1,2,3 UαkUβk∗ e−iΦk |νβ(0, 0)> . (2.32)

The transition amplitude is then

A(να→ νβ) =<νβ(x, t)|να(0, 0)>

= X

k=1,2,3

U_βk∗ UαkeiΦk.

One can now use Φj− Φk = ∆m2jk L

2E and the complex relationship

|z1+ z2+ z3|2= |z1|2+ |z2|2+ |z3|2+ 2<(z1z∗2+ z1z3∗+ z2z3∗) + 2=(z1z∗2+ z1z3∗+ z2z3∗)

to find an expression for the transition probability. [37]

P (να→ νβ) =| <νβ(x, t)|να(0, 0)> |2 = X k=1,2,3 |Uαk|2|Uβk|2 + 2<[X j>k U_βk∗ UαkUβjUαj∗ e −i∆m2 jk2EL _{] + 2=[}X j>k U_βk∗ UαkUβjUαj∗ e −i∆m2 jk2EL _]

Because of unitarity of the mixing matrix: U U† = 1 →P

kUαkUβk∗ = δαβ , the first term can be

rewritten as X k |Uαk|2|Uβk|2=δαβ− 2<[ X j>k UαkUβk∗ Uαj∗ Uβj].

Therefore the general expression for the oscillation probability of a flavor neutrino ναoscillating

into a flavor neutrino νβ is

P (να→ νβ) = δαβ− 4<[ X j>k UαkUβk∗ U ∗ αjUβj] sin2(∆m2jk L 4E)

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Or with natural units recovered and E in GeV, L in km and ∆m2 _{in eV/c}2_. P (να→ νβ) = δαβ− 4<[ X j>k UαkUβk∗ Uαj∗ Uβj] sin2(1.27∆m2jk L E) + 2=[X j>k UαkUβk∗ Uαj∗ Uβj] sin(2.54∆m2jk L E). (2.34)

For anti-neutrinos one can define the flavor state as |¯να>=

X

k

U_αk∗ |¯νk> .

And because <(z∗) = <(z) and =(z∗) = −=(z)

The transition probability for anti-neutrinos can be written as

P (¯να→ ¯νβ) =δαβ− 4<[ X j>k UαkUβk∗ Uαj∗ Uβj] sin2(1.27∆m2jk L E) − 2=[X j>k UαkUβk∗ Uαj∗ Uβj] sin(2.54∆m2jk L E). (2.35)

2.3.3 Electron, muon and tau neutrino oscillations

If one were to conduct an appearance experiment where the flavor of the neutrino in the final state is different from the flavor in the initial state, the exact oscillation probabilities are useful to know. Using equation (2.33) the exact oscillation probabilities for electron, muon and tau neutrinos oscillating into another flavor neutrino can be written as:

P (νe→ νµ) =4c213 h c2₁₂s2₁₂(c2₂₃− s213s 2 23) sin 2_(∆m2 12 L 4E) + s2₁₃s2₂₃ c2₁₂sin2(∆m2₃₁ L 4E) + s 2 12sin 2_(∆m2 32 L 4E) i + 4 cos(δCP)c12c213c23s12s13s23 h (c2₁₂− s212) sin 2_(∆m2 21 L 4E) (2.36) + sin2(∆m2₃₁ L 4E) − sin 2_(∆m2 32 L 4E) i + 2 sin(δCP)c12c213c23s12s13s23 h sin(∆m221 L 2E) − sin(∆m2 31 L 2E) + sin(∆m 2 32 L 2E) i P (νe→ ντ) =4c213 h c2₁₂s2₁₂(s2₂₃− c2 23s 2 13) sin 2_(∆m2 12 L 4E) + c2₂₃s2₁₃ c2₁₂sin2(∆m2₃₁ L 4E) + s 2 12sin 2_(∆m2 32 L 4E) i − 4 cos(δCP)c12c213c23s12s13s23 h (c2₁₂− s212) sin 2_(∆m2 21 L 4E) (2.37) + sin2(∆m2₃₁ L 4E) − sin 2_(∆m2 32 L 4E) i − 2 sin(δCP)c12c213c23s12s13s23 h sin(∆m221 L 2E) − sin(∆m2 31 L 2E) + sin(∆m 2 32 L 2E) i

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P (νµ→ ντ) =4 h c2₂₃s2₁₃s2₂₃(s₁₂4 + c4₁₂) + c2₁₂s2₁₂s₁₃2 (c4₂₃+ s4₂₃) − c2₁₂c2₂₃s2₁₂s2₂₃(1 + s4₁₃) + cos(δCP)c12c23s12s13s23(1 + s213)(c 2 12− s 2 12)(c 2 23− s 2 23) − 2 cos(2δCP)c212c 2 23s 2 12s 2 13s 2 23 i sin2(∆m2₂₁ L 4E) (2.38) + 4c213c223s223 h (s212− c212s213) sin 2 (∆m231 L 4E) + (c 2 12− s212s213) sin 2 (∆m232 L 4E) i + 4 cos(δCP)c12c213c23s12s13s23(c223− s 2 23) h − sin2_(∆m2 31 L 4E) + sin 2_(∆m2 32 L 4E) i + 2 sin(δCP)c12c213c23s12s13s23 h sin(∆m212 L 2E) − sin(∆m 2 31 L 2E) + sin(∆m 2 23 L 2E) i

From these equations it can be seen that an increase(decrease) in the probability P (νe→ νµ)

due to CP violation results in an decrease(increase) in the probability P (νe→ ντ).

Consider the case of no CP violation (δCP = 0). In this case the imaginary terms in eq.(2.33)

vanish. One can quickly see how the oscillation probabilities look like by substituting δCP= 0 in

the above equations. The Compared to each other, the probabilities are displayed in figure 2.4, where the current best known values for the oscillation parameters from [51, 50] have been used.

Figure 2.4: The oscillation probabilities from eq. (2.36),(2.37) and (2.38) plotted for the case of no CP violation(δCP = 0) and normal mass hierarchy. The current best value fits have been used

for the mixing angles and mass squared differences, given in section 4.9 [50]

To illustrate the effect of CP violation on the appearance oscillation probabilities, equations (2.36), (2.37) and (2.38) have been plotted for various values of δCP in figure 2.5. These plots

show the first two oscillations which occur in the region L/E < 200km/GeV. Increasing the region can increase the difference between the oscillation probabilities. However, a higher value of L/E is unrealistic for most neutrino experiments. Additionally, if δCP 6= 0, the oscillation probabilities

in vacuum are different for normal (|∆m2

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(a) The oscillation probability P(νe→ νµ)

(b) The oscillation probability P(νe→ ντ)

(c) The oscillation probability P(νµ→ ντ)

Figure 2.5: The oscillation probabilities P(νe → νµ), P(νe → ντ) and P(νµ → ντ) plotted for

various values of δCP and normal mass hierarchy (|∆m231|2 > 0) and the current best known

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The dependency of the CP violating phase can be visualized by a contourplot of the oscillation probabilities, given in figure 2.6.

Notice that under CPT invariance, P(να→ νβ)=P(¯νβ→ ¯να).

(a) A contourplot of P(νe→ νµ)=P(¯νµ→ ¯νe) (b) A contourplot of P(¯νe→ ¯νµ)=P(νµ→ νe)

(c) A contourplot of P(νe→ ντ)=P(¯ντ → ¯νe) (d) A contourplot of P(¯νe→ ¯ντ)=P(ντ → νe)

(e) A contourplot of P(νµ→ ντ)=P(¯ντ → ¯νµ) (f) A contourplot of P(¯νµ→ ¯ντ)=P(ντ → νµ)

Figure 2.6: Contourplots of the oscillation probabilities P(νe→ νµ), P(νe→ ντ) and P(νµ → ντ)

plotted as a function of δCP and L/E for normal mass hierarchy. 0 < δCP < 2π and 0 < L/E <

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For the disappearance experiment νe→ νe, the probability P(νe→ νe) takes the form P (νe→ νe) = 1 − 4c212c 4 13s 2 12sin 2 (∆m221 L 4E) − 4c2 12c 2 13s 2 13sin 2_(∆m2 31 L 4E) (2.39) − 4c2 13s212s213sin 2 (∆m232 L 4E)

and therefore a disappearance experiment of electron neutrinos is independent of the Dirac phase. This means that disappearance experiments for electron neutrinos are very well suited to measure the mixing parameters without having to worry about CP invariance. No CP dependency also has the consequence that this probability, in vacuum, is independent of the mass hierarchy, i.e. whether ∆m2

31 is larger or smaller then 0.

The disappearance probabilities for muon and tau neutrinos take the following forms: P (νµ→ νµ) = 1 − 4 h c2₁₂s2₁₂(c2₂₃− s2 13s 2 23) 2_{+ c}2 23s 2 13s 2 23(c 4 12+ s 4 12) sin 2_(∆m2 21 L 4E) + c213s223(c223s122 + c212s213s223) sin 2 (∆m231 L 4E) + c2₁₃s2₂₃(c2₁₂c2₂₃+ s2₁₂s2₁₃s2₂₃) sin2(∆m2₃₂ L 4E) i − 8 cos(δCP)c12c23s12s13s23 h (c2₁₂− s2 12)(c 2 23− s 2 13s 2 23) sin 2_(∆m2 21 L 4E) (2.40) + c2₁₃s2₂₃sin2(∆m2₃₁ L 4E) − c2 13s 2 23sin 2_(∆m2 32 L 4E) i + 8 cos(2δCP)c212c 2 23s 2 12s 2 13s 2 23sin 2_(∆m2 21 L 4E) P (ντ → ντ) = 1 − 4 h c2₁₂s2₁₂(s2₂₃− c2 23s 2 13) 2_{+ c}2 23s 2 13s 2 23(c 4 12+ s 4 12) sin 2_(∆m2 21 L 4E) + c213c 2 23(s 2 12s 2 23+ c 2 12c 2 23s 2 13) sin 2 (∆m231 L 4E) + c2₁₃c2₂₃(c2₁₂s2₂₃+ c2₂₃s2₁₂s2₁₃) sin2(∆m2₃₂ L 4E) i + 8 cos(δCP)c12c23s12s13s23 h (c212− s212)(s223− c223s213) sin 2 (∆m221 L 4E) (2.41) + c2₁₃c2₂₃sin2(∆m2₃₁ L 4E) − c2 13c 2 23sin 2_(∆m2 32 L 4E) i + 8 cos(2δCP)c212c 2 23s 2 12s 2 13s 2 23sin 2_(∆m2 21 L 4E)

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Graphically, the disappearance probabilities compared to each other are displayed in figure 2.7.

Figure 2.7: The neutrino oscillation probabilities P(νe→ νe), P(νµ→ νµ) and P(ντ → ντ) plotted

for the case of no CP violation(δCP = 0) and normal mass hierarchy. The current best value fits

have been used for the mixing angles and mass squared differences [50], given in section 4.9. Individually, the disappearance oscillations in the case of normal mass hierarchy are displayed in figure 2.8 for various values of δCP.

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(a) The oscillation probablility P(νe→ νe)

(b) The oscillation probablility P(νµ→ νµ)

(c) The oscillation probablility P(ντ → ντ)

Figure 2.8: The oscillation probabilities P(νe → νe), P(νµ → νµ) and P(ντ → ντ) plotted for

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Notice that for the disappearance probabilities, the effect of CP violations is only significant for sufficiently high L/E. Values like these are unrealistically high for experiments where the neutrino source is on earth.

The dependency of P(νµ → νµ) and P(ντ → ντ) on δCP is displayed in figure 2.9.

(a) A contourplot of P(νµ→ νµ) (b) A contourplot of P(ντ → ντ)

Figure 2.9: Contourplots of the disappearance probabilities plotted as a function of δCP and L/E

for normal mass hierarchy. 0 < δCP < 2π and 0 < L/E < 20000 km/GeV. Lighter colors represent

a higher value of the corresponding probability.

2.3.4 Small L/E approximation

For short baseline experiments, where L/E is small, it is useful to approximate these probabilities. When these oscillation probabilities are considered for small L/E,

sin2(∆m2 31 L 4E) ≈ sin 2_(∆m2 32 L 4E) sin 2_(∆m2 21 L 4E), because ∆m 2 31≈ ∆m232 ∆m221. The

Oscil-lation probabilities given above then reduce to:

P (νe→ νµ) =4c213s 2 13s 2 23sin 2_(∆m2 31 L 4E) (2.42) P (νe→ ντ) =4c213c 2 23s 2 13sin 2_(∆m2 31 L 4E) (2.43) P (νµ→ ντ) =4c413c 2 23s 2 23sin 2_(∆m2 31 L 4E) (2.44) P (νe→ νe) =1 − 4c213s 2 13sin 2_(∆m2 31 L 4E) (2.45) P (νµ→ νµ) =1 − 4c213s 2 23(c 2 23+ s 2 13s 2 23) sin 2_(∆m2 31 L 4E) (2.46) P (ντ → ντ) =1 − 4c213c223(s223+ c223s213) sin 2 (∆m231 L 4E) (2.47)

Therefore, for short baseline experiments the Dirac phase δCP is not tested.

2.3.5 Large L/E approximation

One can evaluate the oscillation probabilities for large L/E. This approximation is made in solar neutrino experiments and occasionally in atmospheric neutrino experiments. In this approximation

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the terms involving ∆m2

31 and ∆m232 are rapidly oscillating and will average out

sin2(∆m2₃₁ L 4E) →< sin 2_(∆m2 31 L 4E) >= 1/2 (2.48) sin2(∆m2₃₂ L 4E) →< sin 2_(∆m2 32 L 4E) >= 1/2 (2.49)

For solar neutrinos it is useful to evaluate electron neutrinos. The probabilities for an electron neutrino to oscillate or not to oscillate reduce in this approximation to

P (νe→ νe) = 1 − 4c212c 4 13s 2 12sin 2_(∆m2 21 L 4E) − 2c 2 13s 2 13 (2.50) P (νe→ νµ) + P (νe→ ντ) = 4c212c 4 13s 2 12sin 2_(∆m2 21 L 4E) + 2c 2 13s 2 13 (2.51)

For some fits on experimental data, θ13is approximated as 0. This can be done for convenience

in the fit, but it was also a standard procedure for solar neutrino experiments that where conducted before the mixing angle θ13 has been proved to be nonzero in 2012 [7]. A comparison between the

exact oscillations and the approximations made for solar neutrinos is displayed in figure 2.10.

(a) The probability P(νe→ νe) (b) The probablity P(νe→ νµ)+P(νe→ ντ)

Figure 2.10: A comparison of the exact oscillation (blue line) according to equations (2.39) (2.36) and (2.37), the approximation where ∆m2

31 and ∆m232 averaged out(red line) and the

approxi-mation where ∆m2

31 and ∆m232 are averaged out and θ13 = 0 (yellow line). The current best

values for the oscillation parameters, given in section 4.9, have been used in the plot of these probabilities.[50]

2.4 The 3 neutrino MSW effect

The method used to calculate the MSW effect for 3 neutrinos is similar to the derivation of the MSW effect for 2 neutrinos.

First of all, the evolution equation for 3 mass eigenstates takes the form

i~_dtd   ν1 ν2 ν3  = H   ν1 ν2 ν3   (2.52)

and the flavour neutrinos are defined as   νe νµ ντ  = UP M N S   ν1 ν2 ν3  . (2.53) Therefore i~_dtd   ν1 ν2 ν3  = U † P M N Si~ d dt   νe νµ ντ  = H   ν1 ν2 ν3  = HU † P M N S   νe νµ ντ  , (2.54)

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where in vacuum the Hamiltonian has the form H =   E1 0 0 0 E2 0 0 0 E3  ≈ E   1 0 0 0 1 0 0 0 1  + c4 2E   m21 0 0 0 m2 2 0 0 0 m2 3  . (2.55)

And since constants will drop out eventually, the Hamiltonian can be written as

H = c 4 2E   m2 1 0 0 0 m2 2 0 0 0 m2₃  . (2.56)

Multiplying on the left in eq. (2.54) with UP M N S gives

i~_dtd   νe νµ ντ  = UP M N SHUP M N S†   νe νµ ντ  . (2.57)

Where the term UP M N SHUP M N S† ≡ Hf

Because in normal matter only νe experience CC interactions during propagation, eq. (2.57)

takes the form

i~_dtd   νe νµ ντ  = " Hf+   VCC 0 0 0 0 0 0 0 0   #  νe νµ ντ  . (2.58)

Where the constant VCC is the potential that arises from the charged current interaction

between electron neutrinos and electrons. This constant equals VCC =

√

2GFNe, where GF is the

Fermi constant and Neis the electron number density.

Because of this potential, the Hamiltonian can be written in the form of a matter modified Hamiltonian. Hm   ν1 ν2 ν3  = c4 2E   m2 1 0 0 0 m2 2 0 0 0 m2 3     ν1 ν2 ν3  + VCC   c2 12c213 c12c213s12 c12c13s13e−iδ c12c213s12 c213s212 c13s12s13e−iδ c12c13s13eiδ c13s12s13eiδ s213     ν1 ν2 ν3   (2.59) The matter modified Hamiltonian Hmcan be diagonalized to the form

Hm= c4 2E   m2 1m 0 0 0 m2 2m 0 0 0 m2 3m  , (2.60) where m2

1m, m22m and m22m are the mass modified eigenstates. These mass eigenstates are the

solution of 0 = m2 1+ ∆V c212c213− λ ∆V c12c213s12 ∆V c12c13s13e−iδ ∆V c12c213s12 m22+ ∆V c132 s212− λ ∆V c13s12s13e−iδ ∆V c12c13s13eiδ ∆V c13s12s13eiδ m23+ ∆V s213− λ , where ∆V = 2EVCC c4 .

This equation can be written as

0 = (m2₁− λ)(m22− λ)(m 2 3− λ) − ∆V h s2₁₃(m2₁− λ)(m22− λ) + c2 s2 (m2− λ)(m2_{− λ)} _(2.61)

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where the three solutions of λ correspond to the mass modified eigenmasses m2

1m, m22mand m22m.

In polynomial form this looks like 0 =λ3− (m2 1+ m 2 2+ m 2 3+ ∆V )λ 2 +hm2₁m2₂+ m2₁m2₃+ m2₂m2₃+ ∆Vc2₁₂c2₁₃(m2₂+ m2₃) + c2₁₃s2₁₂(m2₁+ m2₃) + s2₁₃(m2₁+ m2₂)iλ − m2 1m 2 2m 2 3− ∆V c 2 12c 2 13m 2 2m 2 3+ c 2 13s 2 12m 2 1m 2 3+ s 2 13m 2 1m 2 2. (2.62)

This is a third order polynomial function of the form ax3+ bx2+ cx + d = 0 and can be solved using Cardano’s solution for the cubical formula.[93]

The general solutions for λ are λ1= 1 3(m 2 1+ m 2 2+ m 2 3+ ∆V ) + 2 3 · 22/3 A B +√B2_{+ 4A}31/3 (2.63) − 2 6 · 21/3 B +pB2_{+ 4A}31/3 λ2= 1 3(m 2 1+ m 2 2+ m 2 3+ ∆V ) −1 + i √ 3 3 · 22/3 A B +√B2_{+ 4A}31/3 (2.64) + 1 6 · 21/3 B +pB2_{+ 4A}31/3 λ3= 1 3(m 2 1+ m 2 2+ m 2 3+ ∆V ) −1 − i √ 3 3 · 22/3 A B +√B2_{+ 4A}31/3 (2.65) + 1 6 · 21/3 B +pB2_{+ 4A}31/3 with A = − (m2₁+ m2₂+ m2₃+ ∆V )2 + 3m2₁m2₂+ m2₁m2₃+ m2₂m2₃+ ∆Vc2 12c 2 13(m 2 2+ m 2 3) + c 2 13s 2 12(m 2 1+ m 2 3) + s 2 13(m 2 1+ m 2 2) (2.66) and B = − (m2₁+ m2₂− 2m2 3)(2m 2 1− m 2 2− m 2 3)(m 2 1− 2m 2 2+ m 2 3) + 3∆Vhc213 c212(−2m41+ m42− 4m22m32+ m43+ 2m21(m22+ m23)) + s212(m41− 2m42+ 2m22m32+ m43+ 2m21(m22− 2m23)) (2.67) s2₁₃(m4₁− 4m21m 2 2+ m 4 2+ 2(m 2 1+ m 2 2)m 2 3− 2m 4 3) i − 3(∆V )2h_c2 13 c2₁₂(2m2₁− m2 2− m 2 3) + s 2 12(m 2 1− 2m 2 2+ m 2 3) − s2 13(m 2 1+ m 2 2− 2m 2 3) i − 2(∆V )3_.

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B can also be expressed in terms of mass differences. Which is as follows B = − (∆m231+ ∆m232)(∆m231+ ∆m221)(∆m232− ∆m221) + 3∆Vhc2₁₃ c2₁₂[2(∆m2₃₂)2− (∆m2 31) 2_{− (∆m}2 21) 2_] + s2₁₂[2(∆m2₃₁)2− (∆m2 32) 2_{− (∆m}2 21) 2_] _(2.68) s2₁₃[2(∆m2₂₁)2− (∆m2 32) 2_{− (∆m}2 31) 2_]i − 3(∆V )2h_c2 12c 2 13(∆m 2 31+ ∆m 2 21) + c 2 13s 2 12(∆m 2 32− ∆m 2 21) − s 2 13(∆m 2 31+ ∆m 2 32) i − 2(∆V )3

Notice how the mass eigenstates m2

2m= λ2and m23m= λ3pick up imaginary terms. This however

does not say anything about the number of real roots, since the cube roots can be complex. One can simplify the expressions by expressing the Hamiltonian in eq (2.56) in terms of mass squared differences. The Hamiltonian then takes the form

H = c 4 2E   m2 1 0 0 0 m2 2 0 0 0 m2 3  = c4 2E   0 0 0 0 ∆m2 21 0 0 0 ∆m2 31  − c4_m2 1 2E   1 0 0 0 1 0 0 0 1   (2.69) = c 4 2E   0 0 0 0 ∆m2₂₁ 0 0 0 ∆m231   (2.70)

since constants will drop out. It can easily be seen that equations (2.66) and (2.67) can be written in terms of the two independent mass eigenstate differences ∆m221and ∆m231.

A = − (∆m2₂₁+ ∆m2₃₁+ ∆V )2 + 3∆m2₂₁m2₃₁+ ∆Vc2 12c 2 13(∆m 2 21+ ∆m 2 31) (2.71) and B = − (∆m2₂₁− 2∆m231)(−∆m 2 21− ∆m 2 31)(−∆m 2 21+ ∆m 2 31) + 3∆Vhc2₁₃ c2₁₂((∆m2₂₁)2− 4∆m2 21∆m 2 31+ (∆m 2 31) 2₎ + s2₁₂(−2(∆m2₂₁)2+ 2∆m₂₁2 ∆m2₃₁+ (∆m2₃₁)2) s213(∆m221)2+ 2∆m212 ∆m231− 2(∆m231)2) i − 3(∆V )2hc213 c212(−∆m 2 21− ∆m231) + s212(−2∆m221+ ∆m231) − s2 13(∆m221− 2∆m231) i − 2(∆V )3 _(2.72)

The mass squared differences in matter are now of the form

∆m2_21m= m2_2m− m2 1m= − 3 + i√3 3 · 22/3 A B +√B2_{+ 4A}3 1/3 − 1 B +pB2_{+ 4A}3 1/3 (2.73)

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∆m2_31m= m2_3m− m21m= − 3 − i√3 3 · 22/3 A B +√B2_{+ 4A}31/3 − 1 6 · 21/3 B +pB2_{+ 4A}31/3 _(2.74) ∆m232m= m 2 3m− m 2 2m= 21/3_i √ 3 A B +√B2_{+ 4A}31/3 (2.75)

At first, imaginary terms in the expressions for mass modified eigenmasses and matter modified mass differences don’t seem to make sense, since imaginary mass squared differences will cause the oscillation probability to blow up. However, any third order polynomial function of the form ax3_{+ bx}2_{+ cx + d = 0 will have 3 real solutions if the discriminant D satisfies the condition [93]}

D = 18abcd − 4b3d + b2c2− 4ac3_{− 27a}2_b2_{≤ 0} _(2.76)

If this quantity is equal to 0, then at least two of the matter modified eigenmasses are equal to each other, which leads to at least one mass difference to be 0. If all matter modified eigenmasses are equal to each other, all mass differences will be equal to 0 and therefore there will be no neutrino oscillations. If D > 0, then there will be 1 real solution and 2 imaginary solutions, which are non-physical.

Equation (2.62) can be written in the form of the independent mass differences ∆m21 and

∆m31

0 = − λ3+ (∆m2₂₁+ ∆m2₃₁+ ∆V )λ2

−∆m2₂₁∆m2₃₁+ ∆V (1 − c2₁₃s2₁₂)∆m2₂₁+ c2₁₃∆m2₃₁

λ + c2₁₂c2₁₃∆m2₂₁∆m2₃₁∆V (2.77) If the condition in (2.76) is applied, then it will give a fourth order polynomial function. Notice how this condition is dependent on ∆m231 instead of |∆m231|. Therefore this condition will give

different solutions for normal and inverted mass hierarchy. The easiest way of evaluating this is to fill in the current best known values for θ12, θ13, ∆m221and ∆m231, given in section 4.9. For normal

mass hierarchy, this gives

− 1.86 · 10−4− 0.14∆V − 27(∆V )2_{− 2.88 · 10}−8_{(∆V )}3_{+ 5.93 · 10}−6_{(∆V )}4_{< 0} _(2.78)

The only negative solution is if 0 < ∆V < 2133eV2/c4_{, which gives a constraint for the energy of}

the neutrino multiplied with the electron number density, since ∆V = 2EVCC

c4 =

2√2GF

c4 ENe. In

other words if ENe< 2133eV

2

2√2GF ≈ 8.2 · 10

45_eV/m3

, neutrino oscillations in matter are not possible. For inverted mass hierarchy, ∆m2₃₁< 0, the condition is

− 1.43 · 10−4+ 0.12∆V − 27(∆V )2+ 2.54 · 10−8(∆V )3+ 5.63 · 10−6(∆V )4< 0 (2.79) Which gives a negative solution for 0 < ∆V < 2190 eV2/c4 _{with the exception of a small range}

of ∆V = 23038 · 10−7± 10−8 _eV2_/c4 _{where ∆V becomes positive. Therefore the the condition}

for matter effects to occur is ENe< 2190eV

2

2√2GF ≈ 8.7 · 10

45_eV/m3

. The other negative solution is the small region around ∆V = 2.3 · 10−3 which gives ENe6≈ 9.2 · 1039 eV/m3. Notice that these

numbers can easily change value due to the margins of error on the current best known values of the mixing parameters.

The Sun has an average electron number density of approximately Ne,Sun ≈ 100cm−1NA ∼

1030_m−3 _{[49], where N}

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a neutrino propagating through the sun needs to have an energy less then ∼ 107 _{GeV in order}

to experience matter affected oscillations. In the case of inverted mass hierarchy, there is a small energy range of approximately 105 eV around the energy ∼ 109 eV where oscillations are not possible.

Unfortunately, at values of ENe where the cutoff is predicted, the neutrino oscillations are

strongly suppressed by the matter effects. Figure 2.2 shows that in the two neutrino approximation the mixing angle in matter is almost 0 at the ENe values where the cutoff is predicted.

One can see that the exact expressions for the matter modified eigenmasses are inconvenient to work with. This becomes clear if one tries to calculate the matter modified mixing angles according to the procedure for the two neutrino mixing. Therefore, two different approximations can be suitable for experiments. Approximations can be made in the case of a dominant ∆m2

31,

used for accelerator experiments and the case of an active ∆m2

21, used for solar and to some extent

atmospheric neutrinos.

2.4.1 Dominant ∆m2

31 approximation

Since ∆m2

31 is about a factor 30 larger then ∆m221it is convenient to evaluate the MSW effect for

3 flavor neutrinos as ∆m2

31 ∆m221 with ∆m221≈ 0. Therefore, the Hamiltonian can be written

as H ≈ c 4 2E   0 0 0 0 0 0 0 0 ∆m2 31  . (2.80)

Then, equation (2.62) will be

0 = − λ3+ (∆m2₃₁+ ∆V )λ2−∆V c2₁₂c2₁₃∆m2₃₁λ. (2.81) This gives one trivial solution, λ = 0, and

λ+,−= 1 2(∆m 2 31+ ∆V ) ± 1 2 q (∆m2 31c4+ ∆V )2− 4c213∆m231c4∆V . (2.82)

Therefore the matter modified mass squared difference ∆m231m= m23m− m21m= λ+− λ− is

∆m2_31m= ∆m2₃₁ s cos(2θ13) − ∆V ∆m2 31c4 2 + sin2(2θ13). (2.83)

The matter modified mixing angle will satisfy the condition tan(2θ13m) = tan(2θ13) h 1 − 2EVCC ∆m2 31c4cos(2θ13) i−1 . (2.84)

This result is equal to the case of 2 flavor neutrinos where the mixing is in the 13 sector. This approximation can be useful for accelerator experiments.

2.4.2 Active ∆m2

21 approximation

Consider solar and very long baseline (VLBL) experiments where the small ∆m2

21 is active and

the large ∆m2

31 is averaged out due to the fact that ∆m2₂₁

2

L

E ∼ π. [54]

In order to simplify the evolution equation, define ˆΨα= U13† U † 23Ψαwith Ψα=   νe νµ ντ  . so the evolution equation, eq. (2.58), can be written as

id dt ˆ Ψα= (U12HU12† + U † 13V U13) ˆΨα (2.85) 4  s2 12∆m221+ c213ACC c12s12∆m221 −c13s13e−iδCPACC

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Where ACC = 2EVCC. For solar neutrinos one can consider 2EVCC = 2

√

2EGFNe ≈ 4 · 10−6.

And since |U33| = ∆m231+ s 2

13ACC ≈ ∆m232 is much larger then the other terms in the matrix,

the third state of ˆΨα=

  ˆ Ψα1 ˆ Ψα2 ˆ Ψα3 

can be approximated as decoupled from ˆΨα1and ˆΨα2. The third

eigenvalue is therefore ∆ ˆm3,m= ∆m231.

Therefore, the evolution of ˆΨα1 and ˆΨα2can be written as a two neutrino oscillation.

id dt _ˆ Ψα1 ˆ Ψα2 = c 4 2E s2 12∆m221+ c213ACC c12s12∆m221 c12s12∆m221 c212∆m221 _ˆ Ψα1 ˆ Ψα2 (2.87) = c 4 2E h∆m2₂₁ 2 + 1 2∆m 2 21 − cos(2θ12) sin(2θ12) sin(2θ12) cos(2θ12) + c2₁₃ACC 1 0 0 0 iΨˆ_α1 ˆ Ψα2 . (2.88) Compared to the case of two neutrino flavors, the first term is different, but irrelevant and in the last term, ACC is replaced with c213ACC. Therefore, the evolution equation can be solved through

the analogy ˆΨα1∼ νe2ν and ˆΨα2∼ νµ2ν. Therefore, the mass modified parameters become

∆m2_21m= ∆m2₂₁ s (cos(2θ12) − 2 cos2_(θ 13)EVCC ∆m2₂₁c4 ) 2_{+ sin}2_(2θ 12) (2.89) and tan(2θ12m) = tan(2θ12) h 1 − 2 cos 2_(θ 13)EVCC cos(2θ12)∆m221c4 i−1 . (2.90)

For solar neutrinos it is useful to calculate the oscillations of electron neutrinos. When oscilla-tions due to ∆m231are averaged out and matter effects are taken into account, eq. (2.39) reduces

to P_m3ν(νe→ νe) = 1 − 4c212mc 4 13s 2 12msin 2_(∆m2 21m L 4E) − 2c 2 13s 2 13 = c4₁₃P_m2ν(νe→ νe) + s413. (2.91) Where P2ν

m(νe → νe) is the oscillation probability in the 2 neutrino model with matter modified

oscillation parameters as in eq. (2.26). [54]

2.5 Expanded models

The measurements of the LEP showed that only active 3 neutrinos exist [48]. However, recent measurements, section 4.8, show excessive data at low energy thresholds. One way that might explain these measurements is by adding another neutrino mass into the neutrino model. Because the LEP data state that only 3 flavor neutrinos can interact through the weak force, this extra neutrino must be sterile.

2.5.1 3+1 Model

In this model the mixing among neutrinos remains intact and therefore, for 4 neutrinos there are N (N −1)₂ = 6 mixing angles, namely θ12, θ13, θ23, θ14, θ24 and θ34. In addition, there are

(N −1)(N −2)

2 = 3 physical phases, δ1, δ2 and δ3, where δ1 mimics the CP violating phase in the

3 neutrino model.

The mixing between the 3 flavor neutrinos νe, νµ, ντ and the sterile neutrino νs now has the

form     νe νµ ντ νs     =     Ue1 Ue2 Ue3 Ue4 Uµ1 Uµ2 Uµ3 Uµ4 Uτ 1 Uτ 2 Uτ 3 Uτ 4 Us1 Us2 Us3 Us4         ν1 ν2 ν3 ν4     = U3+1     ν1 ν2 ν3 ν4     . (2.92)

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The mixing matrices can be chosen to be written as: U12=     c12 s12 0 0 −s12 c12 0 0 0 0 1 0 0 0 0 1     (2.93) ˜ U13=     c13 0 s13e−iδ1 0 0 1 0 0 −s13eiδ1 0 c13 0 0 0 0 1     (2.94) U23=     1 0 0 0 0 c23 s23 0 0 −s23 c23 0 0 0 0 1     (2.95) ˜ U14=     c14 0 0 s14e−iδ2 0 1 0 0 0 0 1 0 −s14eiδ2 0 0 c14     (2.96) ˜ U24=     1 0 0 0 0 c24 0 s24e−iδ3 0 0 1 0 0 −s24eiδ3 0 c24     (2.97) U34=     1 0 0 0 0 1 0 0 0 0 c34 s34 0 0 −s34 c34     (2.98)

Therefore the mixing matrix for the 3+1 model can be written as a product of these matrices. U3+1= U34U˜24U˜14U23U˜13U12. (2.99)

The general formula for the oscillation probability derived for 3 neutrinos still holds, with the modification that the flavors now includes the sterile flavor and the sum over the mass eigenstates includes the fourth eigenstate ν4.

P (να→ νβ) =δαβ− 4 X j>k <[UαkUβk∗ U ∗ αjUβj] sin2(∆m2jk L 4E) + 2X j>k =[UαkUβk∗ Uαj∗ Uβj] sin(∆m2jk L 2E). (2.100) where j, k = 1, 2, 3, 4 and α, β = e, µ, τ, s

This implies that for disappearance experiments the oscillation probability is P (να→ να) = 1 − 4 X j>k |Uαk|2|Uαj|2sin2(∆m2jk L 4E). (2.101)

For sterile neutrinos one can assume that the mixing angles θ14, θ24 and θ34 are small such

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write the the appearance and disappearance probabilities in the short baseline approximation in terms of ’sterile mixing elements’.

P (να→ να) ≈1 − 4(1 − |Uα4|2)|Uα4|2sin2(∆m241 L 4E) (2.102) P (να→ νβ) ≈4|Uα4|2|Uβ4|2sin2(∆m241 L 4E) (2.103) Where α 6= β.

For the experiments described in section 4.8 it might be useful to include the sin2(∆m2₃₁) term, because the prefactor of sin2(∆m241) might be very small. The following oscillation probabilities

in the short baseline approximation are useful: P (νe→ νe) ≈ 1 − 4c213s 2 13sin 2_(∆m2 31 L 4E) − 4(1 − |Ue4|2)|Ue4|2sin2(∆m241 L 4E) (2.104) = P (¯νe→ ¯νe) P (νµ→ νµ) = 1 − 4c213s 2 13(1 − c 2 13s 2 23sin 2_(∆m2 31 L 4E)) − 4(1 − |Uµ4|2)|Uµ4|2sin2(∆m241 L 4E) (2.105) = P (¯νµ → ¯νµ) P (νe→ νµ) ≈4c213s 2 13s 2 23sin 2_(∆m2 31 L 4E) + 4|Ue4|2|Uµ4|2sin2(∆m241 L 4E) (2.106) = P (¯νµ → ¯νe) = P (¯νe→ ¯νµ) = P (νµ → νe) 2.5.2 3+2 Model

In some experiments, see also section 4, the best fit values for the data of the neutrino experiments prefers two different values for ∆m2₄₁ for neutrino vs anti-neutrino and appearance vs disappear-ance experiments. This might indicate that there are at least two mass splittings between mostly active and mostly sterile neutrinos. Therefore a 3+2 model has been proposed where there are 2 sterile neutrino flavors along with 3 active neutrino flavors.

In this model there are N (N +1)₂ = 10 mixing angles and (N −1)(N −2)₂ = 6 physical phases.

In the short baseline approximation, the useful probabilities for most sterile neutrino experi-ments, section 4.8, will be of the form

P (νe→ νe) ≈ 1 − 4c213s 2 13sin 2_(∆m2 31 L 4E) − 4 1 − |Ue4|2− |Ue5|2|Ue4|2sin2(∆m241 L 4E) − 4 1 − |Ue4|2− |Ue5|2|Ue5|2sin2(∆m251 L 4E) (2.107) − 4|Ue4|2|Ue5|2sin2(∆m245 L 4E) = P (¯νe→ ¯νe)

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P (νµ→ νµ) ≈ 1 − 4c213s213(c223+ s213s223) sin 2 (∆m231 L 4E) − 4 1 − |Uµ4|2− |Uµ5|2|Uµ4|2sin2(∆m241 L 4E) − 4 1 − |Uµ4|2− |Uµ5|2|Uµ5|2sin2(∆m251 L 4E) (2.108) − 4|Uµ4|2|Uµ5|2sin2(∆m245 L 4E) = P (¯νµ→ ¯νµ) P (νe→ νµ) ≈4c213s132 s223sin2(∆m231 L 4E) + 4h|Ue4|2|Uµ4|2+ Re(Ue5Uµ5∗ Ue4∗Uµ4) i sin2(∆m2₄₁ L 4E) + 4h|Ue5|2|Uµ5|2+ Re(Ue4Uµ4∗ U ∗ e5Uµ5) i sin2(∆m2₅₁ L 4E) − 4Re(Ue4Uµ4∗ Ue5∗Uµ5) sin2(∆m245 L 4E) (2.109) + 2Im(Ue5Uµ5∗ U ∗ e4Uµ4) sin(∆m241 L 2E) + 2Im(Ue4Uµ4∗ Ue5∗Uµ5) sin(∆m251 L 2E) + 2Im(Ue4Uµ4∗ U ∗ e5Uµ5) sin(∆m245 L 2E) = P (¯νµ → ¯νe) P (¯νe→ ¯νµ) ≈4c213s 2 13s 2 23sin 2_(∆m2 31 L 4E) + 4h|Ue4|2|Uµ4|2+ Re(Ue5Uµ5∗ U ∗ e4Uµ4) i sin2(∆m2₄₁ L 4E) + 4h|Ue5|2|Uµ5|2+ Re(Ue4Uµ4∗ Ue5∗Uµ5) i sin2(∆m251 L 4E) − 4Re(Ue4Uµ4∗ U ∗ e5Uµ5) sin2(∆m245 L 4E) (2.110) − 2Im(Ue5Uµ5∗ Ue4∗Uµ4) sin(∆m241 L 2E) − 2Im(Ue4Uµ4∗ Ue5∗Uµ5) sin(∆m251 L 2E) − 2Im(Ue4Uµ4∗ U ∗ e5Uµ5) sin(∆m245 L 2E) = P (νµ → νe)

In these oscillation probabilities CPT conservation is assumed, so that P (να→ νβ) = P (¯νβ → ¯να).

These equations also show that for 3+2 neutrinos, CP invariance can create tension between neu-trino and anti-neuneu-trino oscillations in SBL experiments.

Notice that the sin2(∆m2 45 L 4E) and sin(∆m 2 45 L

2E) terms arise from mixing among the sterile

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2.5.3 3+3 Model

The 3+3 model has a somewhat natural beauty, since for every active neutrino, there is a sterile counterpart. Or, in this model, one can see the active neutrino flavors as the left handed neu-trino components and the sterile neuneu-trino flavors as the right handed components. Furthermore, according to some analyses, this model fixes the incompatibility between some appearance and disappearance data [44].

This model has N (N −1)₂ = 15 mixing angles and (N −1)(N −2)₂ = 10 physical phases. One can see that analyses on experimental data is complicated due to the large amount of parameters.

2.5.4 Matter resonance in the Sun

Because sterile neutrinos cannot interact though the weak interaction, there is not only a asym-metry in the charged current interaction, like in the 2 and 3 neutrino models, but also in the neutral current interaction. For solar neutrinos, one can expand the active ∆m2₂₁ approximation where the mass squared differences larger then ∆m221 are averaged out. Here it is assumed that

∆m2_k1 ∆m2

31, ∆m221where k > 4 and that the mixing angles between the fourth or higher mass

eigenstate and the first three mass eigenstates are small. For 3+1 neutrinos, the evolution equation becomes

HmΨα= c4 2E     0 0 0 0 0 ∆m2 21 0 0 0 0 ∆m2 31 0 0 0 0 ∆m2 41     Ψα+ U3+1†     VCC+ VN C 0 0 0 0 VN C 0 0 0 0 VN C 0 0 0 0 0     U3+1Ψα (2.111) ∼ c 4 2E     0 0 0 0 0 ∆m2 21 0 0 0 0 ∆m2 31 0 0 0 0 ∆m2 41     Ψα+ U3+1†     VCC 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −VN C     U3+1Ψα (2.112) Where Ψα=     νe νµ ντ νs     , Vcc= √ 2GFNe and VN C = − √ 2GFNn/2

One can simplify the evolution equation by defining ˆΨα= U14† U † 24U † 34Ψα, such that HmΨˆα=(U23U13U12HU12† U † 13U † 23+ U † 14U † 24U † 34V U34U24U14) ˆΨα (2.113) ≈c 4 2E     0 (UP M N SHUP M N S† ) 0 0 0 0 0 ∆m2 41     ˆ Ψα + VCC     c2 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0     ˆ Ψα+ VN C     0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −c2 14c224c234     ˆ Ψα (2.114)

Since in this approximation the fourth state of ˆΨα is decoupled from the other states, the mass

squared difference ∆m2

41m= ∆m241− c214c224c234VN C2E_c4 ≈ ∆m241and the evolution equation for the

other three states reduces to the active ∆m2₂₁ approximation for 3 neutrinos with the difference that VCC → c214VCC. Therefore ∆m231m≈ ∆m 2 31and ∆m2_21m= ∆m2₂₁ s (cos(2θ12) − 2c2 13c214EVCC ∆m2 21c4 )2_{+ sin}2_(2θ 12) (2.115)

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and tan(2θ12m) = tan(2θ12) h 1 − 2c 2 13c214EVCC cos(2θ12)∆m221c4 i−1 . (2.116)

One can use a similar approach for the 3+2 model. Then one will find ∆m2

51m ≈ ∆m251, ∆m2 41m≈ ∆m251, ∆m231m≈ ∆m231 and ∆m2_21m= ∆m2₂₁ s (cos(2θ12) − 2c2 13c214c215EVCC ∆m2 21c4 )2_{+ sin}2_(2θ 12) (2.117) and tan(2θ12m) = tan(2θ12) h 1 − 2c 2 13c214c215EVCC cos(2θ12)∆m221c4 i−1 . (2.118)

Therefore a general solution for the matter modified mixing angles in a model with N neutrinos in this approximation is ∆m2_k1m= ∆m2_k1 where 3 ≤ k ≤ N and

∆m2_21m= ∆m2₂₁ v u u u u t cos(2θ₁₂) − 2( N Q i=3 c2 1i)EVCC ∆m2 21c4 !2 + sin2(2θ12) (2.119) with tan(2θ12m) = tan(2θ12) " 1 − 2( N Q i=3 c2_1i)EVCC cos(2θ12)∆m221c4 #−1 . (2.120)

Since it is assumed that the mixing angles between the first three light eigenmasses and the fourth or higher heavy eigenmasses are very small, the product in the matter modified parameters will be approximately

N

Q

i=3

c2

1i ≈ c213. Therefore the matter effects due to mixing parameters of sterile

neutrinos will be very hard to measure. So far there has been no evidence nonzero mixing angles between the light and heavy mass eigenstates.

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3. Dirac vs Majorana neutrinos

All known fundamental fermions other then neutrinos are Dirac particles as a consequence of elec-tric charge conservation. They are described by four component complex spinors that obey the Dirac equation. Because neutrinos do not have electric charge, there is a possibility that they are Majorana spinors.

A nice example of how neutrinos can be Dirac or Majorana particles:

It is an experimental fact that the only experimentally observed neutrinos are left handed neutri-nos, νL, and right handed anti-neutrinos, ¯νR.

Imagine a frame where a left handed neutrino νL is moving in the z-direction. Because neutrinos

have mass, their propagation speed is lower then the speed of light. Therefore there must be a frame of an observer that is moving faster then the neutrino. Because the observer sees the neutrino moving in the −z-direction, the handedness inverts. So the observer sees a right handed neutrino.

To explain the right handed neutrino of the observer, there are 2 possibilities: 1. Postulate 2 more states, νR and ¯νL. In this case the neutrino is a Dirac particle.

2. νR= ¯νRand νL= ¯νL, this indicates that neutrinos are their own antiparticles. In this case the

neutrino is a Majorana particle. [75]

3.1 Dirac masses

For Dirac masses the Lagrangian is of the form

L = −m¯νν = −m(¯νRνL+ ¯νLνR) (3.1)

and the Higgs-lepton Yukawa interaction looks like LH,L= − X α,β=e,µ,τ Y_αβ0lL¯αLφl 0 βR− X α,β=e,µ,τ Y_αβ0νL¯αLφν˜ 0 βR+ h.c. (3.2)

Where the right handed neutrino νR is introduced. These are the sterile neutrinos that do not

interact through the weak force.

Y0l_{is the Yukawa coupling matrix for leptons and Y}0ν_{is the Yukawa coupling matrix for neutrinos,}

which at this point is an unknown matrix. L is the lepton doublet, φ is the Higgs multiplet, lβR

and νβR are the right handed components of the leptons/neutrinos.

In matrix form the Higgs-Yukawa Lagrangian looks like: LH,L= −( v + H √ 2 )[ ¯ ~ l0_LY0ll~0_R+ν~¯_L0Y0νν~_R0 ], (3.3) where ~ ν0_L=   ν_eL0 ν_µL0 ν_{τ L}0   and ~ν 0 R=   ν_eR0 ν_µR0 ν_{τ R}0  

The presence of right handed neutrino fields implies new gauge-invariant interactions in the Yukawa sector L0 Y = − X α,β=e,µ,τ Yαβ0 LαLφνβR, (3.4) where φ = √v

2 is the Higgs expectation value. so

Lmass= − X α,β=e,µ,τ Yαβ0 v √ 2ν¯αLνβR+ h.c. (3.5) = − X α,β=e,µ,τ Mαβν¯αLνβR+ h.c., (3.6)

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where Mαβ is the mass matrix.

In general this matrix is not diagonal, so the fields ναL and νβR do not correspond to the

chiral projections of the fermionic fields. To obtain the physical fields, the matrix needs to be diagonalized with a biunitary transformation

U†M U = m. If the conditions ναL= X k=1,2,3 UαkνkL ναR= X k=1,2,3 UαkνkR

are used, this will lead to

Lneutrino mass = − X α,β=e,µ,τ Mαβν¯αLνβR+ h.c. (3.7) = − X k=1,2,3 mk¯νkLνkR+ h.c. (3.8)

Therefore the fields νk are fields with definite mass mk and are therefore physical particles.

Similary, one can write for leptons

Lleptons mass = −

X

α=e,µ,τ

ml_α¯lαLlαR+ h.c. (3.9)

So the Higgs-lepton Yukawa Lagrangian is

LH,L= − h X αβ=e,µ,τ ml_α¯lαLlαR+ X k=1,2,3 mkν¯kLνkR i + h.c. (3.10) −√H 2 h _X α=e,µ,τ Y_αl¯lαLlαR− X k=1,2,3 Y_kν¯νkLνkR i + h.c. (3.11)

One can now define Dirac neutrino fields as: νk = νkL+ νkR. And because ¯νLνL= 0 = ¯νRνR,

it follows that ¯νkνk = ¯νkLνkR+ ¯νkRνkL. Therefore the Higgs-lepton Yukawa Lagrangian can be

written as: LH,L= − h _X α=e,µ,τ ml_α¯lαlα+ X k=1,2,3 mk¯νkνk i (3.12) −h X α=e,µ,τ Yl α √ 2 ¯_l_αL_l_αR₋ X k=1,2,3 Yν k √ 2ν¯kLνkR i . (3.13)

Where massive Dirac neutrinos couple to the Higgs field through the last term.

Note: The neutrino masses are proportional to the Higgs vacuum expectation value v, like the masses of leptons. However, this model does not explain why neutrino masses are so small. In fact, the SM Higgs mechanism has no explanation for the origin of the values of quark and lepton masses at all. [54]

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3.2 Majorana masses

For Majorana masses the Lagrangian looks like

LM mass= − 1 2m¯ν C LνL+ h.c. (3.14)

where the factor 1

2 is added to avoid double counting (ν C L = C ¯ν

T L).

Therefore the Lagrangian for Majorana neutrinos cn be written as

LM ₌_LM kinetic+L M mass (3.15) =1 2 h ¯ νLi ←→ / d νL+ ¯νLCi ←→ / d ν_LC− m(¯ν_LCνL+ ¯νLνLC) i . (3.16)

Using the Euler-Lagrange equations, the Majorana field equation can be derived.

mC ¯ν_LT = iγµdµνL (3.17)

By using the definition of the Majorana field

ν = νL+ νR= νL+ νLC (3.18)

and the Majorana condition ν = νC_{, the Lagrangian can be written as}

LM ₌ 1

2ν(i¯ ←→

/

d − m)ν. (3.19)

Whereas for Dirac particles

LM _{= ¯}_ν(i←→_/_{d − m)ν.} _(3.20)

Here, one can see that

LM Majorana,L+L M Majorana,R= 1 2 ¯_{(C ¯}_νT L + νL)(i ←→ / d − m)(νL+ C ¯νLT) (3.21) +1 2 ¯ (C ¯ν_RT + νR)(i ←→ / d − m)(νR+ C ¯νRT) (3.22) = ¯(¯νL+ ¯νR)(i ←→ / d − m)(νL+ νR) (3.23) =L_DiracM . (3.24)

Therefore, 2 Majorana fields are equal to 1 Dirac field.

The fields will have to be anti commuting in order to have a Majorana Lagrangian. Since fermionic fields anti-commute, one can write

ν_LTi←→/d ν_LT = ¯νLi

←→ /

d νL. (3.25)

Now the Lagrangian can be rewritten as LM _{= ¯}_ν Li ←→ / d νL− m 2 (−ν T LC †_ν L+ ¯νLC ¯νLC ¯νLT). (3.26) In this formLM

kin has the same form as that of a massless neutrino in the SM.

The Majorana mass Lagrangian can be written as

LM mass= − 1 2m(¯ν C LνL+ ¯νLνLC) (3.27) = −1 2(νLν¯L) 0 m m 0 νC L ¯ ν_LC (3.28) = −1 2 X α,β ΨαLMαβΨ¯βR+ h.c. (3.29)

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Where 0 m

m 0

is the mass matrix in the Majorana basis and ΨβR=

νR ¯ νR = ¯ν C L νC L . In general the mass term of the Lagrangian can be written as

Lmass= −

X

α,β

MαβΨ¯αLΨβR+ h.c. (3.30)

If Mαβhas nonzero diagonal elements, the neutrinos are Majorana particles. Therefore the mass

matrix can be used to identify Majorana or Dirac particles.

For N fields Mαβ is a 2N × 2N matrix divided into blocks of N × N matrices. For a zero

diagonal: Mαβ= 0 MD M_DT 0 (3.31) Where MDis the Dirac mass matrix. If there are any terms in the diagonal of the mass matrix,

then they are called the Majorana mass terms. [54]

Note that the name Dirac mass matrix does not determine whether the mass eigenstates are Dirac or Marojana. This depends on the diagonal of Mαβ.

3.3 Three Majorana neutrinos mixing

Now consider the case where 3 neutrino flavors are Majorana particles. The previous section is now generalized for the mixing of 3 neutrinos.

LM mass = 1 2 X α,β=e,µ,τ ν_αLT C†MαβνβL+ h.c. (3.32) =1 2 X k=1,2,3 X α,β=e,µ,τ ν_kLT C†U_αk† MαβUβkνkL+ h.c. (3.33) =1 2 X k=1,2,3 mkνkLT C†νkL+ h.c. (3.34)

Where the fact is used that Mαβ can be diagonalized using

X αβ U_αk† MαβUβk= mk (3.35) or, using νT kLC† = −CνkLT = ¯νkL. LM mass= 1 2 X k=1,2,3 mkν¯kLνkL. (3.36)

So the total Lagrangian can be written as

LM =LkinM +L M mass= 1 2 X k=1,2,3 ¯ νkL(i ←→ / d − mk)νkL. (3.37) Because LM

mass in not invariant under an U(1) rotation, the mixing matrix for Majorana

particles can be written as

U = UDUM, (3.38)

Beyond the Standard Model with neutrino physics

Master of Science in Physics

Theoretical physics

Master thesis

Beyond the Standard Model with

neutrino physics

by

Robert-Jan Hagebout

6182178

60 ECT

September 2013 - November 2014

Supervisor:

dr. T.M. Nieuwenhuizen

Examiner:

Prof.dr. P.J.G. Mulders

Abstract

Contents

1

Introduction

2.

Neutrino mixing

2.1

2 Neutrino mixing in vacuum

2.2

2 Neutrino mixing in matter

2.3

3 Neutrino mixing

2.4

The 3 neutrino MSW effect

2.5

Expanded models

3.

Dirac vs Majorana neutrinos

3.1

Dirac masses

3.2

Majorana masses

3.3

Three Majorana neutrinos mixing