Universität Bielefeld

Masterarbeit

Massive Neutrinos in the Early Universe

Author:Roman BorgolteMatrikelnr.: 1911778

Gutachter:Prof. Dr. Dominik SchwarzProf. Dr. Dietrich Bödeker

Mai 2015

Contents

1 Motivation 4

2 Neutrinos In The Standard Model 52.1 History and phenomenology . . . . . . . . . . . . . . . . . . . 52.2 The Standard Model . . . . . . . . . . . . . . . . . . . . . . . 62.3 The Weak Interaction . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.1 The Dirac Equation . . . . . . . . . . . . . . . . . . . . 92.3.2 On chirality and helicity . . . . . . . . . . . . . . . . . 112.3.3 The Neutrino Case . . . . . . . . . . . . . . . . . . . . 132.3.4 Interaction Terms . . . . . . . . . . . . . . . . . . . . . 13

2.4 Charge conjugation and antiparticles . . . . . . . . . . . . . . 162.4.1 Charge conjugation C . . . . . . . . . . . . . . . . . . . 162.4.2 On anti-particles . . . . . . . . . . . . . . . . . . . . . 17

2.5 The mass-giving Higgs field . . . . . . . . . . . . . . . . . . . 17

3 Cosmological Fluid 203.1 Cosmological Principle and the energy-momentum tensor . . . 20

3.1.1 Derivation of chemical potentials . . . . . . . . . . . . 233.1.2 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4 Relativistic Neutrino Gas 264.1 Neutrino Decoupling . . . . . . . . . . . . . . . . . . . . . . . 27

4.1.1 e+e−-annihilation . . . . . . . . . . . . . . . . . . . . . 274.1.2 The effective number of neutrinos . . . . . . . . . . . . 29

5 Big Bang Nucleosynthesis 305.1 Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 30

6 Beyond The Standard Model 346.1 Motivation For Neutrino Masses . . . . . . . . . . . . . . . . . 34

6.1.1 Theoretical Motivations For Neutrino Masses . . . . . 346.1.2 Neutrino Oscillations . . . . . . . . . . . . . . . . . . . 356.1.3 The PMNS matrix . . . . . . . . . . . . . . . . . . . . 38

6.2 Neutrino Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . 396.2.1 Mass Terms in Field Theory . . . . . . . . . . . . . . . 396.2.2 General Neutrino Mass Terms . . . . . . . . . . . . . . 40

6.3 Dirac Mass Term . . . . . . . . . . . . . . . . . . . . . . . . . 416.4 Majorana Mass Term . . . . . . . . . . . . . . . . . . . . . . . 42

6.4.1 The Majorana Gedankenexperiment . . . . . . . . . . . 426.4.2 The Majorana Field . . . . . . . . . . . . . . . . . . . 43

6.4.3 Construction of the Mass Term . . . . . . . . . . . . . 456.5 Implementation of massive Neutrinos in the Standard Model . 46

6.5.1 Implementation of the Dirac masses to the StandardModel . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

6.5.2 Implementation of Majorana masses to the StandardModel . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

6.5.3 Seesaw Mechanism . . . . . . . . . . . . . . . . . . . . 486.6 Are Neutrinos Majorana or Dirac? . . . . . . . . . . . . . . . 49

7 Effects of massive neutrinos on cosmology 517.1 Early Universe Observables influenced by neutrino masses . . 517.2 Massive neutrinos and BBN . . . . . . . . . . . . . . . . . . . 517.3 Large Lepton Asymmetry . . . . . . . . . . . . . . . . . . . . 52

7.3.1 Neff of massive neutrinos . . . . . . . . . . . . . . . . . 537.3.2 If neutrinos are Majorana... . . . . . . . . . . . . . . . 55

8 Conclusion 56

Notation

General Notation

3-vectors are denoted by a fat symbol like u,v... or carry a latin index, likei, j, k..., running from 1 to 3.

4-vectors carry a greek index, like α, µ, ... and run from 0 to 3.Throughout the whole thesis, if not said explicitly, all relations are ex-

pressed down in natural units, i.e.

c = ~ = kB = 1 (0.1)

The relativistic notation

In the relativistic theory mass appears to be velocity dependent in the waythat

m =m0√

1− u2, (0.2)

where m0 and u are the rest-mass and the present velocity of the describedsystem. It is convenient to introduce the calculus of four-vectors when regard-ing relativistic processes. For that reason one defines Einstein’s space-timeas the four-vector

Xα = (t,x) , (0.3)

with x =

xyz

. One further defines the proper time asdτ =

√1− u2dt. (0.4)

With Eq. (0.3) and (0.4) we are able to derive the four-velocity as

Uα =dXα

dτ=

1√1− u2

dXα

dt, (0.5)

with which we are able to construct the four-momentum vector

Pα = m0Uα = (m,p) . (0.6)

The energy-momentum tensor is made of the rest-energies of the particlesand their movement components, i.e. the connected currents muiuk. Imag-ine m0 and P

α being the rest-mass and the four-momentum of an observedparticle then PαP β/m0 is the wanted tensor.

1 Motivation

Neutrino physics exists since almost a century and still preserves unexplainedquestions. In this thesis I want to review the main parts of the presentknowledge about neutrinos. Starting with the historical evolution, goingon with phenomenology and theoretical description, I will end up with theimplementation of massive neutrinos into the Standard Model and the effectson (isotropic, homogeneous) cosmology.

The imprint of the neutrino in today’s physics is tremendous. The elec-troweak theory of the Standard Model is formulated in terms of chiral objectsmostly because of the left-handed nature of the neutrino. Cosmologists hopeto find a link between neutrinos and dark matter or structure formation andeven the asymmetry of matter and antimatter in the Universe could havebeen influenced by the physical properties of the neutrinos. This seems tobe a huge responsibility for such a weakly interacting particle.

Even though the Standard Model gave excellent results, recent develop-ments pointed out that it has to be modified. The solar neutrino problemand atmospheric neutrinos revealed that they are oscillating between differ-ent flavours and therefore need to acquire, at least a small, mass. But aswe will see in the progression of this thesis, even the implementation of suchmasses gives rise to new unanswered questions.

One of them affects the nature of the neutrino. As the only particle inthe Standard Model the neutrino could be described as a Majorana particleinstead of, as all other fermions a Dirac particle. Both possibilities end up inunsolved problems. But with the phenomenon of double beta decays, thereexists a promising experiment that could nail down the neutrino’s nature andrule out some of the speculative theories built around the neutrino.

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2 Neutrinos In The Standard Model

2.1 History and phenomenology

In 1896 Henri Becquerel was the first who observed the phenomenon of ra-dioactivity and received, beside Marie and Pierre Curie, the Nobel prize in1903 for his discovery. The fact that during β-decay the electron is ejectedfrom the nucleus of the atom was mentioned by Niels Bohr. James Chadwick,Nobel prize winner of 1935 for his discovery of the neutron in 1932, madealready in 1914, while working under Hans Geiger, the crucial discovery thatthe spectrum of β-radiation is continuous. 1929, still before the discovery ofthe neutron, Wolfgang Pauli proposed the introduction of a neutral masslessparticle of spin one half to save the principles of energy and angular mo-mentum conservation. When Chadwick found the neutron, which then wasmeant to be the particle that emits the electron and the proposed neutrinofrom the nucleus, all cards were on the table. Enrico Fermi wrote down thefour-Fermi Hamiltonian of β-decay by using neutron, proton, electron andneutrino. The weak interaction was born (see [13]).

What was theoretically written down, became reality in 1956, when ClydeCowan and Frederick Reines discovered the neutrino experimentally. Theyreceived the Nobel prize 1995 for that huge discovery. As physics evolves andwith it the technological possibilities, physicists today believe that there arethree types of neutrinos. Each coming with it’s own family of quarks andleptons. Since the neutrino only interacts via the weak force it is very hard tomeasure and to detect. On the one hand this makes it difficult for scientiststo isolate the properties and participation of the neutrino in physics. On theother hand this gives rise to theories of not yet understood physics that couldbe explained in terms of the neutrino. In the past decades the importanceof the neutrino towards the understanding of the universe became more andmore crucial, since it is the most abundant known particle beside photonsin the universe. It plays an important role in Big Bang nucleosynthesis andis therefore significantly responsible for the origin of heavy elements, whichthemselves are the basis of life.

But before getting into the details, let us start with a brief overviewof the known properties of the neutrino. The neutrino is known to be anelectrically uncharged lepton of spin one half in units of ~ and originallybelieved to be massless. The experimental evidence for neutrino oscillationshowever shows that they must have a small mass (compared to the mass ofthe charged leptons). As mentioned before there are three types of neutrinosknown today. When one found that the muon does not decay to an electronand a photon, the first guess was, that there exists an according neutrino.

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The guess was proven right, when the muon neutrino was discovered in 1962by the group of Ledermann, Schwartz and Steinberger. But since one foundanother charged lepton, namely the tau, which was discovered by the groupof Perl in the mid-seventies it was obvious to again postulate an according tauneutrino, which has been detected at Fermilab in the 1990’s. One is inclinedto think that there is no reason why the third neutrino is the end of thetrack. But measurements [1] of the Z boson’s width of LBP have definitelyruled out the existence of a fourth neutrino species, at least if it’s lighterthan 40GeV. If there were a fourth species of neutrinos it would, by analogy,imply a new kind of matter, which would be a profound discovery. As wewill see in this thesis the seesaw mechanism (6.5.3) introduces an additional,very heavy neutrino. In speculative cosmological theories this might be thelink to leptogenesis asymmetries and dark matter.

2.2 The Standard Model

The theoretical manifest of the interactions between subatomic matter is theStandard Model of particle physics. It connects all known elementary parti-cles with the four forces, i.e. gravitation, electromagnetism, weak- and stronginteraction. Apart from gravitation the theory is unified and self-consistent.When constructing new physical models, the principle of symmetry is alwaysthe guide (see [4] 1.2). Since Emmy Noether proposed her famous theoremin 1918, we know that for every continuous symmetry in nature there is arelated conservation law and the other way around. This gives us the possi-bility to include any observed conservation law and on the other hand simplyadd symmetry constraints to the Lagrangian and maybe find “new” physics.Besides symmetries also broken symmetries are essential on the way to theStandard Model. Let us briefly sum up everything one is able to cover withit today:

• matter: all fundamental fermions that we know can be grouped in threefamilies, that behave in the same way:

(νe, e, u, d)

(νµ, µ, c, s) (2.1)

(ντ , τ, t, b) .

Here we used the common short cuts for the quarks (u, d, c, s, t, b) ,the charged leptons (e, µ, τ) and their related neutrinos (νi, i = e, µ, τ).Since all of these particles are fermions, they are all represented byDirac-spinors.

6

– leptons: According to the Dirac equation for a charged massivefermion, there always exists an anti-fermion with the same massand magnitude of spin but opposite sign of charge and oppositemagnetic moment relative to the direction of the spin. Leptonicinteractions seem to preserve electric charge and it is believed thata lepton can only change to another of the same family. Only lep-ton and antilepton of the same family can be created or destroyedtogether. These laws can be exemplified in the following decay:

µ− → νµ + e− + ν̄e (2.2)

Another important fact regarding leptons is, that every interactionconserves the lepton number, i.e. every lepton is counted with +1and every anti-lepton is counted with −1. Adding up all leptonnumbers of the upper decay one finds:

µ− → νµ + e− + ν̄e+1→ +1 + 1− 1

1→ 1. (2.3)

Both sides add up to a lepton number of 1, meaning that leptonnumber is conserved. Note that, in order to be accurate, thelepton number is defined for each flavour separately. One couldthink of adding a baryon to one of the sides, but there is alsoan according conservation law regarding baryon number, whichimmediately rules out this option.

– quarks: There are six known quarks today. Like in the leptoniccase they are fermions, but they carry only fractions of electriccharge. The up-type quarks (u, c, t ) carry (2/3)e and the down-type quarks (d, s, b) (−1/3)e of electric charge. Furthermore theycarry baryon number and can change their flavour via weak in-teractions. Every quark carries a color, making them interact viathe strong force. We drop a detailed look on that topic here, sincewe are essentially dealing with neutrino physics throughout thisthesis.

• interactions: Interactions are believed to be mediated by particles. Themost famous representative of them is the photon, which is the medi-ator of the electromagnetic force. For the remaining three forces onefinds the W±- and Z-bosons (weak interaction), the gluons (strong in-teraction) and the graviton (gravitation). All these particles are bosons

7

carrying spin 1. Since the existence of the graviton was not experimen-tally proven and the gravitation itself has no general accepted quantumtheory yet, we will drop it in this thesis.

• Higgs: The mass-giving Higgs field, which (at least) has one mediatorcalled the Higgs boson, plays an important role in the electroweaktheory and has to be considered. Mass-giving means that all rest massesof massive particles are generated by the interaction with the Higgsfield. We will deal with this topic in detail in 2.5.

To categorize which particle is affected by which interactions one adds anaccording charge to them. In other words: The gauge boson, that mediatesthe interaction, is only able to couple to the particle if the particle carries thecharge connected with this interaction. For example in the electromagneticcase, the charge is called Q and can have positive and negative values interms of the elementary charge e. To restrict the possible interactions inother ways one defines quantum numbers, as the lepton number mentionedabove, as conserved quantities.

In quantum field theory, every observed fermion comes with four possiblestates. Two spin states for both, particle and antiparticle. In case of the neu-trino this is not observed. Since a right-handed neutrino (or a left-handedantineutrino) was never observed, one simply did not include it into the Stan-dard Model. Because of this left-handedness, the neutrino is not invariantunder the parity operation, which interchanges left-handed and right-handedparticles. This is the promoter of the left-handedness of the weak theory,which will be presented in detail the following sections.

However, there exists a strong connection between the electromagneticand the weak theory. For the charged W -bosons and the Z-boson, thus allweak mediators, the according charge is called the weak isotopic spin andis illustrated with the symbol IW3 . Via the weak hypercharge Y

W there isanother possibility to couple to the Z-boson. Electric charge and the twoweak charges are linked via the following relation to each other:

Q = IW3 +Y W

2(2.4)

Table 1 shows all possible particle states regarding the electron-type lep-tonic sector and their according charges. Note that eR and ēL do not coupleto the W±, which is physically logic since there is no connected neutrino.That’s why they are called weak isotopic singlets. Carrying electric and hy-percharge, they are still coupling to the photon and the Z, in contrast totheir hypothetical neutrino counterparts νR and ν̄L. They do not have any

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Particle States IW3 YW Q L

eL −1/2 −1 −1 +1νL +1/2 −1 0 +1ēR +1/2 +1 +1 −1ν̄R −1/2 +1 0 −1eR 0 −2 −1 +1ēL 0 +2 +1 −1νR 0 0 0 +1ν̄L 0 0 0 −1

Table 1: Lepton charges in the Standard Model: With IW3 weak isotopiccharge, Y W weak hypercharge, Q electric charge, L lepton number and therelation Q = IW3 +

YW

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charges and therefore do not couple in any way. These neutrinos are calledsterile neutrinos and are not included in the Standard Model.

2.3 The Weak Interaction

This section is a brief review of the weak interaction, as described by theStandard Model of particle physics.

2.3.1 The Dirac Equation

In the Standard Model fermions are described by solutions of the Dirac equa-tion

(iγµ∂µ −m)ψ = 0, (2.5)

where γµ are the Dirac gamma matrices. The Dirac equation is the covariantform of the Schrödinger equation and in momentum space reads

(γµpµ −m)u(pµ) = 0, (2.6)

where ψ(xµ) = e−ixµpµu(pµ), since the fermion wavefunction is supposed to

be a plane wave. When searching for the solutions, it is easier to return tothe original form

Hu(pµ) = (α · p+ βm)u(pµ), (2.7)

where the coefficients α and β are determined by the requirement that a freeparticle satisfies the relativistic energy-momentum relation. In the following

9

we will suppress the arguments of the spinors and simply write u and ψ.Dirac found that these coefficients are matrices and the lowest dimensionalitythat satisfies all the requirements is of fourth order. In the Pauli-Diracrepresentation they take the form

α =

(0 σσ 0

), β =

(1 00 −1

)(2.8)

where the σ are the Pauli matrices and 1 the 2× 2-unit matrix.Considering a particle at rest in (2.7) one finds (in the Pauli-Dirac rep-

resentation)

Hu = βmu =

(m1 00 −m1

)u, (2.9)

which easily shows that there are the eigenvalues E = (m,m,−m,−m) andthe according eigenvectors

1000

,

0100

,

0010

,

0001

. (2.10)The four are surprising, since one expects two spin states for a half spinfermion. They describe two spin states for a E = m and two spin states forE = −m. The positive solutions represent the particle, while the negativesolutions describe the antiparticle (see 2.4.1. For non-vanishing momentum,(2.7) becomes,

Hu =

(m σ · pσ · p −m

)(uAuB

)= E

(uAuB

), (2.11)

where u has been divided into the two two-component spinors uA, uB. Oneretains the two coupled relations

uA =σ · p

(E −m)uB,

uB =σ · p

(E +m)uA. (2.12)

By taking u(s)A = χ

(s), where

χ(1) =

(10

), χ(2) =

(01

), (2.13)

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for the upper two (E > 0) components, one obtains the lower two componentsvia (2.12), thus

u(s) ∝(

χ(s)σ·pE+m

χ(s)

), for E > 0. (2.14)

For the E < 0 solutions we set u(s)B = χ

(s) and obtain

u(s+2) ∝( −σ·p|E|+mχ

(s)

χ(s)

), for E < 0. (2.15)

u(3) and u(4) describe a positron of energy E = −√m2 + p2 and momentum

p. It is convenient to define v(1)(pµ) ≡ u(4)(−pµ) and v(2)(pµ) ≡ u(3)(−pµ) todescribe positive energy positron states.

2.3.2 On chirality and helicity

In quantum mechanics one defines the spin operator (in this definition wewrite down the ~, which in natural units is set to 1)

Σi ≡~2

(σi 00 σi

), (2.16)

for which one can show, that it does not commute with the Hamilton operatorand therefore its eigenvalues are no conserved quantum numbers. In contrast,the projection of the spin on the direction of the momentum ei(pj) =

pj|pj | ,

Λ ≡ Σiei(pj) =~2

(σiei(pj) 0

0 σiei(pj)

), (2.17)

commutes with the Hamilton operator and therefore it eigenvalues are quan-tum numbers that can be used to label the solutions [7]. The possible eigen-values are ±~

2(or ±1

2in natral units) and are called the helicity of a particle.

In case of massive particles, the helicity is not Lorentz-invariant. It is alwayspossible to find a frame of reference in which the direction of the momentumchanges and therefore influences the helicity.

Additionally one defines the chirality or handedness. These quantities arethe eigenvalues of the chiral operator

γ5 ≡ iγ0γ1γ2γ3 =(

0 11 0

), (2.18)

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and play a crucial role in the electroweak theory. One further defines theoperators,

PL ≡1

2

(1− γ5

)PR ≡

1

2

(1 + γ5

), (2.19)

which fulfil the relations

[PL]2 = PL, [PR]

2 = PR; (2.20)

PLPR = PRPL = 0; (2.21)

PL + PR = 1. (2.22)

These relations show that PL and PR are projectors and that it is possibleto write any Dirac spinor as the sum of its left- and right-handed parts, i.e.

ψ = PLψ + PRψ = ψL + ψR. (2.23)

Note that for antiparticles the signs in (2.19) change, giving

vL(p) ≡(1 + γ5)

2v(p). (2.24)

The split of the Dirac spinor into its chiral parts is an essential feature ofthe Weinberg-Salam theory of electroweak interactions. This is mainly thecase because in nature we only observe left-handed neutrinos, so that in theinteraction term of the weak interaction one only wants the left-handed partsof the neutrino to appear, while the right-handed are suppressed. If one saysthe weak interaction, involving the W are left-handed, one means that theleft-handed components of the particle and the right-handed components ofthe antiparticle are picked out within these interactions.

In the following it is convenient and especially useful to switch fromthe Pauli-Dirac representation to the chiral (or Weyl) representation of thegamma matrices. In this representation γ0 changes to

γ0 =

(0 11 0

), (2.25)

and since γ5 = iγ0γ1γ2γ3 it accordingly changes, i.e.

γ5 =

(−1 00 1

). (2.26)

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Within this representation the order of energy states in the eigenvectorschanges and we find a spinor like

ψ =

e↑

ē↑

e↓

ē↓

, (2.27)where the arrows denote the different spin states and ē denotes the positronstate.

Note that it is convenient to define that the spinor ψ creates an antipar-ticle and destroys an particle. In the chiral representation the spinor ψLcreates a right-handed antineutrino and annihilates a left-handed neutrino.As before the hermitian adjoint spinor ψ̄L has the opposite effect.

2.3.3 The Neutrino Case

In the Standard Model the neutrino is a massless particle with a spin of onehalf. Therefore it is described by the solution of the Dirac equation, too.Rewriting (2.12), one obtains

γ5u(p) =

( p·σE+m

0

0 p·σE−m

)u(p), (2.28)

which becomes, regarding massless particles (E =| p |),

γ5u(p) = Λu(p), (2.29)

revealing that in case of massless particles helicity and chirality are the same.

2.3.4 Interaction Terms

Let us have a closer look at the Lagrange density of the Standard Model,focussing on the neutrino relevant parts. This reminder will make it easier tofind the changes when turning to additional physics like massive neutrinos.

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The full Weinberg-Salam Lagrangian has the form [7],

L = −14WµνW

µν − 14BµνB

µν

+ L̄γµ(i∂µ −

g

2σ ·Wµ − g′

Y

2Bµ

)L

+ R̄γµ(i∂µ − g′

Y

2Bµ

)R

+

∣∣∣∣(i∂µ − g2σ ·Wµ − g′Y2 Bµ)φ

∣∣∣∣2 − V (φ)−(G1L̄φR +G2L̄φCR + h.c.

). (2.30)

L and R denote left-handed and right-handed fermions, where L is a doublet,

for example L =

(νee−

)L

, and R is the singlet R = e−R. σ (the Pauli matrices)

and Y are the generators of the SU(2) and U(1) groups of gauge transfor-mations. The first line then represents the W±, Z and γ kinetic energies andself interactions, the second line introduces the left-handed doublet’s kineticterm and the couplings to the gauge bosons, while the third line does thesame to the right-handed singlets. The fourth line adds the Higgs mechanismto the Standard Model giving mass to the gauge bosons, while the last lineshows the coupling of the Higgs to the leptons and quarks.

The crucial part of the Lagrangian, regarding neutrino interaction, is thesecond line. Let us have a closer look at the interaction terms

L̄γµ(i∂µ −

g

2σ ·Wµ − g′

Y

2Bµ

)L. (2.31)

Neglecting the kinetic part of this expression one ends up with

1

2

(ν†e , e

†L

)( Bµg′ + gWµ,3 g (Wµ,1 − iWµ,2)g (Wµ,1 + iWµ,2) Bµg

′ − gWµ,3

)(γµνeγµeL

)= (g′Bµ + gWµ,3) ν

†eγ

µνe + (g′Bµ − gWµ,3) e†Lγ

µeL

−√

2g(W−µ ν

†eγ

µeL +W+µ e†Lγ

µνe

). (2.32)

To identify the gauge fields with their physical interpretations one defines,as convenient (see [15]), the following relations

W−µ = −1√2

(Wµ,1 − iWµ,2) , W+µ = −1√2

(Wµ,1 + iWµ,2) ,

Zµ = Bµ sin(θ)W +Wµ,3 cos(θW ), Aµ = −Bµ cos(θW ) +Wµ,3 sin(θW ),(2.33)

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where θW is the electroweak mixing angle or Weinberg angle and

sin(θW ) ≡g′√

g′2 + g2, cos(θW ) ≡

g√g′2 + g2

. (2.34)

The necessity of these definitions become clear if one takes a closer look at theright hand side of equation (2.32). If one would identify the electromagneticfield with Y , the first term would include an interaction with the neutrinos.Since these processes are not observed, this is an unwanted result. With theabove definitions one remedies these problems. Note that the two physicalneutral gauge fields Aµ and Zµ are thus orthogonal combinations of the gaugefields W 3µ and Bµ with the mixing angle θW .

With these representations one finds

• Coupling to W±: charged coupling

Lew = −g√2

(jµW−µ +W

+µ j

µ†) , (2.35)where the currents

jµ =∑α

ν†αγµαL

jµ† =∑α

α†Lγµνα, (2.36)

(2.37)

with α = e, µ, τ were introduced.

• Coupling to Z: neutral coupling

LeZ =−e

sin (2θW )(jn)µ Z

µ, (2.38)

where the neutral current

jµn =∑α

ν†α,Lγµνα,L − cos(2θW )α†Lγ

µαL + 2 sin2(θW )α

†Rγ

µαR, (2.39)

with α = e, µ, τ was introduced.

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2.4 Charge conjugation and antiparticles

2.4.1 Charge conjugation C

Of profound importance for the discussion in this thesis is, to understandhow particle and antiparticle states are related and, in particular, how theydiffer from each other. The charge conjugation operator is constructed toturn a particle state into its according antiparticle state. To understand themechanism we just want to represent it here referring to [15].

We start from the Dirac equation of an electron in an electromagneticfield (to obtain an e-dependence) [7],

γµ (i∂µ + eAµ)ψ −meψ = 0, (2.40)

where Aµ is the electromagnetic vector potential field and e the charge of theelectron. Complex conjugation gives

(γµ)∗ (−i∂µ + eAµ)ψ∗ −meψ∗ = 0. (2.41)

With the relation

U (γµ)∗ U−1 = −γµ, (2.42)

(which is proven in [15] on p. 63) where U is a non-singular matrix, onemultiplies (2.41) with U from the left. Taking (2.42) into account and usingthe trivial relation ψ∗ = U−1Uψ∗ one obtains

−γµ (−i∂µ + eAµ)Uψ∗ +meUψ∗ = 0. (2.43)

As a last step, we define the state

ψC ≡ Uψ∗ ≡ Cψ, (2.44)

where we implicit defined the charge conjugation operator C. The effect ofthis operator becomes immediately clear, if one pulls the minus sign into thebrackets in (2.43), thus

γµ (i∂µ − eAµ)ψC +meψC = 0. (2.45)

This result can easily be identified with the positron equation, which mani-fests in the changed sign in front of the electron charge. The charge conjuga-tion operator therefore is defined to change the sign of all charges of a givenparticle state.

Worrying about the representation of the matrix U one finds [15], that,in terms of the Pauli-Dirac matrices, U has the form

U = iγ2. (2.46)

16

2.4.2 On anti-particles

Let us just remind ourselves of the accurate definition of the anti-particle.This will be of enormous importance when we introduce Majorana particles.Historically the anti-particle was found when evolving the non-relativisticquantum theory into a relativistic invariant one. Dirac then found that therewere eigenstates of the relativistic Hamiltonian that gave rise to negativeenergy states.

Today we define an anti-particle as a particle that is identical in mass,life-time and spin but carries the opposite values in all charges. Chargesin this case mean electric charge (or more explicit weak hypercharge andweak isospin), the color charge of the strong interaction and all charge-likequantum numbers, like lepton number and baryon number. These charge-like quantum numbers are defined as the difference between particles andanti-particles in occurring interactions. They regulate interactions like themuon decay

µ− → e− + ν̄e + νµ1µ = 1e + (−1e) + 1µ

where we counted every lepton with +1 and every anti-lepton with −1 andthe subscript shows the family dependence. Since one defines the leptonnumber as the difference of particles and anti-particles, one easily checksthat even the difference is conserved on both sides (+1µ).

2.5 The mass-giving Higgs field

The massiveness of the neutrino is the fundamental change to the StandardModel in this thesis. It is therefore important to have a look at the mechanismthat generates the masses of all known fundamental particles, called the Higgsmechanism. The idea of the Higgs field arises from the fact, that the weakgauge bosons have a mass, in contrast to the photon or the gluon. Let usshortly review the most important facts.

The Higgs mechanism is used to generate the three massive gauge bosonsof the weak interactions, but needs to leave the photon massless. Thereforeone defines a new SU(2)-doublet with Y = 1 and adds it to the Lagrangian,

Φ =

(φ+

φ0

), (2.47)

LHiggs = (DµΦ)†DµΦ− µ2Φ†Φ− λ

(Φ†Φ

), (2.48)

where φ+ = φ1 + iφ2, φ0 = φ3 + iφ4 and Dµ = i∂µ − gσ2Wµ − g

′ Y2Bµ

the covariant derivative of the electroweak theory. Note that through the

17

covariant derivative the gauge fields Wµ and Bµ are introduced and oneadds the dynamic parts ∝ WµνW µν and ∝ BµνBµν to the Lagrangian. Forthe crucial step of symmetry breaking one determines the minimum of thepotential part of the Lagrangian, which is the ground state φground of thefield,

V(Φ†Φ

)= µ2Φ†Φ + λ

(Φ†Φ

)2∂V(Φ†Φ

)∂ (Φ†Φ)

∣∣∣∣Φ†Φ=0

= 0

Φ†Φ =−µ2

2λ≡ v2

φ21 + φ22 + φ

23 + φ

24 = v

2, (2.49)

and has to choose any condition that fulfils it. Obviously there are infinitepossible choices, but for the Standard Model inclusion it is helpful to have alook at the underlying symmetries. The gauge symmetries of the electroweaktheory are SU(2) and U(1)Y , but especially the U(1)em with its generator

Q = T 3 +Y

2. (2.50)

According to [7] any broken symmetry generates a massive gauge boson,meaning that in our case, the choice of the vacuum expectation value φgroundneeds to break SU(2) and U(1)Y , but has to leave U(1)Q invariant. Oneconventionally chooses the first component of the Higgs doublet as φ+. Theplus indicates that this part of the doublet couples to the photon and there-fore is the charged part. The choice of the vacuum expectation value thenneeds to be made in the second component of the doublet, because the pho-ton is meant to be massless. Consequently φ1 and φ2 must have vanishingexpectation values. We ask the field φground to be real and choose

φ1 = φ2 = φ4 = 0,φ3 = v

φground ≡(

0v

). (2.51)

We found that Φ is a state that is shifted from the potential minimum φgroundand can be described by the decomposition Φ = φground + h(x), where thefield h(x) is real. Now one can rewrite the Lagrangian (2.47) in terms of thespontaneously broken Φ. The potential part becomes

V(Φ†Φ

)= m2h2 +

m2h3√2v

+m2h4

8v2= V (h), (2.52)

18

and the covariant derivative yields

DµΦ = i

(0∂µh

)− g

′

2

(0

Bµ (v + h)

)− g

2

(√2W+µ (v + h)−W 3µ (v + h)

). (2.53)

Rearranging then gives the Higgs sector that generates the masses of theweak gauge bosons.

Since this thesis is about neutrinos, we are more interested in the waythe Higgs field generates the masses of the leptons. To do that one includesa SU(2)× U(1)-invariant term in the Lagrangian:

LM = −Ge[(ν̄e, ē

)L

(φ+

φ0

)eR + ēR

(φ+, φ0

)(νee

)L

]. (2.54)

Then one spontaneously breaks and substitutes Φ = 1√2

(0

v + h(x)

), giving

−Ge√2

[v (ēLeR + ēReL) + h(x) (ēLeR + ēReL)] . (2.55)

One easily identifies the chiral form of the Dirac mass term and thereforechooses Ge so that

me =Gev√

2, (2.56)

turning (2.54) into

LM = −meēe−mevēeh. (2.57)

Besides the mass term one receives an interaction term coupling the Higgsscalar to the electron. But since the coupling me

vis very small it has no

detectable effect in electroweak interactions (see [7]). In particular there isno mass term for the neutrinos.

In this review of the Higgs sector we focussed on the electron flavours.The muon and tau flavours act in the same way.

19

3 Cosmological Fluid

3.1 Cosmological Principle and the energy-momentumtensor

One of the basic assumptions of the hot Big Bang model of the Universe isthat it is homogeneous and isotropic [18]. Lemâıtre, Friedman, Robertsonand Walker proposed the following metric which describes a homogeneousisotropic space

ds2 = dt2 −R2(t)(

dr2

1− kr2+ r2dθ2 + r2 sin θdφ2

). (3.1)

Here (t, r, θ, φ) are the coordinates, R(t) is the cosmological scale factor andk ∈ (−1, 0, 1) depends on the curvature, i.e. negative, flat and positivecurvature. Isotropy and homogeneity constrain the energy-momentum tensorto be maximal invariant in its spatial components, thus it obtains the form

T 00 = ρ(t), T 0i = 0, T ij = −p(t)gij, (3.2)

or, in terms of (0.5),

T µν = (ρ+ p)UµUν − pgµν . (3.3)

This is the form of a perfect fluid, if one identifies ρ as the rest energy densityand p as the pressure, which makes clear that thermo- and hydrodynamicsis a proper approximation of the universe.

The energy-momentum tensor T µν obeys the covariant energy-momentumconservation

T µν;µ = 0 (3.4)

which, in terms of (3.1) takes the form [18]

T µν;ν =∂p

∂xνgµν +

1√g

∂

∂xν[√g (ρ+ p)UµUν ] + Γµνλ (p+ ρ)U

νUλ. (3.5)

g is defined as

g ≡ − det gµν (3.6)

and in terms of Eq.(3.1) has the explicit form

g =R6r4 sin2 θ

1− kr2. (3.7)

20

Γνµλ is known as the Christoffel symbol and is defined by the metric tensor(see [2])

Γνµλ =1

2(∂µgαλ + ∂λgµα − ∂αgµλ) gαν . (3.8)

Note that in case of (3.1) Γi00 = 0, i.e. the spatial part of the trace isvanishing. Since the contents of the universe are, on the average, at rest inthe spatial coordinate system of (3.1) [18], the same effect happens to thevelocity four-vector Uµ,

U0 = 1 (3.9)

U i = 0. (3.10)

Taking this into account Eq. (3.5) simplifies to one equation, namely

R3(t)dp

dt=

d

dt

{R3(t) [ρ(t) + p(t)]

}. (3.11)

Introducing the Hubble constant

H =Ṙ(t)

R(t)(3.12)

it is easy to turn this expression into

ρ̇+ 3ρH = −3pH. (3.13)

Let us quickly review some basic thermodynamics in Minkowski space-time before going on. To avoid confusion the momentum will be denoted withq, since the pressure is denoted by p. The number density n, energy density ρand pressure p of a weakly-interacting gas of particles with g internal degreesof freedom (such as spin etc.) is given in terms of its phase space distributionfunction f(q, T, µ):

n(T, µ) =g

(2π)3

∫f(q, T, µ)d3q (3.14)

ρ(T, µ) =g

(2π)3

∫E(q)f(q, T, µ)d3q (3.15)

p(T, µ) =g

(2π)3

∫| q |2

3Ef(q, T, µ)d3q (3.16)

where E2 =| q |2 +m2 and T and µ are the temperature and the chemi-cal potential. For particle species in thermal equilibrium the phase space

21

distribution f(q, T, µ) is given by the Fermi-Dirac or the Bose-Einstein dis-tributions

f(q, T, µ) =1

exp [(E(q)− µ) /T ]± 1(3.17)

with µ the chemical potential of the species and +1 for the fermion case and−1 for the bosons.

One can use these expressions to re-define T µν

T µν =

∫fqµqν

q0d3q

(2π)3(3.18)

which in particular gives

T 00 = ρ =

∫fq0

d3q

(2π)3(3.19)

T ij = pδij =1

3

∫fq2

q0δi0

d3q

(2π)3. (3.20)

Note that, as defined before q0 = E =√q2 +m2 and that homogeneity and

isotropy ensures that the pressure p is the same in all directions.If one defines another four-vector

Nµ =

∫fqµ

q0d3q

(2π)3(3.21)

with the zero component N0 = n, the particle density and N i as the flux ofparticles (which is vanishing because of the cosmological principle), one gainsall thermodynamic equations (3.14) in the compact relativistic notation.

Since it is a very often used quantity, let us just write down the mostgeneral number density for neutrinos. We are focussing on the very earlyuniverse, where the temperature is high enough to, even if they have a verylittle mass, neglect the neutrino mass and set E = q. One then obtains

N0 = n(µ, T ) = 4πg

(2π)3

∫ ∞0

q2

exp(q−µT

)+ 1

dq

= − gπ2T 3Li3

(−eµ/T

), (3.22)

where g denotes the internal degrees of freedom of the particle and Li3 isthe polylogarithm function [12] with basis 3. If the chemical potential µ andis much smaller than the temperature T , this expression yields the familiarform

n(µ, T ) =g

π2T 3

3

4ζ(3) (3.23)

22

3.1.1 Derivation of chemical potentials

In the previous section the theoretical basis, for describing the expanding Uni-verse as a fluid, was introduced. In this section we want to look closer at thechemical potentials of the ingredients of the early Universe. Of special in-terest is the relation between them in chemical equilibrium. This section isstrongly reclined to [18].

Let us take the Universe as an ideal gas of particle species i, with internaldegrees of freedom gi, whose number density n, in thermal equilibrium andin the fluid rest frame u0 = 1, ui = 0, can be described by

N0 = n = gi

∫fd3q

(2π)3, (3.24)

with f either the Fermi-Dirac or the Bose-Einstein distribution function(3.17).

If the species i is in chemical equilibrium, its chemical potential µi isrelated to the chemical potentials of the species which it interacts with. Forexample, i is interacting with j, k and l,

i+ j ↔ k + l,

then

µi + µj = µk + µl. (3.25)

The chemical potentials must be determined from the consideration ofthe conservation laws obeyed by various possible reactions. The basic rule isthat µi is additively conserved in all reactions. According to [18] the followingrules are set for the determination of the chemical potentials:

• Photons can be emitted and absorbed in an arbitrary reaction in anynumber, so µγ = 0. In particular, the photon is its own antiparticle.

• Particle-antiparticle pairs can annihilate into photons, so the chemicalpotentials of a particle and its antiparticle are equal and opposite sign.

• Electrons and muons can be converted into their associated neutrinosνe and νµ by collisions with each other or nucleons, giving reactions as

e− + µ+ → νe + ν̄µ etc.

Due to the additively conserved chemical potentials this lead to

µe− + µµ+ = µνe + µν̄µ (3.26)

23

Altogether there are four independent conserved quantum numbers: chargeQ, baryon number B, electron-lepton number E (e− and νe minus e

+ and ν̄e),muon-lepton number M (as in the electron case) and tau-lepton number T(as in the electron case). The later three are collected in the lepton numberdensity defined as

L ≡∑α

nα + nνα − (nᾱ + nν̄α) , (3.27)

but we will have a closer look at this quantity in section 7.3. Hence thereare only six independent chemical potentials left to determine, which can betaken as µp, µe, µνe , µνµ , µτ , µντ . One can assign to each of these quantumnumbers a density, i.e. NQ, NB, NE, NM .

Taking into account that the average charge density NQ of the universeis zero, or at least very small, and that the baryon density NB is much lessthan the number density of the photons nγ, it seems a reasonable guess toset all conserved quantum number densities equal zero, i.e.

NQ = NB = NE = NM = 0 (3.28)

The chemical potential of particle and antiparticle are equal and oppositesign, so the four densities NQ, NB, NE and NM are odd functions of the fourchemical potentials µp, µe− , µνe , µνµ . Hence the values of the µi are simply

µi = 0, (3.29)

making all equations of (3.14) temperature dependent only.As we will see in section 6, this assumptions are (of course) not accurate,

but rather a good approximation, leading to easier calculations.

3.1.2 Entropy

According to the second law of thermodynamics, the entropy of the particleswithin in thermal equilibrium at temperature T in the volume V is a functionS(T, V ) with

dS(T, V ) =1

T[d (ρeq(T )V ) + peq(T )dV ] (3.30)

so that

∂S(V, T )

∂V=

1

T{ρeq(T ) + peq(T )} (3.31)

∂S(V, T )

∂T=V

T

dρeq(T )

dT(3.32)

24

By mixing up the last expressions one finds

∂

∂T

[1

T{ρ(T ) + p(T )}

]=

∂

∂V

[V

T

dρ(T )

dT

]. (3.33)

This expression can be rearranged, by executing the derivatives, to make itlook like

1

T(ρ(T ) + p(T )) =

dp(T )

dT. (3.34)

Integrating the first equation in (3.31) gives

S(V, T ) =

∫dV

1

T{ρeq(T ) + peq(T )}

=1

T(ρ(T ) + p(T ))V + g(T ), (3.35)

where g(T ) is an arbitrary function due to integration without limits. Theintegrability condition

∂2S

∂T∂V=

∂2S

∂V ∂T(3.36)

relates energy density and pressure:

∂2S

∂V ∂T= −(ρ(T ) + p(T ))

T 2+

1

T

(dρ(T )

dT+dp(T )

dT

),

∂2S

∂T∂V=

1

T

dρ(T )

dT

1

T

dρ(T )

dT= − 1

T 2(ρ(T ) + p(T )) +

1

T

(dρ(T )

dT+dp(T )

dT

)Tdp(T )

dT= ρ(T ) + p(T ) (3.37)

Taking the derivative of (3.35) with respect to T gives

∂S

∂T= −ρ(T ) + p(T )

T 2V +

V

T

(dρ(T )

dT+dp(T )

dT

)+dg(T )

dT. (3.38)

But with the relation (3.37) one finds

∂S

∂T= −ρ(T ) + p(T )

T 2V +

V

T

(dρ(T )

dT+ρ(T ) + p(T )

T

)+dg(T )

dT

=V

T

dρ(T )

dT+dg(T )

dT. (3.39)

Comparing this result with the second equation of (3.31) it becomes clearthat dg(T )/dt = 0 and therefore g(T ) = const. We will drop the integrationconstant and arrive at

S =ρ(T ) + p(T )

TV. (3.40)

25

4 Relativistic Neutrino Gas

Let us remind that H(t) = Ṙ(t)R(t)

(3.12). At early times R(t) is small and thedensity term must have dominated the evolution of the Universe. For thisera we can reasonable approximate the expansion rate by

H(t) =

(Ṙ

R

)2=

8πGN3

ρ(t). (4.1)

This expression plays a crucial role when questioning whether a componentis still coupled or already decoupled from the rest of the Universe. Sincewe have used the conventional distribution functions we implicitly assumedthe components to be in thermal equilibrium. In thermodynamics one refersto an equilibrium if the state of the system (number of particles at a givenenergy level) does not evolve with time. But since in an expanding universethe temperature is constantly changing, the particles have to adjust theirenergy faster than the time it takes to change the temperature. In otherwords: Since the energy of a particle is changed by its interactions withother components, the interaction rate of the particle must be faster thanthe expansion rate of the universe, i.e.

Γint(t) > H(t). (4.2)

If this condition fails to hold at some time, the considered particles are saidto decouple or freeze-out, i.e. falling out of thermal equilibrium.

In the case of neutrinos one can calculate the freeze-out time via a roughcalculation. Since they are only interacting weakly the cross section of theseinteraction is of the order G2FE

2 (at energies small compared to the W -mass).Energy is of the order of T and as calculated in (3.23) the number density isof the order T 3, thus one finds that the typical neutrino interaction rate isof the order

Γνint ' G2FT 5. (4.3)

In the very early universe, when the temperature was higher than the mass ofsome of the massive contents, it is reasonable to count these particles as radi-ation too. Therefore one can neglect the matter energy density contributionin (4.1) and thus write (see [13])(

Ṫ

T

)2=

8πGN3

π2

30g∗T

4 (4.4)

26

where g∗ =∑i

(gb)i +78

∑i(gf )i is the effective number of relativistic degrees

of freedom that contribute to the energy density. The Planck mass is definedthrough the gravitational constant

GN =1

M2Pl(4.5)

and therefore one obtains

H(t) ' 1.66g1/2∗T 2

MPl. (4.6)

4.1 Neutrino Decoupling

Comparing the neutrino interaction rate (4.3) to the approximate expan-sion rate of the early universe (4.6) reveals that Γνint is falling faster thanthe expansion rate H(t) as the temperature of the universe decreases. Theequilibrium condition implies that, below a temperature TD, when the inter-action rate starts to get smaller than the expansion rate, the neutrinos areno more in thermal equilibrium and decouple. This decoupling temperatureTD is given by

G2FT5D ' g1/2∗

T 2DMPl

(4.7)

TD '

(g

1/2∗

G2FMPl

)1/3. (4.8)

Using MPl = 1.2×1019GeV and GF = 1, 166×10−5GeV−2, one obtains TD '1MeV. This means that below a temperature of 1 MeV, or similarly after theuniverse is older than tH ' 1 s, there is (almost) no neutrino scattering. Theneutrinos expand as a free relativistic gas, with their temperature droppingby 1/R. We should mention, that the tau- and muon neutrinos have alreadydecoupled earlier, since they do participate in the e+e−-annihilation [19].

4.1.1 e+e−-annihilation

Electron-positron annihilation appears in two possible reactions

e− + e+ ↔ ν + ν̄ e+ + e− ↔ γ + γ. (4.9)

As the neutrinos decouple 1s after the big bang one of these options freezesout and the annihilation process canalises through the electromagnetic chan-nel. Before photons, neutrinos, electrons and the according antiparticles

27

were in thermal equilibrium and thus had the same temperature (especiallyTγ = Tν). As the positrons disappear after the neutrino decoupling butbefore the universe gets dilute the annihilation heats up the photons (see[3]). One can calculate the difference between the neutrino and the photontemperature by the condition of entropy conservation. Entropy fulfils therelation

Sγ + Se− + Se+ = S′

γ (4.10)

where S′γ denotes the photon entropy after the positron disappearance. Re-

calling (3.40) and taking into account that for the highly relativistic case therelation peq(T ) =

13ρeq(T ) holds, one obtains

S(V, T ) =4

3

V

Tρeq(T ). (4.11)

Since we are still in the highly relativistic era and the significant masses arevery small compared to the temperature one can interpret all involved parti-cles as radiation and use the Stefan-Boltzmann law, which relates radiationenergy to the present temperature, i.e.

ρR ≡g∗

2aSBT

4, (4.12)

where aSB =π2

15the Stefan-Boltzmann constant and g∗ =

∑i

(gb)i+78

∑i(gf )i.

Plugging (4.12) into (4.11) and summing over the degrees of freedom (4.10)becomes [

2 +7

8(2 + 2)

]V T 3 = 2V ′T ′3

V′

V

(T′γ

Tγ

)3=

11

4(4.13)

On the other hand the neutrinos, since they are already decoupled, do notinteract and therefore their entropy stays the same

Sν = S′ν(

T ′νTν

)3V ′

V= 1. (4.14)

Since photons and neutrinos were in thermal equilibrium before the decou-pling of the neutrinos, the relation Tγ = Tν holds. Thus (4.13) and (4.14)

28

give

T ′γ =

(11

4

)1/3× T ′ν = 1.4× T ′ν . (4.15)

This shows that, because of neutrino decoupling and entropy conservation,the photons get a thermal shift after neutrino decoupling. Since we knowthat the cosmic microwave background (CMB) has today a temperature ofTγ,0 ' 2.7K, the temperature of the relic neutrinos now should be Tν,0 '1.9K.

4.1.2 The effective number of neutrinos

In this section the effective number of neutrinos today is calculated, con-veniently denoted by Neff. In the last section we found, that the neutrinodecoupling takes place at a temperature of TD ' 1MeV. In this phase theexpanding Universe is still radiation dominated and if we assume that theneutrino masses are still much smaller than the temperature ([3]),

ρtot = ργ +Nνρν . (4.16)

The factor Nν is the count of different neutrino species. According to (4.12),one finds

ρtot = ργ

(1 +Nν

ρνργ

). (4.17)

Calculating the ratio ργρν

and exploiting the temperature relation betweenphotons and neutrinos, yields

Neff ≡ρνργ×

(8

7

(11

4

)4/3)=g∗νg∗γ

(T ′νT ′γ

)4×

(8

7

(11

4

)4/3)=g∗νg∗γ

= 3. (4.18)

In g∗ν the three known neutrino species and their (active) degrees of freedomwere used (3× (1 + 1)).

29

5 Big Bang Nucleosynthesis

The Big Bang Nucleosynthesis (BBN) is the phase of the Universe, in whichthe primordial abundances of light elements (2H, 3He, 4He, 7Li) are produced.It is one of the observational pillars of the hot Big Bang model and themost robust probe of the earliest phase of the Universe. In this section thestandard approach of the theory is presented. In section 7 we will see how amassive neutrino influences the procedure effectively.

5.1 Initial Conditions

At BBN the temperature of the universe forces the nucleons to form lightnuclei, like helium. In kinetic equilibrium the number density of a nuclearspecies (especially helium) is given by

nA = gA

(mAT

2π

)3/2exp

(µA −mA

T

)(5.1)

where A is the conventional symbol for the mass number of the particularnuclei and µA is the related chemical potential. If we denote the charge of thenuclei with Z, the considered nucleus is made of Z protons and A− Z neu-trons. Assuming that the nucleus forming interaction occur rapidly comparedto the expansion rate H, chemical equilibrium holds and as a consequence thechemical potential µA can be rewritten in terms of the chemical potentials ofthe nucleons

µA = Zµp + (A− Z)µn. (5.2)

Equation (5.1) also applies to protons and neutrons ([9]), thus

nA = gAA3/22−A

(2π

mNT

)3(A−1)/2nZp n

A−Zn exp (BA/T ) , (5.3)

where BA denotes the the binding energy BA ≡ Zmp + (A− Z)mn −mA ofthe nuclear species A(Z). Furthermore it is convenient to define the fractionof particle to nucleon density

Xi ≡ni

(np + nn)=

ninN

for i = p, n. (5.4)

Of special interest is the ratio between neutrons and protons, because itaffects the outcome of primordial nucleosynthesis since essentially all theneutrons get incorporated into 4He. Due to weak interactions the balance

30

between neutrons and protons is maintained. If the expansion rate H ofthe universe is much slower than the rate of these balancing interactions,chemical equilibrium obtains

µn + µν = µp + µe. (5.5)

If one defines Q ≡ mn − mp ' 1.3MeV one finds, in chemical equilibrium,according to (5.1),

n

p=nnnp

= exp

(−QT

+(µe − µν)

T

), (5.6)

with µν and µe the chemical potentials of neutrinos and electron neutrinos.

The pre-factor(mnmp

)3/2is essentially 1 and was suppressed.

Taking the reasonable assumption of charge neutrality of the universeinto account, meaning that in order to neutralize the positive charge of theprotons, the number of the electrons must be of same seize, one finds

nenγ

=npnγ∼ 10−10, (5.7)

from which follows [9] µeT∼ 10−10. The electron neutrino number of the

universe is similarly related to µνT

but in the conventional approach, it isassumed that the lepton number asymmetry is of the same order as thebaryon asymmetry, i.e. � 1, and therefore µν

T� 1. From these assumptions

follow that the equilibrium value of the proton-to-neutron ratio is given by(n

p

)eq

= exp (−Q/T ) . (5.8)

The chemical equilibrium between neutrons and protons is maintained byweak interactions, through the charged-current processes ([11])

νe + n→ e− + p ν̄e + p→ e+ + ne− + p→ νe + n n→ e− + ν̄e + p (5.9)e+ + n→ ν̄e + p e− + ν̄e + p→ n.

The rates for these processes are found by integrating the square of the matrixelement for a given process, weighted by the available phase-space densitiesof particles, while enforcing four-momentum conservation [9]. For examplethe rate for the process ep→ νn is given by

Γep→νn =

∫fe(Ee) [1− fν(Eν)] | M |2ep→νn (2π)

−5 δ4(p+ e− ν − n)

× d3pe

2Ee

d3pν2Eν

d3pn2En

, (5.10)

31

where fi are the according distribution functions. All these processes havein common the factor from the β-decay matrix element

| M |2∝ G2F(1 + 3g2A

), (5.11)

where gA ∼ 1.26 is the axial-vector coupling of the nucleon. It is possible toexpress this factor in terms of the neutron mean lifetime τn ' 880s, since

τ−1n = Γn→pνe =G2F2π3

(1 + 3g2A

)m5eλ0, (5.12)

where

λ0 ≡q∫

1

d��(�− q)2√�2 − 1 ' 1.636 (5.13)

represents a numerical factor from the phase space integral for neutron decay[9]. The neutron-proton mass difference and the electron mass determine thelimits of the integration. Substituting,

q = Q/me � = Ee/me

z = me/T zν = me/Tν , (5.14)

to gain dimensionless quantities, (5.10) turns into

Γep→νn = (λ0τn)−1

∞∫q

d��(�− q)2

√�2 − 1

[1 + exp (�z)] [1 + exp ((q − �)zν)], (5.15)

yielding, in low- and high-temperature limits

Γep→νn →{

τ−1n (T/me)3 exp(−Q/T ) T � Q,me

760π (1 + 3g2A)G

2FT

5 ' G2FT 5 T � Q,me(5.16)

Comparing this to the expansion rate of the Universe H, one finds [9]

Γ/H ' (T/0.8MeV)3 , (5.17)

for T & me. At temperatures higher than 0.8MeV one expects, that chemicalequilibrium between protons and neutrons hold. This implies, that at tem-peratures much higher than an MeV Xp ' Xn. At temperatures below thisvalue, the weak interactions become too slow (Γ < H) to keep the proton-neutron ratio in chemical equilibrium and its value is approximately givenby (

n

p

)fo

= exp (−Q/TFO) '1

6. (5.18)

32

After freeze-out, this ratio only changes due to occasional weak interactions,meaning by neutron decays. Tγ is already smaller than the binding energy ofdeuterium, but since deuterium has a low binding energy and the amount ofhigh energetic photons is still huge, deuterium synthesis starts only when thephotodissociation process becomes ineffective. This is called the deuteriumbottleneck, which is overcome at TBBN ∼ 0.07MeV [11].

Because of the high binding energy of 4He practically all deuterium nucleiare immediately burned into 4He. Helium has the highest binding energy,among the light nuclei, and therefore represents the main outcome of BBN(see [11]). It is a very good approximation to assume that all neutrons, thathave not decayed at TBBN are bound into helium nuclei. This leads to theresult for the helium mass fraction

Yp ≡4n4HenB

∼ 21 + exp(Q/Tfo) exp(

t(TBBN)τn

)∼ 0.25, (5.19)

where t(TBBN) is the age of the Universe at BBN, τn is the neutron lifetimeand Tfo is the temperature of the weak interaction freeze-out.

33

6 Beyond The Standard Model

6.1 Motivation For Neutrino Masses

6.1.1 Theoretical Motivations For Neutrino Masses

Without questioning the big success of the Standard Model in explaining thelow energy weak interactions, one could look at it and find that massless neu-trinos seem quite constructed. For example, there is no fundamental reasonfor not introducing a right-handed neutrino field (νR). If one would do so,the Higgs mechanism could produce a mass term for the neutrinos (see [13]).This makes it even more obvious, that it is just left out, to explain why oneonly observes left-handed neutrinos. In contrast there is the massless photon,whose mass properties follow from a conserved gauge symmetry. For neutri-nos we do not have any symmetry principle that predicts a massless particle.Setting constraints on a theory to get a certain result seems theoreticallyunsatisfactory.

Luckily modern experiments confirm that neutrinos seem to have a mass[5]. The formalism for introducing a mass term into the Standard Modelthat is used for the charged leptons and quarks, can also produce massiveneutrinos. One could simply describe it by a Dirac field interacting with theHiggs field. But the neutrino, as the only uncharged fermion known, couldeven be it’s own antiparticle. This was the basic idea of Majorana.

Adding a mass term of the neutrino to the Standard Model would effecthuge parts of the theory. One of these ‘new’ issues would be the possibility ofneutrino mixing. A phenomenon that is analogous to the mixing of quarks.If neutrino mixing is present the observed neutrino, for example νe at t = 0is a superposition of various mass eigenstates with different masses. As timegoes by, the different components could evolve in a different manner andtherefore, after some time, the νe looks more like a νµ or a ντ . This effect iscalled neutrino oscillation and were experimentally proven, at least indirectly,via neutrino deficits. As a consequence of neutrino oscillations it would bepossible that a νe neutrino couples to a µ after some time, which is notpossible in terms of the Standard Model. The effect of neutrino oscillationswill be presented in the next subsection.

If neutrinos were Majorana particles they must not have any additivequantum number at all, i.e. local or global. This implies that the B − L-symmetry, which is the global symmetry of the Standard Model in presenceof neutrino mixing, must be broken. One should be able to observe B − L-violating processes like the neutrinoless double beta decay. This decay seemsto be the most promising experiment today for proving the neutrino’s nature.

34

The pp Chain

Step 1: Two protons make a deuteron

p+ p→ d+ e+ + νep+ p+ e− → d+ νe (6.1)

Step 2: Deuteron plus proton makes 3He

d+ p→ 3He + γ (6.2)

Step 3: Helium-3 makes alpha particle or 7Be

3He + p→ α + e+νe3He + 3He→ α + p+ p (6.3)

3He + α→ 7Be + γ

Step 4: Berillium makes alpha particles

7Be + e− → 7Li + νe7Li + p→ α + α

7Be + p→ 8B + γ (6.4)8B→ 8Be∗ + e+ + νe

8Be∗ → α + α

Figure 1: The pp chain: how protons make alpha particles in the sun

6.1.2 Neutrino Oscillations

Since the idea of massive neutrinos was mainly catalysed by the discovery ofneutrino oscillations, we will discuss it in short here.

If one tries to calculate what is going on in the interior of the sun, neu-trinos are the perfect probes to study, as they interact weakly and find a fastway out of the sun, in contrast to the even more abundant photons. Thepp-chain (Figure 1) illustrates the possible processes that keep the sun, andother relatively light stars, alive.

Even though this is a very interesting topic itself, for our purpose thedetails are not that important. The point is that in the pp-chain are fivereactions that yield neutrinos and for each one of them the neutrinos come outwith a characteristic energy spectrum (Figure 2). John N. Bahcall calculated

35

Figure 2: The calculated energy spectra for solar neutrinos. J. N. Bahcall2002.

the abundances of the specific solar neutrinos. The majority of them comefrom the initial reaction p + p → d + e + νe, but unfortunately they carryrelative low energy. Most detectors are insensitive in this regime. A shortglance at the energy spectra in Figure 2 shows that the energetic highestneutrinos are the ones produced in the 8B reactions. Even though they arefar less abundant most experiments work with them.

In 1968 Ray Davis reported the first experiments to measure solar neutri-nos. He used a huge tank of chlorine in the Homestake mine in South Dakota.This was necessary because one wanted to eliminate the background of cosmicrays. Chlorine can absorb a neutrino and convert to argon by the reactionνe +

37Cl→ 37Ar + e. Davis collected argon atoms for several month but thetotal accumulation was only a third of what Bahcall predicted. This was thebirth of the famous solar neutrino problem.

The most obvious explanation to the results is of course that the exper-iments were simply wrong. But when other experiments that used differentdetection methods confirmed the deficit, the physic community took the so-lar neutrino problem seriously. Bruno Pontecorvo already in 1968 suggesteda beautifully simple explanation for it. He proposed that the neutrinos pro-duced by the sun are transformed in flight into different species (muon neu-trinos, or even antineutrinos). Davis’ experiment was insensitive to thesespecies and therefore was not able to detect them. This idea by Pontecorvo

36

is what one calls neutrino oscillation today.Theoretically, it is basically the quantum mechanics of mixed states,

meaning that the true stationary states for the system are evidently somelinear combinations of the neutrino types. For an easy example let us firstdrop the tau neutrino and just take the electron and the mu neutrino intoaccount. Then the possible states would be defined as

ν1 = cos θνµ − sin θνe; ν2 = sin θνµ + cos θνe (6.5)

Where we wrote the coefficients as sine and cosine to enforce normalizationin a natural way.

According to the Schrödinger equation, these eigenstates have the simpletime dependence e−iE1t/~:

ν1(t) = ν1(0)e−iE1t; ν2(t) = ν2(0)e

−iE2t (6.6)

Suppose, for example, that the particle that is emitted from the sun is anelectron neutrino, sets the boundary conditions of Eq. (6.5) as

νe(0) = 1, νµ(0) = 0, so ν1(0) = − sin θ, ν2(0) = cos θ, (6.7)

plugging these conditions into Eq. (6.6) one gets

ν1(t) = − sin θe−iE1t and ν2(t) = cos θe−iE2t. (6.8)

Going back to Eq. (6.5) and solve it for νµ, gives

νµ = sin θ cos θ(e−iE2t − e−iE1t

). (6.9)

With this equation it is possible to calculate the probability of an electronneutrino turning into a muon neutrino, after time t. Without showing thewhole calculation, even a glance at the result is enlightening:

Pνe→νµ =| νµ(t) |2=[sin(2θ) sin

(E2 − E1

2t

)]2. (6.10)

This result illustrates why one calls it neutrino oscillations : νe will convertto νµ and then, sinusoidally, back again. So far we showed how it is possiblefor the neutrinos to oscillate between two, or more realistic three, differentstates. But for now there is no direct necessity for a neutrino mass. Having aquick look at the result in Eq. (6.10) one recognizes the energy dependence.

37

The relativistic energy momentum relation is given by

E2 = m2+ | p |2=| p |2(

1 +m2

| p |2

)E ≈| p |

(1 +

1

2

m2

| p |2

)=| p | + m

2

2 | p |(6.11)

⇒ E2 − E1 ≈m22 −m21

2 | p |≈ (m

22 −m21)2E

.

This relation only holds, if one assumes that p1 = p2.Plugging this into Eq. (6.10) we find

Pνe→νµ =

{sin(2θ) sin

[(m22 −m21)c4

2~Et

]}2, (6.12)

meaning, if neutrino oscillations occur one needs unequal - and in particularnon-zero - masses for the neutrinos. In addition there has to be some kind ofmixing (θ). Since this cross-generation mixing already is part of the quarksector it is simple to demand, that it also takes place in the lepton sector.

6.1.3 The PMNS matrix

To describe neutrino oscillation one uses the mechanism that already workedfor the flavour-mixing processes in the quark sector: the observed flavourstates να, are not the ones that are mass states νi. Rather are the flavourstates linear combinations of the mass states, meaning

να =3∑i

Uiανi, (6.13)

where Uiα is an unitary matrix analogue to the CKM-matrix of the quarksector, called Pontecorvo-Maki-Nakagawa-Sakata-matrix (PMNS). As an ex-ample we write Eq. (6.5) in terms of the matrix(

ν1ν2

)=

(cos θνµ − sin θνesin θνµ + cos θνe

)=

(cos θ − sin θsin θ cos θ

)(νµνe

). (6.14)

For the realistic case of three neutrino flavours this matrix can be de-scribed by three angles and one complex phase, thusν1ν2

ν3

= c12c13 s12c13 s13e−iδ−s12c23 − c12s23s13eiδ c12c23 − s12s23s13eiδ s23c13s12s23 − c12c23s13eiδ −c12s23 − s12c23s13eiδ c23c13

νeνµντ

,(6.15)

38

where cij = cos θij, sij = sin θij and 0 ≤ θij ≤ π2 . The PMNS-matrix isa unitary transformation from the flavour space, where we write να, to themass space with νi. Since all observed physics include the flavour states, wewill use them in all relations and calculations, if not explicit said differently.

6.2 Neutrino Mass

In the last section we proofed the necessity of a neutrino mass. The mostobvious approach to include such a mass term, is the one of all other leptons,meaning adding a mass term that is based on the solution of the Dirac equa-tion. It is conventional to call the form of this mass term Dirac mass term.Even though it is possible to construct such an ‘ordinary’ mass, one ends upwith two never observed states of the neutrino. While some physicists exploitthis fact to produce new theories, that for example link those sterile states todark matter, others seem to be unsatisfied with that. Another approach wasintroduced by Ettore Majorana in 1937, in which one has to introduce a dif-ferent kind of particle. It is massive and its own antiparticle. Accordingly theresulting mass term is called Majorana. Last we will look at the differencesof the implementation of Dirac and Majorana mass terms and particles intothe Standard Model.

6.2.1 Mass Terms in Field Theory

Let us have a short look at general mass terms in field theory. For thatpurpose we define an isospin-doublet

Φ =

(φ1φ2

), (6.16)

made of two fields φ1 and φ2 with identical rest mass m0. A mass term inthe Lagrangian has the form

LM = −m02

(φ∗1φ1 + φ∗2φ2) = −

m02

Φ†Φ. (6.17)

This expression is gauge invariant which can easily be shown

Φ→ Φ′ = eiσ·θΦLM ∝ Φ

′†Φ′ = e−iσ·θφ∗1eiσ·θφ1 = Φ

†Φ.

In the case of different masses for each component of the doublet Φ this gaugeinvariance breaks, because the according mass term should look like

LM = −m12φ∗1φ1 −

m22φ∗2φ2, (6.18)

39

which can not be obtained by a simple scalar product of Φ. But if one definesa mass matrix M , one can retain the gauge invariance, since the relation

LM = −1

2

(φ∗1, φ

∗2

)(m1 00 m2

)(φ1φ2

)= −1

2Φ†MΦ, (6.19)

is gauge invariant, if one demands the matrixM to transform under the SU(2)gauge transformation U = eiσ·θ like M ′ = UMU †. Thus, as constructed,

Φ′†M ′Φ′ = Φ†U †UMU †UΦ = Φ†MΦ. (6.20)

However, having a look at the Standard Model mass terms of leptons, wefind

LM = −m(ψ̄RψL + ψ̄LψR

). (6.21)

ψL annihilates a left-handed particle state and creates a right-handed antipar-ticle state, while ψ̄R creates a right-handed particle state and annihilates aleft-handed antiparticle state. A mass term in quantum field theory there-fore changes the particle’s handedness, in this case from left to right. Eventhough the handedness is changed, the mass term preserves the particle’selectric charge and lepton number.

6.2.2 General Neutrino Mass Terms

In case of the neutrinos, the most general mass term that can be introducedto the Lagrangian of the Standard Model is

−L νmass =∑α,β

ν†α,LmαβνβR + h.c., (6.22)

where mαβ is an arbitrary 3 × 3 complex matrix, α, β run over the threeneutrino types and ναL,R are left-handed and right-handed spinor fields. Sinceany complex matrix can be put into a real valued, diagonal form with thehelp of two unitary matrices UL, UR, we can write

mαβ =∑i

UL∗αimiURβi, (6.23)

where mi are three real and positive masses. If one redefines the fields

νiL =∑α

ULαiναL,

νiR =∑α

URαiναR, (6.24)

40

(6.22) receives the form

−L νmass =∑i

mi

(ν†iLνiR + ν

†iRνiL

). (6.25)

Since unitary matrices, like UL,R, satisfy UU † = U †U = 1 it is easy to invert(6.24), i.e.

ναL =∑i

UL∗αi νiL,

ναR =∑i

UR∗αi νiR. (6.26)

As already mentioned in section 6.1.2 the e, µ and τ neutrinos are mixturesof the mass states νi. The unitary matrices therefore can be immediatelyidentified by the PMNS-matrix of (6.15).

6.3 Dirac Mass Term

The convenient fermionic mass term in quantum field theory is called Diracmass term and arises from the Dirac equation (2.40), which has the La-grangian

LD = −(iψ̄γαDαψ +mDψ̄ψ

), (6.27)

where ψ̄ = ψ†γ0 and Dα the covariant derivative according to the electroweaktheory. The mass term is then

Lmass = −mDψ̄ψ, (6.28)

which gives, with the help of (2.19), the relation [15]

Lmass = −mD(ψ̄LψR + ψ̄RψL

). (6.29)

It becomes clear that in the Dirac approach the mass term is the couplingbetween the left- and right-handed helicity parts. In section 6.1.2 we foundthat the introduction of a mass matrix is necessary because the observedneutrino flavour masses are mixtures of the neutrino mass eigenstates νi.

To obtain the needed structure, where left- and right-handed spinors cou-ple in the mass term, the mass matrix, in flavour space, needs to have avanishing diagonal, because

L Dmass ∝(ψ̄L ψ̄R

)( 0 mm 0

)(ψLψR

)= m

(ψ̄LψR + ψ̄RψL

). (6.30)

According to section 6.2.1 we find that the Dirac mass term conserveslepton number. Since it is basically the conventional quantum field theoryapproach this is not surprising.

41

6.4 Majorana Mass Term

As argued above, a mass term needs a left- and right-handed field. Forneutrinos we only observe the left-handed components. Ettore Majoranawondered, if there was a possibility to make a right-handed field from theleft-handed one and form a mass term with that. As a result one yields anew kind of particle, which is self-conjugated.

6.4.1 The Majorana Gedankenexperiment

As we already saw in particle physics the components of a spinor are alwaysconnected to a particle, an antiparticle and their possible states. In the Stan-dard Model the neutrinos, since they move with the speed of light, only haveone possible helicity state. When neutrinos gathering mass, this situationdramatically changes. There is a simple gedankenexperiment to understandwhy. Imagine an moving electron, which is left-handed. The question is:How does the same electron appear to an observer running faster than theobserved electron? To him the electron moves into the negative direction,as its spin does. This makes momentum and spin parallel to each other andthe observed electron is, in the frame of reference of the faster observer, aright-handed object. But since we got two of that kind in our model (eR andēR), how can we distinguish between them? Answering this question becomesimmediately easy if you have a look at the electric charge. The charge of theobserved electron is not affected by the boost to the faster moving frame ofreference. This means that the electron still has a negative charge, whichleads to the answer that the faster observer sees an right-handed electron.

Now imagine the same situation, considering a neutrino. In the StandardModel the neutrino has no mass and therefore it is moving with the speed oflight. If you observe a left-handed neutrino there is no way for any observerto go faster than the observed neutrino and therefore it will, in any frameof reference, stay a left-handed neutrino. But because of neutrino oscillationwe know that neutrinos do have a mass, which implies, thanks to specialrelativity, that they do not travel at the speed of light. This is why you canalways boost into a frame of reference, as in the electron-case, that is fasterthan the neutrino. According to these facts one could observe a neutrino,which is moving in positive direction and with its spin anti-parallel. Thenjust boost to a system that is moving faster and find yourself in the sameposition as in the electron-case. The faster observer sees a right-handedobject. As already mentioned we have experimental evidence that right-handed neutrinos are always antineutrinos. So since we started with a left-handed neutrino there are two possible ways to deal with this new situation:

42

• a) Postulate the two never measured states, e.g. νR and ν̄L

• b) Taking into account that the observer possibly sees ν̄R

Going way a) we will end up in the situation of the electron, apart from thecharge. Meaning we will describe the neutrino with a four-component Dirac-Spinor. Option b) gives rise to the possibility to create a massive (in contrastto the Weyl particle) two-component spinor consisting of the particle and theantiparticle state. In contrast to the electron case, there is no way for theobserver to distinguish between the particle and the antiparticle state, sincethe neutrino carries no electric charge. The only quantum number that iseffected by switching from particle to antiparticle is the lepton number (L).But the lepton number symmetry is global and does not govern the dynamics,which unscares the fact that it is broken in this way. Nevertheless the leptonnumber plays an important role in cosmological models of the early Universeand the formation of light elements (BBN).

Another question that could arise when confronted with this mechanismis: How is it possible to obtain the antiparticle by just boosting the par-ticle? One easily can answer this question since the vocabulary particle orantiparticle are defined by (locally) conserved quantum numbers (see sectionon antiparticles). But there are no conserved quantum numbers which allowus to distinguish between those two in the case of a Majorana particle. TheMajorana idea seem to fit perfectly all properties one needs to describe theneutrino, without having states that are not observed and therefore artifi-cially have to be excluded from the Standard Model.

6.4.2 The Majorana Field

In this section the theoretical description of the Majorana particle is derived.Starting from the solution of the Dirac equation (6.31), we will insist that thesolution of the charge conjugated field is the field itself again. The field wewill obtain by this procedure, will be the one that describes the phenomenologyof the idea described above. This section is strongly influenced by the textbook[13].

Let us start with the free Dirac field ψ(x), which can be written as

ψ(x) =

∫d3p√

(2π)32Ep

∑s=± 1

2

(fs(p)us(p)e

−ipx + f̂s†(p)vs(p)e

ipx)

(6.31)

with, as convenient in quantum field theory, fs(p)(f†s (p)) an operator that

annihilates(creates) a state of momentum p and spin s. The hats symbolizethe same mechanics for the according antiparticles. The spinors us(p) and

43

vs(p) are plane wave solutions of positive and negative energies of the Dirac-equation and therefore satisfy the equations

(γµpµ −m)us(p) = 0(γµpµ +m) vs(p) = 0. (6.32)

In contrast to ψ(x), ψ∗(x) annihilates an antiparticle and creates a particle.In the Majorana case we want particle and antiparticle to be identical, thusit becomes immediately clear that ψ(x) and ψ∗(x) have to be linked in someway to each other. The first guess might be

ψ(x) = ψ∗(x) (6.33)

but this expression is not Lorentz covariant which can simply be shown.Under a Lorentz Transformation that changes coordinates as

x′µ = xµ + ωµνxν (6.34)

the field changes according to the rule

ψ′(x′) = exp

(− i

4σµνω

µν

)ψ(x) (6.35)

with

σµν =i

2[γµ, γν ] . (6.36)

A complex conjugation of this expression leads to

ψ∗′(x′) = exp

(i

4σ∗µνω

µν

)ψ∗(x) (6.37)

which, in general, is not the same transformation law as in Eq.(6.35). Thisis a consequence of the not purely imaginary σµν . It turns out that Eq.(6.33)might be true in one frame of reference, but in general, breaks down whenboosting to another. Since Lorentz covariance is a general physical condition,one defines a conjugated field,

ψ̂(x) ≡ γ0Cψ∗(x), (6.38)

which ensures that both fields transform in the same way under Lorentztransformations, i.e.

ψ̂′(x′) = exp

(− i

4σµνω

µν

)ψ̂(x)

= exp

(− i

4σµνω

µν

)γ0Cψ∗(x). (6.39)

44

If one mixes up Eq.(6.35) and Eq.(6.38) one obtains

ψ̂′(x′) = γ0Cψ∗′(x′)

= γ0C exp(i

4σ∗µνω

µν

)ψ∗(x) (6.40)

Equating the last two expressions and eliminating the field one ends up with

γ0C exp(− i

4σµνω

µν

)= γ0C exp

(i

4σ∗µνω

µν

)(6.41)

To be general we impose that this holds for any arbitrary ωµν , which inparticular implies

γ0Cσ∗µν = −σµνγ0C. (6.42)

We have to admit that an exact definition of the Majorana field startswith a more general expression. One demands

ψ(x) = eiθψ̂(x). (6.43)

The phase can always be absorbed by the definition of the fermion field ψ(x),so we are doing nothing wrong by constraining ourselves to the special casewhere θ = 0. But in some cases the freedom of the phase is convenient.

The plane wave expansion of the Majorana field operator is

ψ(x) =

∫d3p√

(2π)32Ep

∑s=± 1

2

(fs(p)us(p)e

−ipx + λf †s (p)vs(p)eipx)

(6.44)

Notice that the Majorana field operator has a creation term that lookslike ∼ λf †s (p) which, in contrast to the Dirac field operator Eq. (6.31), isthe creation operator of the particle, not the antiparticle. This makes clearthat particle and antiparticle are identical except for a phase λ. This isonly conventional. One could also shift the annihilation part by a phase. Itshould be clear by now that such a spinor will violate the lepton numberconservation.

6.4.3 Construction of the Mass Term

To construct a mass term with the Majorana field, one uses the property

(ψL)C = iγ2ψ∗L = iγ

2PLψ∗ = PRiγ

2ψ∗ = (ψC)R , (6.45)

45

where the relation γ2γ5 = −γ5γ2 was used. Additionally we exploit therelation

ψ̄LψL = ψ̄PRψL = ψ̄PRPLψ = 0, (6.46)

and the analogue one ψ̄RψR = 0 and use them to rewrite ψ as

ψ = ψL + (ψC)L , (6.47)

which satisfies the equation (see [11]),

iγµ∂µψL = m (ψC)L . (6.48)

According to (6.29) we are now able to construct a proper Majorana massterm as

LMmass = −1

2m[(ψ̄C)LψL + ψ̄L (ψC)L

], (6.49)

where the factor 12

was introduced because of the double counting of degreesof freedom.

One should keep in mind that a charge conjugated field, has the sameeffect as the hermitian conjugated of that field, i.e.

ψLh.c.←→ ψ̄L = (ψL)C . (6.50)

This reveals the fact that Majorana particles violate lepton number again.In this formalism it is possible to write down terms like ψLψL. Each ofthese spinors create a right-handed antineutrino state. But since we haveconstructed the Majorana particle to be its own antiparticle, the expressionin turn can be seen as the creation of two right-handed neutrino states. Incase of neutrinos this is equivalent to raising the lepton number by 2. Or if onethinks in terms of antiparticles lowering the lepton number by 2. Either way,there is no mechanism that could remedy this, meaning that the introductionof a Majorana particle includes a violation of the lepton number symmetry.

6.5 Implementation of massive Neutrinos in the Stan-dard Model

As neutrinos interact with other particles via the weak currents, it is reason-able to question whether the form of the currents differ, after introducing amassive neutrino. Since in the Standard Model the charged current (2.36) iswritten down in terms of (6.26), the relation changes accordingly:

jµ =∑α

ν̄αLγµαL =∑α,i

ULαiν̄iLγµαL. (6.51)

In the neutral current the neutrinos are coupling to each other, and becauseof the unitary of the UL,R, it stays simply the same and the U ’s disappear.

46

6.5.1 Implementation of the Dirac masses to the Standard Model

The necessity of a neutrino mass based on the observed effect of neutrinooscillations, was discussed in section 6.1.2. We found earlier that in orderto describe these effects, analogous to the quarks, one has to introduce amass matrix. Taking a look at (6.22) immediately reveals the fact that aneutrino mass matrix needs a right-handed helicity state. But neverthelessone does not observe these states in nature. The right-handed states of theneutrino, according to the right-handed states of the charged leptons, needto be SU(2)-singlets, meaning that they do not couple to the W±, becausethey are invariant under the SU(2) gauge transformations. For that reasonone defines the SU(2)-singlet

νR ≡(νe νµ ντ

). (6.52)

Taking a look at the Higgs sector in section 2.5 the implementation of aneutrino mass reveals another barrier. The coupling of the Higgs to the leptonsector looks like (2.54) and does not give mass to the left-handed neutrino,because the neutrino sits in the first component of the SU(2) spinor. Butthis situation is familiar, since one has to deal with it in the quark sectortoo. To remedy this problem one introduces the conjugated Higgs field

ΦC = −iσ2Φ∗ =(−v − h(x)

0

), (6.53)

which has the same transformation properties under SU(2) as Φ. With thehelp of this relation one can generate a Yukawa coupling for the lepton sectoras

L =Ge√

2

[L̄ΦeR + ēRΦ

†L]

+∑α,β

Gναβ√2

[L̄ΦCνR + ν̄RΦ

†CL]

=(v + h(x))√

2

[Ge (ēLeR + ēReL)−

∑α,β

Gναβ (ν̄LνR + ν̄RνL)

], (6.54)

where α, β = e, µ, τ . (v + h(x))Gναβ as before, can be identified with themass matrix mαβ.

In the Dirac case one therefore has two sterile states (eR and ēL), whichgive rise to speculative theories of new physics.

6.5.2 Implementation of Majorana masses to the Standard Model

In section 6.4.3 we constructed a mass term, that consisted of left-handedstates alone. Table 1 shows that such a mass term has the third component

47

of weak isospin I3 = 1 and correspondingly Y = −2 ([11]). Hence it behaveslike a weak-isospin triplet and not as a singlet, which is required by gaugeinvariance. For this reason one cannot obtain such a term by spontaneoussymmetry breaking from a corresponding Yukawa term involving a singleHiggs doublet. One could use two Higgs doublets, whose product containsa weak-isospin triplet, to construct a coupling term of the mass dimension5, but since this is a huge topic with immense consequences, we will drop ithere.

The seesaw mechanism, presented in the next chapter solves the problemin a miraculous way and additionally gives a natural explanation on thesmallness of the neutrino masses.

6.5.3 Seesaw Mechanism

The only directly detectable neutrino components are the να,L. We saw that,if we introduce the three sterile right-handed singlets νβ,R we are able toconstruct a Dirac mass term like

L Dmass = −ν̄β,RMDβανα,L + h.c, (6.55)

where α and β run over the possible neutrino flavours, so that Mβα is a 3×3complex matrix. Generally it is one could think of more than those threeright-handed states and even think of sterile left-handed states too. It is alsopossible to introduce a Majorana mass term for the νβ,R,

L Rmass =1

2ν̄Cβ,RM

Rββ′νβ′,R + h.c., (6.56)

where MRββ′ again is a 3 × 3 complex matrix. In gene