UNIVERSIDAD NACIONAL AUTÓNOMA DE MEXICO 
POSGRADO EN CIENCIAS FÍSICAS 
 
 
 
 
NEUTRINO MASS GENERATION AND DARK MATTER MODELS 
 
 
 
 
TESIS 
QUE PARA OPTAR POR EL GRADO DE:  
DOCTOR EN CIENCIAS (FÍSICAS) 
 
 
 
PRESENTA: 
JORGE MARIO LAMPREA GARZON 
 
 
 
ASESOR: 
DR. EDUARDO PEINADO RODRÍGUEZ  
INSTITUTO DE FÍSICA – UNAM 
 
 
 
MIEMBROS DEL COMITÉ TUTOR: 
DR. GENARO TOLEDO SÁNCHEZ 
INSTITUTO DE FÍSICA – UNAM 
 
DR. MARIANO CHERNICOFF MINSBERG 
FACULTAD DE CIENCIAS – UNAM 
 
 
 
 
CUIDAD DE MÉXICO, SEPTIEMBRE 2018 
 
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Resumen de la tesis
Las masas de los neutrinos y la existencia de materia oscura son las pruebas más directas
sobre la necesidad de nueva física más allá del Modelo Estándar de partículas elementales.
Sin embargo, la naturaleza de esta nueva física sigue siendo al día de hoy desconocida. Esto
ha imposibilitado determinar cuál es la extensión del Modelo Estándar que de cuenta de
todos o la mayoría de los fenómenos que el Modelo Estándar no puede explicar. Por el lado
de los neutrinos, se desconoce su naturaleza, es decir, si son fermiones de Dirac o Majorana,
su escala de masas, el mecanismo detrás de la generación de sus masas tan pequeñas o si
existe una razón detrás del patrón de masa y mezcla.
Adicionalmente, es interesante considerar que las masas y mezclas de los neutrinos pueden
ser explicados por una teoría que de cuenta de los patrones de masa y mezcla de los todos los
fermiones en el Modelo Estándar. Esto es posible ya que los acoplamientos de Yukawa, de
los cuales dependen las masas y mezclas de los quarks y leptones, son parámetros complejos
totalmente libres en el Modelo Estándar. A dicha arbitrariedad de los acoplamientos de
Yukawa y, por tanto, falta de explicación para las masas y mezclas de los fermiones, es
referido usualmente como el problema del sabor del Modelo Estándar.
Por otro lado, la existencia de materia oscura, inferida a través de sus efectos gravita-
cionales sobre la materia visible, urge por una explicación dentro del escenario de la física de
partículas. Esto debido a que el Modelo Estándar falla en proporcionar un candidato viable
a materia oscura. Además, tales candidatos a materia oscura deben satisfacer constricciones
de experimentos de busqueda directa e indirecta cada vez más fuertes.
La idea de relacionar el mecanismo detrás del origen de las masas de los neutrinos
y la materia oscura es, desde el punto de vista teórico, muy atractivo y puede generar
interesantes desarrollos en ambos sectores. Basados en dicha premisa, la presente tesis está
dedicada al estudio de tres realizaciones teóricas de la conexión entre los neutrinos y la
materia oscura, teniendo en común la mínima extensión del Modelo Estándar a través de la
adición de simetrías discretas.
iv
Esta tesis está ordenada como se describe a continuación. Primero en el capítulo 1,
desarrollamos una breve introdución al Modelo Estándar, la física de neutrinos y la materia
oscura. Esto con el fin de establecer la notación y convenciones usadas a lo largo del
documento. A continuación en el capítulo 2, se presentan dos extensiones al Modelo
Estándar donde la materia oscura y la masa de los neutrinos están relacionadas por medio
de una simetría de sabor no Abeliana. Dicha simetría de sabor es responsable por la
fenomenología de los neutrinos. Además, su rompimiento en una simetría residual, genera
un mecanismo para la estabilizacíon de la materia oscura. Las masas de los neutrinos son
generadas a través del mecanismo de seesaw tipo I. Las predicciones de sabor se deben a
que la matriz de masas de los neutrinos izquierdos posee una textura con dos ceros. A
partir de dicha textura, es posible obtener correlaciones entre los parámetros de oscilación
y, por lo tanto, los observables de los neutrinos. Una de estas correlaciones lleva a una cota
inferior a la masa de Majorana efectiva del neutrino del electrón, la cual es proporcional a
la amplitud del decaimiento beta doble sin neutrinos (0νββ). Dicha cota está en la región
de sensitividad de futuros experimentos.
En el capítulo 3, se estudia una extensión del modelo Escotogénico de Ma, donde las
masas de los neutrinos son generadas a través de correcciones radiativas a un lazo. En
dichas correcciones radiativas la materia oscura participa en el lazo. El modelo de Ma
añade tres neutrinos derechos y un doblete de Higgs inerte al Modelo Estándar, todos ellos
cargados ante una simetría discreta exacta. De tal modo, los neutrinos izquierdos adquieren
sus masas mediante una corrección radiativa a un lazo. Sin embargo, existe una tensión
en el modelo Escotogénico. Cuando el candidato a materia oscura es el neutrino derecho
más ligero y solo su aniquilación es responsable por la densidad reliquia, el espacio de
parámetros que satisface las cotas experimentales de los procesos con violación de sabor es
severamente restringido. De este modo, se muestra que al añadir un campo escalar complejo
singlete bajo el grupo de norma del Modelo Estándar se reduce significativamente dicha
tensión. Esto debido a que el campo escalar puede generar dinámicamente la masa de los
neutrinos derechos, introduciendo un nuevo canal de aniquilación para la materia oscura.
Posteriormente en el capítulo 4, se explora un modelo de simetría de sabor donde los
neutrinos adquieren masas por medio de un mecanismo de seesaw para neutrinos de Dirac.
El modelo resuelve parcialmente el problema de sabor en el Modelo Estándar. Por un
lado, el patrón de masa de los fermiones es explicado por medio de una relación de masas
derivada de la simetría de sabor, entre los leptones cargados y los quarks tipo down. Dicha
relación ha sido propuesta anteriormente en otros trabajos. Por otro lado, el patrón de
v
mezcla de los fermiones está relacionado por la estructura de la simetría de sabor, la cual
puede reproducir los elementos de la matriz CKM, y de este modo, fijar la contribución
de los leptones cargados a la matriz de mezcla leptónica. El modelo tiene predicciones
respecto a la fenomenología de los neutrinos, siendo solo consistente con un ordenamiento
invertido de las masas de los neutrinos y un ángulo de mezcla atmosférico no maximal.
Además, se tiene que la fase de violación de CP es no nula cuando la masa más ligera de
neutrinos menor a 2 meV, mientras que para masas mayores es compatible con un escenario
con conservación de la simetría de CP.
Por último damos nuestras conclusiones y comentarios finales en el capítulo 5.
vi
Abstract
Neutrino masses and dark matter existence are the most direct proves for the need of
new physics beyond the Standard Model (SM). On the neutrino side, we do not know their
nature, that is, whether they are Dirac or Majorana fermions, their mass scale and which
is the mechanism behind their small masses. Furthermore, it is interesting to consider
that neutrino mass and mixing patterns could be explained within a theoretical framework
relating mass and mixing patterns for quarks and leptons in the SM. On the dark matter
side, its existence, inferred through gravitational effects on visible matter, needs to be
explained within particle physics theoretical framework, as the SM fails on providing a
viable dark matter candidate. Additionally, these dark matter candidates have to satisfy
current experimental constraints.
The idea of relating the mechanism behind the neutrino mass generation and the dark
matter is, form the theoretical point of view, quite appealing and could lead to interesting
developments in both sectors. Based on such idea, this thesis is intended to investigate
three possible theoretical realisations of this connection based on minimal SM extensions
using discrete symmetries. Thus, this thesis is organised as follows.
In chapter 1, we develop a brief introduction to the Standard Model, neutrino physics
and dark matter, setting notation and convention used throughout the document. Following,
the chapter 2 is devoted to the study of an A4 symmetric SM extension, where neutrino
mass generation and dark matter are related by the breaking of such non-Abelian discrete
flavour symmetry into a residual symmetry, which stabilises the dark matter. Neutrino
phenomenology follows from the flavour symmetry assignments and its specific breaking.
Correlations between neutrino oscillation parameters and, then observables, are found. One
of such correlations leads to a lower bound on the neutrinoless double beta decay (0νββ)
amplitude, lying within the sensitivity range of near-future experiments.
In chapter 3, we study an extension of Ma’s Scotogenic model, where neutrino mass
generation is due to one-loop radiative correction involving dark matter particles in the
loop. Previous works have shown that when the lightest right-handed (RH) neutrino is
the dark matter and the right relic abundance is coming only from its annihilation, lepton
flavour violating processes (LFV) severely constrain the parameter region for the model.
We have shown that by adding a complex scalar singlet of the SM gauge, which generates
dynamically RH neutrinos masses, an additional dark matter annihilation channel opens
up relaxing the aforementioned tension.
vii
In chapter 4, we study an A4 flavour symmetric realisation of a type-II Dirac seesaw.
This model partially addresses the flavour problem in the SM. First, the fermion mass
patterns are explained by a flavour dependent mass relation between down-type quarks and
charged leptons, proposed in previous works. Secondly, the model can fit the CKM matrix
elements, then fixing the charged lepton contribution to lepton mixing matrix. Regarding
neutrino phenomenology, this model is only consistent with an inverted ordering of neutrino
masses and non-maximal atmospheric mixing angle. The model also predicts a non-zero
CP violating phase when the lightest neutrino mass is less than 2 meV, while for bigger
masses it is consistent with a vanishing value of the CP violating phase.
Finally, we draw our summary and final remarks in chapter 5.
This work is based on the following publications:
• Seesaw scale discrete dark matter and two-zero texture Majorana neutrino mass
matrices. J. M. Lamprea, E. Peinado. Phys. Rev. D 94 (2016).
• Flavour-symmetric type-II Dirac neutrino seesaw. C. Bonilla, J.M. Lamprea, E.
Peinado, J. W. F. Valle. Phys. Lett. B 779 (2018).
• Fermionic dark matter from radiative neutrino mass. C. Bonilla, L. M. Garcia de la
Vega, J. M. Lamprea, R. Lineros, E. Peinado, in prep.

Table of contents
List of figures xi
List of tables xv
1 Introduction 1
2 A4 flavour symmetric models for Majorana neutrinos and dark matter 31
2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2 The models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.3 Neutrino phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.5 Dark matter phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3 Radiative Majorana neutrino mass generation and fermionic dark matter 53
3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4 A4 flavour symmetric model for a type-II Dirac neutrino seesaw 73
4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.2 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.2.1 Lepton sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.3.1 The generalised bottom-tau mass relation . . . . . . . . . . . . . . 78
x Table of contents
4.3.2 CKM fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.3.3 Lepton masses and mixing . . . . . . . . . . . . . . . . . . . . . . . 83
4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5 Summary and final remarks 89
Appendix A The A4 group 93
A.1 A4 irreducible representations product . . . . . . . . . . . . . . . . . . . . 94
Appendix B Radiative one-loop mass calculation for Scotogenic model 97
Appendix C Oblique parameters for a radiative Majorana neutrino mass
generation 101
References 103
List of figures
1.1 Feynman diagram of the type-I seesaw. The tree level completion of the
dimension-5 Weinberg operator is done by the exchange of RH neutrinos
N = νR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.2 Feynman diagram for the type-II seesaw mechanism. The tree level completion
of the dimension-5 Weinberg operator is done by the exchange of a SU(2)L
triplet complex scalar Δ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3 Feynman diagram for the type-III seesaw mechanism. The tree level comple-
tion of the dimension-5 Weinberg operator is done by the exchange of SU(2)L
triplet RH fermions Σ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4 One-loop radiative mass generation for LH Majorana neutrinos in the Zee
model [61, 62]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.5 Two-loops radiative mass generation for LH Majorana neutrinos in the
Zee–Babu model [65, 64]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.6 Feynman diagram for a type-I Dirac neutrino seesaw. The tree level com-
pletion of the generalised dimension-5 operator in Eq. (1.52), is done by the
exchange of Dirac fermions N = NL + NR. . . . . . . . . . . . . . . . . . . 21
1.7 Feynman diagram for a type-II Dirac neutrino seesaw. The tree level com-
pletion of the generalised dimension-5 operator in Eq. (1.52) is done by the
exchange of φ a complex scalar doublet of SU(2)L. . . . . . . . . . . . . . . 22
1.8 Feynman diagram for a type-III Dirac neutrino seesaw. The tree level
completion of the generalised dimension-5 operator in Eq. (1.52) is done by
the exchange of vector-like fermions E0. . . . . . . . . . . . . . . . . . . . . 23
xii List of figures
2.1 Correlation between sin2 θ23 and the sum of the light neutrino masses,
∑
mν,
in model A (with the B3 texture) at the top and model B (with the B4 texture)
at the bottom, where NO (IO) allowed region is in magenta (cyan). The
horizontal red (blue) shaded region corresponds to the 1σ value in sin2 θ23 for
NO (IO) from [121]. The red (blue) horizontal dashed line represent the θ23
best fit value in NO (IO), while the doted horizontal red line represents the
value of local minimum in NO from from [121]. The vertical grey shaded
region is disfavoured by Planck data [124]. . . . . . . . . . . . . . . . . . . 46
2.2 Correlation between sin2 θ23 and the sum of the light neutrino masses,
∑
mν,
in model A (with the B3 texture) at the top and model B (with the B4 texture)
at the bottom, where NO (IO) allowed region is in magenta (cyan). The
horizontal red (blue) shaded region corresponds to the 1σ value in sin2 θ23
for NO (IO) from [122]. The red (blue) horizontal dashed line represents
the best fit value in NO (IO) from [122]. The vertical grey shaded region is
disfavoured by the Planck data [124]. . . . . . . . . . . . . . . . . . . . . . 47
2.3 Correlation between sin2 θ23 and the sum of the light neutrino masses,
∑
mν,
in model A (with the B3 texture) at the top and model B (with the B4 texture)
at the bottom, where NO (IO) allowed region is in magenta (cyan). The
horizontal red (blue) shaded region correspond to the 1σ in sin2 θ23 for NO
(IO) from [123]. The case for IO has two 1σ regions in the data used. The
red (blue) horizontal dashed line represents the best fit value in NO (IO)
from [123]. The vertical grey shaded region is disfavoured by Planck data [124]. 48
2.4 Effective 0νββ parameter |mee| versus the lightest neutrino mass mνlight
in model A (B) at the top (bottom). The mνlight
is m1(m3) for NO (IO).
The model allowed region for NO is in magenta (dark magenta) for the
1σ (3σ) atmospheric mixing angle and for IO in cyan (dark cyan) for the
1σ (3σ) atmospheric mixing angle region from [121]. The yellow (green)
band correspond to the “flavour–generic" IO (NO) neutrino spectra at 3σ.
The horizontal red shaded region is the experimental limit on 0νββ, while
the red (blue) horizontal (vertical) lines are the forthcoming experimental
sensitivities on |mee| (mνlight
) from [125–130]. The vertical blue shaded region
is disfavoured by Planck data [124]. . . . . . . . . . . . . . . . . . . . . . . 49
List of figures xiii
2.5 Effective 0νββ parameter |mee| versus the lightest neutrino mass mνlight
in
model A (B) at the top (bottom). The mνlight
is m1(m3) for NO (IO). The
mνlight
is m1(m3) for NO (IO). The model allowed region for NO is in magenta
(dark magenta) for the 1σ (3σ) atmospheric mixing angle and for IO in cyan
(dark cyan) for the 1σ (3σ) atmospheric mixing angle region from [122]. The
yellow (green) band correspond to the “flavour–generic" IO (NO) neutrino
spectra at 3σ. The horizontal red shaded region is the experimental limit
on 0νββ, while the red (blue) horizontal (vertical) lines are the forthcoming
experimental sensitivities on |mee| (mνlight
) from [125–130]. The vertical blue
shaded region is disfavoured by Planck data [124]. . . . . . . . . . . . . . . 50
2.6 Effective 0νββ parameter |mee| versus the lightest neutrino mass mνlight
in model A (B) at the top (bottom). The mνlight
is m1(m3) for NO (IO).
The model allowed region for NO is in magenta (dark magenta) for the
1σ (3σ) atmospheric mixing angle and for IO in cyan (dark cyan) for the
1σ (3σ) atmospheric mixing angle region from [123]. The yellow (green)
band correspond to the “flavour–generic" IO (NO) neutrino spectra at 3σ.
The horizontal red shaded region is the experimental limit on 0νββ, while
the red (blue) horizontal (vertical) lines are the forthcoming experimental
sensitivities on |mee| (mνlight
) from [125–130]. The vertical blue shaded region
is disfavoured by Planck data [124]. . . . . . . . . . . . . . . . . . . . . . . 51
3.1 One-loop neutrino mass generation in Ma’s Scotogenic model [133]. . . . . 54
3.2 Relevant annihilation channels for scalar DM η0 in Ma’s Scotogenic model. 56
3.3 Relevant annihilation channel for fermionic DM N1 in Ma’s Scotogenic model. 57
3.4 One-loop neutrino mass generation in the model. . . . . . . . . . . . . . . . 59
3.5 New annihilation channels for fermionic DM, N1, in the model. Up: t-
channel. Down: s-channel resonance. . . . . . . . . . . . . . . . . . . . . . 61
3.6 Up: Dark matter velocity averaged annihilation cross section to gammas
〈σv〉γ as a function of the dark matter mass mN1
. The Fermi-LAT [146]
indirect detection exclusion curve is shown in blue. Bottom: Dark matter–
nucleon spin independent scattering cross section σSI as a function of dark
matter mass mN1
. The PandaX–II [145] 54 ton-day exclusion curve is shown
in blue. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.7 Up: SM Higgs total width Γh1
as a function of dark matter mass mN1
.
Bottom: SM Higgs total width Γh1
as a function of the mixing parameter sin θ. 68
xiv List of figures
3.8 Correlation between the of SM Higgs branching ratio to invisible Br(h1 → inv)
and the effective coupling sin θ/vφ. . . . . . . . . . . . . . . . . . . . . . . . 69
3.9 Up: lightest LH neutrino mass mν1
as a function of dark matter mass mN1
.
Bottom: gauge singlet scalar–dark matter Yukawa coupling hN1
as a function
of inert Higgs–neutrinos Yukawa couplings Y ν
i . . . . . . . . . . . . . . . . . 70
3.10 Correlation between the dark matter spin-independent scattering cross section
σSI as a function of the Higgs portal effective coupling hN1
. . . . . . . . . . 71
4.1 Neutrino mass generation in the type-II seesaw for Dirac neutrinos, as
in [69, 153, 154]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.2 The regions in the atmospheric mixing angle θ23 and the lightest neutrino
mass m3 allowed by oscillation data in shaded (green) areas. The horizontal
dashed line represents the best-fit value for sin2 θ23, whereas the horizontal
shaded region corresponds to the 1σ allowed region from Ref. [33]. . . . . . 86
4.3 Correlation between the CP violation and the lightest neutrino mass. Up:
correlation between the Jarlskog invariant J and the lightest neutrino mass
m3 allowed by the current oscillation data [33]. Bottom: Allowed regions
for the correlation between the Dirac CP phase δCP and the lightest neutrino
mass m3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.4 The allowed regions of the atmospheric mixing angle and δCP are indicated
in green shaded. The unshaded contour regions represent the 90 and 99%CL
regions obtained directly in the unconstrained three neutrino oscillation global
fit [33]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
B.1 Feynman diagram for the Scotogenic neutrino mass generation in mass
eigenstates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
List of tables
1.1 Fundamental fermions in the Standard Model arranged by generations. . . . 2
1.2 Fundamental fermions in the Standard Model, where a = 1, 2, 3 is the gener-
ation index. Particle electric charges are given by the Gell-Mann – Nishijima
formula Q = I3 + Y [15–17], where the third component of weak SU(2)L is
I3 = ±1/2 for doublets and I3 = 0 in the case of singlets. . . . . . . . . . . 3
1.3 Fundamental scalars in the Standard Model. Particle electric charges are
given by the Gell-Mann–Nishijima formula Q = I3 + Y [15–17], where the
third component of weak SU(2)L is I3 = ±1/2 for the doublet. . . . . . . . 3
1.4 Summary of relevant particle content and quantum numbers in the Zee
model [61, 62]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.1 Summary of the relevant particle content and quantum numbers for model A. 36
2.2 Summary of the relevant particle content and quantum numbers for model B. 38
3.1 Relevant particle content and quantum numbers in Scotogenic model . . . . 53
3.2 Summary of the relevant particle content and quantum numbers in the model. 57
4.1 Charge assignments for the particles involved in the type-II Dirac neutrino
seesaw realisation, as in [69]. . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.2 Charge assignments for the particles involved in the neutrino mass generation
mechanism, where ω3 = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3 Particle content and quantum numbers for the complete model. . . . . . . . 79
4.4 Experimental and predicted quark masses and mixing parameters from the
CKM fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Chapter 1
Introduction
This chapter is intended as an introduction to the thesis and to survey the notation and
convention we will use throughout the document. Here we will briefly review the Standard
Model of Particle Physics, neutrino physics and dark matter as the necessary background
to our work.
The Standard Model in a nutshell
In this section, we will give a short review of the Standard Model properties. A more
detailed discussion can be found in several textbooks, for instance in [1, 2]. The description
of the fundamental interactions, excluding gravitation, is given quite accurately by the SM.
This is a renormalisable gauge theory [3] based upon the symmetry group
GSM = SU(3)C ⊗ SU(2)L ⊗ U(1)Y ,
where C, L and Y stands for colour, left-handed (LH) chirality and hypercharge, and the
corresponding coupling constants are gs, g and g′ respectively. Each part in this direct
product of symmetry groups is responsible for an interaction. The SU(3)C group describes
the strong interaction in what is known as Quantum Chromodynamics (QCD) [4–8], while
the product SU(2)L ⊗ U(1)Y is responsible for an unified description of the Electroweak
interaction [9–11].
The gauge symmetry group choice in the SM fixes the gauge bosons content as well as
their irreducible representations (irreps.) to be the adjoint irreps., although it does not fix
the content of fermion and scalar fields. There are four Electroweak gauge bosons: three
2 Introduction
weak gauge boson W 1,2,3 µ in a SU(2)L triplet and a Bµ associated with U(1)Y . There are
also eight QCD gauge bosons (or gluons) labelled as G1,··· ,8 µ forming up a SU(3)C octet.
These gauge bosons account for the 12 generators of GSM .
Additionally, as the bare mass terms for gauge bosons and chiral fermions are forbidden by
the gauge invariance, the SM incorporates the Electroweak spontaneous symmetry breaking
mechanism (EWSB) [12–14], or Brout–Englert–Higgs (BEH) mechanism, responsible for
giving mass to the Electroweak gauge bosons W and Z and the whole fermions. This
mechanism leads to the breaking of SU(2)L ⊗ U(1)Y → U(1)Q, with Q the electric charge.
Such spontaneous symmetry breaking is triggered by the field H, a complex scalar doublet
of SU(2)L with hypercharge Y = 1/2, which is referred as the Higgs doublet.
The SM includes three generations of quarks and leptons. These are shown in Tab. 1.1.
Each fermion generation contains 15 two-component spinors1: two charged leptons and 12
quarks, all of them in both chiral components left- (LH) and right-handed (RH) forming up
seven Dirac fermions2, plus one left-handed neutrino. Particle content and gauge symmetry
Family 1st 2nd 3rd
Quarks
u c t
d s b
Leptons
νe νµ ντ
e µ τ
Table 1.1 Fundamental fermions in the Standard Model arranged by generations.
assignments of the fermions and scalar in the SM are displayed in Tabs. 1.2 and 1.3
respectively. The LH fermions transform as SU(2)L doublets, while RH fermions are
SU(2)L singlets, as a consequence that weak interactions violate parity maximally.
The Standard Model Lagrangian can be written as:
L = LGauge + LDirac + LYuk + LHiggs, (1.1)
1The homogeneous Lorentz group irreps. for spin 1/2 particles are labelled as (1/2, 0) and (0, 1/2), which
are known as left-handed (LH) χL and right-handed (RH) χR Weyl or Majorana fields or two-component
spinors.
2Dirac fermions or four-component spinors are made of both homogeneous Lorentz group irreps.
ψ = (χL χR)T . Left and right projectors PL = 1−γ5
2
and PR = 1+γ5
2
are defined through the relations
PLψ := (χL 0)T and PRψ := (0 χR)T .
3
Fields SU(3)C SU(2)L U(1)Y
La =
(
νaL ℓaL
)T
1 2 −1/2
ℓaR 1 1 −1
Qa =
(
uaL daL
)T
3 2 1/6
uaR 3 1 2/3
daR 3 1 −1/3
Table 1.2 Fundamental fermions in the Standard Model, where a = 1, 2, 3 is the generation
index. Particle electric charges are given by the Gell-Mann – Nishijima formula Q =
I3 + Y [15–17], where the third component of weak SU(2)L is I3 = ±1/2 for doublets and
I3 = 0 in the case of singlets.
Fields SU(3)C SU(2)L U(1)Y
H =
(
H+ H0
)T
1 2 1/2
Table 1.3 Fundamental scalars in the Standard Model. Particle electric charges are given by
the Gell-Mann–Nishijima formula Q = I3 + Y [15–17], where the third component of weak
SU(2)L is I3 = ±1/2 for the doublet.
where each term represents a contribution conceptually different. In order, these are the
kinetic energy of the gauge bosons (Yang–Mills theory), kinetic energy of the fermion
content, Yukawa interactions between fermions and scalars and finally Higgs interactions.
Ghost and gauge fixing terms could be excluded working in the unitary gauge.
The gauge part in Eq. (1.1) describes the behaviour of the gauge bosons. This term is
written as:
LGauge = −1
2
Tr(GµνGµν) − 1
2
Tr(W µνWµν) − 1
4
BµνBµν , (1.2)
with Gµν , W µν and Bµν the field strength tensors3 of SU(3)C , SU(2)L and U(1)Y respec-
tively.
The Dirac Lagrangian in Eq. (1.1) contains the kinematic and gauge interactions of the
whole fermions in the SM. Such Lagrangian is given by:
LDirac = ψ̄a(iγµDµ)ψa, (1.3)
3The field strength tensor for a non-Abelian gauge theory is defined as F i
μν := ∂μAi
ν −∂νAi
μ −g cijk Aj
μAk
ν ,
with Ai
μ the gauge boson fields, g the coupling constant and cijk the group structure constants.
4 Introduction
where the index a refers to all the fermion fields and Dµ is the covariant derivative defined
as:
Dµ =: ∂µ − igsG
a
µλa − igW a
µ τa − iY g′Bµ, (1.4)
with λa and τa the SU(3) and SU(2) generators in the same representation that ψ, and Y
the hypercharge of ψ.
The Higgs interactions term in Eq. (1.1) is given by the Lagrangian
LHiggs = DµH†DµH − µ2(H†H) − λ(H†H)2, (1.5)
where the Higgs doublet, H, can be written as4:
H =
⎛
⎝
H+
1√
2
(v + h + i A)
⎞
⎠ . (1.6)
When µ2 < 0, the Higgs potential in Eq. (1.5) has a non-zero minimum value, then the
Higgs vacuum expectation value (vev.) is non-zero. Thus, when H develops a vev., denoted
as 〈H〉 = v/
√
2, it triggers the spontaneous EWSB.
The tree level Higgs boson h mass is given in terms of the parameters in Eq. (1.5) as
m2
h = −µ2λ = 2λv2. (1.7)
This mass has been experimentally determinated approximately as mh ≈ 125 GeV [18, 19].
After EWSB, the fields H± and A will be the pseudo Nambu-Goldstone bosons [20–22]
corresponding to the broken generators of SU(2)L ⊗ U(1)Y . Such fields will become the
longitudinal degrees of freedom of the gauge fields W i µ and Bµ (W and Z in mass basis),
thus generating masses for them. At tree level, the W and Z gauge boson masses yield
mW =
g
2
v ∼ 78 GeV and mZ =
√
g2 + g′2
2
v =
mW
cos θW
∼ 89 GeV, (1.8)
with g and g′ the coupling constants associated with SU(2)L and U(1)Y , v ≃ 246 GeV, the
weak scale, and sin2 θW = g′2
g2+g′2 ≃ 0.23 the sine of the weak mixing angle. This mixing
angle parametrises the mixing between the neutral gauge bosons in the interaction basis
4The Higgs doublet in the unitary gauge is H = 1
√
2
(0 v + h)
T
.
5
W 3µ and Bµ in terms of the physical eigenstates Zµ and Aµ, the latter being the photon
field which remains massless.
Finally, Yukawa Lagrangian in Eq. (1.1) is responsible for giving mass to the fermions
through EWSB. This Lagrangian is given by:
LYuk = Γe
ij L̄i H ℓRj + Γu
ij Q̄i H̃ uRj + Γd
ij Q̄i H dRj + h.c., (1.9)
where i, j are generation indices, Γa are 3 × 3 general complex Yukawa coupling matrices
and H̃ = iσ2H∗, with σ2 the Pauli matrix, is the Higgs doublet charge conjugate which has
Y = −1/2. Notice that fields in Eq. (1.9) are written in the flavour (interaction) basis not
as mass eigenstates.
After EWSB, from Eq. (1.9) fermion mass matrices are give by
Ma
ij = Γa
ij
v√
2
,
where a = {e, u, d}. As Γa are general complex matrices, they can be diagonalised by two
unitary matrices U and V as:
V a † Ma Ua = Ma
D := diag (m1, m2, m3) , (1.10)
with mi the running masses.
Since neutrinos in the Standard Model are massless, one has the freedom to redefine
charged lepton fields making them diagonal. However, such field redefinitions cannot be
done for quarks as u- and d-type quarks couple both to the Higgs, then quark mass matrix
will be non-diagonal in flavour space. Thus, we can redefine quarks in flavour space in term
of the unitary matrices that diagonalise the quark mass matrix, Eq. (1.10), as uL = V uu′
L,
dL = V dd′
L and uR = Uuu′
R, dR = UdU ′
L , where the prime refers to mass eigenstates.
Therefore, the quark weak charged current:
LCC = − g
2
√
2
3
∑
i=1
(ūiLγαdiL) W +α + h.c., (1.11)
will be diagonal in flavour space. We can define the quark mixing matrix or the Cabibbo–
Kobayashi–Maskawa (CKM) matrix [23] from the mismatch between flavour and mass
eigenstates as:
VCKM = V u †V d. (1.12)
6 Introduction
The CKM matrix is parametrised (PDG parametrisation [24]) by three mixing angles θ12,
θ23 and θ13 and one complex phase δ which allows charge-parity (CP) violation in quark
sector. Thus,
VCKM =
⎛
⎜
⎜
⎝
1 0 0
0 c23 −s23
0 s23 c23
⎞
⎟
⎟
⎠
⎛
⎜
⎜
⎝
c13 0 −e−iδs13
0 1 0
eiδs13 0 c13
⎞
⎟
⎟
⎠
⎛
⎜
⎜
⎝
c12 −s12 0
s12 c12 0
0 0 1
⎞
⎟
⎟
⎠
, (1.13)
where sij = sin θij and cij = cos θij.
The CP violating δ phase in VCKM , Eq. (1.13), depends on the chosen parametrisation.
An independent way to have a CP violation quantification is through the Jarlskog invariant
JCP [25]:
JCP = −Im[VusVcdV ∗
csV
∗
ud] = cos θ12 cos θ23 cos2 θ13 sin θ12 sin θ23 sin θ13 sin δ. (1.14)
Finally, it is worth to notice that in its minimal extension the Standard Model has 19
free parameters: nine fermions (quark and charged lepton) masses, four parameters in CKM
matrix, three gauge couplings, the Higgs mass, the weak scale and θQCD (related with the
strong CP problem).
Open questions in the Standard Model
The SM has shown to be a successful theory in the last decades, even the latest LHC results
have confirmed its astounding accuracy describing fundamental interactions. However, we
know this is not a complete theory. There are some open theoretical issues and phenomena
that the SM cannot account for, as the existence of dark matter and neutrino masses.
First, the SM gives no prediction or explanation on the number of generations of fermions,
the mass hierarchy between generations and the mixing patterns of quarks and leptons.
In addition, it does not explain either electric charge quantisation or CP conservation in
the strong interaction (strong CP problem). Moreover, it is necessary a fine-tuning in the
radiative corrections to the Higgs mass to account for the observed one.
There are in addition some phenomena that the SM fails explaining. The most relevant
include the smallness of the neutrino masses, the existence of dark matter (DM), the origin
of the asymmetry between matter and antimatter in the Universe, also called Baryonic
7
Asymmetry of the Universe (BAU), and finally the accelerated expansion of the Universe
or Dark Energy.
The quest for an unified description of the forces in Nature leads to consider the Standard
Model as an effective theory valid at most to the Planck scale MPlanck = G
−1/2
N ∼ 1019 GeV,
when the quantum theory of gravity has to be taken into account. One hopes that such
new physics beyond Standard Model (BSM) be at a sufficiently low scale, near to energy
scale achievable by next-generation experiments, as the latest LHC runs have not spotted
any new physics.
Neutrino Physics
Nowadays it is known that neutrinos have masses, as it has been inferred from neutrino oscil-
lation experiments. Solar, atmospheric, reactor and accelerator neutrino experiments [26–30]
have provided convincing evidence for the oscillation of three flavours of active (LH) neutri-
nos: νeL, νµL and ντL.
Experimental data analysis has shown that such flavour neutrino oscillations are con-
sistent with the mixing of three mass eigenstates νi with masses mi, i = {1, 2, 3}. Such
mixing is given by
νaL =
3
∑
i=1
V †
ai νi, (1.15)
with a = {e, µ, τ} the flavour index and Vai a 3 × 3 unitary mixing matrix. This mixing
matrix, as for quarks, comes from the left-diagonalising matrices for the charged leptons V ℓ
and neutrinos V ν making the weak neutral current flavour diagonal.
We can redefine the LH flavour fields as νL = V νν ′
L and ℓL = V ℓℓ′
L, where prime refers
to mass eigenstates. Thus, from the weak charged current
LCC = − g
2
√
2
3
∑
a=1
(ν̄aLγαℓaL)W +α + h.c., (1.16)
we define the lepton mixing matrix [31] as
V = V ν†V ℓ. (1.17)
In the canonical case, three flavours and three mass eigenstates, the lepton mixing
matrix for Dirac neutrinos can be parametrised by three mixing angles θ12, θ23 and θ13, and
8 Introduction
one CP violating phase δ as
V =
⎛
⎜
⎜
⎝
c12c13 s12c13 s13e
−iδ
−s12c23 − c12s23s13e
iδ c12c23 − s12s23s13e
iδ s23c13
s12s23 − c12c23s13e
iδ −c12s23 − s12c23s13e
iδ c23c13
⎞
⎟
⎟
⎠
, (1.18)
with cij = cos θij, sij = sin θij and 0 ≤ θij ≤ π/2. As for quarks, the magnitude of CP
violation is determined by the rephasing Jarlskog JCP invariant [25]:
JCP = −Im[Vµ3V
∗
e3Ve2V
∗
µ2] =
1
8
cos θ13 sin 2θ12 sin 2θ23 sin 2θ13 sin δ. (1.19)
Neutrino oscillation experiments are only sensitive to mass square differences Δm2
ij =
m2
i − m2
j . From three mass eigenstates, only two independent mass squared differences
could be defined. By convention these are: Δm2
21 and Δm2
3i, with i = 1, 2 depending on the
mass ordering. In summary, neutrino oscillation experiments are sensitive to six parameters.
These are, three mixing angles: solar θ12(∼ 34.5◦), atmospheric θ23(∼ 48◦) and reactor
θ13(∼ 8.49), the CP phase δ(∼ 3π/2) and two mass squared differences Δm2
21(∼ 7.5 × 10−5
eV2) and |Δm2
3i| ∼ (2.5 × 10−3 eV2). However, we do not know what is the sign for the
latter mass squared difference. This enables two possible arrangements for the neutrino
masses:
Normal ordering (NO): m1 < m2 < m3,
Inverted ordering (IO): m3 < m1 < m2,
for which Δm2
31 > 0 in NO and Δm2
32 < 0 for IO.
It is worth to mention that values for δ are not directly measured but inferred from
neutrino oscillation experiment global fits. In addition, as θ23 is quite close to the maximal
mixing value sin2 θ23 ∼ 1/2, precise determination of such mixing angle and the sign of the
corresponding mass squared difference, Δm2
3i is challenging. Currently, both θ23 and δ are
the less precise measured neutrino oscillation parameters. This situation is expected to
improve in forthcoming years when new experiments, as NOvA or Hyper-Kamiokande, start
reporting results. Finally, it is worth to stress that latest global fits on neutrino oscillation
parameters [32, 33] have a preference for a normal ordering of the neutrino masses at 3 σ
confidence level.
9
The neutrino mass scale can be determined from another type of experiments rather than
oscillation experiments. Currently, the strongest limit is obtained from the measurement of
the energy spectrum of electrons near to the end point in 3H β-decay experiments. Such
limit sets [34, 35]
mνe
< 2.05 eV, at 95% CL. (1.20)
However, it is expected that near future experiments improve this bound. The KATRIN
experiment plans to achieve a sensitivity of mνe
∼ 0.20 eV [36]. Also from Cosmology, Planck
collaboration has reported an upper bound on the sum of masses of active neutrinos. This
limit comes from the global fit that combines data from the cosmic microwave background
(CMB) temperature power spectrum anisotropies, polarisation, gravitational lensing effects,
low ℓ CMB polarisation spectrum, supernovae and Baryonic Acoustic Oscillations (BAO)
and assuming three actives neutrinos and ΛCDM as fiducial model. This bound relies
highly on the assumptions made, but gives important information on neutrinos masses [37]:
∑
j
mj < 0.23 eV, at 95% CL. (1.21)
Given the neutrality properties of neutrinos, these could be Majorana fermions. Deter-
mining their nature, whether they are Majorana or Dirac fermions, remains as an open
question in neutrino physics. If LH neutrinos are Majorana fermions νR = νc
L, then two
phases of the lepton mixing matrix, Eq. (1.17), cannot be re-absorbed by the LH fields.
Thus, lepton mixing matrix for Majorana neutrinos has the form [38]
V = V DD, (1.22)
where V D is the lepton mixing matrix for Dirac neutrinos, Eq. (1.18), and
D = diag(1, eiα21/2, eiα31/2), (1.23)
is a matrix of Majorana phases αij.
Regarding nature of neutrinos, there are some processes which can happen only for
Majorana neutrinos, as lepton number L is violated by two units for Majorana neutrinos
while for Dirac neutrinos it is conserved. The neutrinoless double beta decay (0νββ), where
a nucleus undergoes: (A, Z) → (A, Z + 2) + 2e−, is one of such processes where ΔL = 2 [39].
The amplitude for 0νββ in the case of the exchange of three active neutrinos, generated only
10 Introduction
through a (V–A) charged current, long-range contribution, is proportional to the effective
Majorana mass (see e.g. Refs. [40–42]):
|mee| = |m1V
2
e1 + m2V
2
e2 + m3V
2
e3|
= |(m1c
2
12 + m2s
2
12e
iα21)c2
13 + m3s
2
13e
i(α31−δ)|. (1.24)
Finally, the black box theorem [43, 44] states that observation of such process would suffice
to prove the Majorana nature of neutrinos.
Neutrino masses
As mentioned before, neutrinos can be either Majorana or Dirac fermions due to their
neutrality properties. Each case leads to different mass terms for them. In the case of Dirac
neutrino lepton number, L = Le + Lµ + Lτ , is an accidental global symmetry, as their mass
term does not break L, while Majorana neutrino mass term breaks L by two units. With
this on mind, we will examine the mass terms for Dirac and Majorana neutrinos and the
ways to generate such masses.
Dirac neutrino masses
Even though neutrinos are massless in the SM, there is nothing forbidding them of having
Dirac masses. A Dirac mass term can be incorporated to the SM in the same way as for
quarks and charged leptons. For this, it suffices to add three RH neutrino chiral components
νi
R, i = 1, 2, 3, which are SU(3)C ⊗ SU(2)L ⊗ U(1)Y singlet, and use the BEH mechanism.
Thereby, Dirac neutrino fields will be νi
D = νi
L + N i, having four independent degrees of
freedom: νL, νR, νc
L and νc
R. Thus, a Dirac mass term is given by:
− LD = mD(ν̄LνR + ν̄RνL), (1.25)
where the index i has been omitted. Note that if we assign L = 1 to νL,R, thus L = −1 to
ν̄L,R, the mass term in Eq. (1.25) does not violate global lepton number symmetry.
The Dirac mass term, Eq. (1.25), could be generated through the BEH mechanism by
the Yukawa interaction Lagrangian
LY = Γν L̄ H̃ νR + h.c., (1.26)
11
where Γν is a general 3 × 3 complex matrix and yi
ν its eigenvalues. Then, Dirac neutrino
masses will be M i
D = v yi
ν .
Finally, we notice that rather small neutrino masses require quite small Yukawa couplings
compared with Yukawa couplings for the remaining same generation fermions (charged
lepton and quarks), whose Yukawa couplings are approximately of the same order. For
instance, electron neutrino Yukawa coupling is at most of the order O(yνe
)  10−11, for
mνe
< 1 eV. Such Yukawa coupling is at its greater value five orders of magnitude smaller
than the electron Yukawa coupling O(ye) ∼ 10−6; up and down quark Yukawa couplings are
of the order O(yu,d) ∼ 10−5, just an order of magnitude larger than the electron Yukawa,
but at least seven orders of magnitude larger than the neutrino Yukawa. Such disparity
between Yukawa couplings in the same generation is referred to the unnatural value of the
neutrino masses.
Majorana neutrino masses
In the case of Majorana neutrinos, RH components are not independent form the LH
ones. The Majorana condition relates them as: νi = νc i, where the c stands for the charge
conjugation operator5. The LH Majorana fields νi
M have only two independent degrees
of freedom νi
L and νc i
R , and therefore νi
M = νi
L + νc i
R . Then, Majorana mass term for LH
Majorana neutrinos is given by
− LM = MM(ν̄Lνc
R + ν̄c
RνL), (1.27)
where we have omitted the index i.
The bilinear term ν̄Lνc
R in the mass Lagrangian, Eq. (1.27), has weak isospin third
component I3 = 1. Therefore, it cannot be coupled to a SU(2)L doublet as the Higgs
doublet, H, and cannot be generated by EWSB as the Dirac neutrino mass term. However,
this mass term could be generated within the SM via non-renormalisable operators violating
L by two units and whose high energy completion need additional heavy fields to the SM
ones, which we will discuss in more detail later.
5The charge conjugation is defined as:
C : ψc := C(ψ̄γ0)T = Cψ̄T .
12 Introduction
Analogously, the Majorana mass term for RH (or sterile) neutrinos νs = νR + νc
L is
− LS =
Ms
2
(ν̄c
LνR + ν̄Rνc
L). (1.28)
This mass term, Eq. (1.28), is a SU(3)C ⊗ SU(2)L ⊗ U(1)Y singlet and therefore it should be
added to the SM when RH neutrinos are incorporated unless there is some new symmetry
forbidding it. The mass term for sterile neutrinos can be generated by the vev. of a scalar
field singlet under SU(3)C ⊗ SU(2)L ⊗ U(1)Y .
Finally, LH as well as RH Majorana mass terms, Eqs. (1.27) and (1.28), violate L by two
units. As in the SM L is an accidental symmetry, processes with lepton number violation
by two units can occur for Majorana neutrinos. The discovery of one of these processes, the
neutrinoless double beta decay, will imply the Majorana nature of neutrinos as has been
stated by in the black-box theorem [43].
General neutrino mass term
In the case where LH and RH neutrinos are present, neutrinos can have Majorana as well
as Dirac mass terms at the same time. Then, the general mass term takes the form:
− L =
1
2
(
ν̄L ν̄c
L
)
⎛
⎝
MM MD
MT
D Ms
⎞
⎠
⎛
⎝
νc
R
νR
⎞
⎠+ h.c., (1.29)
where the mass matrix defined in this Lagrangian is a complex symmetric matrix.
Neutrino mass models
Now, we will review the ways to naturally generate small masses for Dirac and LH Majorana
neutrinos. In general, one has the freedom to add extra fermions and scalars to the SM, as
the particle content is not fixed by the gauge symmetry. We will focus on generating light
neutrino masses through (non-renormalisable) high dimension effective operators. This can
be done writing down the lowest dimension effective operator generating neutrino masses
using only SM fields and then looking for its possible high energy realisations.
13
Majorana mass models
Using only SM fields, the lowest dimensional operator with lepton number violation is the
dimension-5 Weinberg operator [45]. Such effective operator can be written as
O(5) =
g
Λ
(L̄c ⊗ H ⊗ H ⊗ L). (1.30)
The fact that dimension-5 Weinberg operator violates lepton number by two units has
been exploited in the generation of mass for Majorana neutrinos. The suppression for
LH Majorana neutrino masses can be explained by the combination of a large lepton
number breaking scale Λ and a small coupling g. Higher order operators would lead to
further neutrino mass suppression. For instance, taking a coupling g ≃ O(1) and a scale
Λ ≃ 1013−15 GeV one can generate LH neutrino masses in the range of 1 eV with the
dimension-5 Weinberg operator.
Canonical seesaw mechanisms
The canonical or high energy seesaw mechanisms are specific high energy realisations at
tree level of the dimension-5 Weinberg operator, Eq. (1.30). The idea is that lightness of
LH neutrinos is due to new physics effects at low energy (as the EW scale) of the exchange
of heavy mediators associated with breaking of lepton number. Depending on whether the
mediator is scalar or fermionic and its SU(2)L ⊗ U(1)Y irreps., there are three canonical
ways to complete at three level the dimension-5 Weinberg operator. These are called
type-I [46–52], -II [53, 50, 54, 52, 55] and -III [56–58] seesaw mechanisms. Finally, there
also exists the possibility of tree level completion of the dimension-5 Weinberg operator
with a lower scale than the canonical seesaw mechanisms. This can be done by a further
suppression to the neutrino masses trough additional small couplings in violating lepton
number terms. The inverse and linear seesaw are some examples [59, 60].
Type-I seesaw
The type-I seesaw is the high energy completion of the dimension-5 Weinberg operator
realised by the addition of ns RH neutrinos νR i = Ni, i = {1, . . . , ns}. The Feynman
diagram for the type-I seesaw is shown in Fig. 1.1. The relevant terms in the Lagrangian
are:
L = Y ij
N L̄iH̃Nj + MR N̄ cN + h.c., (1.31)
14 Introduction
νL
〈H〉 〈H〉
N N νL
Fig. 1.1 Feynman diagram of the type-I seesaw. The tree level completion of the dimension-5
Weinberg operator is done by the exchange of RH neutrinos N = νR.
where YN is a 3 × ns general complex Yukawa coupling matrix and MR is the RH
neutrino Majorana mass matrix, which is a ns × ns complex and symmetric matrix. The
addition of the ns RH neutrinos induce a Dirac mass matrix MD = v YN . The MD is a
3 × ns complex matrix. Therefore, in this scenario general neutrino mass matrix MD+M
will be similar to the general case, Eq. (1.29), with a vanishing LH Majorana mass matrix
MM = 0. Then,
MD+M =
⎛
⎝
0 MD
MT
D MR
⎞
⎠ . (1.32)
The mass matrix in Eq. (1.32) can be diagonalised by an unitary matrix, as it is a complex
symmetric (3 + Ns) square matrix. Let W be a unitary matrix, then
M = W †MD+MW, (1.33)
where M is a block diagonal matrix.
One expects that the mass scale Λ ∼ MR to be much greater than MD. Therefore, the
mass matrix in Eq. (1.32) can be approximately block diagonalised as
W †MD+MW ≈
⎛
⎝
Mlight 0
0 Mheavy
⎞
⎠ , (1.34)
with
W =
⎛
⎝
1 − 1
2
M †
D(MRM †
R)−1MD [(MR)−1MD]
†
−(MR)−1MD 1 − 1
2
(MR)−1MDM †
D(M †
R)−1
⎞
⎠ , (1.35)
and the mass sub-matrices Mlight and Mheavy are given by
MT-I
ν = Mlight ≈ −MT
D(MR)−1MD and Mheavy ≈ MR, (1.36)
15
where only terms up to first order in MD(MR)−1 are kept in the expansion.
Finally, it is worth to notice that in order to generate at least two non-zero LH neutrino
masses, as indicated by neutrino oscillation experiments, Mlight has to be a matrix of rank
at least two. Then, the number of RH neutrinos participating in the type-I seesaw has to
be ns ≥ 2.
Type-II seesaw
νL νL
Δ
〈H〉 〈H〉
Fig. 1.2 Feynman diagram for the type-II seesaw mechanism. The tree level completion of
the dimension-5 Weinberg operator is done by the exchange of a SU(2)L triplet complex
scalar Δ.
The type-I seesaw is the only tree level canonical realisation of the dimension-5 Weinberg
operator using SU(2)L singlets as heavy mediators. However, there also exists the possibility
of using SU(2)L triplets as the heavy mediators. In the case in which this iso-triplet is
a scalar Δ carrying hypercharge Y = +1, one has the type-II seesaw mechanism. The
Feynman diagram showing the type-II seesaw mechanism is displayed in Fig. 1.2.
This scalar triplet, Δ, can be represented in the SU(2)L space as the 2 × 2 matrix:
Δ =
⎛
⎝
Δ+/
√
2 Δ++
Δ0 −Δ+/
√
2
⎞
⎠ . (1.37)
Accordingly, the relevant Lagrangian terms are
L = Y ij
Δ L̄c
i σ2Δ Lj + h.c., (1.38)
16 Introduction
with YΔ a 3 × 3 complex and symmetric Yukawa coupling matrix. This Lagrangian leads
to the LH Majorana neutrino mass matrix
MT-II
ν = YΔ
〈
Δ0
〉
. (1.39)
The scalar potential involving the Higgs doublet, H, and the scalar iso-triplet, Δ, is
thus,
V = −m2
hH†H +
λ
4
(H†H)2 + M2
Δ Tr(Δ†Δ) + λ1Tr[Δ†Δ]2 + λ2Tr[(Δ†Δ)2]
+ λ3H
†H Tr[Δ†Δ] + λ4H
†ΔΔ†H + (µHT iσ2Δ†H + h.c.). (1.40)
The minimisation of such scalar potential leads to
〈
Δ0
〉
=
µv2
M2
Δ
, (1.41)
where the iso-triplet vev. has to be 〈Δ0〉 < v, and could be at most of the order of a few
GeV to evade ρ parameter constraints.
Assuming YΔ of the order one, the smallness of the LH neutrino masses, and 〈Δ0〉, could
come from either choosing the scale MΔ large or the coupling µ small. In the first case, if
the seesaw scale is large enough effects of new physics will appear at very high energies
(not in forthcoming experiments). While in the latter, making µ small will reduce the scale
of the seesaw, bringing phenomena associated with the triplet Δ at sight in forthcoming
experiments, e.g. collider signatures associated with the decay of the doubly charged scalar
Δ±±.
Type-III seesaw
Finally, the canonical type-III seesaw mechanism is realised from the three level exchange of
nT SU(2)L triplet RH fermions Σi, with i = {1, · · · , nT } a mass eigenstate index. Fig. 1.3
shows the Feynman diagram for the type-III seesaw. The RH Majorana fermions can be
represented in the same way as the scalar triplet in Eq. (1.37) as:
Σ =
⎛
⎝
Σ0/
√
2 Σ+
Σ− −Σ0/
√
2
⎞
⎠ . (1.42)
17
νL
〈H〉 〈H〉
Σ Σ νL
Fig. 1.3 Feynman diagram for the type-III seesaw mechanism. The tree level completion of
the dimension-5 Weinberg operator is done by the exchange of SU(2)L triplet RH fermions
Σ.
The relevant terms in the Lagrangian are analogous to the type-I seesaw, though field
contractions change accordingly,
L = YΣ L̄cH̃Σ + MΣTr(Σ̄cΣ) + h.c., (1.43)
where YΣ is a 3×nT general complex Yukawa coupling matrix and MΣ is nT ×nT Majorana
mass matrix. The LH Majorana neutrino mass matrix is obtained in a similar fashion as
type-I seesaw. Then, assuming MΣ ≫ v, LH Majorana mass matrix will be
MT-III
ν = −v2 Y T
Σ (MΣ)−1 YΣ. (1.44)
As in type-I seesaw mechanism, at least two RH fermions ΣR are needed to generate two
non-zero masses for LH Majorana neutrinos.
Radiative mass generation
Now we turn our attention to a different class of mass generation mechanisms, where LH
neutrino mass suppression is generated by the combination of loop factors and Yukawa
couplings. This radiative mass generation could be realised at one, two or more loops. In
the following we will review two models: the Zee model [61, 62] and the Cheng–Li–Babu–
Zee [63–65] model. These models were the first works where the dimension-5 Weinberg
operator is completed at one and two loops respectively.
18 Introduction
Li ℓR i φ1 φ2 η+
SU(2)L 2 1 2 2 1
L 1 1 0 0 -2
Table 1.4 Summary of relevant particle content and quantum numbers in the Zee model [61,
62].
The Zee model
The Zee model is one of the simplest setups for radiative neutrino mass generation at
one loop. The particle content in the model enhances the SM adding two scalars: a
singly charged SU(3)C ⊗ SU(2)L ⊗ U(1)Y singlet η+ and an additional SU(2)L iso-doublet
φ2 =
(
φ+
2 φ0
2
)T
. For sake of simplicity in the notation, the SM Higgs will be denoted as φ1.
The two Higgses φ1,2 can develop vevs.
Fig. 1.4 One-loop radiative mass generation for LH Majorana neutrinos in the Zee model [61,
62].
The LH Majorana neutrinos acquire their masses through the Feynman diagram in
Fig. 1.4. One vertex comes from the Yukawa interaction of the LH leptons and the singly
charged scalar, while the other vertex comes from the Yukawa interaction between leptons
and Higgses.
From the particle assignments in Tab. 1.4, the relevant part of the Lagrangian is given
by:
− L = Y φ1,2 L̄ φ1,2 ℓR + fαβ L̄c
α iσ2Lβη+ − µ φ†
1 iσ2 φ∗
2η
+ + h.c., (1.45)
where fαβ is an antisymmetric Yukawa coupling matrix between η+ and LH neutrinos.
In the base where charged leptons are diagonal α, β are flavour indices. The violation in
lepton number, needed to generate Majorana neutrino masses, comes from the last term in
Eq. (1.45).
19
There exists a restricted version of the Zee model, called Zee–Wolfenstein [66], where
only the φ1 scalar iso-doublet couples to leptons. In this model, LH Majorana mass matrix
has zeros in the diagonal, as consequence of the Yukawa matrix fαβ anti-symmetry. After
removing all the unphysical phases, LH neutrino mass matrix is parametrised by three real
parameters as
Mν =
⎛
⎜
⎜
⎝
0 fµe(m
2
µ − m2
e) fτe(m
2
τ − m2
e)
fµe(m
2
µ − m2
e) 0 fτµ(m2
τ − m2
µ)
fτe(m
2
τ − m2
µ) fτµ(m2
τ − m2
µ) 0
⎞
⎟
⎟
⎠
. (1.46)
From Eq. (1.46), the model predicts a pattern for neutrino masses and mixing which are
ruled out by nowadays oscillation parameters.
Even though the Zee–Wolfenstein model is ruled out, there are no conflicts with
experimental oscillation parameter values for the original Zee model. As both Higgses φ1,2
couple to leptons, there are two different Yukawa coupling matrices entering in the right
vertex of Fig. 1.4, then Mν has non-zero diagonal components.
The Zee–Babu model
Fig. 1.5 Two-loops radiative mass generation for LH Majorana neutrinos in the Zee–Babu
model [65, 64].
The Zee–Babu model [65, 64] adds to the SM particle content two SU(3)C ⊗ SU(2)L ⊗ U(1)Y
singlets: a singly charged η+ and a doubly charged k++. The LH Majorana masses are
generated by the two-loop diagram in Fig. 1.5. The relevant part of the Lagrangian is given
by
− L = fαβ L̄c
α iσ2 Lβ η+ + hαβ ℓ̄c
αR ℓβR k++ − µ̃ η−η−k++ + h.c., (1.47)
where fαβ and hαβ are antisymmetric Yukawa coupling matrices. The lepton number
assignment is similar to the Zee model, L(k++) = −2 and L(η+) = −2. The breaking of
lepton number is given explicitly by the µ̃ term in Eq. (1.47).
20 Introduction
The LH neutrino mass is calculated from the diagram, Fig. 1.5, giving:
Mν = f̃ mℓi
h̃∗ mℓj
f̃ Iℓiℓj
, (1.48)
where f̃ = fαβ, h̃ = hαβ, mℓ = diag (me, mµ, mτ ) and Iℓiℓj
is a loop function defined by:
Iℓiℓj
=
∫ d4p
(2π)4
∫ d4q
(2π)4
1
p2 − m2
ℓ1
1
q2 − m2
ℓ2
1
p2 − m2
η
1
q2 − m2
η
1
(p − q)2 − m2
k
. (1.49)
One interesting feature of this model is that, as f̃ is antisymmetric, the determinant of Mν
is zero for three generations. Thus the lightest neutrino mass is zero. This only implies
that at two loops, the lightest neutrino mass is zero, however at higher loop corrections
this mass will be different from zero, but much smaller than the other two.
Dirac neutrino mass models
Turning back to Dirac neutrinos, there are alternative models to generate naturally small
Dirac neutrino masses to the BEH mechanism in the SM extension discussed previously,
Eq. (1.26). The classification of such models is analogous to the Majorana mass cases:
through (non-renormalisable) higher dimensional operators and their corresponding high
energy completions at tree level or involving loops. However, in order to achieve Dirac
neutrino masses, a new conserved symmetry in the model has to be imposed forbidding the
Majorana mass terms for the RH neutrinos, Eq. (1.28). Such symmetry has been realised as
an extra U(1) lepton number symmetry [67, 68] or as the discrete parity Zn (n>2) [69, 70].
In addition, one also has to forbid the usual SM Yukawa coupling, Eq. (1.26), which
has been done by means of a Z2 parity [71], flavour symmetries [72, 73] or even trough an
unconventional U(1)B−L symmetry [74, 68].
In the context of the SM, an effective operator leading to Dirac neutrino masses has the
form:
1
Λ2n
L̄ H̃ νR(H†H)n, n ∈ {0, 1, 2, ...}, (1.50)
as H and L are SU(2)L doublets, the operator only involves odd number of Higgses. The
lowest order operator is the dimension-4 or tree level Dirac mass, Eq. (1.26), while the first
non-renormalisable one is a dimension-6 operator. However, as models for natural Dirac
21
neutrino masses forbid such tree level mass term, then any higher order operator is also
forbidden.
One simple way to generate higher order operators leading to small Dirac neutrino
masses is through non-renormalisable operators involving additional scalar fields. Thus, a
generalised dimension-5 Weinberg operator for Dirac neutrinos could be written as
O(5) =
g
Λ
L̄ ⊗ H ⊗ X ⊗ νR, (1.51)
where X is a scalar field transforming under SU(2)L either as a singlet or a doublet and
zero hypercharge.
We will focus on the high energy completions for the simplest generalised dimension-5
operator, that is, when X = σ is SU(3)C ⊗ SU(2)L ⊗ U(1)Y singlet. Thus, Eq. (1.51) yields
O(5) =
g
Λ
(L̄H̃) σνR, (1.52)
whose tree level completions by heavy mediators can be considered as the Dirac counterpart
of the canonical Majorana seesaws.
νL
〈H〉 〈σ〉
NR NL νR
Fig. 1.6 Feynman diagram for a type-I Dirac neutrino seesaw. The tree level completion
of the generalised dimension-5 operator in Eq. (1.52), is done by the exchange of Dirac
fermions N = NL + NR.
In this context, a type-I Dirac seesaw is the high energy completion of the generalised
dimension-5 operator in Eq. (1.52). This is realised by the tree level exchange of n heavy
fermions, with chiral components N i
L, N i
R, i = {1, · · · , n}, transforming as singlets under
SU(2)L ⊗ U(1)Y . The Feynman diagram for this type-I Dirac seesaw is shown in Fig. 1.6.
The relevant terms in the Lagrangian are
L = Y H L̄ H̃ NR + Y σN̄LνRσ + MN N̄RNL + h.c., (1.53)
22 Introduction
where Y H(σ) are 3 × n (n × 3) general complex Yukawa matrices and MN is a n × n
Dirac mass matrix for the heavy fermions N i
L,R. It is worth to stress that there must be
a symmetry forbidding the Majorana mass terms for NL as well as NR in addition to a
symmetry forbidding the tree level Dirac mass term.
The (3 + n) square mass matrix for the neutrinos and heavy fermions, NL, NR, in the
basis (νL, NL) and (νR, NR)T is given by:
Mν, N =
⎛
⎝
0 vY H
uY σ MN
⎞
⎠ , (1.54)
with 〈H〉 = v and 〈σ〉 = u.
νL νR
φ
〈H〉 〈σ〉
Fig. 1.7 Feynman diagram for a type-II Dirac neutrino seesaw. The tree level completion of
the generalised dimension-5 operator in Eq. (1.52) is done by the exchange of φ a complex
scalar doublet of SU(2)L.
Finally, the Dirac neutrino mass matrix is found in the limit where MN ≫ v, u in a
similar fashion as the expansion in the type-I Majorana seesaw mechanism. This leads to,
MT-I
ν = u v Y σ M−1
N Y H , (1.55)
and the smallness of neutrino masses in due to a large scale MN and small 〈σ〉.
The type-II Dirac seesaw realisation is shown in Fig. 1.7. In this case, the high energy
completion of the generalised dimension-5 operator, Eq. (1.52), is through the tree level
exchange of φ a heavy scalar doublet of SU(2)L with Y = −1/2. Thus, the relevant part of
23
the Lagrangian is
− L = Y ν L̄ φνR + h.c., (1.56)
and the relevant part of the scalar potential include
V (H, φ, σ) ⊃ κ H̃ φ σ, (1.57)
where 〈φ〉 = vφ, Mφ is the scalar field φ mass and κ the coupling in the scalar potential
term involving the fields H, σ and χ.
From Eqs. (1.56) and (1.57), the Dirac neutrino mass matrix is given by:
MT-II
ν = κ
vvφ
M2
φ
Y ν , (1.58)
νL
〈σ〉 〈H〉
E0
R E0
L
νR
Fig. 1.8 Feynman diagram for a type-III Dirac neutrino seesaw. The tree level completion
of the generalised dimension-5 operator in Eq. (1.52) is done by the exchange of vector-like
fermions E0.
Finally, the type-III Dirac seesaw realisation is shown in Fig. 1.8. In this case, the
high energy completion of the generalised dimension-5 operator, Eq. (1.52), is through
the tree level exchange of heavy vector-like fermions, with chiral components EL and
ER, transforming as SU(2)L doublets. In contrast with type-I and type-II Dirac seesaws,
mentioned before, a complete high energy theory using type-III seesaw has not been
explicitly developed. Finally, it is worth to mention that radiative mass generation for
Dirac neutrinos has been shown to exist in several realisations, as for example in [70, 75].
Dark Matter
In the following section, we will review the main aspects of the dark matter (DM), which
together with neutrino masses are one of the compelling evidence for BSM physics.
24 Introduction
Evidence of dark matter
The first indication of DM presence comes from the dynamical study in our galaxy. In
1922, the astronomer James Jeans [76], re-analysed the vertical motion of stars near the
galactic plane [77]. He showed that spatial density of visible stars cannot explain their
vertical motion, it was needed to have two non-visible (dark) stars to each visible star.
The second indication of DM was made by Fritz Zwicky. In 1933, he measured the
galaxy radial velocities of the Coma galaxy cluster. Zwicky [78] found that the galaxy
orbital velocities were larger by a factor of ten than what was expected from the total
mass sum of galaxies in the Cluster. Then, he concluded that the galaxy cluster should
contain larger amounts of non-visible (dark) matter. Later in the 70’s, Vera Rubin and
others [79, 80] analysed the rotation curves of several galaxies, showing that the virial mass
and the observed mass, inferred from spectroscopical observations, did not match in every
single observation suggesting the evidence of dark matter at galactic scales.
Another cosmological evidence of DM is related with the spatial distribution of galaxies
in the Universe. It was shown in the 70’s, that the galaxy spatial distribution is not uniform,
as was assumed earlier. Posterior observation of the redshifts of visible galaxies [81]
showed that such galaxies are clustered in a filamentary cosmic web and the space between
filaments is practically devoid of galaxies. These voids have diameters of the order of ten
Megaparsecs [82]. Nowadays, it is known that structure formation in the Universe is due to
gravitational clustering [83], originated by small initial fluctuations in matter density. This
process is very slow, so in order to achieve the observed large scale structure, initial matter
density amplitude has to be at least 1/1000th of the matter density at the recombination
epoch, when the CMB was originated. Finally, when these matter fluctuations were inferred
from CMB measurements, they showed to be two orders of magnitude smaller than expected
from only baryonic matter density evolution. Therefore, a dark matter component is needed
to explain the structure formation.
The next evidence of DM came from Cosmology in the 80’s. From the expansion rate
of the Universe, it is possible to calculate the critical energy density of the Universe. The
mean energy density has to be close to the critical one, as it is known that the Universe
is flat [37]. This means that the energy density can be estimated from the masses of the
galaxies and the gas between them. Such estimate shows that the baryonic matter (mostly
from stars in galaxies and the interstellar and intergalactic gas) only accounts for a small
percentage of the critical density, which however is consistent with the bounds from Big
Bang Nucleosynthesis of light elements: 1H, D, 3He, 4He, and 7Li [84]. Finally, in the
25
middle 80’s the DM existence was confirmed by other independent sources: the weak lensing
mass estimates [85, 86] and the X-ray studies of galaxy clusters [87].
Currently, the most precise estimation for the dark matter energy density as well as
baryonic energy density, is obtained combining the CMB temperature anisotropies and the
spatial distribution of galaxies [37], which has found that the DM abundance6 is
Ωch
2 = 0.11933 ± 0.00091 at 68% CL.,
and
Ωbh2 = 0.02242 ± 0.00014 at 68% CL.,
where h is the Hubble constant in units of 100 Km (s Mpc)−1.
Production mechanisms of dark matter
The first and more studied dark matter production mechanism is the thermal production.
In this setup, the DM particles are in chemical and thermal equilibrium with ordinary
matter in the early Universe, until the DM annihilation rate falls below the expansion rate
of the Universe [88, 89] and the DM particles "freeze-out". Among the physical processes
that could modify this simple thermal production mechanism, one can mention the co-
annihilation of DM with degenerate particles. There are some alternatives to the thermal
production mechanism as gravitational production, through the decaying of heavy particles,
and scenarios with a non-standard expansion rate of the Universe.
Now, we will explain the thermal freeze-out in detail. Let be X a DM stable particle
interacting with the SM particles Y through some process XX̄ ↔ Y Ȳ or alternatively
XX ↔ Y Ȳ if X is its own antiparticle. In the early Universe, when the temperature was
much larger than the X mass mX, the annihilation and creation processes were equally
efficient, leading to the DM partiwcles X to be in the same amount than the SM particles.
However, when the temperature drops below mX, the XX̄ creation process is exponentially
suppressed, while the XX̄(XX) (co-)annihilation process remains unaffected.
In thermal equilibrium at temperature T , the number density of the non-relativistic
species X (and X̄) is given by
nX, eq = gX
(
mXT
2π
)
e−mX/T , (1.59)
6Where ΩX = ρx/ρcrit, being ρcrit the critical density, i.e. Ωtot = 1, which correspond to a flat Universe.
26 Introduction
with gX the number of internal degrees of freedom of X. If the DM particle X remains
indefinitely in thermal equilibrium, its number density would further suppress as the
Universe cools due to the expansion, diluting them quickly. The annihilation of DM
particles can be countered by the Hubble expansion rate H.
As the expansion and corresponding dilution of the X dominates over the annihilation rate,
the particle number density becomes small enough, then the X interactions cease, surviving
until the present epoch.
The expansion and (co-)annihilation effects are described by the Boltzmann equation:
dnX
dt
+ 3HnX = −〈σXX̄v〉
(
n2
X − n2
X, eq
)
, (1.60)
where nX is the number density of X, H := ȧ/a = (8π3ρ/3MPl)
1/2 is the expansion rate
of the Universe, and 〈σXX̄v〉 is the thermally average annihilation cross section times its
relative velocity.
One can identify two limits in the Boltzmann equation. At high temperatures (T ≫ mX),
the X density is given by its equilibrium value nX, eq. At low temperatures (T ≪ mX), the
equilibrium density is very small, allowing to the terms 3HnX and 〈σXX̄v〉n2
X further reduce
the number density of X. For small enough nX values, the annihilation term becomes
negligible with respect to the Hubble expansion dilution. When this happens, the comoving
number density of X does not change, i.e. it freezes–out.
The freeze out temperature is found numerically solving the Boltzmann equation. Let
x := mx/T and TFO be the freeze out temperature, this is approximately given by
xFO ≈ log
⎡
⎣c(c + 2)
√
45
8
gX
2π3
mXMPl(a + 6b/xFO)
g
1/2
∗ x
1/2
FO
⎤
⎦ , (1.61)
where c ∼ 0.5 is a quantity determined numerically, g∗ is the number of degrees of freedom
(in the SM g∗ ∼ 120 at T ∼ 1 TeV and g∗ ∼ 65 at T ∼ 1 GeV), and a and b are non-
relativistic expansion terms of the annihilation cross section, 〈σXX̄v〉 ≃ a + b〈v2〉 + O(v4).
The DM relic density in the Universe today is approximately
ΩXh2 ≈ 1.04 × 109 GeV −1
MPl
xFO
g∗(a + 3b/xFO)
. (1.62)
Finally, if the DM particle X has a mass scale between GeV and TeV and annihilation
cross section at the scale of the generic weak interaction, the freeze out occurs at xFO ≈
27
20 − 30, leading to a relic density of
ΩXh2 ≈ 0.1
(
xFO
20
)(
g∗
80
)
(
a + 3b/xFO
3 × 10−26cm3/s
)−1
. (1.63)
In other words, if the mass scale of a thermally produced particle with a relic density similar
to the relic density of DM is in the GeV–TeV range, this has to have a thermally average
cross section of the order of 3 × 10−26 cm3/s. This is value is similar to the value of the
cross section for a generic weak interaction, which has been referred as the "WIMP miracle".
Dark matter candidate profile
The dark matter cannot be made of SM particles, therefore it is necessary to postulate new
particles coming from beyond Standard Model theories (BSM) as dark matter candidates.
In fact, a possible connection between DM and BSM physics has proliferated the creation
of DM candidates, which currently are under search in accelerators and direct and indirect
detection experiments.
A particle can be considered as a good DM candidate if it fulfils the following condi-
tions [90]:
• It has the right relic density,
• It is cold,
• It is neutral under colour and electric charges7,
• It is consistent with the BBN limits,
• It is consistent with the stellar evolution,
• It is compatible with the direct and indirect detection searches limits.
Dark matter candidates
The first obvious DM candidate were the SM neutrinos. However, this scenario leads
to troubles with the large scale structure of the universe, as active neutrinos were ultra-
relativistic particles (because of their small masses) at the time of formation of galaxies,
7Rigorously, there exist some electric charge and colour bounds for the DM, though they are model
dependent.
28 Introduction
which leads to few amounts of small scale structure formation, i.e. galaxies. This sets an
upper bound to the free-streaming contribution of neutrinos [91] in structure formation,
which is transformed to a bound on neutrino energy density
Ωνh2 ≤ 0.0062, at 95 % C.L. (1.64)
In the case of non–relativistic DM particles at the beginning of structure formation,
or cold dark matter (CDM), unlike ultra-relativistic particles, or hot dark matter (HDM),
are consistent with structure formation. N-body numerical simulations of the structure
evolution in the Universe show that the filamentary superclusters formation and voids are
consistent with a CDM dominated Universe [92].
The most popular candidates to DM include axions (and axion–like particles), sterile
neutrinos and WIMPS (motivated in BSM physics). Axions were postulated to solve the
strong CP problem in the SM [93, 94]. These are the pseudo Nambu–Goldstone bosons
associated with the spontaneous symmetry breaking of a new global U(1) symmetry, the
Peccei–Quinn symmetry, at the fa scale, where the θQCD is replaced by a dynamic field
that goes to zero at the potential minimum. Axions are CDM candidates, as they can
be produced non–thermally at temperatures larger than the QCD phase. There are two
invisible axion models, the Kim–Shifman–Vainshtein–Zakharov (KSVZ) model [95, 96] and
the Dine–Fischler–Sredniki–Zhitnitky (DFSZ) model [97, 98]. The best Peccei–Quinn (PQ)
scale and axion masses bounds for these models come from astrophysical reasoning. For
instance, bounds in the axion flux in stars which leaves the stellar evolution unaffected.
The strongest bound comes from the (SN) 1987A supernova observations, which sets a
bound to PQ scale fa  4 × 108 GeV.
Sterile neutrinos with masses above the keV could solve the cusp core problem present
in CDM models. This problem comes from structure formation simulations, where in
places with large DM density such value tends to quickly increase, which seems to be in
contradiction with the observation of DM density in galaxy cores. If sterile neutrinos are
non-thermally produced by the mixing with SM neutrinos, this could eventually decay into
an active neutrino and a photon. Though this leads to sterile neutrinos to be not stable.
The WIMPs are particles with masses between 10 GeV and few TeV and annihilation
cross section approximately equal to the weak interactions scale. Its relic density can be
calculated by the standard thermal freeze-out. One WIMP candidate could be a heavy
Dirac neutrino, though a SU(2)L doublet would have too few relic density if its masses
29
exceed the LEP bound mν > MZ/2. The annihilation cross section can be suppressed and
then increase the relic density by, for instance, mixing this heavy SU(2)L doublet with
sterile Majorana neutrinos. However, it has to be required some mechanism that forbids
the decaying of such heavy neutrino.
Another WIMP DM candidate is the lightest super-symmetrical particle (LSP) [99].
This could be either the sneutrino or the neutralino. The negative result in WIMPs searches,
ruled out the sneutrinos as primary components in our galaxy DM halo, leading to the
neutralino as a much more viable candidate. The neutralino relic density can be produced
thermally in right amounts. The neutralino is a mixing of a bino a photino and a Higgsino,
it could be mainly one of them if its mass mχ is below 150 GeV or mχ is close to the mass
of some other sfermion (and the relic density is reduced by the co-annihilations with the
sfermion) or if 2mχ is close to the CP-odd Higgs boson mass present in supersymmetric
models.
There are several non-SUSY SM extensions which provide viable WIMP candidates,
as the lightest T-odd particle (LTP) in little Higgs models [100], where T is a conserved
parity, or techni-baryons in a scenario with additional strongly interacting gauge group
(technicolour or similar). There are also models where the DM particles interact weakly
with ordinary matter but have strong interactions with a dark sector. This kind of models
was motivated by the excess in positrons and electron fluxes from cosmic rays measured
by satellites as PAMELA, ATIC and Fermi. However, such excesses could be due to
oversimplification of the estimates, though if they are real, are too large to be accounted by
WIMPs, they can be explained by astrophysical sources [101].
Axions (and axion-like particles), sterile neutrinos and WIMPS are detectable in principle
with the current and near future technology. However, there are some other DM candidates
whose detection is impossible unless they decay. There exists a lower bound on the mean
time of 1025 to 1026 seconds to a 100 GeV decaying DM particle, which includes the gravitino
and axino [100].

Chapter 2
A4 flavour symmetric models for
Majorana neutrinos and dark matter
In this chapter, we present a scenario where the stability of the DM arises from the family
symmetry explaining neutrino masses and oscillation pattern. A non-Abelian discrete group
is chosen as the family symmetry group, as its breaking could yield to a remanent parity
symmetry Z2. Such parity has been assumed in the literature as the simplest mechanism
behind the stability of the DM. This is the main idea behind the discrete dark matter model
(DDM) [102], a minimal extension of the SM and the basis for the work presented in this
chapter. In particular, we will discuss two realisations of such a scenario.
2.1 Preliminaries
The model studied in [102] assumes A4 as flavour group, being A4 the group of the even
permutations of four elements. The A4 group has order 12 and four irrreps., which are
three one-dimensional 1, 1′ and 1′′ and one three-dimensional 3. A review on the A4 group
properties is given in the Appendix A. The breaking of this flavour symmetry is induced by
means of the EWSB.
The three LH Majorana neutrinos νiL, i = {1, 2, 3}, get their masses through the type-I
seesaw mechanism, for which four RH Majorana neutrinos Nj, j = {1, . . . , 4}, are introduced
in this model. Three of these RH neutrinos transform as a triplet NT ∼ 3 of A4 and the
remaining neutrino N4 is assigned as the (trivial) singlet 1. The scalar sector also has to be
extended because the flavour symmetry requires a way to break it spontaneously without
32 A4 flavour symmetric models for Majorana neutrinos and dark matter
spoiling the EWSB. The scalar H doublet of SU(2)L is assigned to the trivial singlet, 1, of
A4, and is responsible for giving masses to quarks and charged leptons. Three additional
scalar SU(2)L iso-doublets labelled as ηi, i = {1, 2, 3}, are also added forming up a triplet
of A4, η ∼ 3. The SM Higgs is a combination mostly of H and the CP-even component of
η1.
The breaking of the flavour symmetry is driven by the triplet η. The A4 is spontaneously
broken into the sub-group Z2 when this triplet acquires a vev. with alignment 〈η〉 ∼ (1, 0, 0).
Such alignment is consistent with the minimum conditions of the scalar potential and leads
to the breaking of A4 into Z2, the latter will be explained in more detail later. After EWSB,
the residual Z2 charges two components of the A4 triplets. Two scalar iso-doublets η2,3 in
the triplet η as well as two RH Majorana neutrinos N2,3 in the triplet NT will be Z2-odd.
Regarding DM phenomenology, residual Z2 provides the mechanism of stability for the
DM. Then, the DM candidate will be lightest neutral Z2-odd particle. The only neutral
Z2-odd particles are the two CP-even and CP-odd neutral components of the two inert
Higgses and two RH Majorana neutrinos. The inert Higgses masses are around the EW scale
and the RH neutrino masses are expected to be at a larger scale (the seesaw scale). Then,
MNi
> Mηi
and therefore the DM candidate are the lightest neutral scalar combination
arising from the neutral components of the inert Higgses, η2,3.
This first realisation of the DDM with A4 has viable and interesting DM phenomenology,
which was studied in [103]. Unfortunately, it also has strong neutrino phenomenological
predictions as a vanishing reactor mixing angle θ13 = 0, an inverted mass ordering (IO)
with a massless lightest LH neutrino m3 = 01 and no CP violation for the lepton sector.
The vanishing reactor mixing angle and the CP conservation for leptons are in contradiction
with measurements of the reactor mixing angle [104–106] and current hints on the CP
violating phase, δ [33].
In a follow-up work [107], a fifth RH Majorana neutrino N5 is added to the first A4
DDM setup. The fourth RH neutrino changes its flavour assignment from 1 to 1′ while
the fifth RH neutrino is assigned as the singlet 1′′. This new model gives a non-vanishing
reactor mixing angle, besides it predicts a normal ordering (NO) with a non-zero lightest LH
neutrino mass2 and a lower bound for the neutrinoless double beta decay (0νββ) effective
mass, Eq. (1.24), large enough to be in the range of sensitivity of near-future experiments.
1This is a consequence of only two RH neutrinos participate in the seesaw. The LH neutrino mass
matrix, Eq. (1.36), has rank two and therefore only two non-zero neutrino masses can be generated.
2In this model, the addition of the fifth neutrino and the flavour assignments cause that three RH
neutrinos to participate in the seesaw and therefore the LH neutrino mass matrix has rank three.
2.2 The models 33
Nevertheless, even at its maximum predicted value for the reactor mixing angle, θ13, this is
still too small and ruled out by nowadays more precise measurements [104–106].
Other realisations of the DDM with non-Abelian discrete groups as family symmetries
rather than A4 have been also implemented. For instance, in [108] the family group used is
the dihedral group of order four D4 and in [109] the group Δ(54) is chosen. In the former
model, the vanishing reactor mixing angle issue is not solved, while in the latter besides of
addressing this problem the model has further interesting phenomenological features.
2.2 The models
The models studied in [110], hereafter referred to A and B, are extensions of the A4 realisation
of the DDM made in [102]. The relevant particle content and quantum numbers of the models
are shown in Tab. 2.1 for model A and in Tab. 2.2 for model B. The particle content is similar
to the one presented in [107], with some changes in assignments of the flavour symmetry
for the RH neutrinos and the addition of complex scalar SU(3)C ⊗ SU(2)L ⊗ U(1)Y singlets,
also known as flavon fields.
In total there are five RH Majorana neutrinos Ni, i = {1, . . . , 5}, three of them arranged
in a triplet of the family symmetry 3 ∼ NT = (N1, N2, N3)
T , the fourth N4 is assigned as
the singlet 1 and the remaining RH neutrino is assigned as the singlet 1’ or 1” on the model
A or B respectively. The scalar sector contains in addition to SM Higgs doublet, H, which
is flavour blind (1), three extra scalar SU(2)L iso-doublets as a triplet η = (η1, η2, η3)
T
and three flavons also as a triplet φ = (φ1, φ2, φ3)
T of A4. The iso-doublet LH leptons,
Li = (νiL, ℓiL)T , i = {e, µ, τ}, transform under the family symmetry as the irreps. 1, 1’
and 1” and the corresponding RH iso-singlets charged leptons, ℓc
i , as the irreps. 1, 1” and
1’ respectively. Under these assignments of the flavour symmetry, the charged leptons
only couple to the SM-like Higgs doublet, H, and are automatically diagonal by A4. Then,
charged leptons do not contribute to the lepton mixing matrix. Regarding quark sector,
these are assumed to be flavour blind (1) and their masses and mixing could possible arise
from an extra family symmetry, which is not considered in the models. Finally, the LH
neutrino masses arise via type-I seesaw mechanism by the contributions of H and η in the
Dirac mass terms and φ on the RH Majorana masses.
When the φ fields acquire vev. at some scale, for instance around the seesaw scale, they
trigger the spontaneous breaking of the family symmetry into a residual Z2. Unlike the
previous realisations of A4 DDM models, where the flavour symmetry breaking were at EW
34 A4 flavour symmetric models for Majorana neutrinos and dark matter
scale, in the present models the scale changes at the scale set by the flavons, which is larger
than the EW scale. In order to drive the breaking of A4 into the residual Z2, the flavon
triplet φ have to pick up a vev. alignment 〈φ〉 which has to be invariant under Z2 and also
be consistent with the global minimum conditions of the scalar potential. This A4 invariant
scalar potential is the same in both models and is given by
VA4
= VH + Vη + Vφ + VφH + Vφη + VφHη, (2.1)
with
VA4
= µ2
1
(
η†
1η1
)
+ µ2
2
(
h†h
)
+ µ2
3
(
h†η1 + η†
1h
)
+ µ2
4
(
η†
2η2 + η†
3η3
)
+ µ2
5
(
η†
2η3 + η†
3η2
)
+ λ1
(
(η†
1η1)
2 + (η†
2η2)
2 + (η†
3η3)
2
)
+ λ2
(
h†h
)2
+ λ3
(
η†
1η1 η†
2η2 + η†
1η1 η†
3η3 + η†
2η2 η†
3η3
)
+ λ4
(
h†h
) (
η†
1η1 + η†
2η2 + η†
3η3
)
+ λ5
(
η†
2η1 η†
1η2 + η†
3η1 η†
1η3 + η†
3η2 η†
2η3
)
+ λ6
(
η†
1h h†η1 + η†
2h h†η2 + η†
3h h†η3
)
+
[
λ7
(
(η†
3η1)
2 + (η†
1η2)
2 + (η†
2η3)
2
)
+ λ8
(
(η†
1h)2 + (η†
2h)2 + (η†
3h)2
)
+ λ9
(
η†
1h η†
2η3 + η†
2h η†
3η1 + η†
3h η†
1η2
)
+ λ10
(
η†
3η2 η†
1h + η†
1η3 η†
2h + η†
2η1 η†
3h
)
+ h.c.
]
,
and the couplings λ1,...,6 are real and the remaining couplings λ7,...,10 could be complex. For
simplicity, we assume the case of a CP conserving potential, i.e. all the couplings λ are real.
The minimum conditions of the potential, Eq. (2.1), are
∂VA4
∂vi
= 0,
where vi stands for the vevs. of H, ηi and φi, which lead to the vevs. alignment conditions
〈
H0
〉
= vh = 0,
〈
η0
1
〉
= vη = 0,
〈
η0
2,3
〉
= 0, 〈φ1〉 = vφ = 0, 〈φ2,3〉 = 0, (2.2)
where all the vi are real.
The vev. for the A4 triplet 〈η〉 has the alignment vZ2
= (1, 0, 0)T which is Z2 invariant,
and therefore spontaneously breaks A4 into a remanent Z2. In the three dimensional basis
2.2 The models 35
where the Z2 generator S is real and diagonal,
S =
⎛
⎜
⎜
⎝
1 0 0
0 −1 0
0 0 −1
⎞
⎟
⎟
⎠
. (2.3)
Then one can see that vZ2
is manifestly invariant under the Z2 generator S:
S vZ2
= vZ2
. (2.4)
Let a = (a1, a2, a3)T be a generic A4 triplet, then the S generator (a Z2 transformation)
acts over a as:
S a =
⎛
⎜
⎜
⎝
1 0 0
0 −1 0
0 0 −1
⎞
⎟
⎟
⎠
⎛
⎜
⎜
⎝
a1
a2
a3
⎞
⎟
⎟
⎠
=
⎛
⎜
⎜
⎝
a1
−a2
−a3
⎞
⎟
⎟
⎠
. (2.5)
This defines the remanent Z2 symmetry over the triplets NT , η and φ as:
N1 → N1, η1 → η1, φ1 → φ1,
N2 → −N2, η2 → −η2, φ2 → −φ2, (2.6)
N3 → −N3, η3 → −η3, φ3 → −φ3.
Therefore, two RH neutrinos, two Higgs doublets and two flavons will be Z2-odd after the
flavour symmetry breaking. As the remaining Z2 parity is chosen to be conserved, after
EWSB only the Z2-even component of the triplet η could acquire a vev., and the remaining
components will be inert Higgses.
After EWSB and minimisation of the potential, Eq. (2.1), the scalar fields can be written
as:
H =
⎛
⎝
H+
1√
2
(vh + h0 + i A0)
⎞
⎠ , η1 =
⎛
⎝
H+
1
1√
2
(vη + h1 + i A1)
⎞
⎠ ,
η2 =
⎛
⎝
H+
2
1√
2
(h2 + i A2)
⎞
⎠ , η3 =
⎛
⎝
H+
3
1√
2
(h3 + i A3)
⎞
⎠ , (2.7)
φ1 =
1√
2
(vφ + φR1
+ i φI1
) , φ2,3 =
1√
2
(φR2,3
+ i φI2,3
) .
36 A4 flavour symmetric models for Majorana neutrinos and dark matter
The physical spectrum of the scalars is: three charged scalars (H1,2,3), seven CP-even
scalars (h0,1,2,3, φ1,2,3) and seven CP-odd scalars (A1,2,3, φ1,2,3). Integrating out the heavy
scalars φi, whose masses are assumed to be of the order of the seesaw scale, at energies
below the seesaw scale the number of CP-even and CP-odd scalars reduces to four, and the
potential will be Z2-symmetric rather than A4-symmetric.
2.3 Neutrino phenomenology
In this section, the neutrino phenomenology from the models is presented. The models
have similar features, as they arise from similar flavour symmetry assignments, though
predicting different LH neutrino mass matrices textures.
Model A
Le Lµ Lτ ℓe ℓµ ℓτ NT N4 N5 H η φ
SU(2)L 2 2 2 1 1 1 1 1 1 2 2 1
A4 1 1′′ 1′ 1 1′′ 1′ 3 1 1′ 1 3 3
Table 2.1 Summary of the relevant particle content and quantum numbers for model A.
From the particle content and assignments under the flavour group for model A, Table 2.1,
the lepton part of the Yukawa Lagrangian is given by
L
(A)
Y = yeL̄eℓeH + yµL̄µℓµH + yτ L̄τ ℓτ H
+ yν
1 L̄e[NT η̃]1 + yν
2 L̄µ[NT η̃]1′′ + yν
3 L̄τ [NT η̃]1′ + yν
4 L̄e N4 H̃ + yν
5 L̄τ N5 H̃ (2.8)
+ M1 N̄ c
T NT + M2 N̄ c
4N4 + yN
1 [N̄ c
T φ]3NT + yN
2 [N̄ c
T φ]1N4 + yN
3 [N̄ c
T φ]1′′N5 + h.c.,
where [a b]j, stands for the product of the two triplet irreps. a and b contracted into the j
irrep. of A4, and the contributions of the form [a b]3 account for the symmetric part of the
two ways two triplets can be contracted, namely [a b]31
and [a b]32
.
As mentioned previously, the iso-doublet H is flavour blind and responsible for giving
mass to the charged leptons (as well as quarks), which can be calculated straightforward
from Eq. (2.8) giving
Ml =
vh√
2
diag ( ye, yµ, yτ ) , (2.9)
2.3 Neutrino phenomenology 37
where 〈H〉 = vh/
√
2. Then, the lepton mixing matrix V = V ν†V ℓ, Eq. (1.17), comes from
the neutrino part as V ℓ = 1.
The three LH neutrinos obtain their masses through the type I-seesaw. The Dirac
neutrino mass matrix M
(A)
D arises from the contribution of H and η, while the flavon fields
φ contribute to the RH neutrino Majorana mass matrix MR. From Eqs. (2.8), the Dirac
neutrino mass matrix in the basis (ν̄e, ν̄µ, ν̄τ )T , (N1, N2, N3, N4, N5) is given by
M
(A)
D =
⎛
⎜
⎜
⎝
yν
1η0
1 yν
1η0
2 yν
1η0
3 yν
4H0 0
yν
2η0
1 ω∗yν
2η0
2 ωyν
2η0
3 0 0
yν
3η0
1 ωyν
3η0
2 ω∗yν
3η0
3 0 yν
5H0
⎞
⎟
⎟
⎠
, (2.10)
where H0 and η0
i , i = {1, 2, 3} the neutral component of the iso-doublets H and ηi. After
EWSB, Eq. (2.2) leads to
M
(A)
D =
1√
2
⎛
⎜
⎜
⎝
yν
1vη 0 0 yν
4vh 0
yν
2vη 0 0 0 0
yν
3vη 0 0 0 yν
5vh
⎞
⎟
⎟
⎠
. (2.11)
The RH Majorana neutrino mass matrix is calculated from Eq. (2.8) in the basis
(N̄ c
1 , N̄ c
2 , N̄ c
3 , N̄ c
4 , N̄ c
5)T , (N1, N2, N3, N4, N5) as
MR =
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
M1 yN
1 φ3 yN
1 φ2 yN
2 φ1 yN
3 φ1
yN
1 φ3 M1 yN
1 φ1 yN
2 φ2 ω∗yN
3 φ2
yN
1 φ2 yN
1 φ1 M1 yN
2 φ3 ωyN
3 φ3
yN
2 φ1 yN
2 φ2 yN
2 φ3 M2 0
yN
3 φ1 ω∗yN
3 φ2 ωyN
3 φ3 0 0
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
. (2.12)
After EWSB, Eq. (2.2) leads to
MR =
1√
2
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
M1 0 0 yN
2 vφ yN
3 vφ
0 M1 yN
1 vφ 0 0
0 yN
1 vφ M1 0 0
yN
2 vφ 0 0 M2 0
yN
3 vφ 0 0 0 0
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
. (2.13)
38 A4 flavour symmetric models for Majorana neutrinos and dark matter
We can notice from mass matrix in Eq. (2.11) that the RH neutrinos N2 and N3 do not
have any Dirac mass terms, because the breaking of A4 charge them as Z2-odd. Thus, N2
and N3 do not participate in the seesaw, i.e. from the five RH neutrinos only three, N1,4,5,
are active in the seesaw and the remaining two belong to the dark sector. This dark sector
is shown within the lines in the mass matrices in Eqs. (2.11) and (2.13).
From the mass matrices Eqs. (2.11) and (2.13), the LH neutrinos get their Majorana
masses through the type-I seesaw relation, Eq.(1.36), taking the form
M (A)
ν ≃ −
(
M
(A)
D3×5
)T (
MR5×5
)−1
M
(A)
D3×5
=
⎛
⎜
⎜
⎝
a 0 b
0 0 c
b c d
⎞
⎟
⎟
⎠
, (2.14)
where the parameters a, b, c and d are defined as
a =
(yν
4 vh)2
M2
, b =
yν
1 yν
5 vηvh
yN
3
vφ
− yN
2 yν
4 yν
5 v2
h
yN
3
M2
,
c =
yν
2 yν
5 vηvh
yN
3
vφ
, d =
(yN
2 yν
5 vh)2
(yN
3
)2M2
− (yν
5 vh)2M1
(yN
3
vφ)2 + 2
yν
3 yν
5 vηvh
yN
3
vφ
.
(2.15)
This LH Majorana neutrino mass matrix M (A)
ν , Eq. (2.14), has the two-zero B3 texture,
according to the nomenclature in [111]. This mass matrix texture is phenomenologically
favourable, being consistent with both neutrino mass orderings (IO and NO), non-vanishing
lightest neutrino mass and allows the current experimental value for the reactor mixing
angle, as have been proved by several works [111–120].
Model B
In this model the particle content is shown in Tab. 2.2, where the difference with respect to
the previous model is that the fifth RH neutrino, N5, changes its assignment from 1′ → 1′′.
The lepton sector of Yukawa Lagrangian is given by:
Le Lµ Lτ ℓe ℓµ ℓτ NT N4 N5 H η φ
SU(2)L 2 2 2 1 1 1 1 1 1 2 2 1
A4 1 1′′ 1′ 1 1′′ 1′ 3 1 1′′ 1 3 3
Table 2.2 Summary of the relevant particle content and quantum numbers for model B.
2.3 Neutrino phenomenology 39
L
(B)
Y = ye L̄eℓeH + yµ L̄µℓµH + yτ L̄τ ℓτ H
+ yν
1 L̄e[NT η̃]1 + yν
2 L̄µ[NT η̃]1′′ + yν
3 L̄τ [NT η̃]1′ + yν
4 L̄e N4 H̃ + yν
5 L̄µ N5 H̃ (2.16)
+ M1 N̄ c
T NT + M2 N̄ c
4N4 + yN
1 [N̄ c
T φ]3NT + yN
2 [N̄ c
T φ]1N4 + yN
3 [N̄ c
T φ]1′N5 + h.c. .
The mass matrix for the charged leptons is also diagonal as their assignments have not
changed from the model A. The Dirac neutrino mass matrix, from the Lagrangian Eq. (2.16),
in the basis (ν̄e, ν̄µ, ν̄τ )T , (N1, N2, N3, N4, N5) is given by:
M
(B)
D =
⎛
⎜
⎜
⎝
yν
1η0
1 yν
1η0
2 yν
1η0
3 yν
4H0 0
yν
2η0
1 ω∗yν
2η0
2 ωyν
2η0
3 0 yν
5H0
yν
3η0
1 ωyν
3η0
2 ω∗yν
3η0
3 0 0
⎞
⎟
⎟
⎠
. (2.17)
After EWSB, Eq. (2.2) leads to
M
(B)
D =
⎛
⎜
⎜
⎝
yν
1vη 0 0 yν
4vh 0
yν
2vη 0 0 0 yν
5vh
yν
3vη 0 0 0 0
⎞
⎟
⎟
⎠
. (2.18)
The Majorana neutrino mass matrix MR is the same as in Eq. (2.13). Analogously, the LH
neutrino Majorana mass matrix after the EWSB is given by the type-I seesaw relation:
M (B)
ν ≃ −
(
M
(B)
D3×5
)T (
MR5×5
)−1
M
(B)
D3×5
=
⎛
⎜
⎜
⎝
a b 0
b d c
0 c 0
⎞
⎟
⎟
⎠
, (2.19)
with the parameters a, b, c and d defined as:
a =
(yν
4 vh)2
M2
, b =
yν
1 yν
5 vηvh
yN
3
vφ
− yN
2 yν
4 yν
5 v2
h
yN
3
M2
,
c =
yν
3 yν
5 vηvh
yN
3
vφ
, d =
(yN
2 yν
5 vh)2
(yN
3
)2M2
− (yν
5 vh)2M1
(yN
3
vφ)2 + 2
yν
2 yν
5 vηvh
yN
3
vφ
.
(2.20)
Finally, the mass matrix M (B)
ν in Eq. (2.19) has another two-zero texture, B4 according
to the nomenclature in [111]. This B4 texture is also consistent with both neutrino mass
orderings (NO and IO), non-zero lightest neutrino mass and can accommodate the current
reactor mixing angle as shown in several works [111–120].
40 A4 flavour symmetric models for Majorana neutrinos and dark matter
2.4 Results and discussion
In the previous subsection, we show that both models display two different two-zero textures
for the LH Majorana neutrino mass matrices, Eqs. (2.11) and (2.18). The properties of such
two-zero mass matrices have been studied originally in [111] which coined the nomenclature
used. There are seven two-zero textures for Mν out of the fifteen logical possibilities that
are compatible with the current neutrino oscillation data. These are labelled as:
A
1
:
⎛
⎜
⎜
⎝
0 0 ×
0 × ×
× × ×
⎞
⎟
⎟
⎠
, A
2
:
⎛
⎜
⎜
⎝
0 × 0
× × ×
0 × ×
⎞
⎟
⎟
⎠
, (2.21)
B
1
:
⎛
⎜
⎜
⎝
× × 0
× 0 ×
0 × ×
⎞
⎟
⎟
⎠
, B
2
:
⎛
⎜
⎜
⎝
× 0 ×
0 × ×
× × 0
⎞
⎟
⎟
⎠
,
B
3
:
⎛
⎜
⎜
⎝
× 0 ×
0 0 ×
× × ×
⎞
⎟
⎟
⎠
, B
4
:
⎛
⎜
⎜
⎝
× × 0
× × ×
0 × 0
⎞
⎟
⎟
⎠
(2.22)
and
C :
⎛
⎜
⎜
⎝
× × ×
× 0 ×
× × 0
⎞
⎟
⎟
⎠
, (2.23)
with × any nonzero matrix element. Although, the C texture is currently only compatible
with the inverted ordering of masses, see [118].
In order to spot the phenomenological consequences of the two-zero textures, one can
see that in the standard case of three mass and three flavour neutrino oscillations, the
Majorana neutrino mass matrix is given by:
Mν = MT
ν = V diag (m1, m2, m3) V T , (2.24)
with mi the neutrino masses and V = UD, the lepton mixing matrix for Majorana neutrinos,
Eq. (1.22). The Majorana phase matrix being D = diag (eiα1 , eiα2 , eiα3), where one of these
2.4 Results and discussion 41
α phases can be dropped out. Then, Eq. (2.24) can be written as
(Mν)ij =
3
∑
k=1
mkVik Vjk =
3
∑
k=1
mk e2iαk Uik Ujk =
3
∑
k=1
µk Uik Ujk, (2.25)
with µk = mk e2iαk . The two independent zero conditions can be written as
(Mν)ij = 0
(
⇔ (Mν)∗
ij = 0
)
, (2.26)
for some different pair of indices (i, j). Replacing Eq. (2.25) in Eq. (2.26) gives
3
∑
k=1
µk Uik Ujk = 0. (2.27)
This system of equations, Eq. (2.27), for the zeros (i = a, j = b) and (i = c, j = d) is
equivalent system of equations
⎛
⎝
Ua1Ub1 Ua2Ub2
Uc1Ud1 Uc2Ud2
⎞
⎠
⎛
⎝
µ1
µ2
⎞
⎠ = −µ3
⎛
⎝
Ua3Ub3
Uc3Ud3
⎞
⎠ . (2.28)
One can solve the system for µ1 and µ2 in terms of the lightest mass µ3, leading to
⎛
⎝
µ1
µ2
⎞
⎠ = −µ3
D
⎛
⎝
Uc2Ud2 −Ua2Ub2
−Uc1Ud1 Ua1Ub1
⎞
⎠
⎛
⎝
Ua3Ub3
Uc3Ud3
⎞
⎠ , (2.29)
with
D = det
⎛
⎝
Ua1Ub1 Ua2Ub2
Uc1Ud1 Uc2Ud2
⎞
⎠ , (2.30)
which is non-zero within the current 3σ values for the oscillation parameters. Therefore,
finding non trivial solutions of Eq. (2.29) implies that µ3 has to be different from zero, as
µ3 = 0 in Eq. (2.29) implies µ1 = µ2 = 0.
42 A4 flavour symmetric models for Majorana neutrinos and dark matter
Finally, one can use the ratio of the two neutrino squared mass differences, Δm2
21 and
Δm2
3i
3 and divide them by the lightest mass µ3 yielding to
r :=
Δm2
21
Δm2
3i
=
m2
2
m3
3
− m2
1
m2
3
1 − m2
i
m2
3
=
|µ2
2
µ3
3
| − |µ2
1
µ2
3
|
1 − |µ2
i
µ2
3
|
. (2.31)
Thus, one can replace Eqs. (2.29) in Eq. (2.31), obtaining an equation that relates the
oscillation parameters: Δm2
12, Δm2
3i, θ12, θ23, θ13 and δ independently of µ3. This relation
is the basis for the analysis of the parameter regions of the models.
In summary, the two complex zeros impose four independent constrains for the twelve
real parameters in a 3 × 3 Majorana mass matrix, leading to seven independent real
parameters, a global phase can be removed. This seven parameters are the four complex
parameters: a, b, c and d in Eqs. (2.14) and (2.19) minus a global phase. On the other
hand, the neutrino observables are: Δm2
12, Δm2
3i, θ12, θ23, θ13, δ, α21 and α31. Thus, the
models will give us two predictions.
A numerical scan with inputs the 3σ values for the five oscillations parameters: Δm2
12,
Δm2
3i, θ12, θ23 and θ13 from three global fit data [121–123] is performed as follows. From
Eq. (2.31), we can solve for the phase δ, independently of the lightest mass. Then, we have
to give the mass scale as an input in the scan completing the seven independent parameters
and solve for the two remaining Majorana α phases.
As result of the numerical scans, two interesting correlations between the oscillation
parameters are worth to show. The first one between the atmospheric mixing angle, sin2 θ23,
and the sum of light neutrino masses,
∑
mν = mν1
+ mν2
+ mν3
and the second correlation
between the neutrinoless double beta decay effective mass parameter, |mee|, and the lightest
neutrino mass, mνlight
, where mνlight
= mν1
for the NO and mνlight
= mν3
for the IO.
Figs. 2.1, 2.2, and 2.3 show the correlation between the less precise measured mixing
angle, the atmospheric angle sin2 θ23 and the sum of active neutrino masses, for model A
(with B3 texture) at the top panels and model B (with B4 texture) at the bottom ones. In
these plots, the allowed 3σ regions in the oscillation parameters are displayed in magenta for
the normal ordering (NO) and in cyan for the inverted ordering (IO). The 1σ ranges in the
atmospheric mixing angle are represented by the horizontal blue and red shaded bands for
the inverted and normal mass ordering, respectively. The best fit values in θ23 correspond
to the blue and red horizontal dashed lines for the inverse and normal ordering respectively.
3where the index i in ∆m2
3i is i = 2 for the NO, thus ∆m2
32 > 0, and i = 3 for the IO, thus ∆m2
31 < 0.
2.4 Results and discussion 43
The global fit from Forero et al. [121] which has a local minimum in the atmospheric angle
for the IO is shown as the red pointed horizontal line in Fig. 2.1. The global fit from
Capozzi et al. [123] which has two different and separated 1σ regions in the atmospheric
angle for the IO, are shown as two blue shaded horizontal bands in Fig. 2.3. Finally, the
grey vertical band represents the disfavoured region in the sum of active neutrino masses,
∑
mν < 0.23 eV, coming from the cosmological data fits by the Planck Collaboration [124].
The plots in Figs. 2.1, 2.2, and 2.3 show that model A is in agreement with values
for sin θ23 within the 1σ region in both mass orderings, while model B this depends on
what dataset is used. In the data from Forero et al. [121] and Gonzalez-Garcia et al. [122]
only the NO have values for atmospheric angle within the 1σ region. Though, for the
data from [122] this happens for large values of the neutrino masses disfavoured by Planck
constraint. For data from Capozzi et al. [123], only the IO have values within 1σ in the
second octant for the atmospheric angle. Finally, it is worth to mention that the regions for
the NO and IO in model A are the same but flipped in model B, this because both textures
are related by a permutation symmetry between the 2 − 3 rows and the 2 − 3 columns.
The second remarkable correlation between |mee| and mνlight
, is displayed in Figs. 2.4, 2.5
and 2.6 for model A (with B3 texture) at the top panels and model B (with B4 texture)
at the bottom ones. The region for the NO with values of sin2 θ23 within 3σ is in dark
magenta and within 1σ in magenta. Analogously, the region for the IO with values of
sin2 θ23 within 3σ is in dark cyan and within 1σ in cyan. The horizontal red shaded
region corresponds the current experimental limit on the neutrinoless double beta decay
effective mass parameter [130]. The red (blue) lines represent the forthcoming experimental
sensitivities on the parameter |mee| [125–127, 129] (mνlight
[128]). The vertical blue shaded
region displays the disfavoured neutrino mass region set by Planck data [124]. Finally, the
graphics also show in yellow and green bands the corresponding 3σ "flavour-generic" IO and
NO neutrino spectra respectively. Figs. 2.4 and 2.5 show that model B has no 1σ values in
the atmospheric mixing angle, therefore it is only shown the data in the 3σ region in the
IO. The Fig. 2.6, shows that model B does not have values within the 1σ region for the
NO, as mentioned before.
The models have a prediction on the Majorana α phases giving a minimal cancellation
for the parameter |mee|. As can been seen in Figs. 2.4, 2.5 and 2.6 the allowed regions for
the |mee| lie on the upper limits of the generic spectrum bands. The mass parameter of the
two-zero textures B3 and B4 is also sensitive to the value of the atmospheric mixing angle,
in the cases in which this mixing angle prediction overlaps with the experimental value
44 A4 flavour symmetric models for Majorana neutrinos and dark matter
at 1σ, it translates into a localised region for the neutrinoless mass parameter. Another
important feature of the models is related with the parameter |mee|. The flavour structure
sets a lower bound on |mee| for both models, irrespective of the mass ordering, within the
range of sensitivity of the near-future experiments. Finally, it is worth mentioning that a
better measurement of the atmospheric mixing angle is a crucial test for such models.
2.5 Dark matter phenomenology
The dark matter phenomenology arising from the models A and B is different from that
in the original DDM setup, where the limit for large masses (MDM > 100 GeV) was not
allowed. The dark matter phenomenology, in this case, is similar to the inert Higgs doublet
model [131], but with two active and two inert Higgses. Unlike the first A4 DDM model,
there is no inconvenient in generating the right relic abundance even if the mass of the
DM candidate is larger than the mass of the gauge bosons. One can see that the limits
presented in the minimal dark matter model [132] apply, and for those masses the dark
matter annihilates mainly into gauge bosons.
2.6 Conclusions
In this chapter, two models (A and B) based on the discrete dark matter mechanism with A4
as flavour symmetry were presented. The family symmetry is spontaneously broken at the
seesaw scale, into a remanent Z2. In the models, three LH Majorana neutrinos acquire their
masses through the type-I seesaw mechanism, by adding five RH neutrinos to the particle
content of such models. After the flavour symmetry breaking, two RH neutrinos N2,3 will
be Z2-odd and the remaining three N1,4,5 will be Z2-even. The latter three RH neutrinos
are the only which participate in the seesaw. In addition, the scalar sector is enhanced
by the SU(3) ⊗ SU(2)L ⊗ U(1)Y singlet scalar fields φ which trigger the breaking of A4,
in such a way that two-zero textures for the active neutrino mass matrices are achieved.
These textures are in agreement with current experimental data for the reactor mixing
angle and allow the two neutrino mass orderings with non-zero lightest neutrino masses.
Another consequence of the A4 breaking is that these models contain a dark matter
candidate stabilised by the remnant Z2 symmetry. The dark matter phenomenology in the
models is different from the original DDM [103], where exists a limit for large DM masses
2.6 Conclusions 45
(MDM  100 GeV), and will be similar to the inert Higgs doublet model [131] with extra
two active and two inert SU(2)L doublet scalar fields.
In addition, an updated analysis for the two-zero textures mass matrix obtained for both
models, B3 and B4, is presented. The models predict a correlation between the atmospheric
mixing angle and the sum of the active neutrino masses, as well as a lower bound for the
neutrinoless double beta decay effective mass parameter being in the region of sensitivity of
near future experiments.
Finally, if the flavon fields acquire vevs. at some scale larger than the seesaw scale, the
remaining Z2 symmetry at the seesaw scale could lead to a mixing between the three Z2-even
RH neutrinos. Such mixing is crucial in scenarios with matter-antimatter asymmetry, via
leptogenesis, while in the original DDM A4 setup, this is not possible.
46 A4 flavour symmetric models for Majorana neutrinos and dark matter
Fig. 2.1 Correlation between sin2 θ23 and the sum of the light neutrino masses,
∑
mν, in
model A (with the B3 texture) at the top and model B (with the B4 texture) at the bottom,
where NO (IO) allowed region is in magenta (cyan). The horizontal red (blue) shaded region
corresponds to the 1σ value in sin2 θ23 for NO (IO) from [121]. The red (blue) horizontal
dashed line represent the θ23 best fit value in NO (IO), while the doted horizontal red line
represents the value of local minimum in NO from from [121]. The vertical grey shaded
region is disfavoured by Planck data [124].
2.6 Conclusions 47
Fig. 2.2 Correlation between sin2 θ23 and the sum of the light neutrino masses,
∑
mν, in
model A (with the B3 texture) at the top and model B (with the B4 texture) at the bottom,
where NO (IO) allowed region is in magenta (cyan). The horizontal red (blue) shaded region
corresponds to the 1σ value in sin2 θ23 for NO (IO) from [122]. The red (blue) horizontal
dashed line represents the best fit value in NO (IO) from [122]. The vertical grey shaded
region is disfavoured by the Planck data [124].
48 A4 flavour symmetric models for Majorana neutrinos and dark matter
Fig. 2.3 Correlation between sin2 θ23 and the sum of the light neutrino masses,
∑
mν, in
model A (with the B3 texture) at the top and model B (with the B4 texture) at the bottom,
where NO (IO) allowed region is in magenta (cyan). The horizontal red (blue) shaded region
correspond to the 1σ in sin2 θ23 for NO (IO) from [123]. The case for IO has two 1σ regions
in the data used. The red (blue) horizontal dashed line represents the best fit value in NO
(IO) from [123]. The vertical grey shaded region is disfavoured by Planck data [124].
2.6 Conclusions 49
Fig. 2.4 Effective 0νββ parameter |mee| versus the lightest neutrino mass mνlight
in model
A (B) at the top (bottom). The mνlight
is m1(m3) for NO (IO). The model allowed region
for NO is in magenta (dark magenta) for the 1σ (3σ) atmospheric mixing angle and for
IO in cyan (dark cyan) for the 1σ (3σ) atmospheric mixing angle region from [121]. The
yellow (green) band correspond to the “flavour–generic" IO (NO) neutrino spectra at 3σ.
The horizontal red shaded region is the experimental limit on 0νββ, while the red (blue)
horizontal (vertical) lines are the forthcoming experimental sensitivities on |mee| (mνlight
)
from [125–130]. The vertical blue shaded region is disfavoured by Planck data [124].
50 A4 flavour symmetric models for Majorana neutrinos and dark matter
Fig. 2.5 Effective 0νββ parameter |mee| versus the lightest neutrino mass mνlight
in model
A (B) at the top (bottom). The mνlight
is m1(m3) for NO (IO). The mνlight
is m1(m3) for
NO (IO). The model allowed region for NO is in magenta (dark magenta) for the 1σ (3σ)
atmospheric mixing angle and for IO in cyan (dark cyan) for the 1σ (3σ) atmospheric
mixing angle region from [122]. The yellow (green) band correspond to the “flavour–generic"
IO (NO) neutrino spectra at 3σ. The horizontal red shaded region is the experimental limit
on 0νββ, while the red (blue) horizontal (vertical) lines are the forthcoming experimental
sensitivities on |mee| (mνlight
) from [125–130]. The vertical blue shaded region is disfavoured
by Planck data [124].
2.6 Conclusions 51
Fig. 2.6 Effective 0νββ parameter |mee| versus the lightest neutrino mass mνlight
in model
A (B) at the top (bottom). The mνlight
is m1(m3) for NO (IO). The model allowed region
for NO is in magenta (dark magenta) for the 1σ (3σ) atmospheric mixing angle and for
IO in cyan (dark cyan) for the 1σ (3σ) atmospheric mixing angle region from [123]. The
yellow (green) band correspond to the “flavour–generic" IO (NO) neutrino spectra at 3σ.
The horizontal red shaded region is the experimental limit on 0νββ, while the red (blue)
horizontal (vertical) lines are the forthcoming experimental sensitivities on |mee| (mνlight
)
from [125–130]. The vertical blue shaded region is disfavoured by Planck data [124].

Chapter 3
Radiative Majorana neutrino mass
generation and fermionic dark matter
In the following chapter, we will review a model where light Majorana neutrino masses arise
radiatively at an one-loop level and a DM candidate emerges. This model is an extension
of the Ma’s Scotogenic model, where Z2-odd particles enter in the loop generating neutrino
masses and allowing two possible DM candidates: fermionic or scalar.
3.1 Preliminaries
The Scotogenic model [133] is a minimal extension of the SM where an exact Z2 symmetry
is imposed. The symmetry group is therefore enlarged to SM ⊗Z2. This parity only charges
additional particles to the SM particle content, in such way that active neutrinos get masses
only at loop level. The parity also works as a stability mechanism for the particle DM
candidates arising in the model.
L ℓR H η Ni
SU(2)L 2 1 2 2 1
Z2 + + + - -
Table 3.1 Relevant particle content and quantum numbers in Scotogenic model
The model enhances the SM particle content by the addition of an extra SU(2)L scalar
doublet η, and three generations of heavy RH neutrinos νRi = Ni, i = {1, 2, 3}. The particle
54 Radiative Majorana neutrino mass generation and fermionic dark matter
content and its assignments are shown in Tab. 3.1. As η is Z2-odd and Z2 is considered
as a conserved symmetry, η cannot develop a vev, thus it is an inert doublet in the sense
of [134].
Fig. 3.1 One-loop neutrino mass generation in Ma’s Scotogenic model [133].
There are no tree level mass terms for the active neutrinos. The Dirac mass term coming
from the Yukawa interaction between the neutrinos and the SM Higgs H is forbidden by
the Z2-odd assignment of Ni, and the Yukawa interaction between neutrinos and η does
not lead to a mass term as η is an inert doublet. The LH Majorana neutrinos acquire
their masses through radiative corrections involving the Z2-odd particles in the loop. The
Feynman diagram in Fig. 3.1 shows the one-loop process.
From the particle content and assignments in Tab. 3.1, the relevant part of the Lagrangian
is given by:
− LY = Y ℓL̄Hℓ + Y νL̄η̃N +
1
2
MNN̄ cN + h.c., (3.1)
where the scalar SU(2)L doublets can be written as
H =
⎛
⎝
H+
H0
⎞
⎠ and η =
⎛
⎝
η+
η0
⎞
⎠ , (3.2)
and η̃ = iσ2η∗. The Yukawa matrices Y (ℓ,ν) are 3 × 3 general complex matrices and MN is
the RH neutrino Majorana mass matrix.
The LH Majorana neutrino mass matrix is calculated from the one-loop diagram in
Fig. 3.1 leading to:
(Mν)ij =
∑
k
Y ν
ikY ν
jkΛk (3.3)
where Λk is a loop function, whose calculation is shown in the Appendix B.
3.1 Preliminaries 55
The scalar potential, following the particle assignments in Tab. 3.1, is given by:
V = µ2
HH†H + µ2
ηη†η +
1
2
λ1(H
†H)2 +
1
2
λ2(η
†η) + λ3(H
†H)(η†η)
+ λ4(H
†η)(η†H) +
1
2
λ5
[
(H†η)2 + (η†H)2
]
, (3.4)
where all the couplings λi are real, except for λ5 if one considers the case of CP violation in
scalar sector.
After EWSB, the iso-doublet η neutral component η0 ≡ ηR +i ηI and charged component
η± acquire the tree level masses:
m2
η± = µ2
η + λ3v
2, (3.5)
m2
ηR
= µ2
η + (λ3 + λ4 + λ5)v
2, (3.6)
m2
ηI
= µ2
η + (λ3 + λ4 − λ5)v
2, (3.7)
where v = 〈H〉. Notice that the mass squared splitting between the neutral components of
the iso-doublet η is m2
ηR
− m2
ηI
= 2λ5v
2.
The DM candidates in the model are stable as Z2 is a conserved symmetry. The DM
could be either the lightest singlet fermion N1 or the lightest neutral component of the
scalar iso-doublet η. The case where ηR (or ηI) is the DM resembles the inert Higgs doublet
model (IHDM), which has been studied in several works [135, 136]. Fig. 3.2 shows the
relevant annihilation channels for the calculation of the relic abundance in the case of
η0 = ηR or η0 = ηI as DM.
In the case of fermionic DM, the relevant annihilation channels in the calculation of
the relic abundance are shown in Fig. 3.3. It is worth to mention that in order to have the
right relic density, the Yukawa couplings relevant to these processes should be O(Y ν) ∼ 1.
However, lepton flavour violating (LFV) processes as: µ → e γ, τ → µγ and µ → 3e
are generated in the Scotogenic model at one-loop level in a process similar to the way
neutrino mass is generated and depending on the same Yukawa couplings Y ν . Therefore,
experimental bounds on such LFV processes have provided several constraints over the
viable parameter region for the fermionic DM in Scotogenic model. Even forthcoming
experiments potentially could rule out the entire parameter region in the case of fermionic
DM with no η0 − N1 co-annihilations.
56 Radiative Majorana neutrino mass generation and fermionic dark matter
Fig. 3.2 Relevant annihilation channels for scalar DM η0 in Ma’s Scotogenic model.
3.2 The model
The model considered in [137] is an extension of the Scotogenic model motivated by the
issues in generating the right relic density and at the same time being consistent with the
LFV constraints in the case of fermionic DM. In this model a SU(3)C ⊗ SU(2)L ⊗ U(1)Y
singlet complex scalar field φ is added, such that when φ acquires a vev. 〈φ〉 = vφ lepton
number is broken and the RH neutrinos get masses dynamically. The addition of the scalar
field opens up a new annihilation channel for the DM which relaxes the constraints in the
Yukawa couplings for generating the right relic density.
The relevant particle content and quantum numbers are shown in Table 3.2. The scalar
sector in the model consists in addition to the SM-like Higgs doublet, H, and the inert
3.2 The model 57
Fig. 3.3 Relevant annihilation channel for fermionic DM N1 in Ma’s Scotogenic model.
L̄i ℓi H η Ni φ
SU(2)L 2 1 2 2 1 1
U(1)L 1 −1 0 0 −1 2
Z2 + + + − − +
Table 3.2 Summary of the relevant particle content and quantum numbers in the model.
doublet η, of a SM singlet complex scalar field φ. The scalar fields are defined as:
H =
⎛
⎝
H+
1√
2
(vh + h + i A)
⎞
⎠ , η =
⎛
⎝
η+
1√
2
(ηR + i ηI)
⎞
⎠ and φ =
1√
2
(vφ + φR + i φI). (3.8)
The lepton sector remains the same as in the Scotogenic model. This consist of LH
doublets Li, RH charged leptons ℓi and three additional RH Majorana neutrinos Ni,
i = {1, 2, 3}. Where the three RH neutrinos allow the possibility for non-zero masses for
the three active neutrinos. The Z2 symmetry remains the same as in the Scotogenic model,
where it only charges the RH neutrinos Ni and the inert iso-doublet η.
Considering the matter content shown in Table 3.2, the relevant part of Lagrangian is
given by:
LY = Y ℓ
ij L̄iHℓj + Y ν
ij L̄iη̃Nj + hφ
ij φ N̄ c
i Nj + h.c., (3.9)
with η̃ = iτ2η
∗ and i = {1, 2, 3}. The Yukawa coupling matrices Y ν and the singlet scalar–
RH neutrinos Yukawa coupling matrix hφ
ij are hereafter assumed to be real and diagonal.
Then,
Y ν
ij = Y ν
i δij and hφ
ij = hNi
δij.
58 Radiative Majorana neutrino mass generation and fermionic dark matter
The breaking of the global U(1)L is generated dynamically by the gauge singlet scalar
field φ, which has being assigned with L = 2. After the lepton number breaking, φ acquires
a vev. and the RH neutrinos get Majorana masses dynamically. Form the last term in
Eq. (3.9), we have that such masses are
mNi
=
√
2 vφ hNi
. (3.10)
The scalar potential for the model enhances the Scotogenic model scalar potential in
Eq. (3.4) as:
V = µ2
1H
†H + µ2
2η
†η + µ2
3φ
⋆φ + λ1(H
†H)2 + λ2(η
†η)2 + λ3(η
†η)(H†H)
+ λ4(η
†H)(H†η) +
λ5
2
(
(η†H)2 + (H†η)2
)
+ λ6(φ
⋆φ)2 (3.11)
+ λ7(φ
⋆φ)(H†H) + λ8(φ
⋆φ)(η†η).
We have assumed no contribution from the scalar sector into the CP violation. Therefore,
all the quartic couplings λi and vevs. vh,φ are considered real.
After EWSB, the neutral CP-even part of the iso-doublet H and the CP-even part of
the gauge singlet φ mix. Then, the physical fields labelled as h1 and h2, where h1 is the
SM Higgs with mh1
≈ 125 GeV and 〈H〉 = vh ≈ 246 GeV, while h2 is an additional neutral
scalar field. The h − φR mixing is parametrised by a rotation matrix with angle θ as:
⎛
⎝
h
φR
⎞
⎠ =
⎛
⎝
cos θ sin θ
− sin θ cos θ
⎞
⎠
⎛
⎝
h1
h2
⎞
⎠ . (3.12)
The tree level masses of the fields h1 and h2 are calculated from the potential Eq. (3.9)
leading after EWSB to:
m2
h1
= v2
hλ1 + v2
φλ6 +
√
v2
hv2
φλ2
7 + (v2
hλ1 − v2
φλ6)2, (3.13)
m2
h2
= v2
hλ1 + v2
φλ6 −
√
v2
hv2
φλ2
7 + (v2
hλ1 − v2
φλ6)2. (3.14)
3.2 The model 59
Additionally, the tree level masses of the charged and neutral CP-even and CP-odd compo-
nents of the inert doublet η split as:
m2
η± = µ2
2 +
λ3
2
v2
h +
λ8
2
v2
φ, (3.15)
m2
ηR
= µ2
2 +
λ3 + λ4 + λ5
2
v2
h +
λ8
2
v2
φ, (3.16)
m2
ηI
= µ2
2 +
λ3 + λ4 − λ5
2
v2
h +
λ8
2
v2
φ. (3.17)
The η neutral components squared mass splitting is λ5v
2
h = (m2
ηR
− m2
ηI
).
Finally, the CP-odd component of the gauge singlet scalar field, φI , will be the Nambu–
Goldstone boson, also known as Majoron, associated with the symmetry breaking of the
global lepton number, U(1)L. This breaking triggers the neutrino mass generation. The
Fig. 3.4 One-loop neutrino mass generation in the model.
LH neutrinos acquire masses at the one-loop level, as in the Scotogenic model, mediated
by the Z2-odd particles. The neutrino mass matrix can be calculated from the diagram in
Fig. 3.4 giving [133, 138]:
(Mν)ij =
∑
k
Y ν
ik Λk Y ν
jk (3.18)
60 Radiative Majorana neutrino mass generation and fermionic dark matter
with the loop function:
Λk =
mNk
2(4π)2
[
m2
ηR
m2
ηR
− m2
Nk
log
m2
ηR
m2
Nk
− m2
ηI
m2
ηI
− m2
Nk
log
m2
ηI
m2
Nk
]
. (3.19)
The LH Majorana neutrino mass matrix, Eq. (3.18), has three key approximations. If
we denote m2
0 := (m2
ηR
+ m2
ηI
)/2, then we have:
1. If m2
0 ≪ m2
Nk
and the splitting λ5v
2
h ≪ m2
0, then
(Mν)ij ≃ λ5v
2
h
8π2
∑
k
Y ν
ikY ν
jk
mNk
[
log
m2
Nk
m2
0
− 1
]
. (3.20)
2. If m2
0 ≫ m2
k and the splitting λ5v
2
h ≪ m2
0, then
(Mν)ij ≃ λ5 v2
h
8π2
∑
k
Y ν
ikY ν
jk
mNk
m2
0
. (3.21)
3. Finally, if m2
0 ≃ m2
Nk
, then
(Mν)ij ≃ λ5v
2
h
16π2
∑
k
Y ν
ikY ν
jk
mNk
. (3.22)
In the case where the dark matter candidate is the singlet fermion N1, the DM self-
annihilation t-channel is the same as in the Scotogenic model, shown in Fig. 3.3. However,
the addition of the gauge singlet scalar field, φ, to the model opens two new annihilation
channel for the DM. The first one is a new t-channel shown up in Fig. 3.5. Such t-channel
contributes mostly to the self-annihilation of DM into dark radiation, Goldstone boson φI , as
discussed for instance in [139]. Such dark radiation is severely constrained by Cosmological
data to be not very abundant. Thus, we will try to suppress this channel and focus our
attention in the another new annihilation channel. After EWSB, the new scalar mixes
with the SM Higgs, resulting in a s-channel annihilation of dark matter to SM particles
shown bottom in Fig. 3.5. This new annihilation s-channel relaxes the constraint on the
Y ν Yukawa couplings coming from dark matter relic density below the bounds set by LFV
processes without the need of DM-inert scalar (N1 − ηR,I) co-annihilation. We will shown
this in following sections.
3.3 Constraints 61
Ni
Ni
φI,R
φI,R
Ni
N1
N1
φ H0
SM
SM
Fig. 3.5 New annihilation channels for fermionic DM, N1, in the model. Up: t-channel.
Down: s-channel resonance.
3.3 Constraints
In the following section, we will discuss the constraints we have implemented in the analysis
of our model. These come form theoretical as well as experimental considerations.
Theoretical constraints
The perturbative nature of the scalar potential quartic couplings, Eq. (3.11), and the
Yukawa couplings in the Lagrangian, Eq. (3.9), require:
|λi|, |hNi
|2, |Y ν
i |2 
√
4π. (3.23)
62 Radiative Majorana neutrino mass generation and fermionic dark matter
Furthermore, stability of the scalar potential is guaranteed if it is bounded from bellow, then
the vacuum has a minimum. This leads to the couplings of the scalar potential, Eq. (3.11),
must follow [140]:
λ1, λ2, λ6 ≥ 0, (3.24)
λ3 ≥ −2
√
λ1λ2, (3.25)
λ3 + λ4 − |λ5| ≥ −2
√
λ1λ2, (3.26)
4 λ1 λ6 ≥ λ2
7, (3.27)
4 λ2 λ6 ≥ λ2
8. (3.28)
Experimental constraints
The following constraints we considered as experimental, as they are derived from experi-
ments directly or indirectly.
Dark matter relic density: The Planck global fit of CMB temperature anisotropies, using
low multipoles and lensing data combined with spatial distribution of the galaxies (BAO),
constrains the dark matter relic density to be [124]:
ΩDMh2 = 0.1186 ± 0.0020, at 68% c.l., (3.29)
with h the Hubble constant in units of 100 Km/s/Mpc.
Gauge boson widths: Several measurements have precisely determined the gauge boson
widths [24]. In order to avoid altering the value of such widths from their SM values, we
have kinematically forbid the decay of W and Z into the inert scalars: ηR, ηI and η±. This
condition impose that the masses satisfy:
mZ < mηR
+ mηI
, mW < mη± + mηI
,
mZ < 2 mη± , mW < mη± + mηR
.
(3.30)
Scalar mass constraints from LEP: In the context of the inert Higgs doublet model, an
analysis of LEP-II data [141] allows charged and neutral inert scalar field masses above:
mη± > 135 GeV, min {mηR
, mηI
} > 100 GeV. (3.31)
3.3 Constraints 63
Higgs width and branching to invisible decays: From LHC data, the branching ratio of
SM Higgs to invisible decays has an upper bound of 28% and its total width an upper bound
at 13 MeV, both at 95% CL [24]. In our model, the SM Higgs h1 always can decay into pair
of Majorons, h1 → φI φI . When mNi
< mh1
/2 into a pair of RH neutrinos, h1 → Ni Ni,
which contribute to the Higgs invisible decay and enhances its width. In addition, when
kinematically allowed mh2
< mh1
/2, the SM Higgs can decay into a h2 pair, h1 → h2 h2,
leading to a further enhancement of its decay width and depending on the h2 width and
decays could contribute to the SM Higgs invisible channel. Thus, in our analysis of the
Higgs phenomenology we have followed the constraints derived in [139] which lead to a
bound in the mixing parameter
sin θ < 0.2. (3.32)
Electroweak precision parameters S and T: The additional SU(2)L iso-doublet η and
the mixing of the flavon φ with the SM Higgs contribute to the W and Z self-energies. The
deviations from SM are parametrised by the S, T and U [142, 143]. In the appendix C, we
have calculated the S and T parameters, following [144], and used the bounds set by the
electroweak global fits constraints taking U = 0 [24]:
− 0.1 < S < 0.3, −0.1 < T < 0.25, (3.33)
to constrain the model parameter region.
Direct detection of dark matter: Dark matter direct detection experiments have set
bounds over the dark matter-nucleon scattering cross section. For the spin-independent
(SI) process in a dark matter mass range from 10 GeV to 10 TeV, the most stringent
bound is set by PandaX-II 54 ton-day results [145]. The lowest bound value is set at
σSI  8.6 × 10−47 cm2 for a dark matter mass of 40 GeV.
Indirect detection of dark matter: Astronomical gamma ray observations constrain the ve-
locity averaged cross section of dark matter annihilation into gamma rays 〈σv〉γ . The search
for gamma rays with the Fermi-LAT satellite has constrained 〈σv〉γ  10−29cm3s−1 [146].
Lepton flavour violating processes: In the Scotogenic model, it is known that the
cross section of the LFV processes such as the radiative decays µ → eγ and the decay
µ → ee are proportional to the Yukawa coupling between the inert Higgs and RH neutrinos.
Experimental bounds on these LFV processes translate into stringent limits over the Yukawa
couplings Y ν in the case of fermionic dark matter without co-annihilation with the inert
scalars [147, 148]. In our analysis, we have considered the same experimental bounds on
64 Radiative Majorana neutrino mass generation and fermionic dark matter
the LFV processes l′ → γ l and l′ → 3l, with l, l′ = {e, µ, τ} as in [147, 148] for comparison
with the Scotogenic model.
3.4 Results and discussion
In the analysis of the model, we have performed a numerical scan over the model parameter
space using MicrOMEGAS [149]. The model has in total 12 independent parameters, which
have been scanned over the intervals:
10−5 ≤|λ2,8| ≤ 1, 10−6 ≤λ5 ≤ 1, (3.34)
10−4 ≤ sin θ ≤ 0.2, 10 GeV ≤mh2
≤ 1 TeV, (3.35)
100 GeV ≤mη0
≤ 5 TeV, 135 GeV ≤mη± ≤ 5 TeV, (3.36)
10−5 eV ≤mν1
≤ 0.07 eV, 100 GeV ≤mN2,3
≤ 5 TeV, (3.37)
10 GeV ≤mN1
≤ 1 TeV 500 GeV ≤vφ ≤ 100 TeV. (3.38)
Such parameter ranges choice obeys the theoretical and the most straightforward experimen-
tal constraints discussed previously. The remaining experimental constraints, are calculated
for each point and then filtered in our scan. We have also assumed a normal ordering (NO)
for LH neutrino masses and used the best fit values for the mixing angles and squared mass
differences from the global fit given by [24]. In addition, we have taken into account the
Planck limit for the sum of the active neutrino masses (
∑
mν  0.23 eV), which sets the
upper bound on the lightest neutrino mass mν1
 0.07 eV.
Choosing a normal ordering for the LH neutrinos, we obtain the same ordering for the
RH neutrinos. Then, N1 is the lightest dark fermion. In order to have fermionic dark
matter, we have set the inert scalar masses to be heavier than the heaviest RH neutrino,
N3. Furthermore, we have set the dark sector masses to be at least 10% heavier than
N1, avoiding the co-annihilation enhancements. The gauge singlet scalar vev., vφ, range is
chosen so that the Yukawa couplings hNi
in Eq. (3.10) are perturbative and not unnaturally
small.
In Fig. 3.6 we have plotted the dark matter thermal averaged velocity annihilation cross
section to gammas 〈σv〉γ and the dark matter-nucleon spin independent (SI) scattering
cross section σSI as a function of dark matter mass mN1
. We have found that the dark
3.4 Results and discussion 65
matter annihilation into gammas lies below the Fermi bound for most of the generated
points, therefore this observable does not further constrain the model. For the nucleon–dark
matter spin independent scattering cross section, we have found some points above the
experimental bound set by PandaX–II. However, this bound does not represent any stringent
constraint in the model, as one can generate points bellow the bound without any tuning
of the parameters.
In Fig. 3.7 we have plotted the total decay width of the SM Higgs Γh1
as a function of
the dark matter mass mN1
at the top and as a function of the sine of the mixing angle at
the bottom. The 6 MeV value corresponds to the SM Higgs width prediction. Points above
this value represent parameter space points where additional decay channels enhance the
SM Higgs width.
In Fig. 3.8 we have shown the correlation between the SM Higgs branching to invisible
as a function of sin θ/vφ. As mentioned before, the model introduces the decay modes:
h1 → φI φI , h1 → N1 N1 as well as h1 → h2h2. The process is always present, but is
controlled by sin θ/vφ. The other decays modes, when kinematically allowed, depends on
the Yukawa coupling to dark matter hN1
sin θ in the first case and on sin θ cos θ/vφ in the
latter. The SM Higgs branching to invisible is correlated in all the cases to sin θ/vθ as
shown in the Fig. 3.8.
In Fig. 3.9 we have shown at the top the lightest active neutrino mass mν1
against the
dark matter mass mN1
. The full range of neutrino masses is allowed by the constraints and
any DM mass value. At the bottom, we have plotted the inert Higgs – neutrinos Yukawa
couplings, Y ν
i , against the gauge singlet scalar–dark matter Yukawa coupling, hN1
. Notice
that the Yukawas Y ν
i range between 10−6 – 10−2, showing that the addition of the new
dark matter annihilation channel relaxes the LFV constraints over these Yukawa couplings.
Finally, in Fig. 3.10 we show the dependence of σSI with the Yukawa coupling hN1
.
As one can see, this new annihilation channel, Higgs portal in Fig. 3.5, whose effective
coupling is hN1
sin θ, dominates over the Scotogenic channels (Fig. 3.3) the dark matter
spin-independent scattering cross section.
66 Radiative Majorana neutrino mass generation and fermionic dark matter
3.5 Conclusion
In this work, we have extended the Scotogenic model to include an additional gauge singlet
scalar φ which spontaneously gives masses to the RH neutrinos. The lightest RH neutrino
has been considered to be the dark matter candidate. The new scalar mixes with the SM
Higgs doublet, introducing a Higgs portal to dark matter. This portal enhances dark matter
annihilation in the early universe, relaxing the tension between the constraints of LFV
processes and the DM relic density existing in the Scotogenic model. We present a numerical
analysis of the parameter space of the model confirming this hypothesis. Unfortunately,
the addition of this new channel also gives an important contribution of dark matter
annihilating into the Nambu–Goldstone boson φI , associated with the breaking of lepton
number, as well as SM Higgs decaying into φI . In order to avoid these phenomena, we need
a slight fine-tuning in the parameters.
3.5 Conclusion 67
Fig. 3.6 Up: Dark matter velocity averaged annihilation cross section to gammas 〈σv〉γ as a
function of the dark matter mass mN1
. The Fermi-LAT [146] indirect detection exclusion
curve is shown in blue. Bottom: Dark matter–nucleon spin independent scattering cross
section σSI as a function of dark matter mass mN1
. The PandaX–II [145] 54 ton-day
exclusion curve is shown in blue.
68 Radiative Majorana neutrino mass generation and fermionic dark matter
Fig. 3.7 Up: SM Higgs total width Γh1
as a function of dark matter mass mN1
. Bottom:
SM Higgs total width Γh1
as a function of the mixing parameter sin θ.
3.5 Conclusion 69
Fig. 3.8 Correlation between the of SM Higgs branching ratio to invisible Br(h1 → inv) and
the effective coupling sin θ/vφ.
70 Radiative Majorana neutrino mass generation and fermionic dark matter
Fig. 3.9 Up: lightest LH neutrino mass mν1
as a function of dark matter mass mN1
. Bottom:
gauge singlet scalar–dark matter Yukawa coupling hN1
as a function of inert Higgs–neutrinos
Yukawa couplings Y ν
i .
3.5 Conclusion 71
Fig. 3.10 Correlation between the dark matter spin-independent scattering cross section σSI
as a function of the Higgs portal effective coupling hN1
.

Chapter 4
A4 flavour symmetric model for a
type-II Dirac neutrino seesaw
In this chapter, we present an extension of the SM with an underlying flavour symmetry
such that natural small Dirac masses for the neutrinos could be generated in an analogous
way to the type-II seesaw for Majorana neutrinos. The non-abelian discrete group A4 is
chosen as flavour symmetry in addition with the product of discrete abelian cyclic groups
Z3 ⊗Z2. Where the Z3 ensures the "diracness" of the neutrinos at higher order operators and
loop corrections and Z2 forbids the tree level Dirac neutrino mass term. The model has the
interesting feature of partially addressing the flavour problem: explaining the quarks and
leptons mass hierarchies and mixing pattern in each sector. Notably, the model addresses
the former, via the prediction of a flavour-dependent mass relation between charged leptons
and down-type quarks.
4.1 Preliminaries
Given their colour and charge neutrality, neutrinos have been theorised as Majorana [50]
fermions and exploited this fact to construct mechanisms generating natural small Majorana
masses. Such mechanisms include the canonical seesaw realisations [50, 46, 48, 150, 54] and
radiative processes [61, 62, 65, 64, 133]. However, as was mentioned in chapter 1 via the
black-box theorem [43, 44], the neutrinoless double beta decay detection would provide the
only robust way to establish the Majorana nature of neutrinos. Even though, such process
74 A4 flavour symmetric model for a type-II Dirac neutrino seesaw
has not yet been detected [151, 129, 125, 126], suggesting the possibility for neutrinos to
be Dirac fermions.
On the other hand, a full quark–lepton correspondence within the SM and extensions,
suggests neutrinos to be Dirac fermions and the lepton mixing matrix completely analogous
to the CKM matrix. Moreover, the existence of RH states, required for Dirac masses, may
be necessary in order to have a consistent high energy completion, or for realising a higher
symmetry such as the gauged B–L symmetry, present in the conventional SO(10) seesaw
scenarios, see for instance [54].
Dirac neutrinos within the SM can be generated by just adding three RH neutrinos,
and generating neutrino masses though Higgs mechanism via Yukawa interaction term,
Eq. (1.26)
Γν L̄ Hc νR.
However, in this way, one has two issues. The first, as the RH neutrinos are SM singlets,
their Majorana mass terms MRν̄c
RνR are allowed, and then neutrinos will become Majorana.
Therefore, there has to exist some additional mechanism forbidding such Majorana mass
terms and protects the Dirac nature of neutrinos or "diracness". The second issue, as
mentioned in chapter 1, is that there is no explanation about the suppression of neutrino
Yukawa couplings, Y ν  O(10−11).
Nevertheless, there exist natural ways to generate small Dirac neutrino masses in
extensions of the SM, which are analogous to the Majorana seesaw mechanisms. One can
write down a generalised dimension-5 operator, Eq. (1.52), generating Dirac neutrino masses
and finding its high energy realisations. It has been shown that Dirac seesaw mechanisms
exist within the type I [152, 71, 72], as well as type-II [69, 153, 154] realisations, and even
also the possibility of having radiative Dirac neutrino mass models [155–157]. A brief
classification for the Dirac seesaw is shown in [158–160].
In addition, the need of a new unbroken symmetry protecting the diracness of the
neutrinos has several possibilities which involve an extra U(1) [74, 68] or its discrete sub-
groups ZN [69, 155] or flavour symmetries [72, 73]. In the following, we will discuss a
particular realisation for these possibilities.
4.2 The model 75
4.2 The model
In this section, we will explain the details in the construction of the model where we have
considered the possibility of Dirac neutrinos resulting from a family symmetry construction,
in which small neutrino masses arise dynamically via a type-II seesaw mechanism for Dirac
neutrinos. This setup complements the idea of having Dirac neutrinos arising from flavour
symmetries proposed in [161].
The tree level completion of the generalised dimension-5 operator in Eq. (1.52),
O5 =
g
Λ
(L̄H̃) σνR,
by the exchange of a heavy scalar SU(2)L doublet mediator φ is diagrammaftically illustrated
in Fig. 4.1. In order to have such realisation of this type-II Dirac neutrino seesaw, the
φ
H σ
νL νR
Fig. 4.1 Neutrino mass generation in the type-II seesaw for Dirac neutrinos, as in [69, 153,
154].
model has to enhance the SM particle content adding the iso-doublet φ and the scalar
iso-singlet σ. Tab. 4.1 shows the relevant particles and assignments for this type-II Dirac
neutrino realisation.
As mentioned before, a mechanism protecting the diracness of the neutrinos must be
added. In this scenario, we have introduced the symmetry product Z3 ⊗ Z2, such that
it forbids the appearance of Majorana mass operators at the loop level, thus protecting
76 A4 flavour symmetric model for a type-II Dirac neutrino seesaw
L̄ ℓR νR H φ σ
SU(2)L ⊗ U(1)Y (2, 1/2) (1, −1) (1, 0) (2, 1/2) (2, −1/2) (1, 0)
Z3 ω2 ω ω 1 1 1
Z2 + + − + − −
Table 4.1 Charge assignments for the particles involved in the type-II Dirac neutrino seesaw
realisation, as in [69].
the diracness of neutrinos. The A4 flavour assignments of the model are not necessary for
realising this type-II Dirac seesaw but will be relevant for the predictions of the model, as
we will see in the following section.
4.2.1 Lepton sector
The specific particle assignments for the lepton sector and scalars are given in Tab. 4.2. The
L̄ ℓR νR Hd φ σ or σi
SU(2)L ⊗ U(1)Y (2, 1/2) (1, −1) (1, 0) (2, 1/2) (2, −1/2) (1, 0)
A4 3 3 3 3 3 3 or 1i
Z3 ω2 ω ω 1 1 1
Z2 + + − + − −
Table 4.2 Charge assignments for the particles involved in the neutrino mass generation
mechanism, where ω3 = 1
scalar SU(2)L doublets Hd = (Hd
1 , Hd
2 , Hd
3 )T and φ = (φ1, φ2, φ3)T transform as triplets, 3,
under A4. Where each component can be written as follows:
Hd
i =
⎛
⎝
hd +
i
hd 0
i
⎞
⎠ , φi =
⎛
⎝
φ0
i
φ−
i
⎞
⎠ , i = {1, 2, 3}. (4.1)
The vev. of these scalar triplets are given by:
〈
Hd
〉
= (vhd
1
, vhd
2
, vhd
3
), 〈φ〉 = (vφ1
, vφ2
, vφ3
). (4.2)
The SM singlet complex scalars σi, i = {1, 2, 3}, are responsible for inducing the small
vevs. of the φ and could transform as an A4 triplet σ = (σ1, σ2, σ3)
T ∼ 3 or as different
4.2 The model 77
singlets σi ∼ 1i under the flavour group. On the other hand, the LH lepton chiral
components L = (L1, L2, L3)T , RH charged leptons ℓR = (ℓ1R, ℓ2R, ℓ3R)T and RH neutrinos
νR = (ν1R, ν2R, ν3R)T transform as A4 triplets 3.
From the model assignments in Tab. 4.2, one can see that Z3 is a conserved symmetry,
because all scalars developing vevs. are blind (uncharged) under this symmetry. This Z3
symmetry forbids the Majorana mass terms as well as the dimension-5 operators:
MR ν̄c
R νR, L̄cHdLHd, L̄cφ̃Lφ̃ and L̄cHdLφ̃, (4.3)
where φ̃ = iσ2φ
∗ and H̃d = iσ2H
d ∗. The Z3 symmetry also forbids higher order operators
giving rise to Majorana mass terms:
(L̄cHdLHd)(Hd †Hd)n, (L̄cφ̃Lφ̃)(Hd †Hd)n (L̄cHdLφ̃)(Hd †Hd)n, (4.4)
(L̄cHdLHd)(φ†φ)n, (L̄cφ̃Lφ̃)(φ†φ)n, (L̄cHdLφ̃)(φ†φ)n, (4.5)
(L̄cHdLHd)(Hd†φ̃)n, (L̄cφ̃Lφ̃)(Hd†φ̃)n, (L̄cHdLφ̃)(Hd†φ̃)n, (4.6)
ν̄c
R νR σn. (4.7)
Finally, the Z2 charges νR, σ and φ while the remaining particles are Z2-even. Such
symmetry acts in a complementary way to Z3 forbidding the unwanted renormalisable
Yukawa couplings:
L̄ φ̃ ℓR and L̄ H̃d νR, (4.8)
where the first operator allows the extra Higgses φi of giving mass to charged leptons, and
the second operator provides the tree level Dirac neutrino mass term.
Before proceeding, we summarise our model structure by saying that compared with the
minimal SM case, here one has three copies of the Higgs doublet Hd
1,2,3, three extra scalar iso-
doublets φ1,2,3 and three RH neutrinos ν1R, 2R, 3R all of them forming A4 triplets. Moreover,
one has three iso-singlet scalars σ1,2,3 which could be assigned as a triplet or singlets of A4.
After flavour and EWSB, neutrinos get small type-II seesaw masses, Eq. (1.58), as result
of the small vev. 〈φ〉. This vev. is induced by means of the vevs. of the scalar iso-singlet
σ1,2,3, 〈σ〉 ∼ vσ, through the seesaw relation
vφ ≈ κvh
⎛
⎜
⎝
1
λHφ
v2
h
vσ
+ λσφ − 2
µ2
φ
vσ
⎞
⎟
⎠ , (4.9)
78 A4 flavour symmetric model for a type-II Dirac neutrino seesaw
as proposed in [69, 153, 154].
Finally, in accordance with the previous discussion the relevant part of Yukawa La-
grangian for leptons is given as:
LY ⊃ Y i
ℓ
[
L̄ Hd
]
3i
ℓR + Y i
ν
[
L̄ φ
]
3i
νR + h.c., (4.10)
where the symbol [a b]3i
stands for the two ways of contracting two triplets of A4 a and b
into a triplet, as shown in the Appendix A.
4.3 Results and discussion
In this section we will discuss the features of the model. The first one is how the model
leads to a mass formula relating quark and lepton masses, despite the absence of grand
unification, which leads to a flavour dependent generalisation of bottom-tau unification
previously proposed in [162] and studied in [163–166, 72]. Another feature is related with the
specific predictions for the lepton mixing matrix in the neutrino sector. Despite there are no
predictions for the CKM, it can be shown that this can be adequately fitted following [163].
4.3.1 The generalised bottom-tau mass relation
The complete model particle assignment is based in the model [163], shown in Tab. 4.3,
including both SU(2)L ⊗ U(1)Y gauge and flavour transformation properties. In this
complete flavour assignment, only one new SM-like Higgs A4 triplet is added Hu, which is
responsable for giving mass to the u-type quarks. In addition, a new parity is added Zd
2 ,
that ensures d-type quark masses only come from Hd and forbids terms where Hu couples
to d-type quarks and Hd couples to u-type quarks.
From the lepton Yukawa Lagrangian, Eq. (4.10), one sees that the charged lepton as
well as the d-type quarks mass matrices1, following [162–164], can be parametrised as:
mℓ =
⎛
⎜
⎜
⎝
0 aℓαℓe
iθℓ bℓ
bℓαℓ 0 eiθℓaℓρℓ
aℓe
iθℓ bℓρℓ 0
⎞
⎟
⎟
⎠
, (4.11)
1As the d-type quarks have the same flavour assignments than charged leptons, then their masses come
from an analogous term involving Hd: Y i
d [Q̄, Hd]3i
dR + h.c.
4.3 Results and discussion 79
Q̄ L̄ uRi
dR ℓR νR Hu Hd φ σ or σi
SU(2)L 2 2 1 1 1 1 2 2 2 1
Y -1/6 1/2 2/3 -1/3 -1 0 1/2 1/2 -1/2 0
A4 3 3 1i 3 3 3 3 3 3 3 or 1i
Z3 1 ω2 1 1 ω ω 1 1 1 1
Z2 + + + + + − + + − −
Zd
2 + + + − + + + − + +
Table 4.3 Particle content and quantum numbers for the complete model.
where
aℓ = vhd
2
(Y 1
ℓ + Y 3
ℓ ) and bℓ = vhd
2
(Y 2
ℓ + Y 4
ℓ ), (4.12)
are real Yukawa couplings, θℓ is the only unremovable complex phase, which we could
assume θℓ = 0, i.e. CP violation comes entirely from the neutrino and also u-type quark
sectors, and the Hd vev. alignment is parameterised as:
〈
Hd
〉
= (vhd
1
, vhd
2
, vhd
3
) = vhd
2
(ρℓ, 1, αℓ), (4.13)
with
αℓ = vhd
3
/vhd
2
, and ρℓ = vhd
1
/vhd
2
, (4.14)
and vd
hi
(therefore αℓ and ρℓ) real.
Now to determine the generalised bottom-tau mass relation, we can calculate the
bi–unitary invariants of the squared mass matrix in Eq. (4.11), M2
ℓ = mℓm
†
ℓ, where
mD
ℓ = diag (m1, m2, m3) = V †
ℓ mℓUℓ, (4.15)
80 A4 flavour symmetric model for a type-II Dirac neutrino seesaw
with Vℓ and Uℓ unitary matrices, yielding to
Tr M2
ℓ = m2
1 + m2
2 + m2
3 (4.16)
= (a2
ℓ + b2
ℓ)(1 + α2
ℓ + ρ2
ℓ), (4.17)
det M2
ℓ = m2
1m
2
2m
2
3 (4.18)
= (a6
ℓ + b6
ℓ)α
2
ℓρ2
ℓ , (4.19)
(Tr M2
ℓ )2 − Tr (M2
ℓ )2 = 2m2
1m
2
2 + 2m2
2m
2
3 + 2m2
1m
2
3, (4.20)
= a2
ℓb
2
ℓ(1 + α4
ℓ + ρ4
ℓ) + (a4
ℓ + b4
ℓ)
(
ρ2
ℓ + α2
ℓ(1 + ρ2
ℓ)
)
. (4.21)
Under the assumptions ρℓ ≫ αℓ, ρℓ ≫ 1, bℓ > aℓ and ρℓ ≫ bℓ
aℓ
which, at leading order,
ensure adequate family mass hierarchy (m1 < m2 < m3) as well as the Cabibbo mixing
pattern. One can expand Eqs. (4.17, 4.19, 4.21) and use the mass hierarchy on Eqs. (4.16,
4.18, 4.20), yielding at leading order to:
(bℓρℓ)
2 ≈ m2
3, (4.22)
(b3
ℓρℓαℓ)
2 ≈ m2
1m
2
2m
2
3, (4.23)
(aℓbℓρ
2
ℓ)
2 ≈ m2
2m
2
3. (4.24)
Solving the system in Eqs. (4.22-4.24), one can find the approximate expressions 2:
aℓ ≈ m2
m3
√
m1m2
αℓ
, (4.25)
bℓ ≈
√
m1m2
αℓ
, (4.26)
ρℓ√
αℓ
≈ m3√
m1m2
. (4.27)
As previously mentioned, d-type quarks couple only to Hd and hence have the same
flavour structure. This implies that the parameters ρℓ and αℓ in Eq. (4.27) are common to
the charged leptons and the d–type quarks. Therefore,
ρℓ = ρd and αℓ = αd. (4.28)
2These approximations are in 1% agreement with the exact numerical solution.
4.3 Results and discussion 81
Using this fact, we can equate Eq. (4.27) for charged leptons and d-type quarks, and derive
the generalised bottom-tau mass relation, proposed in [167, 162]:
mτ√
memµ
≃ mb√
mdms
. (4.29)
It is worth to mention that this mass relation is scale invariant, as the running of the masses
cancels out. Furthermore, Eq. (4.29) follows only from the flavour group assignments for the
fields in the model; although it has been also obtained in other non-equivalent realisations
of A4 family symmetry [167, 163, 164, 72] and other family groups [166].
4.3.2 CKM fitting
In this section, we focus on the CKM matrix which can be adequately described, providing in
addition an input for the lepton mixing matrix. Although the model has no family symmetry
prediction for the CKM matrix, we can, however, accommodate the CKM elements in the
same way as described in [163]. This fixes the value for the αd parameter which enters also
in the leptonic sector.
A numerical fit for quarks masses and mixing matrix is performed for the model. From
the family assignments in Tab. 4.3, the Yukawa Lagrangian for the quarks sector is:
LY = Y i
u [Q̄ H̃u]1i
uR + Y i
d [Q̄ Hd]3i
dR + h.c., (4.30)
where Y i
u,d are complex Yukawa couplings, and the symbol [a b]i stands for the contraction
of two triplets a and b into the irrep. i = {1, 1′, 1′′, 3} of A4.
From the Lagrangian in Eq. (4.30), the mass matrix for the d-type quarks is:
md =
⎛
⎜
⎜
⎝
0 adαd bdeiθd
bdαd 0 adρd
ad bdeiθdρd 0
⎞
⎟
⎟
⎠
, (4.31)
which has been parametrised in a similar way as the mass matrix for the charged leptons,
Eq. (4.11), and the u-type quark mass matrix can be written as:
mu =
1√
3
⎛
⎜
⎜
⎝
v1
u 0 0
0 v2
u 0
0 0 v3
u
⎞
⎟
⎟
⎠
⎛
⎜
⎜
⎝
1 0 0
0 ω ω∗
0 ω∗ ω
⎞
⎟
⎟
⎠
⎛
⎜
⎜
⎝
Y 1
u 0 0
0 Y 3
u 0
0 0 Y 2
u
⎞
⎟
⎟
⎠
, (4.32)
82 A4 flavour symmetric model for a type-II Dirac neutrino seesaw
with 〈Hu〉 = (vu
1 , vu
2 , vu
3 ) and ω3 = 1.
It is known that such mass matrices can be diagonalised by the bi-unitary transformation:
V u,d†mu,dUu,d = diag (mu,d, mc,s, mt,b) . (4.33)
Therefore, diagonalisation of the squared matices mu m†
u and md m†
d give the left-diagonalising
unitary matrices Vu and Vd, whose product give the CKM matrix, VCKM ,
VCKM = V u† V d. (4.34)
Analytical approximation for V d establishes that in order to generate the Cabbibo mixing
λc ≈ 0.2 in the plane 1 − 2, we need αd ∼ O(1). This also implies that the other mixings,
on the planes 1 − 3 and 2 − 3, are negligible.
Finally, as the contribution from the down sector has been shown to generate the
Cabbibo mixing, the remaining smaller mixings can be generated from the contribution
of the up sector V u. Thus, in [163] is proposed that the structure for M2
u = mu m†
u is in a
hierarchical way
M2
u ∼
⎛
⎜
⎜
⎝
λ8
c λ6
c λ4
c
λ6
c λ4
c λ2
c
λ4
c λ2
c 1
⎞
⎟
⎟
⎠
, (4.35)
which have been proven to generate V u
23 ≈ λ2
c , V u
13 ≈ λ4
c and V u
12 ≈ λ2
c . This could be
achieved if the vevs. of Hu, 〈Hu〉 = (vu
1 , vu
2 , vu
3 ), have the hierarchy:
vu
1 : vu
2 : vu
3 = 1 : λ2
c : λ4
c . (4.36)
Regarding the CP violation in the quark sector, it is possible to generate the right
amount only from the up sector contribution through ω, i.e. one can set the CP-violating
phase form the down sector to zero, θd = 0.
The fitting of the CKM matrix is performed by numerical minimisation of a χ–squared
function. Such χ2 function is defined as
χ2 =
∑
i
(
mob
i − mth
i
σmi
)2
+
∑′
j,k
( |V ob
jk | − |V th
jk |
σVjk
)2
+
(
Job − J th
σJ
)2
, (4.37)
4.3 Results and discussion 83
where the first summation index accounts for all quarks, i = {u, c, t, d, s, b}, the primed
summation refers only to the elements CKM elements3 V12, V13 and V23, and J is the
Jarlskog invariant, Eq. (1.14),
J = Im [VusVcbV ∗
ubV ∗
cs] .
The observed values and uncertainties for the CKM elements, the Jarlskog parameter and
the charged leptons masses are taken from updated references [24], while the quark masses
are taken from [168], at the Z mass scale. The best-fit values are shown in Tab. 4.4 in
accordance with the results of [163].
Observable Exp. value Mod. pred. | Pull | [σ]
mu [Mev] 1.45 ± 0.51 1.51 0.16
mc [Mev] 635 ± 86 585.7 0.12
mt [Gev] 172.1 ± 1.5 172.0 0.04
md [Mev] 2.9 ± 0.51 2.7 0.39
ms [Mev] 57.7 ± 16.8 56.5 0.07
mb [Mev] 2820 ± 80 2835 0.19
|V12| 0.22534 ± 0.00065 0.22504 0.47
|V13| 0.00351 ± 0.00015 0.00359 0.51
|V23| 0.0412 ± 0.008 0.0390 0.28
J × 10−5 2.96 ± 0.20 2.99 0.16
Table 4.4 Experimental and predicted quark masses and mixing parameters from the CKM
fit.
4.3.3 Lepton masses and mixing
In this section, we will focus on the lepton mixing matrix, because it is in this sector that
our model makes non-trivial predictions. In analogy with the previous subsection, Eq. (4.10)
gives the neutrino mass matrix, which can also be parametrised as:
mν =
⎛
⎜
⎜
⎝
0 aναν bνeiθν
bνeiθν αν 0 aνρν
aν bνeiθν ρν 0
⎞
⎟
⎟
⎠
, (4.38)
3We have assumed the unitarity of the CKM matrix.
84 A4 flavour symmetric model for a type-II Dirac neutrino seesaw
where
aν = vφ2
(Y 1
ν + Y 3
ν ) and bν = vφ2
(Y 2
ν + Y 4
ν ) (4.39)
are real Yukawa couplings, θν is the complex phase that cannot be rotated away under
SU(2)L transformations and characterises the strength of CP violation in the lepton sector.
The vev. alignment of the iso-doublets φ can be parametrised as
〈φ〉 = (vφ1
, vφ2
, vφ3
) = vφ2
(ρν , 1, αν), αν = vφ3/vφ2 ρν = vφ1/vφ2. (4.40)
The number of free parameters in the leptonic sector is nine. Four independent parame-
ters (aℓ, bℓ, αℓ and ρℓ) from the charged leptons, Eq. (4.11), and five parameters (aν , bν , αν ,
ρν and θν) from the neutrinos, Eq. (4.38). However, two of them αℓ and ρℓ are fixed from
the quark sector. The remaining seven parameters are used to fit three masses plus two
mass squared splittings, three mixing angles and a Dirac CP violating phase. This is done
using the invariants Eqs. (4.16 – 4.20) of the squared neutrino mass matrix M2
ν = mνm†
ν :
Tr(M2
ν ) = m2
ν1 + m2
ν2
+ m2
ν3
, (4.41)
= (a2
ν + b2
ν)(1 + α2
ν + ρ2
ν) (4.42)
det(M2
ν ) = m2
ν1
m2
ν2
m2
ν3
(4.43)
=
(
a6
ν + b6
ν + 2a3
νb3
ν cos(3θν)
)
α2
νρ2
ν , (4.44)
1
2
[
(TrM2
ν )2 − Tr(M4
ν )
]
= m2
ν1
m2
ν2
+ m2
ν2
m2
ν3
+ m2
ν1
m2
ν3
(4.45)
= a2
νb2
ν(1 + α4
ν + ρ4
ν) + (a4
ν + b4
ν)(ρ2
ν + α2
ν(1 + ρ2
ν)), (4.46)
and performing numerical scan over the parameter regions of solutions of these, Eqs. (4.41
– 4.46), that reproduce the measured elements of the leptonic mixing matrix. The leptonic
mixing matrix, Eq (1.18):
V = V ℓ†V ν ,
have a fixed and calculable contribution from the left-diagonalising unitary matrix of
charged leptons V ℓ; V ℓ is a close to s diagonal matrix. The inputs used in the scan are the
3σ values for three neutrino global fit [33]: the mixing angles (θ12, θ23, θ13) and two mass
squared differences (Δm2
12 , Δm2
3i).
4.3 Results and discussion 85
Neutrino oscillation predictions
In order to determine the model neutrino oscillation predictions, a random numerical scan
is performed over the free parameters ranges:
αν ∈ [−10, 10] , ρν ∈ [−10, 10] θν ∈ [0, 2π] and mlight ∈
[
10−5, 0.23
]
eV. (4.47)
The parameter space points for which the oscillation parameters are within 3σ of the global
fit data [33] are determined. This phenomenological requirement constrains the allowed
regions for the neutrino mass matrix parameters, Eq. (4.38):
0.75  |αν , ρν |  1.25, 0.03  |aν , bν |  0.04, −π/3  θν  π/3. (4.48)
This way, it has been obtained the model–allowed regions for the oscillation parameters.
Correlations among the interesting and poorly determinated atmospheric mixing angle θ23,
the CP phase δCP and the lightest neutrino mass eigenvalue are shown as green shaded
regions in Figs. 4.2, 4.3 and 4.4. Also,t the numerical scan has found that the model is
only compatible with the inverted ordering (IO) of the neutrino masses. The consistent
parameter regions for the atmospheric mixing angle sin2 θ23 vs. m3 the lightest neutrino
mass are given in shaded (green) regions in Fig. 4.2. The green horizontal band represents
the 1σ value for θ23 and the dashed line the best fit value in the data used. It is interesting
to notice that the model is not compatible with a maximal mixing angle θ23 = 45◦.
The allowed regions for the CP violation parameters δCP and JCP vs. m3 are displayed
in Fig. 4.3. From the plots one sees that the allowed region for the lightest neutrino mass
m3 is within the range
6.4 × 10−4 eV  m3  2.7 × 10−3 eV. (4.49)
It is worth to notice that only masses above ∼ 0.002 eV allow JCP = 0, i.e. no CP violation,
while for lower masses such value is always non-zero.
Finally, the parameter regions for the atmospheric mixing angle θ23 vs. δCP the CP
phase are shown as green shaded areas in Fig. 4.4, where the contours lines represent the 90
and 99% C.L. regions obtained directly from the unconstrained three neutrino oscillation
global fit [33].
86 A4 flavour symmetric model for a type-II Dirac neutrino seesaw
   





 	







Fig. 4.2 The regions in the atmospheric mixing angle θ23 and the lightest neutrino mass m3
allowed by oscillation data in shaded (green) areas. The horizontal dashed line represents
the best-fit value for sin2 θ23, whereas the horizontal shaded region corresponds to the 1σ
allowed region from Ref. [33].
4.4 Conclusions
It this chapter, we have shown a model with A4 ⊗Z3 ⊗Z2 flavour extension of the SM where
the small Dirac neutrino masses are generated from a realisation of a type-II Dirac seesaw
mechanism. The model partially addresses both aspects of the flavour problem explaining
the mass hierarchy of quark and leptons and restricting the structure of the mixing matrix.
Concerning the first point, the model leads to a golden mass relation between quark and
lepton masses, which has been proposed in previous works. Regarding the latter point, the
model gives flavour predictions for the lepton mixing matrix.
The model predicts an inverted neutrino mass ordering (IO) with non-maximal atmo-
spheric mixing angle. The model also suggests a slight preference for the higher octant, since
it predicts inverted neutrino mass ordering. This could be at odds with the latest results of
the neutrino oscillation global fit [33], but one could argue that neither preference for normal
ordering nor lower octant are statistically significant, since the general three–neutrino fit
gives four possible closely separated local minima.
The model also has a positive hint for CP violation, δCP = 0, if mνlightest
 0.002 eV,
while bigger masses are consistent with CP-conserving solutions. In addition, the regions
4.4 Conclusions 87
   







 



   





 




Fig. 4.3 Correlation between the CP violation and the lightest neutrino mass. Up: correlation
between the Jarlskog invariant J and the lightest neutrino mass m3 allowed by the current
oscillation data [33]. Bottom: Allowed regions for the correlation between the Dirac CP
phase δCP and the lightest neutrino mass m3.
for the CP phase and the atmospheric angle nicely agree with the currently preferred ones,
though these global fit determinations are not yet very robust.
Regarding the quark mixing matrix, although no predictions are made, the CKM matrix
elements can be fitted which also fix the charged lepton contribution to the lepton mixing
matrix. Finally, it is worth to notice that the Z3 symmetry forbids the neutrino Majorana
mass terms at any order and provides by construction a natural realisation of a type-II
Dirac seesaw mechanism.
88 A4 flavour symmetric model for a type-II Dirac neutrino seesaw
    





	






Fig. 4.4 The allowed regions of the atmospheric mixing angle and δCP are indicated in green
shaded. The unshaded contour regions represent the 90 and 99%CL regions obtained directly
in the unconstrained three neutrino oscillation global fit [33].
Chapter 5
Summary and final remarks
Here we present a very brief concluding summary, while detailed remarks can be found at
the end of each chapter. This thesis collects the results of three projects which investigate
the origin of neutrino masses by the use of abelian and non-abelian discrete symmetries.
Furthermore, two of these works also explore the connection between the neutrino mass
generation mechanism and dark matter. Each work addresses the neutrino mass generation
in different ways, but all have in common the guidance of a predictive bottom-up model
building. That is, a high energy extension of the SM where the particle content and
symmetries are enhanced as minimal as possible and being consistent with existing low
energy constraints.
The first work presents two A4 extensions of the SM. The A4 symmetry is spontaneously
broken down into a remanent Z2, in such a way, this symmetry provides a mechanism for the
dark matter stability. The leptons transform non-trivially under the flavour symmetry, while
the quarks remain blind, and their masses and mixing pattern may be explained by another
complementary mechanism. The models are successful reproducing the neutrino masses and
mixing pattern, which are a consequence of the flavour assignments and specific breaking.
The flavour symmetry leads to mass matrices for light neutrinos with two-zero textures
which accommodate current neutrino oscillation data for both mass orderings with a non-
zero lightest neutrino mass. The models predict correlations among oscillation parameters,
from which we have obtained lower bounds to the 0νββ effective mass parameter in a region
of sensitivity of forthcoming experiments. Regarding dark matter phenomenology, the
models provide an explanation on the dark matter stability mechanism and have as dark
matter candidates: two RH neutrinos not participating in the neutrino mass generation
and two inert Higgs doublets. Considering the last option, the dark matter phenomenology
90 Summary and final remarks
is expected to be analogous to a model with two active and two inert Higgs doublets.
Additionally, as the flavour symmetry breaking is driven by scalar gauge singlet fields, the
scale of such breaking could be chosen to be above the seesaw scale, providing a plausible
scenario for the generation of the matter-antimatter asymmetry via leptogenesis.
In the second work, we have studied a SM extension where an exact abelian discrete
symmetry Z2 is added such that it stabilises the dark matter and in addition is responsible
for the neutrino mass generation. The latter through the one-loop realisation of the
dimension-5 Weinberg operator in which dark matter participates in the loop. We have also
added a gauge singlet scalar field charged under lepton number, such that it gives mass
dynamically to the RH neutrinos. In addition, this gauge singlet scalar field mixes with
the Higgs leading to a new dark matter annihilation channel. We have shown that this
new channel leads to a relaxation on the tension between the dark matter relic abundance
and lepton flavour violation constraints. The mixing between the Higgs and the gauge
singlet scalar field leads to a richer Higgs phenomenology. Further constraints on the model
parameter region can be considered taking into account a detailed analysis of the Higgs
sector phenomenology. Although the model does not provide any prediction on the neutrino
masses nor mixing pattern, it is consistent with current oscillation parameters through the
LFV constraints.
Finally, in the last work we have presented an A4 flavour symmetric realisation of a type-
II Dirac neutrino seesaw. The model can reproduce successfully the patterns for fermion
masses and mixings. The former through a flavour dependent golden mass relation between
the charged leptons and down type quarks, which has been derived in previous works.
Concerning the fermion mixing patterns, the model adequately describes the CKM elements,
which fixes the charged lepton contribution to the lepton mixing matrix. The model flavour
predictions for the leptons are consistent with a non-zero lightest neutrino mass with an
inverted mass ordering, non-maximal atmospheric mixing angle and non-zero CP violating
phase for the lightest neutrino mass < 0.002 eV. In this way, a better determination on the
atmospheric mixing angle and the δCP phase as well as the determination of the neutrino
mass ordering could test this model flavour realisation. This is expected to happen in
the forthcoming years by long baseline experiments as NOνA and atmospheric neutrino
oscillation experiments as Hyper-Kamiokande, etc. The model fails to account for a dark
matter candidate. However, there are several ways to incorporate a dark matter candidate
into this setup, one that has been explored in similar constructions is to enlarge the exact
91
Z3 symmetry responsible for the “diracness” of neutrinos into a larger group as Zn, leading
to a dark matter stability symmetry.
In this thesis, we have assumed that neutrino masses and mixing pattern could be
explained by some underlying discrete symmetry. Also, we have relied upon the hypothesis
that neutrino mass generation mechanism and dark matter are somehow related. In
particular, we have exploited this idea through the use of local discrete symmetries. However,
it is worth mentioning that these bottom-up approaches fail to provide a natural relation
between neutrinos and dark matter, in the sense that at one point one has to assume some
symmetry assignments. It would be more appealing, from the theoretical point of view,
to achieve this relation from a high energy fundamental theory. Thus, in our conservative
model building approach, one could also consider the possibility that these models based
on discrete (flavour) symmetries have to account for more complex interaction not possible
within the SM. It would be interesting to investigate high energy extensions of the SM,
with or without flavour symmetries, where the relation between dark matter and neutrinos
is more natural, and look for its testable predictions. However, one should be cautious
in this unification endeavour. There always exists the possibility that neutrinos and dark
matter are completely independent phenomena.

Acknowledgements
I would like to express my kind gratitude to my advisor Eduardo Peinado for being so
generous with his time and his pragmatism. To Genaro Toledo Sanchez and Mariano
Chernicoff Minsberg, members of my PhD advisory committee. I am also grateful to the
members of the committee of my thesis for generously accepting to review the manuscript
and for their thoughtful comments. I wish to thank the co-authors of all the papers this
thesis is based on for the excellent collaborations.
During my PhD, I had the chance to do a short stay with the Astroparticle and High
Energy Physics (AHEP) group in the IFIC–University of Valencia, Spain. I want to express
my deep acknowledge to Professor Jose W. F. Valle for the financial support during this
staying and the whole people in the group for his kind hospitality and excitement discussions.
Last but not least, I acknowledge the financial support from the Mexico CONACyT fellow-
ship, German-Mexican research collaboration grant SP 778/4-1 (DFG), 278017 (CONACyT)
and Papiit: IA101516, RA101516 and IN107118 grants.

Appendix A
The A4 group
In this appendix, we will review the properties of the smallest non-abelian discrete group
A4. The A4 group is just a realisation of a general class of group called the alternating
group of N elements or AN , which consists of all the even permutations of N elements (i.e.
AN is a sub-group of SN , the permutation group of N elements). The order of AN is N !/2.
One can easily realise that A3 is isomorphic to the abelian group Z3, being ZN the cyclic
group of order N1. On the other hand, the group A4 could be associated with the group of
symmetries of the tetrahedron. The A4 group has twelve elements that can be written in
terms of two generators S and T . These generators satisfy the algebraic relations
S2 = T 3 = (S T )3 = I. (A.1)
The elements of A4 can be written in terms of the generators and the identity e as: e,
S, T , T 2, TS, ST , STS, TST , ST 2, T 2S, TST 2 and T 2ST . These could be classified into
four conjugacy classes as:
C1 :{e}, h = 1, (A.2)
C3 :{S, TST 2, T 2ST}, h = 2, (A.3)
C4 :{T, TS, ST, STS}, h = 3, (A.4)
C ′
4 :{T 2, ST 2, T 2S, TST}, h = 3. (A.5)
1The cyclic group of order N is an abelian finite group which can be defined through one single generator
X, following the relation XN = I.
94 The A4 group
where h is the order of each conjugacy class.
One can see from Eq. (A.1) that A4 contains two sub-groups: Z3 and Z2 each one associated
with the T and S generator respectively. Finally, it is remarkable to mention that A4 is a
finite sub-group of the SO(3) as well as SU(3).
A.1 A4 irreducible representations product
The group A4 has four irreducible representations. Finding those irreps. of A4 requiere to
look at the orthogonality relation
∑
n
mnn2 = NG = 12, (A.6)
where mn is the multiplicity of the irrep. of dimension n and NG = 12 is the order of A4.
On the other hand, we have that
∑
n
mn = 4, (A.7)
because there are four conjugacy classes of A4, Eqs. (A.2)-(A.5). From Eqs. (A.6) and
(A.7), the only solution is: (m1, m2, m3, . . . ) = (1, 0, 1, 0, . . . ). Therefore there exist three
one-dimensional irreps.: 1, 1′, and 1′′ and one three-dimensional irrep. 3.
The one-dimensional unitary irreps. are:
1 : S = 1, T = 1,
1′ : S = 1, T = ω,
1′′ : S = 1, T = ω2,
(A.8)
where ω3 = 1, and the three-dimensional irrep. in the basis where S is real and diagonal is
3 : S =
⎛
⎜
⎜
⎝
1 0 0
0 −1 0
0 0 −1
⎞
⎟
⎟
⎠
and T =
⎛
⎜
⎜
⎝
0 1 0
0 0 1
1 0 0
⎞
⎟
⎟
⎠
. (A.9)
The product rule for the singlets are
1 × 1 = 1′ × 1′′ = 1,
1′ × 1′ = 1′′,
1′′ × 1′′ = 1′,
(A.10)
A.1 A4 irreducible representations product 95
and triplet multiplication rules are
[a b]1 = a1b1 + a2b2 + a3b3,
[a b]1′ = a1b1 + ω a2b2 + ω2 a3b3 ,
[a b]1′′ = a1b1 + ω2 a2b2 + ω a3b3 ,
[a b]31
= (a2b3, a3b1, a1b2) ,
[a b]32
= (a3b2, a1b3, a2b1) ,
(A.11)
where a = (a1, a2, a3), b = (b1, b2, b3) and ω = (1)1/3.

Appendix B
Radiative one-loop mass calculation
for Scotogenic model
Fig. B.1 Feynman diagram for the Scotogenic neutrino mass generation in mass eigenstates.
In Ma’s Scotogenic model, light neutrino masses arise from radiative correction to
neutrino propagator. This correction shifts the pole of the propagator and therefore the
physical mass of the neutrinos. The lowest order quantum correction in the Scotogenic
model is the one-loop process shown in Fig. 3.1. Such loop generation of neutrino masses is
guaranteed to be finite, though the superficial degree of divergence counting of the diagram
leads to believe that this diverges. The reason behind this is that after EWSB there is a
mass splitting between the neutral components of the scalar degree of freedom η0, each
contributing with opposite sign and thus the loop divergence cancels.
For the calculation of the loop, it is crucial to only use the mass eigenstates of the
diagram, as shown in Fig. B.1. In the following, we will denote the Yukawa couplings in
the Lagrangians in Eqs. (3.1) and (3.9) as Y ν
ij = Yij. From the diagram in Fig. B.1, it is
98 Radiative one-loop mass calculation for Scotogenic model
straightforward to see that the integral has the form:
−i Σij = −
∑
k
YikYjk
∫ d4k
(2π)2
[
i(/k + mNk
)
k2 − m2
Nk
i
(p − k)2 − m2
ηR
− i(/k + mNk
)
k2 − m2
Nk
i
(p − k)2 − m2
ηI
]
(B.1)
=
∑
k
Yik Yjk
∫ d4k
(2π)2
(/k + mNk
)
⎡
⎣
1
(k2 − m2
Nk
)
(
(p − k)2 − m2
ηR
)
− 1
(k2 − m2
Nk
)
(
(p − k)2 − m2
ηI
)
⎤
⎦ ,
(B.2)
where the iε prescription for the poles is assumed in all squared masses. Then, we have
been left with two one-loop integrals, one for the real and one for the imaginary part of η0,
where the minus sign in the second term of Eq. (B.1) comes from such splitting.
As we are interested in the mass correction, it is convenient to take zero total external
momentum. Then, p = 0 and Eq. (B.2) takes the form:
− i Σij =
∑
k
YikYjk Ik, (B.3)
where
Ik =
∫ d4k
(2π)4
(/k + mNk
)
[
m2
ηR
− m2
ηI
(k2 − m2
Nk
)(k2 − m2
ηR
)(k2 − m2
ηI
)
]
. (B.4)
In order to calculate this integral, we use the Feynman parameter prescription (see for
instance [169]), and combine the n propagators labelled as An appearing mn times in the
integral as:
1
Am1
1 Am2
2 ...Amn
n
=
Γ(m1 + m2 + ... + mn)
Γ(m1)Γ(m2)...Γ(mn)
∫ 1
0
dx1 xm1−1
1
∫ 1
0
dx2 xm2−1
2 ...
∫ 1
0
dxnxmn−1
× δ (1 − x1 − x2... − xn)
(x1A1 + x2A2 + ... + xnAn)m1+m2+...+mn
. (B.5)
99
In our case, the Feynman parametrisation technique yields to
Ik = (m2
ηR
− m2
ηI
)
∫ d4k
(2π)4
i (/k + mNk
)
[
Γ(3)
∫ 1
0
dx
∫ 1−x
0
dy
× 1
(
x(k2 − m2
Nk
) + y(k2 − m2
ηR
) + (1 − x − y)(k2 − m2
ηI
)
)3
⎤
⎥
⎦ . (B.6)
Rearranging the integral denominator, we have
Ik = 2(m2
ηR
− m2
ηI
)
∫ 1
0
dx
∫ 1−x
0
dy
∫ d4k
(2π)4
[
(/k + mNk
)
(k2 − Δ)3
]
, (B.7)
where
Δ = x (m2
Nk
− m2
ηI
) + y (m2
ηR
− m2
ηI
) + m2
ηI
.
Notice that the integral of k in Eq. (B.7) is zero by symmetry. The remaining integral
is performed using the Wick rotation: k0 = i l0, so that k2 = −l2 and d4k = −i d4l. In this
way, we can perform the integral in a four-dimensional spherical space as:
Ik = 2i
(m2
ηR
− m2
ηI
)mNk
(2π)4
∫ 1
0
dx
∫ 1−x
0
dy
∫
d4l
[
1
(l2 + Δ)3
]
(B.8)
= 2i
(m2
ηR
− m2
ηI
)mNk
(2π)4
∫ 1
0
dx
∫ 1−x
0
dy
∫
dΩ3
dl l3
Δ3 (1 + l2/Δ)3 (B.9)
= 2i
(m2
ηR
− m2
ηI
)mNk
(2π)4
∫ 1
0
dx
∫ 1−x
0
dy (2π2)
∫ dl l3
Δ3 (1 + l2/Δ)3 (B.10)
= i
(m2
ηR
− m2
ηI
)mNk
4(2π)2
∫ 1
0
dx
∫ 1−x
0
dy
1
x (m2
Nk
− m2
ηI
) + y (m2
ηR
− m2
ηI
) + m2
ηI
(B.11)
= i
mNk
4(2π)2
∫ 1
0
dx
1
x (m2
Nk
− m2
ηI
) + y (m2
ηR
− m2
ηI
) + m2
ηI
. (B.12)
The final integral, Eq. (B.12), can be performed by taking the derivative under the integral
sign leading to:
Ik = i
mNk
16π2
[
m2
ηR
m2
ηR
− m2
Nk
log
m2
ηR
m2
Nk
− m2
ηI
m2
ηI
− m2
Nk
log
m2
ηI
m2
Nk
]
. (B.13)
100 Radiative one-loop mass calculation for Scotogenic model
Finally, from the propagator correction Eq. (B.3), we have that the mass matrix for the
LH Majorana neutrinos is of the form −Σij = Mij. Then,
Mij =
∑
k
YikYjk
mNk
16π2
[
m2
ηR
m2
ηR
− m2
Nk
log
m2
ηR
m2
Nk
− m2
ηI
m2
ηI
− m2
Nk
log
m2
ηI
m2
Nk
]
. (B.14)
Appendix C
Oblique parameters for a radiative
Majorana neutrino mass generation
In this appendix, we give more details on the calculation of the electroweak parameters for
the model in chapter 3. Following the notation in [170, 144], the oblique parameters T and
S for a SM extension with n extra SU(2)L iso-doublets with hypercharge −1/2 are given
by:
T =
g2
(8π)2α m2
W
[
F
(
m2
η± , m2
η0
)
+ F
(
m2
η± , m2
ηA
)
− F
(
m2
η0
, m2
ηA
)
+3 sin2 θ
(
F
(
m2
W , m2
h1
)
− F
(
m2
Z , m2
h1
)
− F
(
m2
W , m2
h2
)
+ F
(
m2
Z , m2
h2
)
)
]
,(C.1)
and
S =
g2 sin2 θW
6(4π)2α
[
(
2 sin2 θW − 1
)2
G
(
m2
η± , m2
η± , m2
Z
)
+ G
(
m2
η0
, m2
ηA
, m2
Z
)
+ log
(
m2
η0
m2
ηA
/(m2
η±)2
)
+ sin2 θ
(
log
(
m2
h2
/m2
h1
)
+ Ĝ
(
m2
h2
, m2
Z
)
(C.2)
− Ĝ
(
m2
h1
, m2
Z
)
)
]
,
102 Oblique parameters for a radiative Majorana neutrino mass generation
where the functions F , G and Ĝ are defined as:
F (I, J) ≡
⎧
⎪
⎪
⎨
⎪
⎪
⎩
I + J
2
− IJ
I − J
log
I
J
, if I = J,
0, if I = J.
(C.3)
G (I, J, Q) ≡ −16
3
+
5 (I + J)
Q
− 2 (I − J)2
Q2
+
3
Q
[
I2 + J2
I − J
− I2 − J2
Q
+
(I − J)3
3Q2
]
log
I
J
+
r
Q3
f (t, r) . (C.4)
Ĝ (I, Q) ≡ −79
3
+ 9
I
Q
− 2
I2
Q2
+
(
−10 + 18
I
Q
− 6
I2
Q2
+
I3
Q3
− 9
I + Q
I − Q
)
log
I
Q
+
(
12 − 4
I
Q
+
I2
Q2
)
f (I, I2 − 4IQ)
Q
. (C.5)
and f is defined as
f (t, r) ≡
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
√
r ln
∣
∣
∣
∣
∣
t − √
r
t +
√
r
∣
∣
∣
∣
∣
, if r > 0,
0, if r = 0,
2
√
−r arctan
√
−r
t
, if r < 0.
(C.6)
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