Massive Neutrinos and Leptogenesis

David de Sousa Seixas

Dissertacao para a obtencao de Grau de Mestre em

Engenharia Fısica Tecnologica

Juri

Presidente: Jorge Romao

Orientador: Gustavo Castelo Branco

Vogais: David-Emmanuel Costa

Setembro 2009

Abstract

We review here the recent developments in the lepton sector since neutrino masses and leptonic mixings

have been found. We will give a brief account of neutrino oscillations which are a consequence of

neutrino mass and we will present ways to give mass to the neutrinos in the Standard Model and the

Left-Right Symmetric Model. We will then address the basics of generating the baryon asymmetry of the

universe through these new couplings in the neutrino sector, a mechanism which is called leptogenesis.

Contents

1 Introduction 3

2 Massive Neutrinos 4

2.1 Fermions in the Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.1 Gauge Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.2 Masses and Mixings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Massive Neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Majorana Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.2 Properties under discrete space-time symmetries . . . . . . . . . . . . . . . . . . . 13

2.2.3 Majorana and Dirac mass terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.4 Experimental tests on neutrino mass . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.5 Neutrinoless double-beta decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.6 Neutrino Mass Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.7 Seesaw Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 Neutrino Oscillations 29

3.1 Neutrino oscillations in vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.1.1 Two-flavour case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.1.2 Three-flavour case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.1.3 Majorana mass and neutrino oscillation . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.1.4 Do charged leptons oscillate? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.2 The MSW effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2.1 Constant density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2.2 Oscillations in non-uniform matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3 Solar Neutrino Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

1

4 Neutrino Mass Models 48

4.1 Seesaw models in the Left-Right Symmetric model . . . . . . . . . . . . . . . . . . . . . . . 48

4.1.1 The Gauge Sector and Symmetry breaking . . . . . . . . . . . . . . . . . . . . . . . 48

4.1.2 The seesaw mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.2 Neutrino masses in GUT theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.2.1 SU(5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.2.2 S0(10) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5 Leptogenesis 63

5.1 Baryon asymetry of the universe (BAU) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.1.1 Evidence for BAU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.1.2 Basic Ingredients and direct baryogenesis . . . . . . . . . . . . . . . . . . . . . . . . 65

5.2 Leptogenesis in the single flavour approximation . . . . . . . . . . . . . . . . . . . . . . . 66

5.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.2.2 CP violation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.2.3 Out-of-equilibrium dynamics and wash-out . . . . . . . . . . . . . . . . . . . . . . 70

5.2.4 Lepton asymmetry and anomalous B + L violation . . . . . . . . . . . . . . . . . . . 78

5.2.5 Baryogenesis through leptogenesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.2.6 Dependence on initial conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6 Conclusion 82

A Feynman rules for Majorana spinors 84

B Boltzmann equations for leptogenesis 85

B.0.7 Boltzmann equation for N1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

B.0.8 Boltzmann equation for B−L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

C Calculation of the strength of the CP violation 88

C.1 Tree-level contribution to Nk → ` φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

C.2 Vertex contribution to the CP asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

C.3 Self-energy contribution to the CP asymmetry (1) . . . . . . . . . . . . . . . . . . . . . . . . 93

C.4 Self-energy contribution to the CP asymmetry (2) . . . . . . . . . . . . . . . . . . . . . . . . 95

2

Chapter 1

Introduction

The proposal of neutrinos goes back to the 1930’s when Wolfgang Pauli postulated their existence as a

mean to hold conservation laws. However, at that time he confessed that such doing was in fact obscene

from a point of view of testability, since such a particle would be undetectable. The establishment of the

Standard Model at CERN and Fermilab has helped us to understand and measure neutrino properties

as being a particle playing a role in a new interaction – the weak isopsin interaction – along with the

electron. Then several other neutrinos have been found, fuelling a new problem called the flavour

problem (which is yet unsuccesfully solved) which culminated with the discovery of neutrino masses

and neutrino oscillations. At first, the full importance of the discovery of neutrino masses didn’t strike

the scientific community, although it was nontheless seen as undeniable proof of physics beyond the

Standard Model. Nevertheless, Fukugita and Yanagida [27]-[28] would link in the late years of the

century this discovery to another puzzling mystery of the universe: the origin of matter. One knows that

the Standard Model is correct up to a incredible accuracy, but it nevertheless can’t be correct as a theory of

matter because it does not provide an answer to how the universe at some point chose matter over anti-

matter. Or, more truthfully, it does qualitatively, thanks to baryon and lepton number non-conserving

non-perturbative effects, but not quantitatively. The work of the two japanese physicists is based on the

premise that without imposing any new symmetries on the Lagrangian function of the Standard Model,

the simplest way (the same way as for the other fermions) to give mass to the neutrinos includes a term

which violates lepton number by two units. A class of models called leptogenesis grew from here and

were able to predict the baryon number asymmetry of the universe.

The first part of this thesis discusses neutrinos. We will go from their role in the Standard Model to

neutrino mass models, taking special interest on the phenomena of neutrino oscillations. The second

part concerns leptogenesisin the single flavour approximation . A selection of calculations performed to

study these topics is appended to this thesis.

3

Chapter 2

Massive Neutrinos

2.1 Fermions in the Standard Model

2.1.1 Gauge Interactions

Our knowledge nowadays goes down to the Fermi scale at a very high accuracy. The widely ap-

plauded Glashow-Weinberg-Salam model of electroweak interactions is based on a generalisation of a

principle from which electromagnetism can be derived: gauge invariance. Indeed by imposing that

L(φ(x), ∂φ(x)) = L(eiθ(x)φ(x))[29] in a covariant theory, Maxwell’s equations can be obtained. The same

premise applies to the so-called standard Model of particle physics, except that it makes use of the

SU(2)L×U(1)Y group where probability amplitudes of fermion fields are grouped in (iso)doublets which

bear a (hyper)charge Y. The three generators Tk of SU(2) in the doublet representation are the Pauli

matrices divided by 2. The covariant derivative has to be corrected in order to erase the terms which are

not gauge invariant (basically which depend on the amplitude and not the probability).

The relation between the third weak isospin generator, the hypercharge and the electric charge is

Q = T3 + Y (2.1)

If g is the coupling constant for SU(2)L and g′ is the coupling constant for U(1)Y then the covariant

derivative is

Dµ = ∂µ − igWµk Tk− ig′BµY = ∂µ − i

esw

(W+µT+ + W−µT−) + ie[QAµ−

1swcw

Zµ(T3 −Qs2w)]

=

∂µ + ieQA − i ecwsw

Z( 12 −Qs2

w) −i W+√

2−i W−√

2∂ + ieQA + i e

cwswZ( 1

2 + Qs2w)

(2.2)

where Wµ, Bµ and Aµ are gauge bosons, W± ≡W1 ∓W2, the neutral gauge bosons being redefined as

4

(B

W3

)=

(cw sw−sw cw

) (AZ

)(2.3)

where c2w + s2

w = 1 and define

g =e

sw(2.4)

g′ =e

cw(2.5)

The fermions are put in representations of SU(2)L ×U(1)Y. It has been found experimentaly that only

the left chirality of fermion fields have isospin. Therefore we may write the fermion spinors as

ψ =1 + γ5

2ψ +

1 − γ5

2ψ = ψL + ψR (2.6)

and only the first term is to be introduced in a isodoublet and the second one in an isosinglet. Hence,

QL =

(uLdL

)`L =

(eLν

)uR, dR, eR. (2.7)

No right-handed neutrino RHνexists in the GWS model. The covariant derivatives of the lepton fields

are immediately derived from (2.2)

DµeL =[∂µ + ieAµ

− ie

cwswZµ(

12− s2

w)]

eL − ie

sw√

2W+ν

Dµν =[∂µ + i

ecwsw

Zµ(12

+ s2w)

]ν − i

e

sw√

2W−eL

DµeR = (∂µ + ieAµ) eR

DµνR = ∂µνR

(2.8)

We included the last derivative for an hypothetical right handed neutrino to state the obvious that it

does not have any gauge interactions. The electroweak interaction is vectored by three different currents,

the electromagnetic, the charged and the neutral ones which are obtained from the conserved Noether

currents for each symmetry

−Lint = gWaµJaµ + g′BµJYµ

=g√

2(J+µW+

µ + J−µW−µ )(2.9)

with

5

Jaµ = ψLγµ

τa

2ψ,

JYµ = −ψLγµYψ

J±µ = J1µ ± iJ2

µ = νLγµeL + uLγµdL

(2.10)

From here, the gauge fields can be rewritten in this new basis upbringing two interactions, the charged

and neutral current interactions:

Lem = −eAµJµem

LCC =e

cw√

2(W+

µ uLγµdL + W−µ eLγ

µν)

LNC = eJemµ Aµ +

gcosθW

J0µZµ

(2.11)

where

Jemµ = −ψγµQψ

J0µ = J3

µ − sin2 θW Jemµ

=12

∑ψ

[gLψγµ(1 − γ5)ψ + gRψγµ(1 + γ5)ψ

] (2.12)

Unfortunately, the gauge symmetry we have shown above is not observed in nature at current

temperatures (T = 2.735 K). The only observed gauge symmetry are SU(3)C ×U(1)Q.

In fact, the absence of quadratic terms of any gauge boson fields shows that this theory is incomplete,

since LEP and Fermilab experiments have established that gauge bosons are massive.

the previous discussion tells us is that a pure gauge theory predicts no masses for either bosons

or fermions at tree-level, so all masses must be generated at a higher radiative order with a certain

mechanism which upon minimization yields solutions which are not gauge invariant even though its

Lagrangian respects the symmetry invariance of the theory.

2.1.2 Masses and Mixings

The solution to this problem was presented by several authors in parallel in 1964[33] with the introduction

of a scalar field based on a the same mechanism that gives mass to soft photons[32]. The simplest way to

break the SM into electromagnetism and QCD is to introduce a scalar field which is an weak isodoublet

φ =

(ϕ+

ϕ0

)(2.13)

6

The additional terms to the SM Lagrangian are

LH = (Dµφ)†(Dµφ) +LYuk + V(φ) (2.14)

with

Dµϕ+ =[∂µ + ieAµ

− ie

cwswZµ(

12− s2

w)]ϕ+− i

e

sw√

2W+ϕ0

Dµϕ0 =[∂µ − i

ecwsw

Zµ(12− s2

w)]ϕ0− i

e

sw√

2W−ϕ0 (2.15)

where V(φ) = −µ2φ†φ +µ2

v (φ†φ)2 (λ > 0 and µ2 < 0). Higher order terms in φ aren’t included to

preserve the theory’s renormalizability. When the Higgs field develops a vacuum expectation value

(VEV) along its uncharged component1

⟨φ0

⟩=

v√

2(2.16)

At spontaneous symmetry breaking (SSB) point, the covariant derivative of the Higgs field yields the

following masses for the gauge bosons

MW =gv2

MZ =MW

cw(2.17)

However, no fermion masses have been generated yet. These come fromtree level diagrams of the

Yukawa couplings which are the most general gauge invariant couplings between the fermion and Higgs

fields:

−LYuk = muQLφuR + mdQLφdR + me ¯LφeR (2.18)

But since masses are still undefined, we can add as many fields as we want with different coefficients

mu,md and me. Indeed, experiment has shown that fermions come in three xerox copies (generations):

there are three up-type quarks (up, charm and top), three down-type quarks (down, strange and bottom),

three charged leptons (electron, muon and tauon) and three neutrinos (νe, νµ and ντ) which have the

exact same gauge SM interactions but have different masses.

−L =∑a,b

[QL

aM(u)a,bφuR

b + QLaM(d)

a,bφdRb + ¯L

aM(`)a,bφ`R

b]

(2.19)

1The charged one can’t have a non-zero VEV because that would break electromagnetic gauge invariance giving mass to thephoton.

7

where the indices run over the three generations of fermions. When the Higgs develops a VEV, each

fermion gets a M( f )a,b = h( f )

a,bv/√

2 mass. The h(u) and h(`) matrices may be chosen freely to be diagonal.

However, because of the CC interaction term, the matrix h(d) cannot be rotated in order to give a diagonal

matrix in this basis. We have a unitary transformation

uL = ULu0L uR = VRu0

R dL = ULd0L dR = VRd0

R (2.20)

The M(d) is a complex non-diagonal matrix whose eigenvalues are the masses of the physical states,

by which we mean the masses of the linear combinations of different fermions fields in a basis where the

mass matrices are diagonal. These can be diagonalized by the following biunitary transformation

ULM(d)V†R = D(d) (2.21)

In the new basis where the mass matrices are diagonal, the fermion mass eigenstrates are

uct

L

→ UuL

u0

c0

t0

L

,

dsb

L

→ UdL

d0

s0

b0

L

,

eµτ

L

→ U`L

e0

µ0

τ0

L

,

νeνµντ

L

→ UνL

ν0

eν0µ

ν0τ

L

uct

R

→ VuR

u0

c0

t0

R

,

dsb

R

→ UdR

d0

s0

b0

R

,

eµτ

R

→ U`R

e0

µ0

τ0

R

(2.22)

A unitary transformations which takes a linear combination of the fermion fields and leaves the

gauge interactions and the kinetic terms unchanged is called a weak basis transformation (WBT). So let’s

picture a general WBT:

M′u = W†LMuWuR = Uu

LDuV†R

M′d = W†LMdWdR = Ud

LDdV†R (2.23)

By choosing WL = UuL, Wu

R = UuR and Wd

R = UdR, we get

Mu = Du

M′d = Uu†L Ud

LDd

(2.24)

So in this particular basis, one reduces the 36 real free parameters formely contained in the Mu and

Md matrices down to six masses and four mixing parameters from VCKM ≡ Uu†L Ud

L. The CKM matrix2

2owed to the names of Cabbibo-Kobayashi-Maskawa

8

has 18 real parameters for 3 generations of fermions. Its unitarity implies 9 constraints. Additionally,

5 phases unphysical can be cancelled through redefinitions of the fermion fields. Therefore, one has in

the end 4 parameteres, of which 3 are mixing angles and one is a phase (with physical meaning). One

common parametrization is[34]

VCKM =

c12c13 s12c13 s13e−iδ

−c23s12 − s23s13c12eiδ c23s12 − s23s13s12eiδ s23c13s23c12 − c23s13c12eiδ

−s23c12 − c23s13s12eiδ c23c13

(2.25)

where ci j ≡ cosθi j and si j ≡ sinθi j. Experimental values for the magnitudes of the 9 CKM elements

are[30]

|VCKM| =

0.97383+0.00024

0.00023 0.2272+0.001−0.001 (3.96+0.009

−0.009) × 10−3

0.2271+0.001−0.001 0.97296+0.00024

−0.00024 (42.21+0.1−0.8) × 10−3

(8.14+0.32−0.64) × 10−3 (41.61+0.12

−0.78) × 10−3 0.9991+0.000034−0.000004

(2.26)

The introduction of this subtilty of multiple generations changes the form of the interactions. The

weak charged current for quarks becomes

LqCC =

g√

2uiLγ

µ(VCKM)i jd jLW+µ + h.c. (2.27)

which violates flavour, meaning through this very same interaction there can be transitions between

different eigenstates of a particular weak basis, such as transitions between different mass eigenstates.

One interesting aspect of the multiple generation issue is that three generations imply CP violation. In

fact, one can show[36] that there is a quantity which is independent of the chosen weak basis and must

be zero for CP conservation[40]

Im(Vi jVklV∗ilV∗

kj) = JCKM

3∑m,n=1

εikmε jln, (i, j, k, l = 1, 2, 3) (2.28)

which in general is of course not zero. The value of this so-called Jarlskog invariant in the previsouly

introduced parametrization is

JCKM = c12s12c213s13c13s23 sin δ (2.29)

where δ is the only unknown parameter and is called the CP violating phase.

It is important to emphasize that flavour violation only occurs in the quark sector. The major difference

lies in the absence of neutrino masses. Indeed, the leptonic charged current is

9

LCC =g√

2¯eiL(VL)∗iαγ

µναLW−µ + h.c. =g√

2¯eiL(VL)∗iα(VL)α jγ

µν jLW−µ + h.c. =g√

2¯eiLγ

µνiLW−µ + h.c. (2.30)

where να = (UL)α jν j and we can chooseUνL = U`

L because there are no right-handed neutrinos. This

explains the absence of leptonic mixing processes.

2.2 Massive Neutrinos

2.2.1 Majorana Mass

A mass term is a second-order coupling of two fields which is Lorentz invariant. The following quantity

is invariant under Lorentz transformations [8]:

χTL (iσ2)ψL → χT

LSTL (iσ2)SLψL = χT

L (iσ2)ψL (2.31)

where ψL and χL are two left handed spinors. If one chooses χL = (iσ2)ψ∗R = ˆ(ψR) this term becomes

(iσ2ψ∗R)T(iσ2)ψL = ψ†RψL (2.32)

which is the so-called Dirac mass term which all matter bear (ψ now stands for a four-component

spinor):

m(ψ†RψL + ψ†LψR) = mψ†γ0ψ = mψψ = m(ψLψR + ψRψL) (2.33)

The free Dirac field operator ψ(x) is

ψ(x) =

∫d3p√

(2π3)2Ep

∑s=±1/2

[fs(p)us(p)e−ip·x + fs

†

(p)vs(p)e−ip·x]

(2.34)

where fs(p) and fs†

(p) are the annihilation and creation operators respectively of a Dirac particle with

momentum p and spin s along the momentum direction. The u and v functions are plane wave solutions

of positive and negative energy respectively, given by the Dirac equation

(/p −m)us(p) = 0 (2.35)

(/p + m)vs(p) = 0 (2.36)

(2.37)

10

In the Dirac representation, and when they when they are normalized according to

u†s (p)u†s′(p) = v†s (p)v†s′(p) = 2Epδss′ (2.38)

these functions are

us(p) =√

E + m(

χsσ·p

E+mχs

)(2.39)

vs(p) =√

E + m( σ·p

E+mχ′s

χ′s

)(2.40)

where

χ↑ = −χ′↓ =

(10

)χ↓ = −χ′↑ =

(01

)(2.41)

Now let’s imagine a Dirac field such an electron propagating along the x-direction. It has four degrees

of freedom which are the four spinors above used to describe it. For both electron and positron states

there are two helicity states: therefore the spinors are two with positive energy, with left-handed eL and

right-handed eR helicities, and two of negative energy eL and eR. Now let’s have an observer moving

slower than the electron along the same direction. Supposing it measures a positive spin along this

direction, the field it finds is eR, because the result of his measurement is that spin and momentum are

parallel. But another observer moving faster than the electron actually sees a left-handed field because

now the momentum will look reversed and spin is now anti-parallel to it. Still, there are two fields LH

fields, eL and eL. The right choice is eL of course, because it has the same charge as eR which is a Lorentz

invariant, so can’t be changed by a boost transformation.

Things are a little less clear in the case of a neutrino because it is neutral. There is no way to choose

– and no way to distinguish by measurement – between these two RH states. So instead of somewhat

unaturally postulating the existence of two extra states (νL and νR), we can wonder whether the neutrino

is not a particle with just two degrees of freedom, therefore imposing νL = νL and νR = νR; the neutrino is

its own anti-particle. Such a fermion field was first introduced by Majorana in 1937, and took his name.

To build such a field one would be tempted to write

ψ(x) = ψ∗(x) (2.42)

but these quantities do not transform the same way under a general Lorentz transformation. For

instance, under a transformation

11

x′µ = xµ + ωµνxν (2.43)

the spinor field changes

ψ′(x′) = exp(−

i4σµνω

µν)ψ(x) (2.44)

where

σµν =i2

[γµ, γν] (2.45)

and

ψ′∗(x′) = exp( i4σ∗µνω

µν)ψ∗(x) (2.46)

= exp( i4σ∗µνω

µν)ψ(x) (2.47)

so unless Re(σµν) = 0, equality (2.43) is not covariant. This is not true for the Dirac representation.

However the conjugate field

ψ(x) ≡ γ0Cψ∗(x) (2.48)

transforms the same way as ψ(x) if γ0Cσ∗µν = σµνγ0C. In the Dirac representation,

C = iγ2γ0 =

(0 iσ2

iσ2 0

)(2.49)

We can also define the a representation, called the Majorana represetation, where the γ-matrices are

purely imaginary and the relation ψ and ψ∗ transform the same way. In this representation obviously,

C = −γ0 =

(0 σ2σ2 0

)(2.50)

Thus, the general definition of a Majorana field is

ψ(x) = eiθψ(x) (2.51)

Its operator in the plane wave expansion is

12

ψ(x) =

∫d3p√

(2π3)2Ep

∑s=±1/2

[fs(p)us(p)e−ip·x + eiθ f †s (p)vs(p)e−ip·x

](2.52)

Relation (2.51) is easily verified.

2.2.2 Properties under discrete space-time symmetries

Charge conjugation Under the charge conjugation operation C, a free Majorana field transforms as

Cψ(x, t)C−1 = e−iφγ0Cψ∗(x, t) = e−iφφ(x, t) = e−i(θ+φ)φ(x, t) (2.53)

where φ is a phase. Using the last result and (2.52)

C fs(p)C−1 = e−i(θ+φ) fs(p) (2.54)

with φ = −θ. In a C-symmetric vacuum, (2.54) imples that

C

∣∣∣p, s⟩ = ei(θ+φ)∣∣∣p, s⟩ = ±

∣∣∣p, s⟩ (2.55)

showing that a free Majorana particle is an eigenstate of C3.

Charge-Parity conjugation A free Majorana field transforms under P as

Pψ(x, t)P−1 = eiϕγ0ψ(−x, t) (2.56)

Together with equation (2.53), we get

(CP)ψ(x, t)(CP)−1 = −ei(ϕ−φ)γ0ψ(−x, t) (2.57)

The plane wave expansion reads

γ0ψ(−x, t) =

∫d3p√

(2π3)2Ep

∑s=±1/2

[fs(−p)γ0us(−p)e−ip·x + eiθ f †s (−p)γ0vs(−p)e−ip·x

]=

∫d3p√

(2π3)2Ep

∑s=±1/2

[fs(−p)us(p)e−ip·x

− eiθ f †s (−p)vs(p)e−ip·x] (2.58)

Consequently,

3This is not true in general as an interacting Majorana particle might violate C.

13

(CP) fs(p(CP)−1 = −ei(φ−ϕ−θ) fs(−p) (2.59)

with φ − ϕ − θ + π = 0. Therefore, again,

CP

∣∣∣p, s⟩ = ±∣∣∣−p, s

⟩(2.60)

Thus, if interactions are CP conserving then a Majorana neutrino is an eigenstate of this transforma-

tion. This statement can also be used as a good approximation when CP is only slightly violated, such

as it is by the weak interaction in the quark sector and hopefully in the leptonic sector too.

Charge-Parity-Time reversal A Majorana field transforms under CPT as

(CPT )ψ(x)(CPT )−1 = −eiξγT5ψ∗(−x)

= −e−i(ξ+θ)γT5 C−1γ0ψ(−x)

(2.61)

The expansion in terms of creation and annihilation operators is

γT5 C−1γ0ψ(−x) =

∫d3p√

(2π3)2Ep

∑s=±1/2

(−1)s+1/2[

f−s(p)u∗s(p)eip·x− eiθ f †−s(p)v∗s(p)e−ip·x

](2.62)

where we have used

γT5 C−1γ0us(p) = (−1)s−1/2u∗−s(p) (2.63)

γT5 C−1γ0vs(p) = (−1)s+1/2v∗−s(p) (2.64)

from which we get

(CPT ) fs(p)(CPT )−1 = ei[(s−1/2)π−θ−ξ] f−s(p) (2.65)

(CPT ) f †s (p)(CPT )−1 = ei[(s+1/2)π+θ−ξ] f †−s(p) (2.66)

with ξ = ±π/2. Finaly, we prove once again that

CPT

∣∣∣p, s⟩ = ei[ξ+θ+π(s−1/2)] (2.67)

To say that a Majorana field is an eigenstate ofCPT is a more profound statement as it is believed that

interactions which can be described in field theory are CPT -invariant, so this statement (which means

that neutrino is its own anti-particle) is also valid for a physical particle in an interacting medium which

may not conserve C or CP.

14

2.2.3 Majorana and Dirac mass terms

As already stated above, let’s consider two spinors ψ and χ. Then one can build the following term

χTL (iσ2)ψL (2.68)

which is invariant under Lorentz transformations for which we can choose χL = (iσ2)ψ∗R = ˆ(ψR), hence

yielding the Dirac mass term

mψψ (2.69)

But by choosing χL = ψL one gets this other term which is also Lorentz invariant

m2

[ψTL (iσ2)ψL + ψ†L(iσ2)ψ∗L] (2.70)

The first term is called the Dirac mass term which the mass term all fermions bear. It is U(1) invariant

so it conserves all quantum numbers carried by the fields. However, the second term is called a Majorana

mass term and is not U(1) invariant. It therefore breaks any quantum number carried by the field. Thus,

no particles carrying charge or colour can have a Majorana mass term because of the forbidance imposed

by the SU(3)C × U(1)Q symmetry. This is the case of all fermions except for the neutrino which is both

neutral and black (colourless).

A Majorana field has only half the components of a Dirac field. Hence, it might be possible to describe

a Dirac field in terms of two Majorana spinors. Consider the field ψL and its CPT conjugate ψR which

can form a Majorana spinor. Similarly, another one can be formed out of χL and χR. The possible mass

terms are

12

mψψLψR +12

mχχLχR +12

mψLχR +12

m′χLψR + h.c. (2.71)

Now let’s suppose

Ψ =

(ψLχR

)and Ψ =

(χLψR

)(2.72)

The mass lagrangian can be rewriten as

−Lmass =12

(ψL χL

) ( mψ/2 mm′ mχ/2

) (ψRχR

)+ h.c. (2.73)

15

Let’s write it in a more compact way as

−Lmass =12

∑i, j

ΨLi Mi jΨ

Rj + h.c.

=12

∑i, j

ψTi C−1Mi jPRγ0CγT

0ψ∗

j + h.c.

=12

∑i, j

ψ†jγ0

(C−1PRC

)TMi jψ j + h.c.

=12

∑i, j

ΨLj M jiΨ

Ri + h.c.

(2.74)

from which under permutation of the dummy indices Mi j = M ji. So clearly, m = m′. Removing the

diagonal terms which break phse symmetries in the mass matrix, the final mass Lagrangian is

12

(ΨLΨR + ¯ΨLΨR

)+ h.c. (2.75)

which the mass term for a Dirac field, proving that it can be constructed with two Majorana

fermions[12].

2.2.4 Experimental tests on neutrino mass

Given current experimental results, at least two neutrinos are for sure massive. We present here a list of

experimental tests of the neutrino mass hypothesis.

Kinematic tests

• Nuclear β-decay: The energy distribution of electrons emitted in this decay can be calculated for

mνe = 0. Existence of electron neutrino mass will reduce the peak’s energy. Studies of tritium decay

have been conducted throughout the world to look for deviations of the Kurie plot. The most

stringent bound on neutrino mass from tritium experiments came from a russian group in Troisk

that obtained mνe < 2.5 eV[54].

• Pion decay: accurate measurements of the π+→ µ+νµ decay rate can determine the mass of the νµ.

One recent experimental bound sets mνµ < 190 keV[55].

• Tau decay: There are several decay modes which can be used to determine the mass of ντ, some of

them which are even semileptonic thanks to the tauon’s heavy mass. Experiment has set mντ < 18.2

MeV[56].

16

Exclusive tests The following processes only happen if neutrino masses exist. Therefore a valid sig-

nature of these processes is an undeniable proof of neutrino masses. These tests are based on neutrino

mixing which isn’t necessary for neutrino masses but is more than natural in gauge models.

• neutrino oscillation: this phenomenon which might explain the Solar Neutrino Problem according

to the Sudbury Neutrino Observatory data, changes the flavour of neutrinos as they propagate. This

is due tothe fact that the mass basis is different from the weak basis and different mass components

in a given neutrino flavour will evolve differently.

• neutrino decay: massive neutrinos with flavour mixing can decay in flavour violating processes

such as να → νβ + γ which arise as an effective interaction lagrangian of the form[24]

Lint =12νiσαβ(µi j + εi jγ5)ν jFαβ + h.c. (2.76)

where µi j and εi j are the magnetic and electric transition moments. If neutrino masses are in a

sub-eV scale, cosmic microwave background (CMB) simulations set the bounds on the neutrino

radiative lifetime at 1011− 1012yrs[21].

Two other tests are also important. Massless neutrinos only have one electromagnetic form factor,

its charge radius. Measurement of other properties such as a magnetic moment would imply a mass.

Neutrinoless double beta decay –where two neutrons in the nucleus decay into two protons and two

electrons – is an important signature of Majorana-like neutrinos since the process requires the Majorana

field’s invariance under CPT , making the process impossible with Dirac-type neutrinos.

2.2.5 Neutrinoless double-beta decay

Neutrinoless double-beta decay (ββ0ν) is a lepton number violating process which can only occur if

neutrinos are Majorana. For Dirac neutrinos or massless neutrinos in the Standard Model the only

similar process that occurs is

n + n→ p + p + 2e− + 2νe (2.77)

which has a lifetime of 1018−1021yr depending on what nucleus is involved, and the enegies released

in the beta rays are of about a few MeV. But, thanks to the Majorana neutrino’s CPT invariance, the

diagram in figure 2.2.5 is possible (c.f. Appendix A for the Feynmann rules of Majorana spinors).

17

p

pn

n

νL

νL

W

W

e−L

e−L

Figure 2.1: ββ0ν diagram possible thanks to the CPT invariance of the Majorana fermion propagator.

This process is the number one test to assess the type of fermion mass the neutrinos bear. We will do

a somewhat short kinematical estimation of the ratio between τββ0ν and τββ0ν[13]. Each interaction rate

can be written as

Γ2e2ν = 2π∫|M2e2ν|

2δ(Mi −M f − E1 − E2 − E3 − E4) dΠ1 dΠ2 dΠ3 dΠ4

∝

∫dE1p1E1

∫dE2p2E2

∫dE3p3E3

∫dE4p4E4|M2e2ν|

2δ(Mi −M f − E1 − E2 − E3 − E4)

∝

∫ ∆

0dy

∫ y

0dy2|M2e2ν|

2p1p2E1E2δ(∆ − y)

∼ Q5|M2e2ν|

2 (2.78)

where

∆ = Q/me = (Mi −M f − 2me)/me (2.79)

x = (E2 −me)/me (2.80)

y = (E1 + E2 − 2me)/me (2.81)

Similarly,

Γ2ν0β ∝

∫ ∆

0dy

∫ y

0dy2|M2e2ν|

2p1p2E1E2δ(∆ − y)

∼ Q5|M2e2ν|

2 (2.82)

18

We have assumed for this several crude approximations for which a short discussion is needed. We’ve

taken the amplitude matrix as being independent of the phase-space coordinates. For the case of a light

Majorana neutrino exchange as shown in figure 2.2.5 leads a amplitude matrix proporcional to

M = Mµνhadron

[ue(p1)γµPL]α[

/q + mν

q2 −m2ν + iε

C]αβ[ue(p2)γνPL]β − (1↔ 2)

= Mµνhadron

ue(p1)γµPL

/q + mν

q2 −m2ν

ue(p2)γνPRve(p2) − (1↔ 2)

=mν

q2 −m2ν

Mµνhadron

ue(p1)γµPLue(p2)γνve(p2) − (1↔ 2)

(2.83)

where we have written the full matrix as a product of a hadronic amplitude matrix times a leptonic

one (which we have explicited). This bit turns out to be more-or-less proportional to the energy of the

fermions. That’s why we have introduced a factor pEdE to mimic the dependence of the matrix on the

phase-space coordinates.

From our discussion so far, we’ve estimated that

Γ2e0ν

Γ2e2ν∼

1Q6|M2e0ν|

2

|M2e2ν|2(2.84)

A relation between these two amplitudes must be found. For that matter, picture the neutrino-

less diagram as the standard double-beta decay but where the two neutrinos have the same product

momentum k, plus some lepton violation. That way, we can empirically write

M2e0ν ∼ ε∆L

∫d3k

(2π)3M2e2ν (2.85)

and therefore

Γ2e0ν

2e2ν∼ ε2

∆ε(ωQ

)6 (2.86)

Q is usually of order ∼ 2 − 3MeV. On the other hand if we consider the nucleus to be a Fermi gas of

nucleons in a sphere of radius R = R0A1/3 = (1.2fm)A1/3, where A is the total number of neutrons then in

the simple Fermi-gas approximation, the state density is

n =2V

(2πh)3

∫ pF

0d3p =

Vp3F

3π2~3 (2.87)

which conversely gives

19

pF = ~(3π2 n

V

)=

pF,n = ~r0

(9πN4A

)1/3for the neutron

pF,n = ~r0

(9πZ4A

)1/3for the proton

(2.88)

where N and Z are the number of neutrons and protons respectively, and from which we can deduce

the average Fermi kinetic energy per nucleon (for N = Z = A/2)

〈E〉p =

∫ pF

0 Ed3p∫ pF

0 d3p=

35

p2F

2m=

310m~2

r20

(9π8

)2/3≈ 200MeV (2.89)

(where m is the mass of a nucleon) so that the energies carried by the neutrino lines will be of about

200MeV. At present, the experimental bounds on this process are Γ2e0ν/Γ2e2ν < 10−4, or more importantly

ε∆L . 10−8.

2.2.6 Neutrino Mass Terms

In regard to this new phenomenology one has to extend in some way the SM to include neutrino masses.

But the mass term for neutrinos does not have to be as constrained as the mass term for the other charged

fermions. Mass terms have to be both invariant under the Lorentz and gauge groups.

The Dirac mass is the one we have used for the fermion propagation so far. It is well behaved under

a gauge symmetry and therefore conserves any quantum number associated with the dirac field. On the

other hand, the majorana mass terms violate any quantum number the majorana field may carry for they

are not U(1) invariant. Since all other fermion fields are charged, they should only have dirac mass terms

because we know – up to a good experimental certainty and theoretical one4 – that the universe is neutral

and charge non-conservation in the theory is greatly undesired. Nevertheless, neutrinos lack this charge

which is casted upon every other known fermion and they can endure a Majorana mass term, but one

that one still violate lepton number (L) by two units. One such process that violates this number is the

neutrinoless double β-decay (i.e.next section for further analysis)whose detection stands in the front line

for majorana mass measurements. Now there are three ways to introduce neutrino masses with minimal

change to the current picture of the SM. All terms in the Lagrangian must be gauge invariant so we want

singlet operators. Possible extensions of the SM can be categorized crudely as:

• extend the fermion sector

• extend of the higgs sector

4It has been show by Landau and Lifschitz that for an spherically symmetric closed space net charge of the universe mustvanish[5]

20

• extend both

To take a better look at this matter we need to know what these new operators mean in terms of the

SM gauge group’s SU(3) × SU(2) ×U(1) representations [4]:field label SM representation

lepton doublet lL (1, 2,−1/2)charged lepton singlet eR (1, 1,−1)

Higgs doublet φ (1, 2,−1/2)Higgs doublet φ (1, 2, 1/2)

Extension of the fermion sectorProduct Operator SM rep. required field mass type

lL ⊗ lL (1, 1,−1) singlet scalar η++ LH MajoranalL ⊗ lL (1, 3,−1) triplet scalar ∆ LH MajoranaeR ⊗ eR (1, 1,−2) singlet scalar k++ RH MajoranaνR ⊗ νR (1, 1, 0) singlet scalar k0 RH Majorana

Extension of the Higgs sectorlL ⊗ φ (1, 1, 0) lepton singlet νR DiraclL ⊗ φ (1, 3, 0) lepton triplet Σ LH Majorana

Table 2.2.6 shows what products one can build and the fields one has to introduce in order to make

gauge invariant mass terms. The simplest extension of the SM is to include add RH neutrinos that couple

to the LH component via the usual Yukawa mass term MDlLφνR + h.c.. But in doing so one rejects the

possibility of lepton number violation through majorana mass terms. Plus, the coupling would have

to be extremely small in order to suppress the neutrino masses. One way to naturally suppress the

neutrino masses are the seesaw models which come in three different types and to which we will focus

our particular care because of their importance for leptogenesis.

One might argue that including RHνis not new physics because it is simply a matter of making neutrinos

just like the other fermions. However, if we include RHνthe Majorana mass term has to be included

as we have shown above, because it is allowed by both Lorentz and gauge symmetries. So, if RHνare

indeed included, no matter what experimental outcome might be proven, there is always new physics in

the neutrino sector, which will be either a new symmetry that forbids Majorana terms or, in the opposite

case, lepton number violation. And even if RHνaren’t found, new physics will appear not in the fermion

sector but in the scalar Higgs sector, with the introduction of scalar triplets. So the discovery of neutrino

masses is a very important sign of new physics.

21

2.2.7 Seesaw Models

Type I Seesaw

Motivation from the SO(10) group symmetry The simplest non-abelian group that can accomodate

the Standard Model is SU(5). Nevertheless, besides experimental bounds on the proton stability ruling

out the minimal SU(5) Grand Unification Model (GUT), nothing indicates that grand unification should

stop at SU(5) [8]. SU(5) ×U(1) is in its turn a subgroup of SO(10) and since SU(3) is a subgroup of SO(6)

and SU(2) is a subgroup of SO(4) this group can clearly accomodate the SM in this orthogonal group.

One way to assign fermions of the SM to representations of SO(10) is to first link a representation of

SO(10) to the 5 + 10 reps. of SU(5) (where one SM fermion family neatly fits).

It is striking that one fundamental representation of SO(10) alone can provide for all the Standard

Model fermions plus one SU(5) singlet which one can associate with the right-handed neutrino. The

simple Yukawa mass term that couples one 16 to another only hands a Dirac mass for the neutrino. Even

so, other couplings using the 120 and 126 reps. do allow Majorana masses. The fist type of seesaw deals

with the inclusion of a right-handed neutrino with both Dirac and RH Majorana mass terms.

Singlet fermions This model[38] introduces electroweak singlet right-handed neutrinos as one already

has for the other fermions. The only difference is that Yukawa couplings can also give RH majorana

masses whereas for the other fermions the only possible mass is the Dirac one. Because we want to link

them to some GUT scale at which B + L symmetry is broken it is only natural that these new particles

have fairly large masses. The extended neutrino mass Lagragian becomes then:

Lν = −Mi jDν

iLν

jR + 1

2 Mi jR

¯(νiR)cν

jR + h.c.

= − 12

¯(nL)c

(0 M∗D

M†D M∗R

)︸ ︷︷ ︸

CM∗

σ2nL + h.c. (2.90)

where nL = (νL, (νR)c)T we have used the identity:

νLνR =12

(νLνR + ¯(νR)c(νL)c

)(2.91)

The 3× 3 Dirac mass matrix MD is unitary mixing matrix times the Higgs VEV MD = λi jv/√

2, where

v = 〈φ〉 ≈ 174GeV. We now would like to find the mass eigenstates of M in the electroweak basis to write

the charged and neutral current interaction Lagrangians. Since M is symmetric, it is then diagonalized

by an orthogonal transformation such as [2]:

22

VTM∗V = D, V =

(K3×3 R3×3S3×3 T3×3

), D = Diag(d3×3,D3×3) (2.92)

The neutrino weak-eigenstates are twofold and related to the mass eigenstates by:

νi = Ki jνjL + Ri jνR

j

Ni = Si jνjL + Ti jνR

j (2.93)

Assuming MR MD, the first eigenstates will have a mass d and the second D. The first ones are the

light neutrinos mostly composed of the left-handed component as implied from low-energy observations.

Its mass is given by the usual seesaw relation; from (2.92)

S†MTDK∗ + K†MDS∗ + S†MRS∗ = d (2.94)

S†MTDR∗ + K†MDT∗ + S†MRT∗ = 0 (2.95)

T†MTDR∗ + R†MDT∗ + T†MRT∗ = D (2.96)

(2.95) leads to

S† ≈ −K†MDM−1R (2.97)

which holds up to a very high degree of accuracy. Together with (2.94) we get the usual seesaw

formula[39]

−K†MDM−1R MT

DK∗ = d (2.98)

Similarly,

D ≈ T†MRT∗ (2.99)

where the suppression mechanism becomes obvious become as MR gets bigger, d gets smaller, which in

view of a GUT scenario in which MD MR gives a natural explanation for the smallness of the observed

neutrino masses.

Let’s examplify what was written above for the single generation case. In this case, leptons only come in

one flavour and the mass matrix reads

M =

(0 mD

mD mR

)(2.100)

23

where the entries in this matrix are simply real positive numbers for the time being. An orthogonal

matrix can be paramatrized as

V =

(cosθ − sinθsinθ cosθ

)(2.101)

where tan 2θ = 2mD/mR and which then diagonalizes the mass matrix as

VTM∗V =

(−m1 0

0 m2

)(2.102)

where

m1,2 =12

(∓mR +

√m2

R + 4m2D

)(2.103)

Since both these values are either real or complex, they can’t be the physical value of the mass. So the

previous has to be rewritten as

VTM∗V =

(m1 00 m2

)·

(−1 00 1

)= dK2 (2.104)

and

M = VdK2VT (2.105)

Now, the mass eigenvectors are

(n1Ln2L

)= O

(νLNL

)=

(cosθ − sinθsinθ cosθ

) (νLNL

)(2.106)

and

(n1Rn2R

)= K2O

(νRNR

)=

(− cosθ sinθsinθ cosθ

) (νRNR

)(2.107)

In this basis the mass Lagrangian is diagonal. The one generation two eigenstates are

n1 = n1L + n1R = cosθ(νL − νR) − sinθ(NL −NR)

n2 = n2L + n2R = sinθ(νL + νR) + cosθ(NL + NR)(2.108)

with masses m1 and m2 respectively. Now let us see how this helps in explaining naturally the

smallness of neutrino masses. In a natural explanation, mD should be of the order of the fermion masses,

whereas mR being linked to a high-scale energy at which a GUT theory becomes evident then mR mD

and

24

m1 ∼m2

D

mRm2 ∼ mR (2.109)

and

| cosθ| ∼ 0.92

| sinθ| ∼ 0.38(2.110)

and it follows neatly that m1 mD, rendering the left handed neutrinos (n1 ∼ (νL − νR) - hence

much smaller than the usual fermion scale. The seesaw mechanism can hence explain the smallness of

the fermion scale. However, the cost of it is the same as in any GUT theory which is the insertion of a

high scale. Since cosmological observations now constrain the masses of the light neutrinos down to a

sub-eV scale, the mass of the RHνmust be of about 106TeV, which is far above the reach of any present

accelerator, and probably also of any future ones.

CC interaction The leptonic charged current interaction changes with the addition of these heavy mass

states. It bears now one extra term

LCC = −g

2√

2( ¯iKi jγ

µν j + ¯iRi jγµN j)W−µ + h.c. (2.111)

where `i are the charged leptons with flavour i = e, µ, τ and ν j and N j are the light and heavy mass

eigenstates respectively, with j = 1, 2, 3. In the total decoupling limit, R may be neglected and the

conventional CC lagrangian is recovered, with the light neutrino states interactions with the charged

leptons fields being described by the K matrix.

Parametrizations The light neutrino mass matrix is diagonalized by the K matrix[14]. The SM already

has nine low-energy parameters (three light masses, three mixing angles and three CP violating phases),

the insertion of three heavy RH neutrinos adds nine new parameters: three heavy masses, three mixing

angles and three other CP violating phases. Then there yet another three parameters which are the CP

parities of the RH neutrinos. Therefore there are in total 21 real degrees of freedom for the lepton sector.

There are several ways to parametrize these, namely the top-down approach in which one fixes the high-

energy sector to reconstruct the low-energy one or the other way round, the bottom-up parametrization

where the low-energy sector is fixed instead, or yet the Casas-Ibarra parametrizations[37] which is an

intermediate one and often very handy for calculations. The latter uses the fact the since K is a unitary

hermitian matrix it can be diagonalized by a 3 × 3 orthogonal matrix O which is parametrized in terms

25

of three real angles c(s)i j = (1− δi j)cos(sin)θi j and one Dirac-type CP violating phase δ and another two of

the Majorana-type α and β:

K =

c12c13 s12c13 s13e−iδ

−c23s12 − s23s13c12eiδ c23s12 − s23s13s12eiδ s23c13s23c12 − c23s13c12eiδ

−s23c12 − c23s13s12eiδ c23c13

·

1 0 00 e−iα 00 0 e−iβ

(2.112)

Out of the 18 parameters (3 of the 21 are CP parities choosable at will) three of them are fixed from

solar and atmospheric neutrino observations[1]:

rl∆m2 ≡ m2

2 −m21 =

(7.9+1.1−0.89

)× 10−5eV2 (2.113)

s212 = 0.31+0.07

−0.05 (2.114)

∆m2A ≡ |m

32 −m1

1| =(2.6+0.6−0.6

)× 10−3eV2 (2.115)

s223 = 0.47+0.17

−0.15 (2.116)

The following parameters remain free: neutrino mass scale mmin (smallest neutrino mass), the mass

ordering (sign of m2A), θ13, δ, α and β, and 3 RH masses.

Among the nine unconstrained parameters on the list, the first four are expected to be measured soon,

but the other eleven are beyond our hopes for a near future, which is a major predictive handicap for the

theory. Nonetheless, there are several ways to reduce this high number.

A set of invariants equivalent to the Jarlskog invariant, measuring CP violation in the leptonic sector can

be found by defining[41]:

sαi j = Im[UαiU∗α jR∗

i R j]

tαiβ j = Im[UαiUβ jU∗α jU∗

βi](2.117)

where U = UPMNS and R = diag(1, e−iα, e−iβ) are the two elements composing the K matrix. The

minimal set of independent rephasing invariant quantities is JCP = tαi13, J1 = s113, J2 = s123, which has the

advantage that JCP mimics the Jarlskog invariant for the CKM matrix and only enters processes which

violate CP, whereas J1 and J2 enter only lepton number violating interactions. After parametrization its

values are

JCP = −c12c23c213s12s23s13 sin δ

J1 = −c12c13s13 sin(α − δ) (2.118)

J2 = −s12c13 sin(α − β − δ)

26

which shows that there can still be lepton-number violating processes even if there is no CP violation

(δ = 0); on the other hand, if α = δ and β = 0 implies that there is no CP violation in any lepton number

conserving processes like neutrino oscillations.

One should still notice that current experiments only state that at least two neutrinos are massive.

Therefore, only two RH neutrinos are required. This aprticular hypothesis can be seen as the limit where

the third heavy neutrino decouples away because of its very high mass or relatively small couplings. In

such a case, the lightest neutrino can be aknowledged as effectively massless, thus removing another two

parameters from the theory (mmin and α).

Before moving ahead to the next seesaw mechanism, let’s study how WBTs happen in the leptonic sector.

These are

n′L = WLnL, (2.119)

`′L = WL`L, (2.120)

`′R = WR`R (2.121)

The mass matrices become

M′` = W†LM`WR, (2.122)

M′ν = WTL MνWL (2.123)

If we wish these to be the diagonal basis then

M` = U`LD`U`†

R , (2.124)

Mν = Uν∗L D`Uν†

L (2.125)

where Dν and D` are real and positive. The WBT used here is WL = U`L and W`

R = U`R

M′` = D`,M′ν = K∗DνK† (2.126)

It is important to notice that RHνrotations aren’t free. Hence, negative eigenvalues of the charged

lepton mass matrix are permitted since we are allowed to rotate it at will, but the same is not true for the

neutrinos, since any phases held by the eigenvalues will appear in the Majorana phases.

27

Type II Seesaw

In the previous section we kept ourselves from introducing left-handed Majorana masses because – i.e.??

– the product of two LH lepton fields breeds a singlet operator as well as an SU(2) triplet which must be

made gauge invariant thanks to some coupling to a new triplet Higgs field ∆ with Y=2:

σ · ∆ =

(∆+

√2∆++

√2∆0

−H+

)(2.127)

which has a non-zero VEV along its neutral component⟨H0

⟩= v∆/

√2. This triplet gives rise to the

additional Yukawa coupling and trilinear scalar coupling:

f ¯Lσ · ∆lL + µφσ · ∆φ + h.c. σ · ∆ ≡ σ2σ · ∆∗ (2.128)

which results in a v∆ f ˆLνL Majorana mass term for the neutrino as ∆ develops a VEV. It has been

shown [15] that for a general Higgs multiplet with weak isospin T and weak hypercharge Y:

ρ ≡M2

W

M2Zcos2θW

=ΣT,Y|vT,Y|

2[T(T + 1) − Y2/4]2ΣT,YY2/4

=v2φ + 2v2

∆

v2φ

+ 4v2∆

(2.129)

where the last result holds for our scalar triplet. Since current experiments have constrained the

value of rho to be ρ = 0.998 ± 0.005 which forces (v∆/vφ) < 0.17 (at 1σ) or v∆ < 30GeV which explains the

suppression for the neutrino masses. This model has eleven additional free parameters out of which nine

can be determined from the light neutrino parameters, while the remaining two (the triplet’s complex

coupling to the higgs doublets). The motivation for adding a different Higgs multiplet comes from GUT

theories where often one has to use higher representations of the Higgs to break the symmetry; e.g., only

SU(5) models with (no SUSY and) scalar representations of higher dimension than 5 allow the proton

decay predictions to behave decently in regard to observational constraints. Soft CP violation – where

the latter arises from SSB – also requires more complex multiplets to create the complex VEV.

Type III Seesaw

The third and last type of seesaw introduces the adjoint SU(2) representation triplet fermions Σ. The

relevant Lagrangian is:

λΣ lφΣ −12

MΣΣ (2.130)

and its contribution to the neutrino masses is ∝ v2λΣi jM−1j λΣkj, which is similar to the type I formula.

Again, this model has, like the first type, eighteen parameters beyond the SM’s.

28

Chapter 3

Neutrino Oscillations

3.1 Neutrino oscillations in vacuum

Neutrino oscllations mean that a beam of neutrinos of a specific flavour can change along their trajectory

into another flavour[16]. As stated earlier, if neutrinos are massive, their mass matrix linking weak and

mass eigenstates will be non-diagonal:

να =∑

i

Uαiνi (3.1)

where the greek index run the flavours and the latin one covers the mass eigenstates. U is a unitary

matrix which can be parametrized as the CKM matrix. Usually one assumes that all neutrinos have a

fixed momentum p1. Let’s suppose a neutrino in a beam which is born in a definite flavour α at time

t = 0, then the wave function will be

|να(x, t = 0)〉 =∑

i

eipxUαi |νi〉 (3.2)

and evolves with time as

|να(x, t)〉 =∑

i

ei(px−Eit)Uαi |νi〉 (3.3)

Since the masses are small p mi we may write x ≈ t and Ei =√

p2 + m2i ≈ p + (m2

i /2p):

|να(t)〉 ≈∑

i

e−im2

i2p tUαi |νi〉 (3.4)

From this expression the probability of finding a flavour β at distance x is readily obtained:

1This assessment is true as long as the number os oscillation lengths is not two big. Anyway, after too many lengths theneutrinos in the beam have become decoherent and oscillations have long since ceassed.

29

|φαβ(p, x)|2 =∑

i

U2βiU

2αi +

∑j,i

U∗βiUαiUβ jU∗α j cos 2πxli j

(3.5)

where we have introduced the oscillation length

li j =4πp

|m2i −m2

j |(3.6)

The interest of the phenomenon is its quantum mechanical nature and only exists thanks to mea-

surement theory. Let’s suppose the neutrino beam is created from a decaying beam of pions through

π+→ µ+νµ. The muon and pion’s momenta are measured so that the mass of the product neutrino be

determined and its flavour assessed. To do so, we must determine the mass with an error smaller than

|m2i −m2

j |. The error in the mass of the neutrino is given by

m2i = E2

i − p2⇒ δ(m2

i ) =√

4E2i (δEi)2 + 4p2(δp)2 (3.7)

the momentum error must then be

δ(m2i ) < |m2

i −m2j | ⇒ δp <

|m2i −m2

j |

2p(3.8)

and from Heisenberg’s uncertainty principle

δx &p

|m2i −m2

j |=

li j

4π(3.9)

The uncertainty in the interaction vertex becomes wider than the oscillation length and the oscillation

pattern is therefore lost.

We will first analyse in detail the two flavour case and three flavour one, and conclude with a brief review

of the effects of the majorana mass in neutrino oscillations. Neutrino oscillations are the best probe of

neutrino mass; solar and supernova experiments which aim at studying them can discover neutrino

mass differences down up 10−5 eV.

3.1.1 Two-flavour case

In this case the mixing matrix has only one mixing angle and can be written as

U =

(cosθ0 sinθ0− sinθ0 cosθ0

)(3.10)

and the weak and mass eigenstates are related by

30

|νe〉 = cosθ0 |ν1〉 + sinθ0 |ν2〉 (3.11)∣∣∣νµ⟩ = − sinθ0 |ν1〉 + cosθ0 |ν2〉 (3.12)

From (3.5) and because of CP (or T) symmetry, the transition (|φ(νe → νµ)|2 and |φ(νµ → νe)|2) and

survival probabilities (|φ(νe → νe)|2 and |φ(νµ → νµ)|2) are:

P(νe → νµ; t) = P(νµ → νe; t) ≡ Peµ(µe)(t) = sin2 2θ0 sin2(∆m2

4Et)

= sin2 2θ0 sin2(π

xl

)(3.13)

P(νe → νe; t) = P(νµ → νµ; t) ≡ Pee(µµ)(t) = 1 − sin2 2θ0 sin2(∆m2

4Et)

(3.14)

where ∆m2 = m22 −m2

1 and the oscillation length is

l =4πE∆m2 ≈ 2.48m

E(MeV)∆m2(eV2)

(3.15)

3.1.2 Three-flavour case

In this case the mixing matrix can be parametrized as the CKM matrix for quark mixing angles

νeνµντ

=

c1 s1c3 s1s3−s1c2 c1c2c3 − s2s3eiδ c1c2s3 + s2c3eiδ

−s1s2 c1s2s3 + c2s3eiδ c1s2s3 − c2c3eiδ

ν1ν2ν3

(3.16)

where ci ≡ cosθi and si ≡ sinθi. For a beam of neutrinos produced in the initial state νe the survival

and transition probabilities are

After many oscillation lengths it is natural to assume that the beam becomes decoherent and that the

oscillation pattern is lost. Indeed, when x li j the harmonics are smoothed off and only the average

intensity will be observable[15]:

〈Pee〉 = 1 − 2c21s2

1c23

[1 − cos

(2πxl12

)]− 2c2

1s21s2

3

[1 − cos

(2πxl13

)]− 2s4

1s23c2

3

[1 − cos

(2πxl23

)](3.17)⟨

Peµ⟩

= 2c21s2

1c22 + 2s2

1s23c2

3(s22 − c2

1c22) + 2s2

1s2s3c1c2c3 cos δ(s23 − c2

3) (3.18)

〈Peτ〉 = 2c21s2

1s22 + 2s2

1s23c2

3(c22 − c2

1s22) + 2s2

1s2s3c1c2c3 cos δ(s23 − c2

3) (3.19)

31

3.1.3 Majorana mass and neutrino oscillation

We have discussed neutrino oscillations in the case of Dirac neutrinos. If neutrinos are Majorana rather

than Dirac, the mass term in the lagrangian has to be replaced:

(mD)i jνiLν jR + h.c.→ (mM)i jνTiLCν jR + h.c. (3.20)

This term violates the total lepton number in addition to the individual lepton flavours. The Majorana

mass matrix is diagonalized by the transformation UTL mMUL = (mM)diag, so the problem becomes equiv-

alent to the Dirac one, just with a different matrix, but still with the same number of mass eigenstates as

weak ones. The oscillation pattern and probabilities have the same formulae. Therefore it is impossible

to test the Majorana hypothesis against the Dirac one with neutrino oscillation.

Things somewhat change if the most general case is analysed, where the Dirac and both Majorana terms

are considered. In this case the full mass matrix is a 2n × 2n matrix and a new kind of oscillations is

possible which convert weak-interacting neutrinos into sterile neutrinos να → νβ. These oscillations

violate the total lepton number L. Thus, one can distinguish the mixed Dirac-Majorana case from the

other two pure ones.

3.1.4 Do charged leptons oscillate?

The answer is obviously ’no’. The point here is that e, µ, τ refer to mass eigenstates. Neutrino mixing only

happens because we chose charged leptons to be in the mass basis, because neutrinos can’t be measured

directly but only by interactions such as CC or NC, so what we are really measuring are charged leptons

and associating the missing energy to an invisible particle called ’a neutrino’. And since mass eigenstates

don’t oscillate, charged leptons can’t oscillate. The proof of this goes as follows.[22]

Consider a muon being created at time t0 = 0 and position x0. After a time t and a distance x the state

evolves into

|Ψ(t, x)〉 = e−ipµxµ∣∣∣µ⟩ (3.21)

with a survival probability of

Pµµ = | < µ|Ψ(t, x) > |2 = 1 (3.22)

proving that no oscillation happens. Even if we consider a superposition of two states

32

|Ψ(0)〉 = cosθ∣∣∣µ⟩ + eiα sinθ |e〉 (3.23)

After propagation the state will be

|Ψ(t, x)〉 = e−ipµxµ cosθ∣∣∣µ⟩ + e−ipµxµeiα sinθ |e〉 (3.24)

and the survival probabilities will be again

Pµµ = cos2 θ (3.25)

Pee = sin2 θ (3.26)

and once again we see that charged leptons do not oscillate. But one may ask now: can we design an

experiment where a coherent superposition of e, µ, τ and then also detects their coherent superposition

rather than individual mass-eigenstate charged leptons? If this were possible, one would be able to

observe oscillations between such mixed charged lepton states.[23] In fact, the reason why neutrinos

oscillate and not charged leptons (since the CC interactions is symmetric in respect to neutrinos or

charged leptons) lies in the mass scale difference and in the decoherence properties of charged leptons.

3.2 The MSW effect

The neutrino oscillation problem doesn’t quite stop here. Matter may greatly enhance the oscillation

amplitude. The reason for this is that the interaction hamiltonian may be also flavour dependent. If such

is the case, then extra phases between the states might appear, inducing further levels of oscillation. For

example, ordinary matter is usually mainly made of electrons, protons and neutrons, whereas no taus ou

muons are present. Being so, only electron neutrinos will interact with the matter through charged current

showing an obvious assymetry in the Hamiltonian between the different flavours. Like electromagnetic

waves travelling through a medium and reducing their speed by acquiring an effective mass, neutrinos

also change their effective mass according to the interactions they suffer along their crossing of the

medium. Such processes are neutrino absorption and scattering by the matter constituents which change

both their energy and momentum. However, these processes are proporcional to the Fermi constant

G2F and are thus pretty small. Nevertheless, neutrinos can be forward scattered and their momentum

is conserved. This generates an effective potencial which varies between the electron neutrino and the

other flavoured neutrinos. Despite their being of order GF, their contribution to neutrino oscillation if

33

significant when they are similar or greater than ∆m2/2E. This enhancement can go as far as reaching

unity, unlike the vacuum phenomenon which is never higher than sin2 θ0. The study of neutrino

oscillations in matter starts with Wolfenstein[17] when he pointed out that in ordinary matter electron

neutrinos interact with electrons via charged and neutral current whereas tau and muon neutrinos only

interact through neutral current. The effect of oscillation enhancement in matter is owed to the work

of Mikheyev, Smirnov and Wolfenstein[18] (the MSW effect) and is one main part in solving the solar

neutrino problem.

In matter neutrinos of all three flavours interact with electrons, protons and neutrons through NC

interaction. Electron neutrinos react additionally with the medium’s electrons through CC interactions.

Check diagrams for these interactions in fig.3.2. The effective Hamiltonian for the CC interaction is the

W±e

νe

νe

e

Z0

p, n, e

p, n, e

νe,μ,τ

νe,μ,τ

Figure 3.1: Neutrino scattering diagrams via weak charged and neutral interactions.

usual V − A hamiltonian. The CC and NC effective hamiltonians for the electrons are:

HCC =GF√

2

[eγµ(1 − γ5)νe

] [νeγ

µ(1 − γ5)e]

(3.27)

HNC =GF√

2

∑f=p,n,e

[fγµ

( I3

2(1 − γ5) −Q f sin2θW

)f] [νeγ

µ (1 − γ5

)νe

](3.28)

For the other flavours only the NC interaction happens and the Hamiltonian is the same:

HNC =ρGF√

2

∑f=p,n,e

fγµ

I f3

2(1 − γ5) −Q f sin2θW

f

[ ¯νµ,τγµ(1 − γ5

)νµ,τ

](3.29)

In forward scattering the neutrino’s momentum is left unaffected so:

34

HCC =GF√

2

[¯e(p1)γµ(1 − γ5)νe(p2))

] [¯νe(p3)γµ(1 − γ5)e(p4)

](3.30)

=GF√

2

[¯e(p1)γµ(1 − γ5)νe(p))

] [¯νe(p)γµ(1 − γ5)e(p)

](3.31)

=GF√

2

[¯e(p)γµ(1 − γ5))e(p)

] [¯νe(p)γµ(1 − γ5)νe(p)

](3.32)

The coherent forward scattering contribution to the energy of νe in matter is equivalent to an effective

potencial which is found upon integrating over all the variables that correspond to the electron field:

HCCe f f (νe) = 〈HCC〉e (3.33)

=GF√

2

[¯νe(p)γµ(1 − γ5))νe(p)

]〈eγµ(1 − γ5)e〉 (3.34)

=GF√

2

[¯νe(p)γµ(1 − γ5))νe(p)

]〈eγµ(1 − γ5)e〉 (3.35)

=GF√

2

[¯νe(p)γµ(1 − γ5))νe(p)

] [〈e†e〉 + 〈e†−→α e〉 − 〈e†γ5e〉 − 〈e†−→αγ5e〉

](3.36)

=GF√

2

[¯νe(p)γµ(1 − γ5))νe(p)

] [Ne + 〈−→ve〉 − 〈

−→σe ·−→pe

Ee〉 − 〈−→σe〉

](3.37)

(3.38)

where Ne is the electron number density. In a neutral, unpolarized medium of zero total momentum

the only surviving term in the latter expression is Ne. Let us now calculate the NC contribution for the

effective potencial:

HNCe f f (νe) = 〈HNC〉p,n,e (3.39)

=GF√

2

[νeγ

µ (1 − γ5

)νe

] ∑f=p,n,e

〈 fγµ( I3

2(1 − γ5) −Q f sin2 θW

)f 〉 (3.40)

= GF√

2[νeγ

µ (1 − γ5

)νe

]〈pγµ

(12

(1 − γ5) − 2 sin2 θW

)p (3.41)

+ eγµ(−

12

(1 − γ5) + 2 sin2 θW

)e + nγµ

(−

12

(1 − γ5))

n〉 (3.42)

= GF√

2[νeγ

µ (1 − γ5

)νe

] [(12− 2 sin2 θW

)Np +

(−

12

+ 2 sin2 θW

)Np +

(−

12

)Nn

](3.43)

= GF√

2[νeγ

µ (1 − γ5

)νe

] (−

12

)Nn (3.44)

because Np = Ne in neutral matter. Hence, the effective potencial for the electron is:

35

He f f (νe) ≡ ¯nueVee = 〈HNC〉p,n,e + 〈HCC〉e ⇒ Ve =√

2GF

(Ne −

Nn

2

)(3.45)

The same works for tau and muon flavours except that Nτ = Nµ = 0 so:

Vµ = Vτ = −GF√

2Nn (3.46)

For anti-neutrinos one has to replace V → −V.

Let us consider the evolution of oscillating neutrinos. In vacuum the evolution is obviously easily tracked

in the mass basis whereas in matter – because the effective potencials are flavour-dependent – it is rather

favorable to follow the evolution of the system in the flavour basis. The flavour basis and the mass basis

are related by:

ν f l = Uνm (3.47)

and the Schrodinger equation is:

iddt|νm〉 = Hm |νm〉 , Hm = diag(E1,E2) (3.48)

iddt

∣∣∣ν f l⟩

= UHmU†∣∣∣ν f l

⟩(3.49)

iddt

(νeνµ

)≈ p + U

(m2

1/2E 00 m2

1/2E

)U†

(νeνµ

)(3.50)

= p +m2

1 + m22

4E+

∆m2

4E

(− cos 2θ0 sin 2θ0sin 2θ0 cos 2θ0

) (νeνµ

)(3.51)

For the modified Hamiltonian we must add the effective potencials for both flavours[1]:

iddt

(νeνµ

)= p +

m21 + m2

2

4E−

GF√

2Nn +

∆m2

4E

− cos 2θ0 + 4√

2EGF∆m2 Ne sin 2θ0

sin 2θ0 cos 2θ0

( νeνµ

)(3.52)

=∆m2

4E

− cos 2θ0 + 4√

2EGF∆m2 Ne sin 2θ0

sin 2θ0 cos 2θ0

( νeνµ

)(3.53)

The common diagonal terms to both flavours do not insert any phase difference between the neutrino

states and thus don’t contribute to the neutrino oscillations because they can only change the global

phase of the doublet and can be ommited for the purpose of this study. We shall now study the constant

matter density case for this oscillating system.

36

3.2.1 Constant density

In this case Ne = const. along the neutrino trajectory (so constant along both time and space). Upon

diagonalization, the effective Hamiltonian yields the following eigenstates:

νA = νe cosθ + νµ sinθ (3.54)

νB = −νe sinθ + νµ cosθ (3.55)

and the mixing angle is given by:

tan 2θ =2H12

H22 −H11=

sin 2θ0

cos 2θ0 − cos 2θ0 + 4√

2EGF∆m2 Ne

(3.56)

The difference between the two eigenenergies is:

∆E =∆m2

2E

√(cos 2θ0 −

2√

2GFE∆m2 Ne

)2

+ sin2 2θ0 (3.57)

It is now straight forward to find the transistion probability in matter:

P(νe → νµ; x) = sin2 2θ sin2(∆Ex

2

)= sin2 2θ sin2

(π

xlm

)(3.58)

where the oscillation length is now given by:

lm =2π

EA − EB=

4πE

∆m2

√(cos 2θ0 −

2√

2GFE∆m2 Ne

)2+ sin2 2θ0

(3.59)

The expressions are the same as for the vacuum case, except with a correction for the oscillation length

lm and the mixing angle θ which are given in terms of the vaccum ones losc, θ0 and Ne.

The oscillation amplitude is resonant when:

√

2GFNe =∆m2

2Ecos 2θ0 → sin2 2θ = 1 (3.60)

This is called the MSW resonance condition and when it is fulfilled the maximum value for θ = π/4

is achieved without regard to how small the vacuum mixing angle may be. Thus, one may have large

neutrino flavour transistions in matter even when they are very small in the vaccum.

Anyway, any matter enhancement requires:

37

∆m2

2Ecos 2θ0 =

12E

(m22 −m2

1)(cos2 θ0 − sin2 θ0) > 0 (3.61)

If ν2 is heavier than ν1 then θ ∈ [−π/4, π/4] ∪ [3π/4,−3π/4]. This means that the lighter mass-

eigenstate must have the largest νe component. If one chooses the convention cos 2θ0 > 0 then this

condition is merely a hierarquic one ∆m221 > 0. One the other hand anti-neutrinos have the opposite

condition ∆m221 < 0 which means that neutrinos and anti-neutrinos can’t experience both MSW enhanced

oscillations.

One may think that as long as the MSW condition is satisfied one can go as far as to have θ0 → 0

but if we do so lm → ∞ and the oscillation pattern is destroyed. Usually the neutrinos propagate in

polychromatic beams, so the the resonance condition A = 1 is almost certainly satisfied by some part of

the energy density as long as ∆m2 is of the right order of magnitude. Nevertheless, solar 7Be neutrinos

are monochromatic, so one may ask, can the resonance condition still be satisfied? The answer is yes,

provided that the corresponding resonance density is within the density range of the matter density

distribution in which neutrinos propagate.

3.2.2 Oscillations in non-uniform matter

For practical purposes let us define[12]:

A ≡ 2√

2GFENe = 1 −tan 2θ0

tan 2θ(3.62)

Now we wish to study the effect of matter density changes along the neutrino’s path in the oscillation

pattern. Let’s suppose they are emitted in a very dense medium like the core of the sun, where A is far

above the resonance value. In this region A ∆m2 cos 2θ0 θ→ π/2, the lighter state is almost purely

νµ whereas the heavier one is close to being plain νe. Now in vaccum the exact opposite happens. All this

is pretty clear when one looks at fig.3.2.2. The phenomenon that inverts the effective masses of νµ and νe

from νµ being lower than νe to νe being higher than νµ with the inscrease of A is called level-crossing.

At the core of the sun A 1⇒ θ ∼ πwhich means that the neutrino oscillation is strongly suppressed

by matter. Assuming that the Sun’s density decreases along its radius, A also decreases as the neutrino

propagates out of the star. The mixing then increases at first, reaching its maximum value θ = π/2

when A = 1 and then decreasing again towards vacuum values until θ = θ0. As said earlier, at first the

neutrinos mass eigenstates are almost pure, and if the system was to propagate only in the core of the

sun, any neutrino produced in one mass eigenstate will remain in that very same eigenstate. However,

the fact that it has to go through thinner matter changes the particle’s flavour composition, as long as

38

Figure 3.2: Neutrino energy levels versus electron density in the medium, where the dashed line standsfor the pure states and the solid line stands for the mixed states.

the density variation is made adiabatically so that the neutrino can adaptıtself to the changes. At the

final state νB’s transistion probability is P(νe → νµ) = cos2θ0 so for small θ0 the transition probability is

almost one; the adiabatic convertion is almost complete for νe to νµ which is the impressiveness of the

level-crossing mechanism. This is similar to the Landau-Zener mechanism in atomic physics that governs

the transition probability between two states separated by a time-dependent energy difference.

It is important to note once again that even though the smaller θ0 is the bigger the adiabatic conversion

works and therefore θ0 = 0 may seem – in a very unintuitive way – like a mean of full oscillation

improvement, when the vacuum mixing angle is null the conversion is no longer adiabatic and the

previous reasoning is no longer valid.

It is therefore important to set the validity of this adiabatic approximation. For that purpose we must

switch back to a quantitative description of the neutrino conversion in the adiabatic regime. The effective

Hamiltonian in the flavour basis can be diagonalized by a unitary transformation such as:

U(t)†H f l(t)U(t) = Hd(t) = diag (EA(t),EB(t)) (3.63)

and EA(t) and EB(t) are the instantaneous energies of νA(t) and νB(t). The evolution equation becomes:

iddt

(νAνB

)=

[Hd − iU†

(dUdt

)]=

(EA(t) −iθ(t)iθ(t) EB(t)

) (νAνB

)(3.64)

The adiabatic approximation is valid when the transitions between the instantaneous mass eigenstates

are suppressed and that happens when |θ| |EA − EB|. Only the energy difference counts because as

39

we’ve stated before, any term proporcional to the unity matrix is irrelevant for oscillations.

γ−1≡

2|θ||EA − EB|

=sin 2θ0

∆m2

2E

|EA − EB|3|

√

2GFNe| 1 (3.65)

This γ parameter is called the adiabacity parameter. In the adiabatic limit γ→ 0 the Hamiltonian in

(3.64) becomes diagonal and the time evolution of the matter eigenstates is then a simple matter of phase

factors. To set it more clearly, consider a neutrino born at time t = ti in the state:

ν(ti) = νe = cosθiνA + sinθiνB (3.66)

The adiabatic evolution leads at time t = t f to the state:

ν(t f ) = cosθi exp(−i

∫ t f

ti

EA(x)dx)νA + sinθi exp

(−i

∫ t f

ti

EB(x)dx)νB (3.67)

P(νe − νµ) =12−

12

cos 2θi cosθ f −12

sin 2θi sin 2θ f cos∫ t f

ti

(EA(x) − EB(x)) dx (3.68)

where x ≈ t for relativistic neutrinos. It becaomes evident now that only phase differences are relevant

for neutrino oscillations since:

Φ ≡

∫ t f

ti

(EA(x) − EB(x)) dx =

∫ t f

ti

m2A −m2

B

2E)

dx (3.69)

The only term oscillating with time is the last one. This term vanishes if the matter density at

production vertex is too far above the MSW resonance (sin 2θi ∼ 0), in which case the non-oscillatory

transition takes place with P(νe − νµ) = 12 (1 − cos 2θ f ) = cos2 θ f . If the final medium is just vacuum

θ f = θ0, the formula is exactly like the one we stated in our previous semi-quantitative discussion of

adiabatic conversion. Now let’s suppose some violation of adiabacity happen leading to transitions

between νA and νB. The probability of jumping from one state to another is the same as the Landau-

Zener-Stuckelberg probability denoted PLZS. In case no jump occurs, the transition probability is the

same as in the adiabatic case, and its conditional probability is then (1−PLZS)Pad(νe − νµ). One has to add

now the probability PLZSPad(νe − νe) that the νe survives but a jump occurs . The total probability is:

40

P(νe − νµ) = (1 − PLZS)Pad(νe − νµ) + PLZSPad(νe − νe) (3.70)

=12

(1 − PLZS)(1 − cos 2θi cosθ f − sin 2θi sin 2θ f cos Φ

)(3.71)

+12

PLZS

(1 + cos 2θi cosθ f + sin 2θi sin 2θ f cos Φ

)(3.72)

=12

[1 − (1 − 2PLZS)

(cos 2θi cosθ f + sin 2θi sin 2θ f cos Φ

)](3.73)

We can ommit the oscillating terms that average to zero:

12

[1 − (1 − 2PLZS)

(cos 2θi cosθ f

)](3.74)

If γ is not too small, the semi-classic approximation for the PLZS is valid and is given by:

PLZS ≈ e−2πΓ (3.75)

where

Γ = Φ∗ −Φ = −2Im∫

∆m2

2Edt = −Im

∫ Am

AMSW

∆m2

EdxA

(3.76)

where AMSW is the value of A at resonance and Am is the value of A when ∆m2 = 0. If A = const. as it

is approximately in the Sun, then

PLZS ≈ e−π2 γr (3.77)

where γr is the adiabacity parameter taken at the MSW resonance point. In the adiabatic limit,

γr 1, PLZS ∼ 0; conversely, in the opposite limit γr 1, PLZS ∼ 1 and the both survival and transition

probabilities swap values.

Let’s now discuss the adiabaticity condition. Since the energy differences are minimal and equal to

(∆m2/2E)sin2θ0 at resonance point, if adiabaticity is valid for AMSW then it is valid anywhere else. Let’s

write thus

γr =

(∆m2

2Esin 2θ0

)2 1√

2GFNe|MSW= tan 2θ0 sin 2θ0

∆m2

E

∣∣∣∣∣∣Ne

Ne

∣∣∣∣∣∣−1

MSW(3.78)

The resonance condition is approximately satisfied for ∆m2 sin 2θ ≥ 1/2 and within a range of electron

density

41

δNe ≈∆m2 cos 2θ

2√

2GFE(3.79)

and

δx ≈δNe

|Ne|r=

∆m2 sin 2θ0

|Ne|r2√

2GFE=

√2π

|Ne|rLrGF(3.80)

where Lr is the oscillation length at resonance point

Lr =4πE

∆m2 sin 2θ0(3.81)

and

γr = πδxLr

(3.82)

Therefore the adiabaticity condition sums up to δx Lr.

3.3 Solar Neutrino Problem

The Solar Standard models since the eighties have been widely successful in predicting and explaining

a vast range of experimental data. In these models the main idea is that thermonuclear reactions are at

the base of the solar combustion, and in our sun – being part of a family of stars called main sequence

stars – it comes mainly from a fusion of hydrogen and helium:

4p + 2e− → 4He + 2νe + 26.73MeV (3.83)

The main chain reaction happening in the sun is the pp cycle. The second major cycle is the CNO cycle

(carbon-oxygen-nitrogen) which is only responsible for 2% of the energy produced in the sun. Neutrinos

are emitted in both reaction schemes. Six of the eleven reactions in the pp chain produce neutrinos, which

are either nuclear beta decays or electron capture reactions. The pep and electron capture into 7Be reactions

emit beams of monochromatic electron neutrinos whereas the neutrinos bred in the other pp reactions

have non-discrete energy spectra. The fluxes of neutrinos are calculated based on these thermonuclear

reactions, on the assumption that the sun is in local hydrostatic equilibrium, that the energy is transfered

between different regions of the sun through radiation and convection. Additionally, the models are

calibrated in order to obtain the present values of solar radius, luminosity and He/H ratio. Some recent

42

models even incorporated diffusion processes of helium and heavy elements. The number of existent

models based on these premises in quite large (about 20), but all of them – except one – agree with each

other within a 3σ acceptance level when it comes to predicting fluxes of elements such as 8Be or 7Be[42]

and the most recent models also managed to reproduce succesfully with a very high accuracy the solar

sound velocities inferred from the heliosismological measurements[43]. The solar standard models are

hence widely accepted as accurate.

The neutrino fluxes predicted from these models are shown in table 3.3[44].

On the one hand, the pp flux is known to a good certainty ∼ 1% because it depends on the sun’s

luminosity which is relatively well known. On the other hand, higher energy fluxes are less well known:

∼ 20% for 8B and only the order of magnitude is known for the hep neutrinos. Nevertheless, these fluxes

are quite smaller than the pp ones, so the total flux is pretty much left untouched; however, it can affect

the high energy spectrum.

Several experiments have tried to measure the flux of solar neutrinos. The pioneering experiment is Ray

Davis’s 600 tonne chlorine tank in the Homestake mine[50], South Dakota. It’s based on the reaction:

νe + 37Cl→ 37Ar + e− (3.84)

The energy threshold of this reaction is 0.184MeV, so only the 8B (largest contributor), 7Be and pep

neutrinos can be detected. The Argon is extracted from the tank using chemical methods and counted

in proporcional counters. Davis’ radio-chemistry assay, begun in 1967, already finds evidence for only

one third of the expected number of neutrino events. In 1986, a light water Cherenkov experiment at

Kamioka, Japan, is altered to detect solar neutrinos through scattering with electrons:

νa + e− → νa + e− (3.85)

The only energy threshold E > 7.5MeV for this experiment comes from the background cuts. The later

43

upgrade of this experiment, Super-Kamiokande, has a better threshold of E > 5.5MeV. Because of these

cuts, both versions of the experiment are only sensitive to 8B neutrinos. This experiment also presents a

proof that the detected neutrinos come from the sun because when E me the angular distribution of

(3.85) is peaked around 180o degrees which proves finally that the flux is coming from the sun, unlike

in previous experiments where it was all a matter of common sense that no other source besides the sun

was intense enough to produce netrinos in such amounts of strength and energy. Like in Homestake, the

Kamioka detectors find one half of the expected events for the part of the neutrino spectrum for which

they are sensitive. There are yet another two important experiments contributing to the solar neutrino

problem (SNP), SAGE and GALLEX, both gallium detectors employing the reaction:

νe + 71Ga→ 71Ge + e− (3.86)

whose energy threshold is 0.234 MeV allowing these experiments to harvest neutrinos from the plen-

tiful pp flux. They also find about 60-70% of the expected rate[45]. Anyhow, the clear trend is that the

measured flux is found to be dramatically less than is possible for our present understanding of the

reaction processes in the sun and comes to odds with a widely accepted family of models which so far

appeared to be consistent with each other. The solutions to the SNP are multifold:

• astrophysical solution: insufficient knowledge of solar physics or an error in some input parameter;

• experimental solution: miscalculation of detection efficiencies or cross sections in the puzzling ex-

periments;

• particle physics solution: unknown neutrino physics.

If the first hypothesis is the solution, then only the total fluxes of each nuclear reaction may be differ-

ent from those predicted by the SSM because the energy spectra of the various components of the solar

neutrino flux are well known from standard nuclear physics. Unfortunately, the SNP has more than

just flux deficiencies: the results of different experiments seem to be inconsistent with each other. For

example, one can infer the flux of 8B directly from the Kamiokande and SKamiokande data, and find the

corresponding to the Homestake detection rate. If done so, the contribution is found to be larger than

the total detection rate, which means that the contribution appears to be negative! If we admit that the

solar spectra are undistorted for each reaction, then the SNP remains no matter what we do without new

neutrino physics. We remind again that the SSMs are well grounded, agree with each other and with

44

Solution Status Requirement Dependence Other features Original references

Resonant conver-sion to muon or tauneutrinos

a good fit, wellmotivated[49]

mixing of neutri-nos, mass O(10-3eV), see plot.

day night effect forlarge mixing

large angle solutionbest fit

Mikheyev andSmirnov[18] (1986),Wolfenstein (1978).

Resonant con-version to sterileneutrinos

disallowed by SNOdata[48]

mixing of neutri-nos, mass O(10-3eV)

day night effect forlarge mixing

vacuum oscillation not a good fitinconsistent withSNO[49]

mixing of neutri-nos, mass O(10-5eV)

annual variations Pontecorvo (1967)

helicity flip bad fit magnetic momentO(10-11 µB)

anticorrelation withsunspots

solar magnetic fieldunknown

Voloshin, Vysotskyand Okun[51](1986)

resonant spin-flavor conversion

still alive (hard tokill) unmotivated

magnetic momentO(10-11 µB), massO(10-3 eV)

anticorrelation withsunspots

taking the solarmagnetic field asfree parameter fitsall the results[53]

Akhmedov (88) Limand Marciano (88)

neutrino decay inconsistent lifetime < 8 min static constrained bySN1987A

Bahcall, Cabibboand Yahil (72)Pakvasa andTennakone (72)[52]

solar astrophysics desperate new physics insidethe sun

model dependent Conflicts helioseis-mology

Table 3.1: Different solutions to the SNP and their experimental counterparts.

heliosismological observations.

The second hypothesis is also very unlikely for most of the cited experiments. All of them but Homestake

have been calibrated and found to be in very good agreement with expectations. Even the Homestake

detector’s argon extraction efficiency was checked by doping it with a small number of radioactive argon

atoms, but no real calibration was caried out since no artificial sources of neutrinos exist with a suitable

energy spectrum. Also, to prove this hypothesis one would have to explain why experiments of different

unrelated types (chlorine, gallium or water Cherenkov) have been mislead, which is either very unlikely

or very depressing for a whole generation of experimentalists.

We are then left with the last hypothesis that there is some unknown neutrino physics dimming the

electron-flavoured flux. Several mechanisms have been proposed to explain the SNP with special re-

gards to neutrino oscillations. The proposed alternative solutions and their experimental validation

are summed up in table 3.1. The SNP can be considered almost as a solved problem by now. The

results from SNO clearly indicate that the total neutrino flux from the sun is in accordance with the

solar models of Bahcall and Pisconneault. The Kamland reactor neutrino experiment has measured the

electron antineutrino flux from nearby reactors and the results are consistent with the solar neutrino

oscillations. Neutrino oscilaltions can convert a fraction of solar νe into the other two flavours. Since

the energy of solar neutrinos is limited to a few MeV (which is already a very thin flux coming from

45

hep neutrinos) no CC reactions can happen in the τ and µ flavours because the lepton masses are two

high (mµ ∼ 1GeV and mτ ∼ 100MeV) and their corresponding neutrinos can’t therefore be detected in

chlorine- or gallium-based experiments. They can however still interact through NC interaction which

is visible in water Cherenkov detectors, but has a cross-section about six times smaller than CC, which

explains why K and SK experiments saw a deficit in neutrino flux.

The MSW effect greatly enhances oscillations inside the sun which allows the solar data to be fitted even

with a very small vacuum mixing angle, but no matter what the vacuum mixing angle is, in most cases it

will change the oscillation pattern. The most important of these experiments in validating the neutrino

flavour oscillation is most certainly SNO. SNO is an imaging Cherenkov detector that uses heavy water

(D2O) as both the interaction and detection medium. It has a spherical geometry covered with photo-

multipliers (PMTs). It can accomodate charged current (CC), neutral current (NC) and electron scattering

(ES) interactions. The comparison of each of these reactions (separating events through angular and

cross-section filtering) allows a flavour identification of the measured fluxes. The final results for the

fluxes are:

φCC = 1.76+0.06−.0.05(stat.)+0.09

−0.09(syst.) × 106cm−2s−1, (3.87)

φNC = 2.39+0.24−.0.23(stat.)+0.12

−0.12(syst.) × 106cm−2s−1, (3.88)

φES = 5.09+0.44−.0.43(stat.)+0.46

−0.43(syst.) × 106cm−2s−1, (3.89)

φe = 1.76+0.05−.0.05(stat.)+0.09

−0.09(syst.) × 106cm−2s−1, (3.90)

φµτ = 3.41+0.45−.0.45(stat.)+0.48

−0.45(syst.) × 106cm−2s−1, (3.91)

(3.92)

Adding the statistical and systematic errors in quadrature, we find that φµτ is 5.3σ away from its

null hypothesis value of zero. The total neutrino flux is on good agreement with the SSM’s predictions

for the 8B flux and the combined flux results are consistent with neutrino flavor transformation with no

distortion in the 8B neutrino energy spectrum[19] (cf. Fig.??). The comparison of day and night fluxes

in SNO provides a way to separate the pure-vacuum oscillation and MSW-enhanced oscillation theories

because in the second case one would see a difference between day and night fluxes since in the latter

the neutrinos must travel through the earth’s matter whereas if the MSW effect has no significance one

wouldn’t see any difference between fluxes measured in these two seperate day times. The day-night

assymetry is found to be:

46

Figure 3.3: The flux of νµ + ντ vs. the flux of νe. The fluxes as determined by the SNO CC, NC and ESreactions are shown, as well as the predictions of the Standard Solar Model (SSM). The errors representedare ±1σ, and the best fit values for ϕe and ϕµτ are shown.

Ae = 7.0 ± 4.9(stat.)−1.2+1.3%(syst.) (3.93)

which shows that the no-MSW effect hypothesis is inconsistent with SNO data. The results of all

solar neutrino data (SNO, SK, ...) and the Kamland experiment point to the values:

∆m212 ≈ 8 × 10−5eV2, sin22θ12 ≈ 0.8 (3.94)

47

Chapter 4

Neutrino Mass Models

4.1 Seesaw models in the Left-Right Symmetric model

All known interactions – gravitational, electromagnetic and strong – but the weak interaction respect

parity. If we expect to find some sort of common rule for all of Nature’s forces, it is only tempting to

suppose that at some high energy scale parity is conserved before it is somehow broken and from there

on maximally violated by the weak interaction, as observed nowadays in experiment. The left-right

symmetric model as we introduce it here is a minimal extension to the SM nothing is changed except for

the inclusion of an additional SU(2)R symmetry for the RH fermions which is the mirror of the known

SU(2)L making the model symmetric for left and right chiralities. As we will see in later on sections of

this review, the RHν – which we have already introduced to generate neutrino masses – is needed in

GUTs such as SO(10), and in this case is required in order to complete the left-right symmetry (LRS),

since all fermions must have an partner of opposite chirality. The LRSM is in fact a subgroup of SO(10).

Its attractiveness comes from the fact that parity is now a spontaneaously broken symmetry.

4.1.1 The Gauge Sector and Symmetry breaking

The simplest way to go from the SM to a LRSM is therefore SU(2)L × SU(2)R × U(1)Y where the lepton

and quark fields are assigned to the following irreducible representations1:

LL =

(νLlL

): (2, 1,−1), LR =

(NRlR

): (1, 2,−1);QL =

(uLdL

): (2, 1, 1/3), QR =

(uRdR

): (1, 2, 1/3).

(4.1)

We now require that the lagrangian of the model1We have safeguarded the possibility of the RH neutrino being of Majorana type and having a different mass from the LH

one by writing NR instead of νR

48

L = −14

WLµν ·W

µνL −

14

Bµν · Bµν + LLγµDµLL + QLγ

µDµQL

+ Tr∣∣∣Dµφ

∣∣∣2 + Tr∣∣∣Dµ (σ · ∆L)

∣∣∣2 + (L→ R) − V(φ,∆L,∆R, φ) − LY (4.2)

be invariant under P transformation. The covariant derivative is

∂ − i3∑

k=1

(gLWL

k TLk + gRWR

k TRk

)− ig′BY (4.3)

where WL,R and B are the gauge bosons for SU(2)L,R and U(1)Y respectively and TL,R are the SU(2)L,R

generators. The fields transform as

LL ↔ LR (4.4)

QL ↔ QR (4.5)

WLk ↔WR

k (4.6)

which needlessly to say implies gL = gR ≡ g. The electric charged is given analogously to the SM by

Q = TL3 + TR

3 +Y2

(4.7)

The quark doublets must have hypercharge Y = 1/3 and the lepton doublets Y = −1 since they

all bear a unit charge but with opposite signs. The formula then becomes physically meaningful with

Y = B − L. This formula has very important implications. At a scale where the LRSM has been broken

into SU(2)L ×U(1)B−L we have the relation[57]

∆TR3 = −

12

∆(B − L) (4.8)

If the interactions are baryon number conservating as they are in the SM, then |∆L| = 2∣∣∣∆TR

3

∣∣∣. If

symmetry breaking is chosen as to violate TR3 by one unit then L is violated by two units implying

Majorana neutrinos. Even if TR3 is only violated by half a unit Majorana neutrinos can still arise at higher

orders. The same happens in the hadronic sector if we choose leptons to be conserved: B is violated by

two units and neutron-antineutron oscillation happens.

The Higgs sector of this sector must be LRS so that the symmetry is maintained; the first candidate is the

bidoublet[20]

49

φ =

(ϕ0

1†

ϕ+2

−ϕ−1 ϕ02

), (2, 2, 0) (4.9)

as well as its conjugate field

φ ≡ σ2φ†T =

ϕ02†

ϕ+1−ϕ−2ϕ

01 (4.10)

This field can couple to the fermion bilinears QLQR and LLLR, which after symmetry breaking

⟨φ⟩

=

(k 00 k′

)(4.11)

can give masses to both quarks and leptons. Nevertheless, since the bidoublet is B − L neutral the

remaining symmetry is U(1)TL3 +TR

3×U(1)B−L and not U(1)Q. More scalar multiplets are therefore required.

One might choose simple SU(2) doublets for each chirality but they would only lead to Dirac neutrinos

and since our interest lies in neutrino masses and for the purpose of explaining their smallness we choose

two doubly-charged triplets[58]

∆L : (3, 1, 2), ∆R : (1, 3, 2) (4.12)

which lead to Majorana masses. To represent the triplet in a somewhat simpler and easier-to-compute

way, we note that the tensor product of two doublets is equal to 2⊗2 = 3 + 1, which means that the triplet

can be written down as a 2 × 2 traceless matrix (the product of two doublets minus a singlet). Hence,

∆L =

(∆+

L/√

2 −∆++L

∆0L −∆+

L/√

2

), ∆R =

(∆+

R/√

2 −∆++R

∆0R −∆+

R/√

2

)(4.13)

which develop a vacuum expectation value

⟨∆L,R

⟩=

(0 0

vL,R 0

). (4.14)

It is now useful to define

W± ≡W1 ∓ iW2√

2, T± ≡

T1 ∓ iT2√

2,

T+ =1√

2

(0 10 0

), T− =

1√

2

(0 01 0

). (4.15)

and a zero-mass eigenstate which is given by

50

A = −sw(W3

L + W3R

)+

√c2

w − s2wB (4.16)

where

cW = cosθw = −g′√

g2 + 2g′2

sW = sinθw =

√g2 + g′2

g2 + 2g′2

e = gsw.

(4.17)

We also define

Z = cwW3L −

s2w

cwW3

R +sw

cw

√c2

w − s2wB

Z′ =

√c2

w − s2w

cwW3

R −sw

cwB (4.18)

The covariant derivative becomes

g(WL3 + WR3TR3) + g′BY = −eAQ +g

cwZ(T3

L) −Qs2) +g√

c2w − s2

w

cw

(T3

R − Ys2

w

c2w − s2

w

)Z′ (4.19)

Hence the doublets yield at T=0

g2

2

∣∣∣W+L k1 −W+

Rk∗2∣∣∣2 ,

g2

2

∣∣∣W+L k2 −W+

Rk∗1∣∣∣2 ,

g2|k1|

2

4c2w

∣∣∣∣∣−Z +

√c2

w − s2wZ′

∣∣∣∣∣2 ,g2|k2|

2

4c2w

∣∣∣∣∣−Z +

√c2

w − s2wZ′

∣∣∣∣∣2 (4.20)

Likewise, the triplets yield

|gW+L vL|

2 , |gW+RvR|

2 , |g

cw

Z +s2

w√c2

w − s2wZ′

vL|2 , |

gcw√c2

w − s2w

Z′vR|2 (4.21)

The mass matrix for the neutral gauge bosons is

g2

c2w

(Z Z′

) |k1|

2+|k2|2

4 + |vL|2

−|k1|

2+|k2|2

4

√c2

w − s2w +

s2w√

c2w−s2

w

|vL|2

−|k1|

2+|k2|2

4

√c2

w − s2w +

s2w√

c2w−s2

w

|vL|2 |k1|

2+|k2|2

4 (c2w − s2

w) +s4

w|vL|2+c4

w|vR|2

c2w−s2

w

(

ZZ′

)(4.22)

51

The charged boson mass matrix is also obtained as

g2(

W−L W−R) |k1|

2+|k2|2

2 + |FL|2

−k∗1k∗2−k1k2

|k1|2+|k2|

2

2 + |FR|2

( W+L

W+R

)(4.23)

The charged gauge bosons mix. Since k1k2 has T3L = −1 and T3

R = 1 we may choose the mixing to be

real and the diagonalization of the matrix gives two physical charged bosons

(W1W2

)=

(cos ζ − sin ζsin ζ cos ζ

) (WLWR

)(4.24)

If the Higgs potential is chosen as to give |vL| , |vR|, parity is spontaneously broken. Morover, if

|vR| is assumed to be much smaller than |k1|, |k2| and |vL| (which strictly speanking is not required from

phenomenological considerations), the mixing angle and the charged boson masses are given by

ζ ≈k1k2

|vL|2,

m2W1≈ g2

(|vL|

2 +|k1|

2 + |k2|2

2

)(4.25)

m2W2≈ g2

(|vR|

2 +|k1|

2 + |k2|2

2

)Since the mixing angle is very small, W1 and W2 coincide to a good approximation with WL and WR:

M2WL≈ cos2 ζM2

W1+ sin2 ζM2

W2

M2WR≈ sin2 ζM2

W1+ cos2 ζM2

W2(4.26)

The charged current interaction takes the form in the LRSM

Lcc =g√

2

[(uLγµdL + νLγµeL

)Wµ

L(uRγµdR + NRγµeR

)Wµ

R

]+ h.c. (4.27)

For MWR MWR , the charged current interactions will violate almost maximally parity at low

energies. Therefore, any deviation from the V − A structure may constitute evidence for right handed

CC currents in a LRSM.

Similarly, Z and Z′ may be considered approximate eigenstates of mass

52

m2Z =

g2

c2w

(|k1|

2 + |k2|2

2+ 2|vL|

2)

m2Z′ =

g2

c2w

(|k1|

2 + |k2|2

2(c2

w − s2w) + 2

s4w|vL|

2 + c4w|vR|

2

c2w − s2

w

)(4.28)

The mixing is given by

(Z1Z2

)=

(cos ξ sin ξ− sin ξ cos ξ

) (ZZ′

)(4.29)

where for |vL| 1 and |k1|2 + |k2|

2 |vL|

2 the mixing angle is given by

tan 2ξ ≈(c2

w − s2w)3/4

2c4w

|k1|2 + |k2|

2

|vR|2≈ 2

√c2

w − s2wM2

Z/M2Z′ (4.30)

In the limit where |vR| 1, the mixing angle becomes zero. In this limit, the standard model relation

mW1 = MZ1 cosθw is changed into (assuming |k1|2 |k2|

2)

mW1 =

√|k1|

2 + 2|vL|2

|k1|2 + 4|vL|2

MZ1 cosθw (4.31)

Now, the squared value of the coeficient on the RH side of (4.31) has been constrained experimentally

down to 0.998 ± 0.050 which restricts |vL|/|k1| < 0.17. The VEV of a LH gauge interaction must indeed be

very small compared to the VEV the either one of the doublets.

The neutral current interaction is

Lnc =g

cW

KµLZµ +1√

c2w − s2

w

(sin2 θWKµL + cos2 θWKµR

)Z′µ

≈

gcW

LµL − ξ√c2

w − s2w

(sin2 θWKµL + cos2 θWKµR

)Z1µ +

1√c2

w − s2w

(sin2 θWKµL + cos2 θWKµR

)Z2µ

(4.32)

where

KµL,R =∑

f

fγµ(T3

L,RPL,R −Q sin2 θW)

f (4.33)

Phenomenology implies certain constraints on the masses of the bosons. Experimental bounds from

neutrino neutral current data set the Z2 mass above 389 GeV[60]. Further analysis of pp collider data

leads to a bound of MZ2 ≥ 445GeV[61]. Bounds for the Z2 mass are much easier to set than the ones for

MW2 and ζ because the NC interaction doesn’t depend on the RHν field (c.f. (4.32)), which would require

53

a better knowledge of the RHν mass. Unfortunately, such is needed to set bounds on MZ2 . If the RHν

is much lighter than the muon, then measurements of the ξ-parameter in µ-decay using fully stopped

polarized muons tell us that the mass of this charged boson must be higher than 432 GeV.[62] However,

the result is useless if the RHνis very heavy because these cease to have any detectable contribution to

muon decay. The best limits in this latter case are set by considering the mass difference of KL−KS which

gives[63]

M2WL

M2WR

<1

430⇒MWR ≥ 1.6TeV (4.34)

4.1.2 The seesaw mechanism

As stated before, two exclusive doublets (one for each SU(2) group) could have done the job of breaking

the LRSM into U(1)em but the RHνneutrinos would be of Dirac type and – as shown before in chap.3, the

seesaw mechanism which requires Majorana masses for the neutrinos provides a natural way to explain

the smallness of ν mass. Thus we wish to recreate it here, and such is obtained thanks to the two triplets

we have incorporated in our model. The most general LRSM-invariant Yukawa couplings for leptons are

−LY =∑

i, j

hi jLiLΦL jR + hi jLiLΦL jR (4.35)

+ fi j

(LT

iLC−1ε(τ · ∆L)L jL + LTiLC−1ε(τ · ∆L)L jL

)+ h.c. (4.36)

At a first stage, the right triplet acquires a VEV vR , 0 giving heavy Majorana masses to the RHνgiven

by the matrix fi jvR. At the second stage, the bidoublet Φ acquires a VEV but vL is set to zero for the

purpose of simplicity in our present discussion, so that we obtain a type-I seesaw formula with the

following 2ng × 2ng mass matrix

M =

(0 mD

mTD f vR

)(4.37)

where (mD)i j = hi jk1 + hi jk2. The matrix can be approximately (up to terms. O(ρ2)) block diagonalized

by an orthogonal matrix O[40]

O =

(1 − 1

2ρρT ρ

−ρT 1 − 12ρ

Tρ

)(4.38)

The mass matrix on the lower corner of OTMO can be associated with the heavy RHνbecause

54

mheavy = f vR +1

vR

(mT

DmD f−1 + f−1TmT

DmD

)= f vR + O

m2l

vR

≈ f vR (4.39)

where ml is the mass of the charged leptons. The other one is thus the light mass matrix given by

mlight = −1

vRmD f−1mT

D (4.40)

This formula reveals an additional interest for the LRSM as it connects V + A suppression with the

smallness of the neutrino mass since as vR →∞, mν → 0. Now, if we neglect the mixings and set mD ≈ ml,

we obtain

mνl ≈m2

l

MNl

(4.41)

If the heavy RHνmass is generation independent, this means

mνl −mν′l∝ m2

l −m2l′ (4.42)

This last result brings out the problems of our lack of knowledge about the Dirac mass term in (4.40).

If we wish to test this theory in soon-to-come colliders such as the LHC, the WR mass has to be in the

TeV scale. That will unfortunately imply by taking mD ≈ ml that for mνe ∼ 1eV, we get mνµ ∼ 40keV and

mντ ∼ 12MeV which is a spectrum absolutely ruled out by current data which sets . Therefore we may

conclude that a LRSM at the TeV scale where neutrino Dirac masses arise at Lagrangian level is ruled

out. Nevertheless, Dirac masses may still arise at higher loop levels because then mD ∼ αm f /4π, which

safeguards phenomenological possibilities for the model. This can occur in some SO(10) and E6 GUT

models which gives the model restored credibility.

4.2 Neutrino masses in GUT theories

The Standard Model is a wonderfully predictive model which so far works very well with present

experimental data. Even so, it has failed in blindly predicting any fermion masses. The truth is that any

fermion mass or mixing still is a mystery to the SM, which is related to our unsophisticated knowledge

of the Higgs mechanism. The solution to this and to other problems such as the strong CP problem yet

lies ahead of us. The discovery of neutrino mass and our doubts about how to recreate it within the SM’s

framework have only proven this ignorance of ours. Along with this obvious promise of new physics,

goes another hope for new physics which is more closely related to conceptual aesthetics rather than

actual phenomenological requirement, which is the unification of all forces of nature. The first successful

55

attempt was Maxwell’s unification of the electric force together with the magnetic one; the second one

is the actual SM which unites the weak and electromagnetic interactions. The idea of gauge theories

combined with a symmetry breaking mechanism is a very permisive theoretical groundbasis which has

allowed us to wonder – just like SU(3)C×U(1)em is only part of SU(3)C×SU(2)L×U(1)Y – whether the SM

is just a bit of something bigger such as SU(5), SO(10) or E6. Enlarging the group has many advantages

linked with reducing the model’s parameters but on the other hand also increases the SSB’s complexity

which is exactly what we lack knowledge about. The phenomenological interest of these models is that

they allow baryogenesis and predict charge quantization. We would therefore like to see them also

explain the smallness of neutrino masses or the greatnes of neutrino mixings. We will briefly present

how neutrino masses arise in SU(5) and SO(10).

4.2.1 SU(5)

SU(5) is the smallest group that can contain the SM. Its representations can be expressed in terms of

SU(3) × SU(2) ×U(1)’s representations as shown in Table4.2.1.

dimension (SU(3),SU(2))Y5 (3, 1)−2/3 ⊕ (1, 2)110 (3, 2)1/3 ⊕ (3∗, 1)−4/3 ⊕ (1, 1)215 (6, 1)−4/3 ⊕ (3, 2)1/3 ⊕ (1, 3)224=24∗ (8, 1)0 ⊕ (3, 2)−5/3 ⊕ (3∗, 2)5/3 ⊕ (1, 3)0 ⊕ (1, 1)045 (8, 2)1 ⊕ (6∗, 1)−2/3 ⊕ (3, 3)−2/3 ⊕ (3∗, 2)−7/3

⊕(3, 1)−2/3 ⊕ (3∗, 1)8/3 ⊕ (1, 2)1...

...

one sees quite immediately that all SM representations fit in a 5∗ + 10:

5∗ + 10 = (3∗, 1) + (3, 2) + (1, 1) + (1, 2∗) (4.43)

Let’s rewrite this with explicit reference to the SM’s one-generation family of fermions:

5 : (ψi)L = (d1, d2, d3, e−,−ν)L (4.44)

5∗ : (ψi)R = (d1, d2, d3, e+,−ν)R (4.45)

56

10 : (χi j)L =1√

2

0 uc

3 −uc2 −u1 −d1

−uc3 0 uc

1 −u2 −d2uc

2 −uc1 0 −u3 −d3

u1 u2 u3 0 −ec

d1 d2 d2 ec 0

L

(4.46)

Writing down the Yukawa interactions requires breaking down the group SU(5)→ SU(3)colour×U(1)em

with the simplest (fundamental) scalar representation, i.e., 5, along with 24. This is called the minimal

SU(5) model. Other symmetry breaking mechanisms which induce fermion masses may involve 10, 45

or 50. The mass structure of the fermion sector depends on that. The Higgs representation involved are

H = (h1, h2, h3, h+,−h0) = (3, 1) + (2, 1)under SU(3) × SU(2) (4.47)

Σ = 5 × 5∣∣∣traceless = φ5 ⊗ φ5 −

15φ5 · φ5 (4.48)

When Σ acquires a VEV it breaks SU(5) down to SU(3)C × SU(2)L ×U(1)Y as follows

〈Σ〉 = VDiag (1, 1, 1,−3/2,−3/2) (4.49)

giving masses to the gauge bosons[8]

M2X = M2

Y =258

g2V3 (4.50)

Later on, the 5 scalar acquires in its turn a VEV breaking the remaining symmetry into SU(3)C×U(1)Q

〈H〉 =(0, 0, 0, 0, v/

√

2)T

(4.51)

giving masses to the Z0 and the W± bosons

MW =gv2

MZ =gv

2 cosθW(4.52)

where θW is the usual Weinberg angle now expected to give at unification scale tanθW =√

3/5.

Unfortunately, as pretty as this model may seem, it has been ruled out by LEP and SLC data. The

unification scale V is found by running the coupling constants at low energies and having them unite

into only one. In the one-loop approximation we have

dgn

d lnµ= bn

g3n

16π2 (4.53)

57

where bn = 4/3ng − 11/3n for n = 2, 3 neglecting the Higgs contribution. For g =√

4πα,

dgn

d lnµ= bn

g3n

16π2 ⇔dαn

d lnµ= −

bn

2πα2

n (4.54)

which is solved by

α−1n =

1

α−1n (MU) + bn

2π ln(

MUMZ

)α−1

1 =1

α−11 (MU) − 7

2π ln(

MUMZ

)α−1

2 =1

α−12 (MU) − 19

12π ln(

MUMZ

) (4.55)

α−13 =

1

α−13 (MU) + 41

20π ln(

MUMZ

)For a successful unification one needs to have all αn (MU) equal. This condition is equivalent to:

α−11 − 3.08α−1

2 + 2.08α−13 = 0 (4.56)

which is satisfied by the LEP and SLC data for the gauge couplings:

α−11 (MZ) = 58.89 ± 0.11

α−12 (MZ) = 29.75 ± 0.11 (4.57)

α−13 (MZ) = 0.121 ± 0.004 ± 0.001

Nevertheless, even though the minimal non-supersymmetric model has been ruled out by accelerator

data, it is always interesting to analyse it because it is the simplest GUT and many techniques used

for higher–rank groups come out when studying SU(5). Fermion transform as 5 + 10. The product

representation of two fermion fields gives

(5 + 10) × (5 + 10) = 5 + 5 + 10 + 15 + 45 + 45 + 50 (4.58)

Clearly, there cannot be any bare mass terms or they would explicitly break the gauge symmetry.

Thus fermion masses can only arise via SSB through gauge invariant couplings to Higgs scalars. The

possible Higgs representations are shown in (4.58). Since in minimal SU(5) only the 5 and 24 Higgs

58

representations are taken into account, the most general Yukawa couplings are between the fermion fields

and the fundamental Higgs representation. Σ does not couple to fermions. The Yukawa Lagrangian is

hχTijC−1ψiH j + h′εi jklmχT

ijC−1χklHm + h.c. (4.59)

At SSB, the fermions acquire masses given by

mu = MU11v

mc = MU22v

mt = MU33v

md = me = MD11v (4.60)

ms = mµ = MD22v

mb = mτ = MD33v

The neutrinos remain massless because there are no RHν, so Dirac or RH Majorana terms cannot

form. Neither do LH Majorana mass terms arise at higher loop levels, because the Higgs bear no B − L

quantum number. To see this let’s define an operatorF to which we assignF (H) = −2/3,F (ψ) = 1/3 and

F (χ) = 1. Clearly, the Yukawa couplings given in (4.59) are invariant under U(1)F . This transformation

is related to B − L by the relation

B − L =35F −

2√

15λ24 (4.61)

which brings B − L values of h1,2,3 = −2/5 and h+,0 = 0. The adjoint representation does have a B − L

non-zero value, but it doesn’t couple to the fermion fields. Since the Higgs component which develops

a VEV has no B − L quantum number, it cannot generate any Majorana mass terms.

One might argue that the RHνwas left out of the picture purposefully and that nothing forbids us from

adding an SU(5) singlet to the fermion representations. A Dirac mass term can then arise by a coupling

of the 5 fermion representation with the 5 of Higgs and the fermion singlet νR. But this term would

naturally be expected to have mass at quark or lepton scale which is ruled out by experimental bounds.

It would not explain the smallness of neutrino masses. A seesaw mechanism would come in handy here.

The introduction of a symmetric 15-dimensional Higgs boson Πi j. The Yukawa couplings for this scalar

are

59

L15Y = fψT

i C−1ψ jΠi j + h.c. (4.62)

According to Table 4.2.1, the multiplet Π contains a doubly charged isotriplet ∆L which could induce

a type-II seesaw. We assign a non-zero VEV to

〈Π55〉 =u√

2(4.63)

If we assign F (Π) = 2 then the Lagrangian has exact B− L symmetry which is broken by the vaccum.

However, in the most general gauge-invariant potential B − L is explicitly broken by

AφTε (σ · ∆)φ + h.c. (4.64)

where φ is the isodoublet contained in H. This term has to be forbidden in order to have B−L broken

by SSB, which imposes the existence of a pseudo-scalar particle J produced by the VEV of ∆. Such a

model has already been ruled out by measurements of the Z-width. Yet if we allow B− L to be broken at

lagrangian level in the Higgs potential by

V = µΠHiH jΠi j + h.c. (4.65)

U(1)F is explicitly broken in this model and hence B − L as well. The VEV is

〈Π55〉 ≈ µΠv2

V2 (4.66)

Even if µΠ is of order V, we get u ∼ 10−12v ∼ 10−1eV. The smallness of neutrino mass (which is a

Majorana particle) comes naturally in this model.

4.2.2 S0(10)

The SO(10) GUT is very appealing to us interested in neutrino masses because the most fundamental

representation of this gauge group contains exactly all known fermions plus an overall singlet – the RHν.

Thus, unlike the SM or SU(5), neutrino masses arise naturally in this model. Fermion masses are tighted

up to the choice of the Higgs sector and the latter is tied with how we intend to break the symmetry.

Because SO(10) is of rank-5, there are many ways to break it down to SU(3)c × SU(2)L × U(1)Y. Its two

maxinmal continuous subgroups are [12]

• G224 = SU(2)L × SU(2)R × SU(4)C or G224D = SU(2)L × SU(2)R × SU(4)C ×D

60

• G5 = SU(5) ×U(1)

where D-parity is a discrete symmetry which interchanges (2, 1, 4) ↔ (1, 2, 4) in the SO(10) spinor;

its existence imposes gL = gR and ηB = ηB (ηB is the number density of baryons in the universe). SU(5)

has already been proved to be phenomenologically inviable unless we ”pollute” it with inumerable

Higgs representations or introduce supersymmetry, but it’s the simplest route and the philosophy of

it is basically the same for any route. Nevertheless, we still present other pathways to the SM, which

interestingly use the LRSM introduced in the previous above. Still, several roads lead down to the SM

from here. We state two important ones for neutrino masses2:

•

SO(10)MU→54 G224D

Mp→210 G224

MC=MW+R

→210 G2113

MZ′→126 G321 (4.67)

where MU ≈ 1016.6GeV, MWR = MC ≈ 105− 107GeV with MZ′ ≤ 1TeV, leading to a prediction

of sinθW ≈ 0.227[64]. This model is very seductive because it may be tested in soon-to-come

accelerators such as the LHC because of the TeV-scale of B − L breaking-

•

SO(10)MU=MC=Mp

→45 G3221

MWR ,MZ′→126 G321 (4.68)

where MU ≈ 1015.4GeV for sin2 θW ≈ 0.23 and MWR ≈MB−L ≈ 1012GeV.

Fermion masses are generated through SSB like in all previous gauge theories introduced. The

product of two fermion representations gives

16 ⊗ 16 = 10 ⊕ 120 ⊕ 126 (4.69)

Hence, the fermion masses arise through Yukawa couplings with the 10-, 120- and 126-dimensional

Higgs when these acquire VEV. The φ10 and φ126 couplings are symmetric in family indices while the

φ120 are antisymmetric. Under SU(5) these representations transform as

10 = 5 + 5

126 = 1 + 5 + 10 + 15 + 45 + 50 (4.70)

126 = 5 + 5 + 10 + 10 + 45 + 45

2We have indicated under the arrows the Higgs representation which is used for SSB and above it the masses which itgenerates, ie, the scale of SSB.

61

The components which can have non-zero VEV must be colour and charge singlets. The only Higgs

representations with such components are the 1, 5, 5, 45, 45. Their couplings to the fermions are

ψTLσ

2ψLφ10 = φ10(5) (uRuL + νRνL) + φ10(5)(dRdL + eReL

)ψT

Lσ2ψLφ126 = φ126(1)νT

Rσ2νT

R + φ126(15)νLσ2νL + φ126(5) (uRuL − 3νRνL) + φ10(5)

(dRdL − 3eReL

)(4.71)

ψTLσ

2ψLφ10 = φ120(5)νRνL + φ120(45)uRuL + φ120(5)(dRdL + eReL

)+ φ120(45)

(dRdL − 3eReL

)The mass matrix for the neutrinos is then

(νLνR

) (φ126(15) φ10(5) + φ120,5(−)3φ126(5)

φ10(5) + φ120(5) − 3φ126(5) φ126(1)

) (νL νR

)(4.72)

The φ126(1) breaks SO(10) into SU(5) × U(1) and therefore its VEV leaves the content of SU(5) intact

and its mass is expected to be higher than MX. Both Dirac and left and right Majorana mass terms

appear and the high scale of φ126(1) explains via the seesaw mechanism the small neutrino mass. But the

minimal breaking scheme involves no φ126 and is

SO(10)M′X→16 SU(5)

MX→45 SU(3) × SU(2) ×U(1)

MW→10 SU(3) ×U(1) (4.73)

However, B − L is not conserved and one can still generate RH Majorana masses with two-loop

diagrams at O(α2/π2). The RH-Moajorana mas is of order mu/MW × α2/π2MX′ which is heavy and can

still account for the LHν’s small mass. Indeed, the Dirac masses are given by[8]

mνl = mul

π2

εα2MW

MX′(4.74)

where ε is the associated with magnitude of the two-loop diagram couplings. For neutrino masses at

eV,sub-eV scale, ε & 10−2 which is reasonnable.

62

Chapter 5

Leptogenesis

5.1 Baryon asymetry of the universe (BAU)

Since Dirac’s equation which provided equally probable solutions of positive and negative energy (parti-

cle and anti-particle), the fact that the universe was mainly made of ordinary matter – and not anti-matter

– became puzzling, and because of particle-antiparticle anihilation the very reason why any bit of the

universe still exists is at least enigmatic. The next two sections focus on a new attempt to solve this

mistery.

5.1.1 Evidence for BAU

Within its reach, mankind has found that the world is mainly made of matter. On earth, the only evidence

of anti-matter’s existence comes from accelerators or nuclear decays which produce the greatest known

amounts, which still only add up to a trillionth of a gram. So all matter on earth is ordinary. The rest of

our solar system is also composed of the same kind of matter, because many probes have been sent to

many of the system’s orbiting objects and none of them desintegrated into light. Analysis of solar fluxes

allow us to probe the baryon-to-antibaryon ratio of the sun’s composition, and there again we find a

proton-to-antiproton ratio of and even this small amount of anti-protons appears to be consistent with

the assumption that it results as a byproduct of cosmic ray collision with the interstellar medium. The

same analysis is valid for the rest of the galaxy from which experiments such as Auger have detected

high energy rays coming from very far in the galaxy and the same results apply.

Studies of the large scale structure of the universe and of the cosmic microwave background’s (CMB)

anisotropy from WMAP have given us an estimate of the baryon asymetry. Because the universe went

through an e± anihilation era, it is important to compare the baryon number density to the photon density,

and the ratio of these two densities – since each one of them evolves with the inverse of a comoving

volume R−3 – is a conserved quantity as long as the B-violating reactions are occuring slowly.

63

ηCMBB ≡

ηB

ηγ= (6.1 ± 0.2) × 10−10 (5.1)

This result concurrs with the result from Big Bang Nucleosynthsis (BBN) where the abundances of3He, 4He,D,6Li and 7Li can be measured from astrophysical observations and depend crucially on the

value of the baryon-to-photon ratio[9]:

ηBBNB ≈ (5.6 ± 0.9) × 10−10 (5.2)

which is perfectly consistent, hence demonstrating the validity of BBN and standard cosmology.

For the rest of our universe things become a little more inferrative, because there is no way of telling

whether a galaxy is made of matter or antimatter just by looking at it. Nevertheless, X-Ray analysis of

the intergalactical medium has revealed the existence of hydrogen gas clouds bathing the galaxies in the

local cluster, and a non-neglegible presence of antimatter would certainly produce an accute emission

of gamma rays on the border between the antimatter patches and the gas, but no such flux has been

measured so far. Virgo, for instance, a 1013− 1014M⊙ nearby cluster of galaxies shows no abnormally

strong γ-ray flux. Therefore, if there is a significant amount of antimatter in the universe, it must be

parted from the rest of the universe in a scale of 1012M⊙ or even larger, which seems quite unreasonnable

and by no means indicates a baryon symmetric universe. More precisely, in a locally-baryon-symmetric

universe nucleons and antinucleons remain in chemical equilibrium at nb/s = nb/s ≈ 7 × 10−20 down to

temperatures of ∼22 MeV.[9] These estimated densities are several orders of magnitude smaller than the

observed values of baryon density nb/s ≈ 8× 10−11. In this symetric scenario, the universe would require

a segregation mechanism acting above 38 MeV, when the density was the observed one. Unfortunately,

the horizon at that time only contained 10−7M⊙, and causality obviously rules out by far separating

masses of 1012M⊙. We can without much further ado, say that antibaryons are nowhere to be found in

the universe compared to baryons.

Since B-violating interactions are unknown to us in the present state of our knowledge (not entirely

true at the non-perturbative level), the most reasonnable conclusion is that the universe already possessed

this asymmetry at early times. In the primordial soup (T & 1GeV, t . 10−6s) thermal quark-antiquark

pairs were highly abundant and the present maximal B-asymmetry was at the time a very small q − q

asymmetry:

nq − nq

nq≈ 3 × 10−8 (5.3)

64

In the past, the Universe was sought to have been created with this small yet crucial number right from

the start. With the advent of Grand Unification Theories (GUTs) which predict B-violating interactions,

a loophole was opened for a initially symmetric universe evolving later into this slightly asymmetric

one. But before we get into that, let’s have a short overview of what’s needed for the generation of this

number, a.k.a., baryogenesis.

5.1.2 Basic Ingredients and direct baryogenesis

Long before GUTs ever came to be in 1967, Sakharov had already laid down the threefold essential

features of a theory which aims at producing dynamically the BAU. These conditions are:

• B violation,

• C and CP violation,

• departure from equilibrium.

The technical and physical nature of each mechanism for baryogenesis may vary, but they must all

without exception contain these four crucial ingredients.

B Violation This requirement is fairly obvious; without it the present baryon asymmetry can only reflect

an initial one. GUTs provide an elegant framework for microphysics where B and L violating interactions

arise naturally thanks to putting together fermions and leptons in a reduced number of representations.

In such theories, gauge bosons mediate interactions which transform quarks into leptons and vice-

versa. The lifetime of the proton in these scenarios should be of order τp ∼ α−2GUTM4m−5

p & 1031 yrs

(current experimental limit[30]) which implies that these gauge bosons must have a mass over 1014GeV,

which is the reason why they are so invisible nowadays but could have been relevant in the past at

extreme temperatures. The present Standard Model’s Lagrangian has no such interactions, although the

symmetry is purely accidental and should neutrinos turn out to be Majorana-like, they would violate

lepton number. Also, as indicated by ’tHooft, the SM conserves B and L at the perturbative level but

there is an anomalous very small violation coming from non-perturbative transistions in the theory’s non-

trivial vacuum which is irrelevant at∼ 3K but then again could have been significant at early times thanks

to the high temperatures short after the Big Bang as noted by Kuzmin, Rubakov and Shaposhnikov[65].

C and CP violation Both are provided in the present state of our microscopic knowledge. C is maximally

violated in the electroweak sector. CP was first found to be violated in the neutral kaon system and since

65

it is linked to spontaneous symmetry breaking (SSB) which is still crudely understood one could expect

to see it in all sectors of the theory.

The reason why it is so necessary for baryogenesis seems pretty obvious at first, since we wish to have

a higher amount of baryons compared to antibaryons. Even so, let’s prove it is inevitable. Suppose, for

example, that the X baryons decay into two channels with branching ratios r and 1−r and baryon number

B1 and B2; X bosons decay with different branching ratios r, 1 − r into chanels with baryon number −B1

and −B2. The average baryon number produced by decays of X and X is

∆B = ∆B(X) + ∆B(X)

= B1|M(X→ B1)| + B2|M(X→ B2)|2 − B1|M(X→ B1)| − B2|M(X→ B2)|2 (5.4)

=12

(r − r) (B1 − B2)

One imediately concludes that baryon number (B1 , B2), C and CP (r , r) must be violated in order

to obtain a baryon number generation in the process.

Departure from thermal equilibrium Throughout the history of the universe many species have de-

parted from equilibrium and have left us with precious relics, namely, neutrino decoupling, decoupling of

the CMB, primordial nucleosynthesis and, hopefully, baryogenesis, inflation, decoupling of relic WIMPs

(weakly-interacting massive particles), etc. When a given species is in thermal equilibrium its state is

easy to follow. The densities of a particle species and its antiparticle in equilibrium are given by

nb ∝

∫∞

0

1

e(√

p2+m2b−µ)/T

+ 1dp

nb ∝

∫∞

0

1

e(√

p2+m2b−µ)/T

dp(5.5)

In chemical equilibrium the entropy is maximal when chemical potentials vanish for all nonconserved

quantum numbers. Furthermore, CPT invariance of gauge theories forces both particle and antiparticle

masses to be equal. Hence, in equilibrium, nb = nb.

5.2 Leptogenesis in the single flavour approximation

5.2.1 Overview

Because leptogenesis relies so heavily upon neutrino masses – on their various mass models and their

high parameter flexibility – as well as on many approximation routes, the number of approaches is quite

66

extensive. In this chapter we set ourselves to present a very simple model based on the decay of the lightest

RHνwhose mass is given by the type-I seesaw formula. The is called the one-flavour approximation

leptogenesis and often provides good estimates of the BAU generated, but most importantly illustrates

the main ideas behind the theory.

In this model, first introduced by Fukugita and Yanagida[66], the mass of the neutrinos is generated by

the type-I seesaw mechanism. In such a model, the RHνare very heavy particles which are SU(5)-sterile,

so their interactions are summed up to Yukawa couplings with the Higgs and leptons. Also, because

of their heaviness, the RHνdecouple from the primordial thermal bath and decay away into leptons,

anti-leptons and scalar bosons. The strength of CP-violation in the lepton sector will determine the

excess of leptons over anti-leptons at this stage. We also assume that the RHνhave a hierarchical mass

spectrum and therefore any lepton number created by the two heavier neutrinos N2,3 will be eventually

washed-out by the later decay of the lightest one N1. Once the decay reaction freezes-out, the lepton

asymmetry will evolve without being affected as long as all other processes present are CP-conserving.

At this stage, non-perturbative B + L violating processes that are in equilibrium partly convert this lepton

asymmetry into a baryon asymmetry. So let’s state it again[14]:

• During thermalization RHνare created thanks to inverse decays ` jφ→ Nk and Nk → ¯jφ. A lepton

asymmetry is generated

YL ≡nL − nL

s(5.6)

where YL is the comoving number density. At this stage it is ∼ −ε/CYN where

εkj ≡Γ(N→ ` jφ) − Γ(N→ ¯jφ)

Γ(N→ ` jφ) + Γ(N→ ¯jφ)(5.7)

and C is a wash-out factor greater than unity accounting for the partial depletion of the lepton

asymmetry during thermalization. The equilibrium density of RHνat the end of the thermal epoch

is roughly YN ∼ 10−3.

• When the temperature of the universe becomes . MRHν the heavy neutrinos decouple from the

thermal bath and decay giving way to a lepton asymmetry ∼ εYN. The total asymmetry is

YL ≈ ε(1 −

1C

)︸ ︷︷ ︸

η

YN (5.8)

In thermal leptogenesis, η ∼ 0.1.

67

• Finally, the baryon asymmetry is owed to the work of sphalerons which partly convert the lepton-

asymmetry into a baryon-asymmetry. This factor is of about 1/2. Hence, in thermal leptogenesis

YB ∼ 10−5ε which requires ε ∼ 10−6− 10−7 in order to mimic the observed BAU of YB ≈ 8.7 × 10−11

5.2.2 CP violation

The CP asymmetry is the first parameter which we wish to compute. In the mass eigenbasis the

Lagrangian for the heavy RHνis

LY = yαβ ¯αφeβ − h jk ¯jφNk −12

NkMkNk + h.c. (5.9)

where α, β, j are flavour indices and k = 1, 2, 3 is a hierarchical (from lightest to heaviest) index for the

RHνmass eigenstates. The relevant processes are

Nk →

` j + φ (with rate Γ)¯j + φ (with rate Γ) (5.10)

where L is violated by one unit. Because of CPT invariance, there is no difference between the rate

one process and its CP-opposite because Γ = |h jk|2I0 where I0 is a Lorentz phase space integral and is

therefore invariant under CP transformation. However, at higher radiative orders[4]

Γ =

∫[D]|M0 +M1 +M2 + . . . |2

=

∫[D]

(|M0|

2 +M†0M1 +M†1M0

)(5.11)

= |h jk|2I0 + h∗jkh jmhnmh∗nkI1 + h jkh∗jmh∗nmhnkI∗1 + O(h6)

where∫

[D] is a phase space integral and

Γ = |h jk|2I0 + h jkh∗jmh∗nmhnkI1 + h∗jkh jmhnmh∗nkI∗1 + O(h6) (5.12)

where I1 = I1 for the same reasons as I0 = I0. Consequently,

ε =Γ + Γ

Γ + Γ

= −4

Γ + ΓIm(h∗jkh jmhnmh∗nk)Im(I1)

(5.13)

From (5.13) we see that the couplings must be complex and that two heavy particles must exist in

the model. Furthermore, the mass of the RHνis required to be greater than the sum of the lepton and

68

Higgs mass together because the imaginary part of the integral corresponds to the interference with the

intermediate on-shell part of the one-loop integral. In this simplest model we assume that L evolves

regardless of flavour. Therefore our CP-violation parameter must be summed over the product leptons’

flavour, and since all asymmetry due to N2,3 is washed out by the later decay of N1 the relevant parameter

is

ε1 =∑

j

εkj =1

8π

∑m,1

Im[(h†h)21m]

(h†h)11]

f

M2m

M21

+ g

M2m

M21

(5.14)

where

f (x) =√

x[1 − (1 + x) ln

(1 + xx

)]and g(x) =

√x

1 − x(5.15)

The explicit calculation of ε1 is done in Appendix C. Since we’re dealing with a highly hierarchical

spectrum M1 M2,M3 we can approximate

f (x) + g(x) →x1 −3

2√

x⇒ |ε1| ≈

316π

M1

(h†h)11

∑j

Im[(h†h)1 j]M j

(5.16)

Further, this result, together the seesaw formula

mαβ = v2hαM−1k hβk (5.17)

where v is the isodoublet Higgs VEV, gives

|ε1| ≈3M1

16π(hTm∗h)11

v2(h†h)11≈

316π

(hTm∗h)11

m1(5.18)

where

m1 = 8πv2

M1ΓN1 = (h†h)11

v2

M1(5.19)

It was shown by Davidson and Ibarra [14] and further improved that

|ε1| .3M1

16πv2 (mmax−mmin) ×

1 − mmin

m1if mmin

mmax

1 −(

mmin

m1

)2 (5.20)

Because the baryon asymmetry produced by leptogenesis should at least reproduce the observed one

YB ≈ 1.38 × 10−3ηε1 & YobsB ≈ 8.7 × 10−11 (5.21)

69

it restricts M1 to

M1 & 6.5 × 108GeVη−1×

YB

YobsB

( matm

α(mmax,mmin, m1)

)(5.22)

where α(mmax,mmin, m1) ≈ 1 is anything in (5.20) apart for the 3M1/16πv2 factor. For now we may say

that η . 1. Such a boundary pushes the RHνmass scale far beyond the electroweak scale for leptogenesis

to work, which conversely makes the type-I seesaw mechanism a hardly (if not impossible) testable

model in laboratory experiments. Even so, our aim is to learn the basics of leptogenesis so we will

proceed with our study of this simply model.

5.2.3 Out-of-equilibrium dynamics and wash-out

The previous section shows that by adjusting a few parameters |ε1| can me made great enough to

generate the correct baryon-to-photon ratio ηB ≈ 10−10. However, such an analysis is incomplete; the

very solution to the problem must track the evolution of both N1 density and created B − L asymmetry.

There are various reasons for this, namely, the N1 density might be insufficient (or even completely

absent as could be implied by inflation scenarios) one has to find out whether processes involving the

SM particles can build up enough heavy neutrinos for a successful letogenesis; also, inverse decays

(`φ → N1) might be too strong and wash away any produced lepton asymmetry. It all depends upon

when processes couple or decouple from the thermal bath. Usually there is a good rule of thumb which

states that a given species is[9]

Γ & H (coupled) (5.23)

Γ . H (decoupled) (5.24)

where Γ is the interaction rate per particle for the species that keep the species in thermal equilibrium

and H is the Hubble constant. While this rule is usually very accurate, a proper study requires following

statiscally the microscopic processes involved which is done by solving the Boltzmann equations.

Dynamics of an expanding Universe Since the expansion of the universe is the key for decoupling

processes and leaving relic abundances, one must understand its mechanism. The evolution of the

Universe is described by the Einstein equation which relate the geometry of the Universe to its content:

Rµν −12

gµνR = 8πGTµν + Λgµν . (5.25)

70

In this equation R is the Ricci scalar, Rµν the Ricci tensor, Tµν the stress-energy tensor, gµν the space-

time metric, Λ is the cosmological constant and G is the Newton constant,

G =1

M2pl

,

Mpl ' 1.221 × 1019GeV . (5.26)

Assuming local homogeneity and isotropy (because energy and momentum are conserved locally),

the space-time metric is given by the maximally-symmetric Friedmann-Robertson-Walker (FRW) metric

ds2 = dt2− R2(t)

(dr2

1 − kr2 + r2dθ2 + r2 sin2 θdφ2)

(5.27)

where (t, r, θ, φ) are spherical comoving coordinates, R(t) is the cosmic scale factor and k = −1, 0,+1

gives the type of metric (hyperbolic, flat or spherical respectively).

Assuming further that the Universe content is a perfect fluid, we can write the stress-energy tensor as:

Tµν = −pgµν + (p + ρ)uµuν , (5.28)

where p is the pressure and ρ the energy density of the perfect fluid. The velocity vector of the fluid,

uµ is given in the rest frame of the plasma by u = (1, 0, 0, 0), so that Tµν = diag(ρ,−p,−p,−p).

With both metric and universe content informations we can now rewrite the 0-0 component of the Einstein

equations as

(RR

)2

+k

R2 =8πG

3ρ +

Λ

3(5.29)

known as the Friedmann equation. Setting Λ = 0 and defining H ≡ (R/R) and

Ω ≡ρ

ρC≡

3H2

8πG(5.30)

the Friedmann equation can be recast as

Ω =k

H2R2 + 1 (5.31)

WMAP has recently made measurements of the CMB and determined the density parameter at

present time to be[67]

Ω0 = 1.02 ± 0.02 (5.32)

71

which means that one can to a very good approximation assume that the universe’s curvature is flat

(k = 0). That being so, the Friedmann equation gives

H =

√8πρ

3m2Pl

≈1.66√

g∗T2

mPl(5.33)

where g∗ is the effective number of degrees of freedom contributing to the energy-density and is given

by

g∗ =78

∑f ermions

g f

(T f

Tγ

)4

+∑

bosons

gb

(Tb

Tγ

)4

(5.34)

Equilibrium thermodynamics The number density n is a particle species is given by

n =g

(2π)3

∫f (~p)d3p (5.35)

with

f (~p) =

[exp

(E − µ

T

)+ a

]−1

(5.36)

where a = −1, 0,+1 for Bose-Einstein (BE), Maxwell-Boltzmann (MB) or Fermi-Dirac (FD) distribu-

tions respectively. For MB statistics the equilibrium density is

neqMB =

gT3

2π2 z2K2(z) z = m

T , 0, µ = 0gT3

π2 m = 0.(5.37)

where g is the particle’s internal number of degrees of freedom and the modified Bessel function is

approximated by

z2K2(z) =

∫∞

zxe−x√

x2 − z2dx→

2 z 1(158 + z

) √πz2 e−z z 1

(5.38)

For FD statistics

neqFD =

gT3

2π2

(32ζ(3) +

µTζ(2) + . . .

)m T

neqMB µ T, m T

(5.39)

where ζ(2) = π2/6 and ζ(3) = 1.202 and ζ(4) = π4/90. For BE statistics

neqBE =

gT3

π2

(ζ(3) +

µTζ(2) + . . .

)m T

neqMB µ T, m T

(5.40)

72

The internal numbers of degrees of freedom g for the particles taking part in the relevant pprocesses

for leptogenesis are gN = 2 for the RHνbecause they are Majorana fermions, geR = 1 for the weak isosinglet

and g`,H = 2 for both fermion and scalar weak isodoublets.

Out-of-equilibrium thermodynamics Out-of-equilibrium dynamics require the study of the Boltz-

mann equations. The general Boltzmann equation can be written as

L[ f ] = C[ f ] (5.41)

where f (pµ, xµ) is the phase-space distribution of the species. The covariant Liouville operator is

given by

L[ f ] = pµ∂ f∂xµ− Γ

µνρpνpρ

∂ f∂pµ

(5.42)

where Γµνρ is the Christoffel symbol. Applying this to a Friedman-Robertson-Walker metric we get

L[ f (E, t)] = E∂ f∂t−

RR|−→p |2

∂ f∂E

(5.43)

Since

n(t) =g

(2π)3

∫d3p f (E, t) (5.44)

therefore with an integration by parts the Boltzmann equation becomes

dndt

+ 3Hn =12

∫C[ f ] dΠ (5.45)

where H is the Hubble constant and

dΠ ≡g

(2π)3

d3p2E

(5.46)

The collision term on the r.h.s of (5.45) for the process X + a + b + . . .↔ i + j + . . . is

12

∫C[ f ] dΠ = −

∫dΠX dΠa dΠb . . . dΠi dΠ j . . . (5.47)

× (2π)4δ4(pX + pa + pb + . . . − pi − p j − . . .) (5.48)

×

[|M|

2X+a+b+...→i+ j+... fa fb . . . fX(1 ± fi)(1 ± f j) . . . (5.49)

|M|2i+ j+...→X+a+b+... fi f j . . . fX(1 ± fa)(1 ± fb) . . . (1 ± fX)

](5.50)

73

We will assume an approximate Maxwell-Boltzmann distribution which allows us to simply

(1 + ± f ) ≈ 1 (5.51)

which is valid in absence of Bose condensation or Fermi degeneracy, so

nX + 3Hn = −

∫dΠX dΠa dΠb . . . dΠi dΠ j . . . (2π)4δ4(pX + pa + pb + . . . − pi − p j − . . .) (5.52)

×

[fX fa fb . . . |M(X + a + b + . . .→ i + j + . . .)|2 − fi f j . . . |M(i + j + . . .→ X + a + b + . . .)|2

](5.53)

Assuming that all particles are in kinetic equilibrium so that f = (n/neq) f eq with f eq = exp(−E/T) the

r.h.s. of (5.53) is

∫dΠX dΠa dΠb . . . dΠi dΠ j . . . (2π)4δ4(pX + pa + pb + . . . − pi − p j − . . .) (5.54)nX

neqX

f eqX

na

neqa

f eqa . . . |M(X + a + b + . . .→ i + j + . . .)|2

−ni

neqi

f eqi

n j

neqj

f eqj . . . |M(i + j + . . .→ X + a + b + . . .)|2

It is usually useful to write these equations in terms of the number density per comoving volume

which is done by normalizing it to the entropy s ∝ R−3 which is a conserved quantity. As so, let’s define

Y ≡ns

(5.55)

Since entropy is conserved in a comoving volume, the l.h.s of the Boltzmann equation can be rewritten

as

nX + 3HnX = sYX (5.56)

Let’s also write it in terms of z = mX/T. This can be done by cleverly noting that during a radiation-

dominated epoch, z and t are related by

t = 0.301g−1/2∗

mPl

T2 = 0.301g−1/2∗

mPl

m2X

z2≡ 2/H(z) = 2

mPl

1.66√

g∗m2 z−2 (5.57)

and that

74

s =2π2g∗s

45T3 (5.58)

where the internal number of degrees of freedom which contributes for the entropy is

g∗s =78

∑f ermions

g f

(T f

Tγ

)3

+∑

bosons

gb

(Tb

Tγ

)3

(5.59)

so that the Boltzmann equation becomes

dYdz

= −z

H(m)C (5.60)

where H(x) = H(m)/z2 = 1.66√

g∗m2/mPl. At leptogenesis temperatures, g∗s = g∗. At present state,

the main contribution to the entropy comes from photon, neutrino and anti-neutrino backgrounds at

Tν ≈ (4/11)1/3Tγ. Thus s0 ∝ g∗sT3γ with g∗s ∼ 3.9.

The collisional term in (??) also transforms as

dYX

dz=

YiY j . . .

(sYeqi )(sYeq

j ) . . .γXa...

i j... −YXYa . . .

(sYeqX )(sYeq

a ) . . .γ

i j...Xa... (5.61)

where the interaction density is defined as

γXa...i j... ≡ γ(X + a + . . .→ i + j + . . .)

=

∫dΠX dΠa . . . dΠi dΠ j . . . f eq

X f eqa . . . |M(X + a + . . .→ i + j + . . .)|2(2π)4δ4(pX + pa + . . . − pi + p j + . . .)

(5.62)

Now, the relevant processes are two-body decays and scatterings. In a decay, the four possible final

states, νβφ0,eβφ+,νβ and e−φ, the rates are (c.f. Appendix C)

|M(N1 → νβφ0)|2 = 2(h††)11(pNp`) = (h††)11M2

1 (5.63)

and with

∫(2π)4δ4(pX − pi − p j)dΠidΠ j =

∫|~pi|

16π2√

sdΩi =

|~pi − ~p j|

8π√

s=

√(pi · p j)2 −m2

i m2j

4πs(5.64)

we obtain for the decay rate density (check Appendix C for notation)

75

γ(N1 → `βφ, ¯βφ) = γNφ`β

+ γNφ ¯β

=

∫d3p

2E(2π)3 e−E/T4(h†h)211M2

1

∫(2π)4δ4(pX − pi − p j)dΠidΠ j

=1

8π3 T2M21(h†h)11

∫∞

ze−x√

x2 − z2 dx

= sYeqK1(z)K2(z)

Γ(N1 → `βφ, ¯βφ)

=gNT3

2π2 z2K1(z)Γ(N1 → `βφ) (5.65)

whereK1 is a modified Bessel function [25]:

zK1(z) =

∫∞

ze−x√

x2 − z2 dx→

1 z 1√π2 ze −z z 1

(5.66)

A more extensive discussion of the inclusive two-body decay rate density was evaluated in [26].

The equivalent result for two-body scatterings is given by

γXaij =

∫dΠX dΠa f eq

X dΠi dΠ j f eqa

∫|M(X + a→ i + j)|2(2π)4δ(pX + pa − pi − p j)

= 4gXga

∫dΠX dΠae−(EX+Ea)/T

√(pX · pa)2 −m2

Xm2a σ((pX + pa)2)

= gXga

∫dQ0 d3Q

(2π)4

e−Q0/T

πs[(pX · pa)2

−m2Xm2

a]σ(Q2)

=gXga

32π5

∫s ds dΩ

∫√

sdQ0e−Q0/T

√Q2

0 − s

1 −

m2X + m2

a

s2

2

− 4m2

Xm2a

s2

σ(s)

=T

64π4

∫∞

(mX+ma)2ds√

sK1

( √s

T

)σ(s) (5.67)

where s = p2X + p2

a and sσ(s) = 8[(pµXpaµ)2−m2

Xm2a]σ(s).

So we’re finally ready to approach leptogenesis with this new formalism. Let’s first parametrize the

decay rates to better reflect the CP violation:

γ(N1 → `φ) ≡ γ( ¯φ→ N1) = (1 + ε1)γDγ(N1 → ¯φ) ≡ γ(`φ→ N1) = (1 − ε1)γD (5.68)

where γD is the tree-level decay density

γD = neqXK1(z)K2(z)

ΓD =(h†h)11

16πM1neq

XK1(z)K2(z)

(5.69)

76

This parametrization garantees that

εN ≡γ(N1 → `φ) − γ( ¯φ→ N1)γ(N1 → `φ) + γ( ¯φ→ N1)

= ε1 (5.70)

Assuming that all particles except N1, ` and ¯ are in thermal equilibrium1,

nN1 + 3HnN1 = −2

nN1

neqN1

− 1

(γD + γφ,s + γφ,t) + O(ε1,

µ`T

)(5.71)

where µ` is the chemical potential for `. The other relevant variable is nB−L2:

nB−L + 3HnB−L = −2ε1

nN1

neqN1

− 1

γD −nB−L

neq`

γW + O(ε2

1,µ`T

)(5.72)

where

γW = γD +nN1

neqN1

γφ,s + 2γN,s + 2γN,t (5.73)

At first order – neglecting the neutrino-top scattering rates – the Boltzmann equations are written in

their final form as

dYN

dz= −D(z)K1

(YN(z) − Yeq

N (z))

(5.74)

dYL

dz= −ε1D(z)K1

(YN(z) − Yeq

N (z))−Wid(z)K1YL(z) (5.75)

where

z = M1/T, (5.76)

D(z) = zK1(z)K2(z)

, (5.77)

Wid(z) =12

YeqN

Yeq`

D(z) =14

z3K1(z), (5.78)

K1 =ΓN1

H(M1)(5.79)

The densities in equilibrium are

1c.f. Appendix C2c.f. Appendix C

77

YeqN (z) =

34ζ(3)

452π4g∗

z2K2(z) ' 3.9 × 10−3 for z 1 (5.80)

where the factor 3ζ(3)/4 is a corrective factor to the Maxwell-Boltzmann to make it more like the

Fermi-Dirac behaviour at high energies.

There are two possible regimes: either K1 1 meaning a very strong wash-out because decays are

in-equilibrium and lepton asymmetries are strongly washed out, whereas in the opposite case where

K1 1 the decays go quickly out of equilibrium and wash-out is much weaker. Two solutions are shown

in fig.5.2.3[4].

Figure 5.1: Evolution of comoving number densities as functions of z = M1/T . The grey dashed linerepresents Yeq

N1, the number density of N1 in thermal equilibrium, whereas the black line stands for YN,

the solution of the Boltzmann equation. In red (light grey) is depicted the lepton asymmetry YL. For allthese plots the CP asymmetry is taken equal to ε1 = 10−6, while the washout parameter is K1 = 0.01(100)on the left (right)..

It is important to note that all of Sakharov’s conditions for dynamical baryogenesis are respected in

this model. The production term for L – which is the first term on the r.h.s of the equation – depends on

ε1 , 0 and YN(z) , YeqN (z).

5.2.4 Lepton asymmetry and anomalous B + L violation

The plots in fig.5.2.3 show that there are basically three phases of leptogenesis:

• A thermalization phase where RHν’s are produced by scatterings and inverse decays of SM particles.

This process evolves until equilibrium is reached. In a strong washout regime (SWR), RHνare

thermalised fast and equilibrium is reached at high temperatures; conversely, in the weak washout

78

regime (WWR), equilibrium is only reached at lower energies. Anyhow, both cases produce a

lepton asymmetry which is ∝ YN.

• In the next phase the RHνdensity decreases. In the SWR YN tracks closely its equilibrium value due

to fast decays while in the WWR the decays come later, when YN has already left its equilibrium

value. Another asymmetry of opposite sign is created which threatens to cancel out the first one.

• Eventually, temperatures decrease and the RHν’s become too diluted freezing out the processes that

involve them. A small residual asymmetry survives. In the SWR, the resulting lepton asymmetry

is accurately given by[4]

YL ' ε1

0.4K1.16

1

YeqN (zin) (5.81)

ηs = 0.4/K1.161 (5.82)

in the WWR, the resulting lepton asymmetry is

YL ' 1.3ε1K21Yeq

N (zin)

ηw = 1.3K21

(5.83)

5.2.5 Baryogenesis through leptogenesis

In the SM there is no B or L violation at Lagrangian level. Still, it was pointed out by t’Hooft that non-

perturbative effects called instantons can lead to the violation of B+L – yet with B−L conservation. The is

due to the complex topology of the ground-state in a non-abelian gauge theory. When transitions between

degenerate vacua occur they violate B + L. Such tunneling involves going through field configurations

called sphalerons. These arise from a non-zero divergence of the leptonic and baryonic currents in the

ABJ triangle anomaly[15]:

∂µJBµ = ∂µJL

µ =ng

32π2

(−g2Wa

µνWµνa + g′2BµνWµν

)(5.84)

where g and g′ are the gauge couplings of SU(2)L and U(1)Y respectively, with Waµν and Bµν being the

corresponding field tensors and Fµν ≡ 12εµνλρFλρ . Obviously:

79

∆(B − L) =

∫d4x∂µ

(JBµ − JL

µ

)= 0

∆(B + L) =

∫d4x∂µ

(JBµ + JL

µ

)=

∫d4x

ng

16π2

(−g2Wa

µνWµνa

)+ g′2BµνWµν) (5.85)

= 2ngθ

where ∆Ncs is an integer called the Chern-Simons number. At T = 0 between two vacua with distinct

topological charges is exponentially suppressed, with

Γ ≈ exp(−

8π2

g2

)(5.86)

However at T & TEW Kuzmin, Rubakov and Shapovnikov showed that Higgs and gauge boson field

configurations can cause ”leap-overs” from one vacuum to another. The amplitude for this process is

roughly exp(−Esph/T) with

Esph ≈8πv

g(5.87)

being the height of the energy barrier between two vacua with different winding numbers. Hence, at

T Esph, the transition amplitude is non-neglegible and occurs rapidly. In the SMthere are two different

gauge groups with topologically non-trivial vacua, and therefore there will be effective interactions

mediated by either strong (SU(3)C) or weak sphalerons (SU(2)L):

vacuum ∑

generations qL + qL + uR + dR ⇒ ΓQCD/V ' 250α5s T4 at T . 1013 GeV∑

generations qL + qL + qL + `L ⇒ ΓQCD/V ' 250α5s T4 at T . 1012 GeV

(5.88)

where rates’ temperatures are calculated so that the sphaleron processes occur faster than Hubble

expansion. From here, one might be tempted – because of B − L conservation – to associate a B variation

to a B + L variation. Actually, things are a bit more complicated and require determination of which

interactions are in kinetic equilibrium when the lepton asymmetry is created. The actual formula is[59]

YB = −8ng + 4nH

14ng + 9nHYL (5.89)

where ng = 3 is the number of fermion generation and nH = 1 is the number of Higgs doublets in the

model. Going back to our leptogenesis scenario, as the universe cools down with t &M1/100, the lepton

asymmetry in the single flavour approximation is[4]

YB = −2851

YL ≈ −1.38 × 10−3ε1η (5.90)

80

Choosing ε1 ∼ 10−6 and a global parametrization η = (η−1w + η−1

s )−1' (0.8K−2

1 + 2.5K−1.161 )−1, the

observed value YobsB ≈ 8.7 × 10−11 implies K1 ∼ 3 × 0.3.

5.2.6 Dependence on initial conditions

We have assumed in the discussion above that RHνare inexistent before zin and are produced by inverse

decays and scatterings. This production is motivated by the wish for an initially symmetric universe.

Nevertheless, it is important to track an initial RHνdensity, because we have no idea what could have

happenned before leptogenesis, some GUT mechanism (or something even more exotic) could generate

RHνat a higher temperature scale. Considering three different scenarios, the earlier one YN(zin) = 0, the

equilibrium case YN(zin) = YeqN (zin) and the dominant YN(zin) > Yeq

N (zin), a strong washout scenario is the

one that depends the least on initial conditions and is thus the most robust in terms of predictions. Also,

the more RHνare present in an initial state the greater the efficiency, a difference which is maximal for

the dominant case in a weak washout regime.

81

Chapter 6

Conclusion

From the 1980’s till today, a considerable set of experiments of solar, atmospheric and reactor neutrinos

has established that the once massless neutrinos of the Standard Model have in fact mass (two of them at

least), though in a sub-eV scale. The simplest extension of the Standard Model as presented in the seesaw

mechanism provides a natural explanation for the smallness of their mass. Furthermore, it also provides

for the baryon asymmetry of the universe. The remarkable coincidence that this simple extension would

solve two apparently unrelated problems makes it a very attractive one. One might in fact say that the

discovery of neutrino masses is one of the most important experimental discoveries of recent times.

The main issue remains, whether such a theory of the BAU can be tested. Such a test would involve a

direct measurement of the CP asymmetry produced in heavy neutrino decays, which are far too heavy

to be produced in any even dreamed accelerator. And even if they’re lighter, their couplings become

smaller making it also impossible to measure anything. Indirect tests in cosmology, by measuring the

asymmetries produced by leptogenesis, are also ruled out there are two many high energy parameters

which we have have no idea yet on how to measure them. In the most optimistic case, these ar four:

• the BAU, known to a good accuracy;

• three neutrino cosmic flavour asymmetries, to which no experimental test has been proposed so

far.

Such information would yield information to us on either the flavour CP asymmetries (if leptogenesis

ocurred in the unflavoured regime) or else on a combination of the flavour CP asymmetries and on the

flavour depend washouts. Unfortunately, we haven’t even measured the cosmic neutrino background,

and the possibility of measuring O(10−10) asymmetries in the background is beyond our wildest hopes.

Nevertheless, even though no direct or indirect tests can prove leptogenesis is right, there are still tests that

can make it likely. The first one would obviously be the discovery of Majorana masses by measurement

82

of neutrinoless double-beta decay. A measurement of such a lepton number violating process will prove

for the first time the fulfillment of the first of the three Sakharov conditions. A failure in the discovery

of this process cannot however disprove leptogenesis alone, because if the neutrinos are found to be of

dirac type there is still a chance for leptogenesis with a normal hierarchic mass spectrum. So a complete

rejection of the theory requires data from neutrino oscillation experiments which can measure the mass

differences. In fact, if νe ↔ νµ oscillations are enhanced and νe ↔ νµ are suppressed then the mass state

involved in atmospheric oscillations is the heaviest one, which corresponds to a normal hierarchy. The

opposite goes for an inverted one. So no evidence of ββ0ν at |mee| . 10meV and an establishment of an

inverted hierarchy spectrum will disprove leptogenesis.

The second of Sakharov’s conditions requires the discovery of CP violation in the leptonic sector. Pro-

posed experiments such as SuperBeam and NOνA only probe the Dirac phases of the mixing matrix,

without any sensibility to the Majorana phases. So if ββ0ν is found and no CP violation is measured in

the Dirac sector, leptogenesis can’t be falsified by this test.

Another test which may falsify standard leptogenesis (type-I seesaw, unflavoured regime, heavy Majo-

rana decay) is the establishment of light neutrino masses ∼ 0.1− 0.2eV in which case leptogenesis would

fail to produce enough baryon asymmetry. The LHC at CERN may also help the plead of leptogenesis by

disproving electroweak baryogenesis making leptogenesis the most satisfying proposal for explaining

the BAU. Conversely, discovery of new physics such as leptoquarks or triplet Higgses which will estab-

lish that the neutrino masses don’t come from the seesaw mechanism will leave leptogenesis deprived

of its best motivational feature and probably discard it. So to conclude, the discovery of neutrino masses

is a very important fact which will be refined in the years to come by several experiments of various

types, closing the chapter on the electroweak standard model of interactions by giving mass to the once

awkwardly massless neutrinos and providing a non-supersymmetric chance for the SM to generate the

BAU. One also expects the LHC to give us a better understanding (which is still very crude) of the

symmetry breaking mechanism and of flavour physics, which if yielding a satisfactory and predictive

theory of flavour might help to establish leptogenesis as the standard model of matter generation, just

as in the past Big Bang Nucleosynthesis was for chemical elements.

83

Appendix A

Feynman rules for Majorana spinors

The self-conjugacy of the Majorana particles makes their treatment in Feynman integrals somewhat

delicate. Some rules remain unchanged when going from Dirac to Majorana spinors. For example, the

propagator of a Majorana field is the same:

〈0| T (ψA(x)ψB)(y) |0〉 =

∫d4p

(2π)4eip·(x−y)

[i

/p + mp2 −m2 + iε

]AB

(A.1)

where A and B are spinor indices andT is the time-ordering operator. This easily derived considering

the lagrangian for a Majorana field νR = νR + eiφ/2νR

Lν = iνRj/∂νRk −12

¯νRjM jkνRk −12

ˆνRjM∗jk ˆνRk

=12

[iNRk /∂NRk + i ¯NRk /∂NRk − e−iφk ¯NRkDNRk − eiφkNRkDNRk

]=

12[iNk /∂Nk −DkNkNk

]= −

12

e−iφkNTk C†[i/∂ −Mk]Nk

(A.2)

where NR are the mass eigenstates of the right handed neutrino and νRj = eiφk/2V†jkNRk. The propagator

is readily derived from the last result.

As for the vertices, they are easily defined by the Yukawa Lagrangian defines them

Nk → `Lj φ : N`

φ= −i h jk PR (A.3)

Nk → ¯Lj φ : N`

φ= i h∗jk C†PL (A.4)

For the Majorana fermions in external lines we used the following convention: u(p) for incoming

neutrinos and u(p) for outgoing neutrinos.

84

Appendix B

Boltzmann equations for leptogenesis

B.0.7 Boltzmann equation for N1

Starting with the Boltzmann equation for N1 and writing down all the relevant interactions on the RHS,

we obtain

dnN1

dt+ 3HnN1 = −

nN1

neqN1

γ(N1 → `φ) +n ¯

neq`

γ( ¯φ→ N1) −nN1

neqN1

γ(N1 → ¯φ)

+n`neq`

γ(`φ→ N1) −nN1n`neq

N1neq`

γ(`N1 → tRqL)s + γ(tRqL → `N1)s

−nN1n ¯

neqN1

neq`

γ( ¯N1 → tRqL)s + γ(tRqL → ¯N1)s −nN1

neqN1

γ(N1qL → `tR)t

−nN1

neqN1

γ(N1qL → tR ¯)t +n`neq`

γ(`tR → N1qL)t +n`neq`

γ(`qL → N1tR)t

−nN1

neqN1

γ(N1tR → qL ¯)t −nN1

neqN1

γ(N1tR → qL`)t +n ¯

neq`

γ( ¯tR → N1qL)t

+n ¯

neq`

γ( ¯qL → N1tR)t , (B.1)

where the subscripts s and t denote s- and t-channel processes respectively. In writing down (B.1), we

have assumed that neq`≡ neq

¯ and nφ,tR,qL ≡ neqφ,tR,qL

. Assuming that γ(A→ B) ≡ γ(B→ A) for all processes,

(B.1) becomes

dnN1

dt+ 3HnN1 = −

nN1

neqN1

(1 + ε1)γD +n ¯

neq`

(1 + ε1)γD −nN1

neqN1

(1 − ε1)γD +n`neq`

(1 − ε1)γD

+ 2γφ,s −nN1(n` + n ¯)

neqN1

neq`

γφ,s −4nN1

neqN1

γφ,t +2(n` + n ¯)

neq`

γφ,t , (B.2)

where

γφ,s = γ(`N1 → tRqL)s = γ(tRqL → `N1)s = γ(tRqL → ¯N1)s = γ( ¯N1 → tRqL)s , (B.3)

85

and

γφ,t = γ(N1qL → `tR)t = γ(N1qL → tR ¯)t = γ(`tR → N1qL)t = γ(`qL → N1tR)t ,

= γ(N1tR → qL ¯)t = γ(N1tR → qL`)t ,= γ( ¯tR → N1qL)t = γ( ¯qL → N1tR)t . (B.4)

Simplifying (B.2), we get

dnN1

dt+ 3HnN1 = −

2nN1

neqN1

γD +

n` + n ¯

neq`

γD +n ¯ − n`

neq`

ε1γD + 2γφ,s

−

n` + n ¯

neq`

nN1

neqN1

γφ,s −4nN1

neqN1

γφ,t + 2

n` + n ¯

neq`

γφ,t , (B.5)

= −2nN1

neqN1

γD + 2γD + 2γφ,s −2nN1

neqN1

γφ,s −4nN1

neqN1

γφ,t + 4γφ,t + O(ε1,

µ`T

), (B.6)

= − 2

nN1

neqN1

− 1

(γD + γφ,s + 2γφ,t) + O(ε1,

µ`T

), (B.7)

where in (B.6), we have used the definition of density

n` + n ¯

neq`

=

(g`

2π2

∫f eq`

(E) E2 dE)−1 (

g`2π2

∫ [f`(E) + f ¯(E)

]E2 dE

), m` T ,

=

(∫e−E/T E2 dE

)−1 (∫ [e−(E−µ`)/T + e−(E+µ`)/T

]E2 dE

),

=

(∫e−E/T E2 dE

)−1

2 cosh(µ`

T

) ∫e−E/T E2 dE ,

= 2 + O(µ`

T

). (B.8)

where we have used Maxwell-Boltzmann distribution for the phase space densities and imposed the

condition for kinetic equilibrium µ` ≡ −µ ¯.

86

B.0.8 Boltzmann equation for B−L

Writing down the evolution equation for particle density n ¯:

dn ¯

dt+ 3Hn ¯ =

nN1

neqN1

γ(N1 → ¯φ) −n ¯

neq`

γ( ¯φ→ N1) +n`neq`

γ(`φ→ ¯φ)s

−n ¯

neq`

γ( ¯φ→ `φ)s +n`neq`

γ(`φ→ ¯φ)t −n ¯

neq`

γ( ¯φ→ `φ)t

+ γ(tRqL → ¯N1)s −nN1n ¯

neqN1

neq`

γ( ¯N1 → tRqL)s +nN1

neqN1

γ(N1tR → qL ¯)t

−n ¯

neq`

γ( ¯qL → N1tR)t +nN1

neqN1

γ(N1qL → tR ¯)t −n ¯

neq`

γ( ¯tR → N1qL)t , (B.9)

=nN1

neqN1

(1 − ε1)γD −n ¯

neq`

(1 + ε1)γD +n`neq`

(γN,s + ε1γD) −n ¯

neq`

(γN,s − ε1γD)

+n`neq`

γN,t −n ¯

neq`

γN,t + γφ,s −nN1n ¯

neqN1

neq`

γφ,s +nN1

neqN1

γφ,t −n ¯

neq`

γφ,t +nN1

neqN1

γφ,t

−n ¯

neq`

γφ,t , (B.10)

dn ¯

dt+ 3Hn ¯ =

nN1

neqN1

(1 − ε1)γD −n ¯ − n` ε1

neq`

γD −n ¯ − n`

neq`

γN,s −n ¯ − n`

neq`

γN,t + γφ,s

−nN1n ¯

neqN1

neq`

γφ,s +2nN1

neqN1

γφ,t −2n ¯

neq`

γφ,t . (B.11)

Similarly, we can write down the equation for n` as

dn`dt

+ 3Hn` =nN1

neqN1

(1 + ε1)γD −n` + n ¯ ε1

neq`

γD −n` − n ¯

neq`

γN,s −n` − n ¯

neq`

γN,t + γφ,s

−nN1n`neq

N1neq`

γφ,s +2nN1

neqN1

γφ,t −2n`neq`

γφ,t . (B.12)

Subtracting (B.12) from (B.11), we have

dnB−L

dt+ 3HnB−L = − 2ε1γD

nN1

neqN1

+

n` + n ¯

neq`

ε1γD −nB−L

neq`

γD −2nB−L

neq`

γN,s

−2nB−L

neq`

γN,t −nN1nB−L

neqN1

neq`

γφ,s −2nB−L

neq`

γφ,t , (B.13)

where we have defined n ¯ − n` ≡ nB−L. Using (B.8) to simplify, we then get

dnB−L

dt+ 3HnB−L = −2ε1

nN1

neqN1

− 1

γD −nB−L

neq`

γD − 2γN,s − 2γN,t −nN1

neqN1

γφ,s − 2γφ,t

,= −2ε1

nN1

neqN1

− 1

γD −nB−L

neq`

γW + O(ε2

1,µ`T

), (B.14)

87

Appendix C

Calculation of the strength of the CPviolation

C.1 Tree-level contribution to Nk → ` φ

The Feynman diagram for this process is shown in Fig. C.1a. We can immediately write down the

amplitude for this decay as

M = iu j(−i h jkPR)uk ,

= u j(−i h jkPR)CuTk . (C.1)

|M|2 = u j(−i h jkPR)CuT

k

[u j(−i h jkPR)uc

k

]†,

= u j(−i h jkPR)CuTk (i h∗jk)(−uT

k C†PLu j) ,

= −(h∗jkh jk)u jPRCuTk uT

k C†PLu j , (C.2)

|M|2 = −(h∗jkh jk) PR C

12

∑s

ukuk

T

C† PL

∑s′

u ju j (C.3)

When the universe was hot enough, ` j and φ are strongly relativistic, so m` j ,mφ ≈ 0 and

|M|2 = −

(h∗jkh jk)

2Tr

[PR(−/p + Mk)PL( /p′)

], (C.4)

=12

(h∗jkh jk) Tr[PR/p /p′

], (C.5)

= (h∗jkh jk)(p · p′) . (C.6)

The four-momenta in the centre-of-mass frame are given by

p = (Mk , ~0) , p′ = (Mk/2 , −~q) , q = (Mk/2 , ~q) , (C.7)

and one can quickly deduce that |~q| = Mk/2 and p · p′ = p · q = p′ · q = M2k/2.

88

(b)

Nk

(V ∗h )jk

j

φ

pp′

q

Nk

(Vh)jk

j

φ

(a)

pp′

q

Figure C.1: (a) The Feynman graph for the process Nk → ` jφ. (b) The graph for Nk → ¯jφ. Here q = p−p′,and (Vh) jk ≡ −i h jk PR and (V∗h) jk ≡ i h∗jk C†PL are the vertex factors.

Therefore, we obtain

|M|2 = (h∗jkh jk)

M2k

2,

= (h†h)kkM2

k

2. (after summing over j) (C.8)

The decay rate for Nk → ` φ is then

Γ(Nk → ` φ) = 2 ×|~q|

8πE2cm|M|

2 ,

= 2(h†h)kkM2

k

21

8πMk

21

M2k

, (Ecm ≡Mk) ,

=(h†h)kk

16πMk , (C.9)

where the factor of 2 comes from the fact that there are two possible decay channels: Nk → νφ0 and

Nk → e−φ+.

C.2 Vertex contribution to the CP asymmetry

This contribution comes from the interference between the one-loop vertex graph in Fig. C.2 and its

tree-level counterpart in Fig. C.1a.

Im[(Mtree)†Mloop

]∝ εvertex . (C.10)

For the diagram in Fig. C.2, the orderings are:

[C]→ [A]→ [B] and [D]→ [E]→ [F]→ [B] . (C.11)

89

NmNk

n

j

φ

(V ∗h )nk

(Vh)jm

(Vh)nm

[B]

[A][E]

q3

p

p′

q1

q2

q

[C]

[D][F ]

Figure C.2: One-loop vertex correction graph for the process Nk → ` jφ. On the left: (Vh)ab ≡ −i hab PRand (V∗h)ab ≡ i h∗ab C†PL are the vertex factors; on the right: we have included the momentum flows andspinor indices [X], where q = p − p′, q2 = p − q1 and q3 = q1 − q.

From this, the amplitude of the interference term in index form is given by

I′vertex =

∫d4q1

(2π)4(−ih jm)(−ihnm)(ih∗nk)

[u j

]1C

[PR]CA[SNm(q3)

]AB [PR]FB

×[S`(−q1)

]EF [C†PL]DE

[uc

k

]D1

[D(q2)

]11

[−ih∗jkuT

k C†PLu j

]11︸ ︷︷ ︸

(Mtree)†

, (C.12)

where all symbols are as defined previously. So letting Ah = h∗jkh jmhnmh∗nk, this becomes (in matrix

notation)

I′vertex = Ah

∫d4q1

(2π)4

u j PR(−i)( /q3 + Mm)C PTR i(−M)TPT

LC∗uck(i)(−1)uT

k C†PLu j

(q23 −M2

m + iε)(q21 + iε)(q2

2 + iε),

= −iAh

∫d4q1

(2π)4

u j PR( /q3 + Mm)C C†PRC C†MCC†PLCC∗ CuTk uT

k C†PLu j

(q23 −M2

m + iε)(q21 + iε)(q2

2 + iε),

I′vertex =iAh

2

∫d4q1

(2π)4

PR( /q3 + Mm) PRMPL C (∑

s ukuk)T C†PL∑

s′ u ju j

(q23 −M2

m + iε)(q21 + iε)(q2

2 + iε),

=iAh

2

∫d4q1

(2π)4

Tr[PR( /q3 + Mm) PRMPL C

(/pT + Mk

)C†PL /p′

](q2

3 −M2m + iε)(q2

1 + iε)(q22 + iε)

,

...

= iAh Mk Mm

∫d4q1

(2π)4

q1 · p′

(q23 −M2

m + iε)(q21 + iε)(q2

2 + iε). (C.13)

Let us concentrate on the integral:

I′ = i Mk Mm

∫d4q1

(2π)4

q1 · p′

(q23 −M2

m + iε)(q21 + iε)(q2

2 + iε). (C.14)

To pick out the discontinuity, we apply Cutosky’s cutting rules[31]. Firstly, we note that of the three

possible ways to cut the diagram, only one of them can simultaneously put both cut propagators on-

90

shell, due to the heaviness of Nm: This only way (leftmost diagram) corresponds to cutting through the

propagators associated with momenta q1 and q2 (see Fig. C.2). Thus, we make the replacement

1q2

1 + iε→ −2πiδ(q2

1)Θ(E1) ,

1q2

2 + iε→ −2πiδ(q2

2)Θ(E2) = −2πiδ((p − q1)2)Θ(Mk − E1) , (C.15)

in (C.14), where q1 = (E1, ~q1) and q2 = (E2, ~q2). Using the definitions in (C.7) for p, p′ and q, we can evaluate

q1 · p′ = E1Mk

2− ~q1 · (−~q) = E1

Mk

2+ |~q1||~q| cosθ =

Mk

2(E1 + |~q1| cosθ) , (C.16)

where θ is the smaller angle between ~q1 and ~q. Putting all these together and substituting q3 = q1 − q, we

obtain (ε→ 0):

Disc(I′) =i M2

kMm

2

∫d4q1

(2π)4

(−2πi)2(E1 + |~q1| cosθ)δ(q21)δ((p − q1)2)Θ(E1)Θ(Mk − E1)

(q1 − q)2 −M2m

,

=−i M2

kMm

8π2

∫dE1d3q1 δ(E2

1 − |~q1|2) δ

[(Mk − E1)2

− |~q1|2]Θ(E1)Θ(Mk − E1)

×(E1 + |~q1| cosθ)

(E1 −Mk2 )2 − |~q1 − ~q|2 −M2

m

. (C.17)

Applying the identity: δ(x2− a2) = [δ(x − a) + δ(x + a)] /2|a|, we can rewrite δ(E2

1 − |~q1|2) as

δ(E21 − |~q1|

2) =1

2|~q1|

[δ(E1 − |~q1|) + δ(E1 + |~q1|)

]. (C.18)

Integrating over E1, the terms corresponding to the unphysical energy option of E1 = −|~q1|will drop out

automatically because of the step function Θ(−|~q1|) = 0 and one obtains

Disc(I′) =−i M2

kMm

16π2

∫|~q1|

2d|~q1|dΩ1

| − 2Mk|δ[|~q1| −

Mk

2

]Θ(Mk − |~q1|)

×1 + cosθ

(|~q1| −Mk2 )2 − |~q1|

2 −M2

k4 + |~q1|Mk cosθ −M2

m

, (C.19)

where we have used the identity δ(ax) = δ(x)/|a| and the fact that

−|~q1 − ~q|2 = −[|~q1|

2 + |~q|2 − 2|~q1||~q| cos(θ)]

= −|~q1|2−

M2k

4+ |~q1|Mk cosθ . (C.20)

Note that θ is again the smaller angle between ~q1 and ~q. We perform the d|~q1| integral in (C.19) to obtain

Disc(I′) =−i Mk Mm

32π2

∫dΩ

M2k

41 + cosθ

−M2

k4 −

M2k

4 +M2

k2 cosθ −M2

m

,

=−i Mk Mm

32π2

∫dΩ

1 + cosθ−1 − 1 + 2 cosθ − 4z

, (C.21)

91

where z ≡M2m/M2

k . We then evaluating∫

dΩ:

Disc(I′) =−i MkMm

32π2

∫dφ

∫d(cosθ)

1 + cosθ−2(1 − cosθ) − 4z

,

=i MkMm

32π2 (2π)∫ 1

−1dx

1 + x2(1 − x) + 4z

,

=i MkMm

16π[−z ln(−2z) − ln(−2z)

+z ln(−2(z + 1)) + ln(−2(z + 1)) − 1] ,

=−i M2

k

16π√

z(1 − (z + 1) ln

[z + 1z

]). (C.22)

Therefore, the imaginary part of I′ is given by

Im (I′) =12i

Disc (I′) ,

= −M2

k

32π√

z(1 − (z + 1) ln

[z + 1z

]). (C.23)

The two last pieces of information we require before evaluating the CP asymmetry is the total decay rate,

which we can get from (C.9)

Γtot = Γ + Γ = 2 ×(h†h)kk

16πMk =

(h†h)kk

8πMk , (C.24)

and the 2-body phase space factor which may be readily read off using (??) as

Vϕ = 2 ×︸︷︷︸two channels

|~q|8πE2

cm= 2 ×

18π

Mk

2 M2k

=1

8πMk. (C.25)

Putting all these together and summing over all heavy Majorana neutrino species m , k, as well as the

internal lepton species n, the expression for the CP asymmetry due to the vertex contribution is therefore

εvertex = −4

Γtot

∑m,k

∑n

Im(Ah) Im(I′Vϕ) , (C.26)

where we have used Im(I′Vϕ) ≡ Im(I′)Vϕ as Vϕ ∈ R, and so

εvertex = 4 ×8π

(h†h)kk Mk

∑m,k

∑n

Im(h∗jkh jmhnmh∗nk)

×M2

k

32π√

z(1 − (z + 1) ln

[z + 1z

]) 18πMk

,

=1

8π

∑m,k

Im[h∗jkh jm(h†h)km

](h†h)kk

√z(1 − (z + 1) ln

[z + 1z

]), (C.27)

with z ≡M2m/M2

k .

92

[D]

[C]

q

p′

p

[E] [F ][B]

[A]

p

q2

q1

j

φ

Nm

Nk

n

(Vh)jm

(Vh)nm

(V ∗h )nk

Figure C.3: One-loop self-energy correction graph (1) for the process Nk → ` jφ. On the left: (Vh)ab ≡

−i hab PR and (V∗h)ab ≡ i h∗ab C†PL are the vertex factors; on the right: we have included the momentumflows and spinor indices [X], where q = p − p′ and q2 = p − q1.

C.3 Self-energy contribution to the CP asymmetry (1)

The first self-energy contribution is given by interference between one-loop graph in Fig. C.3 and the

tree-level diagram. We have for the interference term:

Iself-(1) =

∫d4q1

(2π)4(ih∗jk)(−ih jm)(−ihnm)(ih∗nk)

[u j

]1C

[PR]CA[SNm(p)

]AB

× [PR]FB[S`(−q1)

]EF [C†PL]DE

[uc

k

]D1

[D(q2)

]11

[−uT

k C†PLu j

]11, (C.28)

where we have shown all spinor indices explicitly. Letting Ah = h∗jkh jmhnmh∗nk, this then becomes (mφ,m` j ≈

0)

Iself-(1) = Ah

∫d4q1

(2π)4

u j PR(−i)(/p + Mm)C PTR i(−M)TPT

LC∗uck(i)(−1)uT

k C†PLu j

(p2 −M2m + iε)(q2

1 + iε)(q22 + iε)

,

= −iAh

∫d4q1

(2π)4

u j PR(/p + Mm)C C†PRC C†MCC†PLCC∗ CuTk uT

k C†PLu j

(p2 −M2m + iε)(q2

1 + iε)(q22 + iε)

,

=iAh

2

∫d4q1

(2π)4

PR(/p + Mm) PRMPL C (∑

s ukuk)T C†PL∑

s′ u ju j

(p2 −M2m + iε)(q2

1 + iε)(q22 + iε)

,

=iAh

2

∫d4q1

(2π)4

Tr[PR(/p + Mm) PRMPL C

(/pT + Mk

)C†PL /p′

](p2 −M2

m + iε)(q21 + iε)(q2

2 + iε),

...

= iAh Mk Mm

∫d4q1

(2π)4

q1 · p′

(p2 −M2m + iε)(q2

1 + iε)(q22 + iε)

. (C.29)

To pick out the discontinuity of the integral

I(1) = i MkMm

∫d4q1

(2π)4

q1 · p′

(p2 −M2m + iε)(q2

1 + iε)(q22 + iε)

, (C.30)

93

we note that there is only one sensible way to cut the diagram, namely, through the propagators associated

with q1 and q2. So, with the replacement:

1q2

1 + iε→ −2πiδ(q2

1)Θ(E1) , and1

q22 + iε

→ −2πiδ(q22)Θ(E2) , (C.31)

we have (using q2 = p − q1)

Disc(I(1)) = i MkMm

∫d4q1

(2π)4

(−2πi)2(q1 · p′)δ(q21)δ((p − q1)2)Θ(E1)Θ(Mk − E1)

p2 −M2m

. (C.32)

Simplifying this

Disc(I(1)) =−i MkMm

4π2(M2k −M2

m)

∫d4q1

[E1

Mk

2− (−~q · ~q1)

]δ(E2

1 − |~q1|2)

× δ[(Mk − E1)2

− |~q1|2]Θ(E1)Θ(Mk − E1) ,

=−i M2

kMm

8π2(M2k −M2

m)

∫dE1d3q1(E1 + |~q1| cosθ)

12|~q1|

δ(E1 − |~q1|)

× δ[(Mk − E1)2

− |~q1|2]Θ(E1)Θ(Mk − E1) ,

=−i M2

kMm

16π2(M2k −M2

m)

∫|~q1|

2d|~q1|dΩ |~q1|(1 + cosθ)1|~q1|

× δ[(Mk − |~q1|)2

− |~q1|2]Θ(Mk − |~q1|) ,

=−i M2

kMm

16π2(M2k −M2

m)

∫|~q1|

2d|~q1|dΩ (1 + cosθ) δ[M2

k − 2Mk|~q1|]Θ(Mk − |~q1|) ,

=−i M2

kMm

16π2(M2k −M2

m)

∫|~q1|

2d|~q1|dΩ(1 + cosθ)| − 2Mk|

δ[|~q1| −

Mk

2

]Θ(Mk − |~q1|) ,

=−i MkMm

32π2(M2k −M2

m)

∫dΩ

M2k

4(1 + cosθ) ,

=−i M3

kMm

32π2(M2k −M2

m)2π4

∫ 1

−1dx (1 + x)︸ ︷︷ ︸

=2

,

=−i M3

kMm

32π(M2k −M2

m). (C.33)

From this the imaginary part is given by

Im[I(1)

]=

12i

Disc[I(1)

]= −

M2k

64π

MkMm

M2k −M2

m

. (C.34)

The 2-body decay phase space for this case is given by

V′ϕ = 2 × 2 ×|~q|

8πE2cm

= 2 × 2 ×1

8πMk

2 M2k

=1

4πMk, (C.35)

94

where one of the factor of 2 is to account for the two channels of final decay products while the other

is to account for the two types of intermediate state (νφ0 or e−φ+) inside the self-energy loop. Putting

all these together and summing over all heavy Majorana neutrino species m , k, as well as the internal

lepton species n, we get a contribution to the asymmetry due to this interference as

εself-(1) = −4

Γtot

∑m,k

∑n

Im(Ah) Im(I(1)V′ϕ) , (C.36)

εself-(1) = 4 ×8π

(h†h)kk Mk

∑m,k

∑n

Im(h∗jkh jmhnmh∗nk)M2

k

64π

MkMm

M2k −M2

m

14πMk

,

=1

8π(h†h)kk

∑m,k

Im[h∗jkh jm(h†h)km

] MkMm

M2k −M2

m

,=

18π(h†h)kk

∑m,k

Im[h∗jkh jm(h†h)km

] √z

1 − z, z ≡

M2m

M2k

. (C.37)

C.4 Self-energy contribution to the CP asymmetry (2)

The interference term can be readily written down as

Iself-(2) =

∫d4q1

(2π)4(ih∗jk)(−ih jm)(ih∗nm)(−ihnk)

[u j

]1C

[PR]CA[SNm(p)

]AB

× [C†PL]BE[S`(q1)

]EF [PR]FD

[uc

k

]D1

[D(q2)

]11

[−uT

k C†PLu j

]11, (C.38)

where we have again shown all spinor indices explicitly. Letting Bh = h∗jkh jmh∗nmhnk, and putting into

matrix form, we get

Iself-(2) = Bh

∫d4q1

(2π)4

u j PR(−i)(/p + Mm)CC†PL i(M)PRuck(i)(−1)uT

k C†PLu j

(p2 −M2m + iε)(q2

1 + iε)(q22 + iε)

,

= −iBh

∫d4q1

(2π)4

u j PR(/p + Mm)PL (M)PR CuTk uT

k C† PLu j

(p2 −M2m + iε)(q2

1 + iε)(q22 + iε)

,

=−iBh

2

∫d4q1

(2π)4

PR(/p + Mm)PL (M)C[∑

s ukuk]T C† PL

∑s′ u ju j

(p2 −M2m + iε)(q2

1 + iε)(q22 + iε)

, (index form)

=−iBh

2

∫d4q1

(2π)4

Tr[PR(/p + Mm)PL (M)(−/p + Mk) PL /p′

](p2 −M2

m + iε)(q21 + iε)(q2

2 + iε),

=iBh

2

∫d4q1

(2π)4

Tr[PR/pM/p /p′

](p2 −M2

m + iε)(q21 + iε)(q2

2 + iε),

= iBh

∫d4q1

(2π)4

2(p · p′)(p · q1) − p2(p′ · q1)

(p2 −M2m + iε)(q2

1 + iε)(q22 + iε)

. (C.39)

95

We now concentrate on the integral

I(2) = i∫

d4q1

(2π)4

2(p · p′)(p · q1) − p2(p′ · q1)

(p2 −M2m + iε)(q2

1 + iε)(q22 + iε)

. (C.40)

Like before, there is only one way to cut the diagram (through q1 and q2), and the discontinuity is given

by (ε→ 0)

Disc(I(2)) = i∫

d4q1

(2π)4(−2πi)2δ(q2

1)δ((p − q1)2)Θ(E1)Θ(Mk − E1)

×2(p · p′)(p · q1) − p2(p′ · q1)

p2 −M2m + iε

,

=−i

4π2(M2k −M2

m)

∫dE1d3q1δ(q2

1)δ((p − q1)2)Θ(E1)Θ(Mk − E1)

×

2 × M2k

2MkE1 −M2

k

(Mk

2E1 + |~q1||~q| cosθ

) ,=

−i4π2(M2

k −M2m)

∫dE1d3q1δ(E2

1 − |~q1|2)δ

((Mk − E1)2

− |~q1|2)Θ(E1)

×Θ(Mk − E1)

M3kE1 −

M3k

2(E1 + |~q1| cosθ

) ,...

=−i M3

k

16π2(M2k −M2

m)

∫|~q1|

2d|~q1|dΩ (1 − cosθ)1

| − 2Mk|δ(|~q1| −

Mk

2

)×Θ(Mk − |~q1|) ,

=−i M4

k

32π2(M2k −M2

m)2π4

∫ 1

−1d(cosθ) (1 − cosθ)︸ ︷︷ ︸

=2

,

=−i M4

k

32π(M2k −M2

m). (C.41)

Hence, the imaginary part is given by

Im[I(2)

]=

12i

Disc[I(2)

]= −

M2k

64π

M2k

M2k −M2

m

. (C.42)

96

[D]

[C]

q

p′

p

[F ] [E][B]

[A]

p

q2

q1

j

φ

Nm

Nk

n

(Vh)jm

(V ∗h )nm

(Vh)nk

Figure C.4: One-loop self-energy correction graph (2) for the process Nk → ` jφ. On the left: (Vh)ab ≡

−i hab PR and (V∗h)ab ≡ i h∗ab C†PL are the vertex factors; on the right: we have included the momentumflows and spinor indices [X], where q = p − p′ and q2 = p − q1.

Therefore, asymmetry due to this interference term is given by

εself-(2) = −4

Γtot

∑m,k

∑n

Im(Bh) Im(I(2)V′ϕ) ,

= 4 ×8π

(h†h)kk Mk

∑m,k

∑n

Im(h∗jkh jmh∗nmhnk)M2

k

64π

M2k

M2k −M2

m

14πMk

,

=1

8π(h†h)kk

∑m,k

Im[h∗jkh jm(h†h)mk

] M2k

M2k −M2

m

.=

18π(h†h)kk

∑m,k

Im[h∗jkh jm(h†h)mk

] ( 11 − z

), z ≡

M2m

M2k

. (C.43)

It should be noted that upon summing over j, the above expression vanishes (because h∗jkh jm(h†h)mk is

real), thus in the one-flavor approximation, this term is absent. Nonetheless, combining this with result

(C.38), we get the full contribution due to the self-energy correction graphs:

εself =1

8π(h†h)kk

∑m,k

Im[h∗jkh jm

(h†h)km

√z

1 − z+ (h†h)mk

11 − z

], (C.44)

=1

8π(h†h)kk

∑m,k

Mk

M2k −M2

mIm

[h∗jkh jm

(h†h)km Mm + (h†h)mk Mk

]. (C.45)

97

Bibliography

[1] E. Kh. Akhmedov and W. Rodejohann, JHEP 06 (2008), 106.

[2] Gustavo C. Branco, Takuya Morozumi, B. M. Nobre, M. N. Rebelo, A Bridge between CP violation at

Low Energies and Leptogenesis, Nucl.Phys. B617 (2001) 475-492.

[3] R. N. Mohapatra and B. Sakita, SO(2N) grand unification in an SU(N) basis, Phys. Rev. D, 21, 1062-1066

(1980).

[4] F.-X. Josse-Michaux, Recent developments in thermal leptogenesis, Univ. Paris-Sud XI, 2008.

[5] L. Landau et E. Lifshitz, Theorie du Champ, Ed. Mir, (1951).

[6] Sakharov, A. D., JETP Lett. 5, 24 (1967);

[7] S.Barr, G.Segre e H.A. Weldon, Phys. Rev. D, 20 (1979), 2494-2498;

[8] G. Ross, Grand Unified Theories, Benjamin/cummings Publications (1984);

[9] E. W. Kolb, M. S. Turner, The Early Universe, Perseus Publishing, Massachussets (1990);

[10] Ed. W. Kolb, S. Wolfram, Nuclear Phys. B, 172 (1980), 224-284;

[11] G.Segre, M.S. Turner, Phys.Letters B, 99 (1981), 399-404.

[12] R.N. Mohapatra, P.B. Pal, Massive Neutrinos in Physics and Astrophysics, World Scientific Lectrure

notes in Physics – Vol.72, 2004

[13] H Klapdor-Kleingrothaus, S Stoica (ed): Double-Beta Decay and Related Topics, (World Scientific 1995)

[14] Sacha Davidson, Enrico Nardi and Yosef Nir, Physics Reports, Volume 466, Issues 4-5, September

2008, Pages 105-177

[15] Ta-Pei Cheng and Ling-Fong Li, Gauge Theory of Elementary Particle Physics, Oxford Science Publica-

tions, 1984.

98

[16] JETP 6 (1957) 429.

[17] L. Wolfenstein, PR D17 (1978) 2369.

[18] S.P.Mikheyev, A.Y.Smirnov: NuCim C9 (1986) 17.

[19] Carsten B Krauss et al 2006 J. Phys.: Conf. Ser. 39 275-277

[20] G.C. Branco, L. Lavoura, J.P. Silva, CP Violation, Oxford Science Publications, 1999.

[21] A. Mirizzi, D. Montanino and P. D. Serpico, Journal of Physics: Conference Series 120 (2008) 022010

[22] Ev. Akhmedov, JHEP0709, 116 (2007)

[23] S. Pakvasa, Lett. Nuovo Cim. 31 (1981) 497

[24] Carlo Giunti, Alexander Studenikin, arXiv:0812.3646v3 [hep-ph]

[25] Wolfram products, www.wolframalpha.com, Mathematica 7.0

[26] G.F.Giudice, A. Notari, M.Raidal, A.Riotto, A.Strumia, Towards a complete theory of thermal leptogenesis

in the SM and MSSM, Nucl.Phys. B685 (2004) 89-149

[27] M. Fukugita, T. Yanagida, Phys.Rev.Lett. 89 (2002) 131602

[28] M. Fukugita, T. Yanagida, Phys.Lett. B174 (1986) 45

[29] Aitchison, I. J. R. and Hey, A. J. G., Gauge Theories in Particle Physics, Institute of Physics Publishing,

1996

[30] C. Amsler et al. (Particle Data Group), Phys. Lett. B667, 1 (2008) and 2009 partial update for the 2010

edition;

[31] Michael E. Peskin and Daniel V. Schroeder, 1995, Addison-Wesley Advanced Book Program;

[32] Lev Landau, Evgeni Lifchitz, L. Pitaevsky, and V. Berestetsky, Electrodynamique quantique, Editions

MIR, 1989

[33] Guralnik, G.S., C.R. Hagen, and T.W.B. Kibble (1964) Global Conservation Laws and Massless Particles,

Phys. Rev. Lett. 13: 585-87.

[34] L. L. Chau and W.-Y. Keung, Physical Review Letters 53 1802 (1984).

99

[35] L. Wolfenstein, Physical Review Letters 51 1945 (1983).

[36] Class notes for Introducao as Teorias de Unificacao, lectured in 2007.

[37] J.A. Casas and A. Ibarra, Oscillating neutrinos and µ → e, γ, Nucl. Phys. B 618 (2001) 171 [hep-

ph/0103065].

[38] T. Yanagida, Proceedinds of the workshop on unified theories and Baryon number in the universe, A. Sawada

and A. Sugamoto eds.,Tsukuba, Japan 1979;

[39] R.N. Mohapatra and G. Senjanovic, textitNeutrino mass and spontaneous parity nonconservation,

Phys. Rev. Lett. 44 (1980) 912.

[40] G.C. Branco, L. Lavoura and M.N. Rebelo, Majorana neutrinos and CP-violation in the leptonic sector,

Phys. Lett. B 180 (1986) 264.

[41] Utpal Sarkar, Particle and Astroparticle Physics, Taylor and Francis Series in High Energy Physics,

Cosmology and Gravitation, 2008

[42] Fiorentini, G.; B. Ricci (2002). What have we learnt about the Sun from the measurement of the 8B neutrino

flux?, Physics Letters B 526 (3-4): 186-190.

[43] D.Kazakov and G.Smadja, Particle Physics and Cosmology: the Interface, NATO Science Series, Vol. 188

(2005)

[44] K. et al. (Kamiokande Coll.). Solar neutrino data covering solar cycle 22, Phys. Rev. Lett., 77:1683, 1996.

[45] J. et al. (SAGE Coll.)., Phys. Rev. Lett., 77:4708, 1996

[46] Q. et al. (SNO Coll.)., Phys. Rev. Lett., 87:071301, 2001

[47] Q. et al. (SNO Coll.)., Phys. Rev. Lett., 89:011301, 2002

[48] Q. et al. (SNO Coll.)., Phys. Rev. Lett., 89(011302), 2002.

[49] Q. et al. (SNO Coll.)., Phys. Rev. C, 72:05502, 2005.

[50] Martin, B.R. and Shaw, G (1999) Particle Physics (2nd ed.). Wiley. p. 265.

[51] M. B. Voloshin and M. I. Vysotsky, Institute for Theoretical and Experimental Physics (Moscow)

Report No. 86-1, 1986;

100

[52] Barger et al., Phys. Rev. Lett. 82 2640 (1999)

[53] J. Boger , R. L. Hahn , and J. B. Cumming, The Astrophysical Journal, 537:1080-1085, 2000 July 10

[54] AI Belesev et al., PL B350 (1995) 263.

[55] A Assamagan et al., PR D53 (1996) 6065.

[56] R Barate et al., EPJ C2 (1998) 395.

[57] R N Mohapatra, R E Marshak: PL B91 (1980) 222.

[58] R N Mohapatra, G Senjanovic, PRL 44 (1980) 912 and PR D23 (1981) 165.

[59] S. Law, Neutrino mass models and leptogenesis.

[60] V Barger, J Hewett, T Rizzo, PR D42 (1990) 152.

[61] F Abe et al., PR D51 (1995) 949.

[62] A Jodidio et al., PR D34 (1986) 1967.

[63] G Beall, M Bander, A Soni, PRL 48 (1982) 848.

[64] N G Deshpande, E Keith, P B Pal, PR D46 (1992) 2261.

[65] V. Kuzmin, V. Rubakov and M. Shaposhnikov, PL B155 (1985) 36.

[66] M. Fukugita, T. Yanagida, PL B174 (1986) 45.

[67] D.N. Spergel et al.[WMAP collaboration], ApJ Suppl. 148 (2003) 79.

101