neutrinos, symmetries and the origin of matter · in particular, gauge symmetries are at heart of...
TRANSCRIPT
Neutrinos, Symmetries and the Origin of Matter
João Tiago Neves Penedo
Thesis to obtain the Master of Science Degree in
Engineering Physics
Examination Committee
Chairperson: Prof.a Doutora Maria Teresa Haderer de la Peña Stadler
Supervisor: Prof. Doutor Filipe Rafael Joaquim
Members of the Comittee: Prof. Doutor Gustavo da Fonseca Castelo Branco
Prof. Doutor Ricardo Jorge Gonzalez Felipe
November 2013
“Each piece, or part, of the whole of nature is
always merely an approximation to the complete
truth, or the complete truth so far as we know it.
In fact, everything we know is only some
kind of approximation, because we know
that we do not know all the laws as yet.
Therefore, things must be learned only to be
unlearned again or, more likely, to be corrected.”
– Richard P. Feynman (1918-1988)
The Feynman Lectures on Physics, Vol. I
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Acknowledgements
This thesis is not a work of mine alone: behind the stage curtain, a larger cast hides. To them I thank
for not only helping me construct a symmetric work, in the Vitruvian sense, but also for keeping me sane
in the process (well, as sane as possible at least).
I would like to start by thanking my supervisor, Professor Filipe Joaquim, for his guidance, patience,
and incessant encouragement. Being one of the few great professors I had, he has been responsible for
introducing me to the world of particle physics research, the do’s and don’ts of the field, and for providing
all the assistance needed in solving problems and answering questions, big or small, without exception.
I am additionally indebted to Fundacao para a Ciencia e Tecnologia (FCT) and Centro de Fısica
Teorica de Partıculas (CFTP), thanking, in particular, the kindness of Claudia Romao, who has lent
a helping hand whenever needed, and the support of Professor Jorge Romao, who worked towards my
participation in the ICTP Summer School on Particle Physics, earlier this year. There, I was given the
chance to learn from the leading experts in the field, and would like to express my gratitude towards
Doctor Alejandro Ibarra for a most helpful lightning-discussion as well as a saving reference.
I would also like to thank my family for their support and encouragement, and for enabling my crazy
endeavors, without understanding, most of the time, what the heck I am doing.
I thank my friends for the chaotic yet enjoyable sequence of events one calls a physics course. A special
mention must be made to Pedro Boavida and Antonio Coutinho, to whom I thank for our spontaneous
(and crucial) discussions about physics, Copernicus and chameleons.
I end by expressing my thanks to Joao Loureiro, who was there even if two thousand kilometers away,
and to Sılvia Conde, for a superposition of all possible reasons.
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Este trabalho foi financiado pela Fundacao para a Ciencia e Tecnologia,
sob o projecto PTDC/FIS/102120/2008.
This work was supported by Fundacao para a Ciencia e Tecnologia,
under the grant PTDC/FIS/102120/2008.
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Resumo
As simetrias como leis de invariancia desempenham um papel fundamental na construcao de teo-
rias fısicas. Em particular, as simetrias de gauge estao na base do presente conhecimento do mundo
subatomico, que assenta no Modelo Padrao da fısica de partıculas. Apesar de repetido sucesso, este
modelo tem que ser necessariamente expandido a luz da existencia de massas e mistura de neutrinos.
Na presente dissertacao sao exploradas extensoes do Modelo Padrao baseadas no mecanismo seesaw
onde a supressao da massa dos neutrinos e naturalmente explicada. Massas de neutrinos nao nulas
conduzem a mistura leptonica, cuja estrutura se aproxima a um padrao tribimaximal, apontando para a
possıvel presenca de simetrias discretas na teoria a altas energias – como a invariancia sob transformacoes
do grupo A4, considerado neste trabalho.
O Modelo Padrao revela-se igualmente insuficiente na explicacao da assimetria barionica do Universo.
Nos modelos seesaw e possıvel gerar dinamicamente essa assimetria atraves dos decaimentos dos novos
estados pesados (fora de equilıbrio termico) mediante o mecanismo de leptogenese, cuja eficiencia e de-
terminada numericamente resolvendo o sistema de equacoes de Boltzmann adequado. Nesta dissertacao,
apresenta-se a analise de um modelo particular para violacao espontanea da simetria CP onde se explicam
as massas e mistura de neutrinos impondo uma simetria discreta A4. A implementacao do mecanismo
de leptogenese neste contexto e discutida em detalhe.
Palavras-chave: Assimetria barionica do Universo; Leptogenese; Massa e mistura
de neutrinos; Mecanismo seesaw; Simetrias; Violacao de CP
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Abstract
Symmetries, understood as laws of invariance, play a fundamental role in the development of physics.
In particular, gauge symmetries are at heart of our current understanding of the subatomic world, which
relies on the Standard Model of particle physics. Despite its repeated successes, this model must neces-
sarily be extended to accommodate the experimental observation of nonzero neutrino masses and mixing.
In this thesis, we explore seesaw extensions of the Standard Model, where heavy states mediate
neutrino mass generation and the smallness of these masses is naturally accounted for. Nonvanishing
neutrino masses allow for leptonic mixing, whose structure strongly differs from that of quark mixing.
The closeness of the lepton mixing matrix to the tribimaximal pattern points to the presence of discrete
symmetries in the underlying high-energy theory – such as invariance under transformations of the A4
group, considered in this work.
The Standard Model also fails to provide a satisfactory mechanism for the generation of the baryon
asymmetry of the Universe. A remarkable feature of the seesaw extensions is the possibility that the
out-of-equilibrium decays of the new heavy states are responsible for the dynamical generation of this
asymmetry. This corresponds to the leptogenesis mechanism, whose efficiency is here determined by
numerically solving a system of Boltzmann equations. Additionally, a particular model for spontaneous
leptonic CP violation is analysed where neutrino masses and mixing are explained imposing an A4 discrete
symmetry. The implementation of the leptogenesis mechanism in this context is discussed in detail.
Keywords: Baryon asymmetry of the Universe; CP violation; Leptogenesis; Neu-
trino masses and mixing; Seesaw mechanism; Symmetries
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Contents
Acknowledgements iii
Resumo vii
Abstract ix
List of Figures xiv
List of Tables xv
List of Abbreviations xvii
1 Symmetries and Asymmetries in Nature 1
1.1 Evolution of the Concept of Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Groups and Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Symmetry in Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.1 From Classical to Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.2 New Kinds of Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.3 The Discrete Symmetries C, P and T . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.4 Symmetry Breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 A Philosophical Interlude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.5 The Asymmetry of Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.5.1 Experimental Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.5.2 The Tuning of Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.5.3 The Possibility of a B-Symmetric Universe . . . . . . . . . . . . . . . . . . . . . . 16
2 The Standard Model of Particle Physics and (slightly) Beyond 17
2.1 Recap of the Electroweak Sector of the SM . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1.1 Neutral and Charged Electroweak Currents . . . . . . . . . . . . . . . . . . . . . . 19
2.1.2 The Higgs Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.1.3 Fermion Masses and Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2 Neutrinos Beyond the SM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.1 The Neutrino Mass Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.2 The Seesaw Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
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3 Lepton Mixing and Discrete Family Symmetries 35
3.1 Lepton Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Discrete Family Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2.1 Symmetries of the Mass Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2.2 Direct vs. Indirect Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3 An A4 Model with Spontaneous CP Violation . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3.1 Spontaneous CP Violation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.3.2 Neutrino Masses and Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3.3 Nonzero Reactor Neutrino Mixing Angle . . . . . . . . . . . . . . . . . . . . . . . . 47
4 Baryogenesis through Leptogenesis 49
4.1 Topics of Cosmology and Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.1.1 Cosmological Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.1.2 Equilibrium Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.1.3 Expansion, Entropy and Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . 52
4.1.4 Brief Thermal History of the Universe . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2 The Sakharov Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.3 Is the SM Enough? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.4 Thermal Leptogenesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.4.1 CPT, Unitarity and CP Asymmetries . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.5 Boltzmann Equation(s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.5.1 Two-body Decays and Inverse Decays . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.5.2 2↔ 2 Scatterings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5 Type II Seesaw Leptogenesis 69
5.1 Flavoured CP Asymmetries from Triplet Decays . . . . . . . . . . . . . . . . . . . . . . . 69
5.2 Boltzmann Equations for Type II Seesaw . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.3 Scattering Reaction Densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.4 Leptogenesis in an A4 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Conclusions 79
Bibliography 88
A Computing Diagrams with Majorana Fermions 89
B Clebsch-Gordan Coefficients for A4 91
B.1 General Description of the Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
B.2 Choice of an Explicit 3D Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
B.3 The Tensor Product Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
B.4 Computing the CGCs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
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List of Figures
1.1 Selected symmetry drawings of M. C. Escher (1941) . . . . . . . . . . . . . . . . . . . . . 1
1.2 Results of subjecting an artificial “quasi-lattice” based on a Penrose tiling to optical
diffraction (left), obtained by A. Mackay in 1982, and its physical analogue (right): elec-
tron diffraction patterns of an aluminium-based icosahedral quasicrystal, published by D.
Shechtman et al. in 1984 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.1 Mass hierarchy of the elementary fermions observed in Nature. Mass values and uncer-
tainties are obtained from J. Beringer et al. (Particle Data Group) 2012 and references
therein (light quarks present the highest relative mass uncertainties) . . . . . . . . . . . . 25
2.2 Exchange interactions which in the effective theory give rise to the Weinberg operator of
(2.45). Seesaw types I and III correspond to the exchange of fermion fields NR and ΣR,
respectively (left diagram), while the type II seesaw mechanism is implemented through
the exchange of scalar fields ∆ (right diagram) . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3 Vertex contributions from the interaction Lagrangian (2.51) for type I seesaw . . . . . . . 29
2.4 Vertex contributions for type II interactions relevant for effective neutrino mass generation 32
3.1 Depiction of lepton mixing for both a normally ordered and an inverted neutrino mass
spectrum, where the global fit data of Table 3.1 has been considered (left), to be compared
with the tribimaximal ansatz (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 Predicted values for neutrino masses as a function of the CP-violating angle β . . . . . . . 46
3.3 Scatter plot of the experimentally allowed regions in the(ε1, ε2
)plane (left), where exact
TBM is seen to be excluded, and corresponding regions of the(JCP, β
)plane (right) . . . 48
3.4 Values for the neutrinoless double beta decay parameter∣∣mee
∣∣, in the exact TBM and
perturbed cases, as a function of β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.1 Brief thermal history of the Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.2 Schematic representation of the vacuum structure of the electroweak theory (left) and
effective diagram for the transition between vacua (right) . . . . . . . . . . . . . . . . . . 57
4.3 Diagram for the expansion of a ‘true vacuum’ bubble . . . . . . . . . . . . . . . . . . . . . 58
4.4 Effect of electroweak sphalerons on the quantum numbers B and L . . . . . . . . . . . . . 60
4.5 Tree-level and one-loop diagrams for the process X → ` ` whose interference generates a
CP asymmetry when compared to the conjugate process . . . . . . . . . . . . . . . . . . . 61
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5.1 Tree-level diagrams for the decays of type II seesaw scalar triplets and one-loop diagrams
contributing to the decay process ∆i → `α `β . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.2 Scalar triplet interactions relevant to the BE out-of-equilibrium analysis, where one con-
siders the diagrams presented for decays, inverse decays and s- and t-channel scatterings
and their charge conjugates, as well as gauge scattering reactions . . . . . . . . . . . . . . 72
5.3 Contours of the (magnitude of the) maximum CP asymmetries in the decays of hierarchical
scalar triplets ∆1,2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.4 Scatter plot of the baryon asymmetry generated in randomly-chosen perturbed versions of
the model, for M1 = 1012 GeV (black) and M1 = 5× 1012 GeV (cyan) . . . . . . . . . . . 77
5.5 Reaction densities normalized to the product H(T )nγ(T ) (left) and evolution of the various
densities considered in the BE network (right) . . . . . . . . . . . . . . . . . . . . . . . . . 78
A.1 Rules for writing propagator, vertex and spinor contributions obtained from the article of
Denner et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
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List of Tables
1.1 Effect of some S4 permutations on an array of different circles . . . . . . . . . . . . . . . . 6
2.1 Number of parameters contained in complex matrices depending on their properties . . . 23
3.1 Global fit results for the three-neutrino oscillation parameters (mass differences, mixing
angles and Dirac phase) and for both ordering possibilities (see text) . . . . . . . . . . . . 36
3.2 Representation assignments of the various fields under the action of the groups A4, Z4,
and gauge SU(2)L×U(1)Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
B.1 Character table for the group A4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
B.2 Explicit decomposition of the elements of A4 in terms of s and t . . . . . . . . . . . . . . 92
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List of Abbreviations
BAU Baryon Asymmetry of the Universe
BBN Big-Bang Nucleosynthesis
BE Boltzmann (transport) Equation
CC Charged Current
CGC Clebsch-Gordan Coefficient
CP Charge Conjugation and Parity
EWPT Electroweak Phase Transition
EWSB Electroweak Symmetry Breaking
FRW Friedmann-Robertson-Walker (metric)
GUT Grand Unified Theory
NC Neutral Current
QCD Quantum Chromodynamics
QED Quantum Electrodynamics
QFT Quantum Field Theory
RIS Real Intermediate State
SM Standard Model
SMC Standard Model of Cosmology
SSB Spontaneous Symmetry Breaking
SUSY Supersymmetry
TBM Tribimaximal
VEV Vacuum Expectation Value
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Symmetries and
Asymmetries in Nature 1It was from his second visit to the Moorish palace of Alhambra (Granada, Spain), in 1936, that Maurits
C. Escher (1898-1972) drew inspiration to produce sketchbook after sketchbook of patterned drawings in
a style which he called “regular division of the plane” [1]. Unbound by Islamic artistic tradition, which
forbids the depiction of human and animal figures, this Dutch graphic artist, known for his graphical
paradoxes and representation of impossible worlds, was able to produce spectacular and intricate plane
filling motifs as the ones shown in Fig. 1.1.
(a) Symmetry drawing no. 45 (b) Symmetry drawing no. 67 (c) Symmetry drawing no. 88
Figure 1.1: Drawings of M. C. Escher (1941) which exhibit different types of geometrical symmetry. Aside
from translational symmetries, these patterns present (if one ignores colour): a) reflection and four-fold
rotational symmetries, b) glide reflection symmetry, and c) two-fold rotational symmetry.
All M.C. Escher works c© 2013 The M.C. Escher Company - the Netherlands.
All rights reserved. Used with permission. www.mcescher.com
Escher shared his early attempts at plane division with his brother, a geologist at the University of
Leiden, who referred him to the existing work of crystallographers and mathematicians [2]. These were
people who systematically studied the regularities of the flat worlds Escher was trying to recreate. The
artist’s “regular division of the plane” is associated to the mathematical concept of tessellation – which
corresponds to the tiling of a two-dimensional surface – and refers, in particular, to periodic tilings with
translational symmetry in two independent directions. He became especially interested in a 1924 article
by George Polya [3], in which the author proves that any of these tilings can be taken to belong to one
of 17 mathematical classes, corresponding to the so-called wallpaper groups.
1
Figure 1.2: Results of subjecting an artificial “quasi-lattice” based on a Penrose tiling to optical diffraction
(left), obtained by A. Mackay in 1982 [4], and its physical analogue (right): electron diffraction patterns
of an aluminium-based icosahedral quasicrystal, published by D. Shechtman et al. in 1984 [5]. Shechtman
was awarded the 2011 Chemistry Nobel prize for the discovery of quasicrystals.
A wallpaper group is defined by the possible operations one can apply to the aforementioned tilings
while leaving them invariant, i.e. indistinguishable from themselves prior to performing the operation.
This classification of tilings is thus based on the symmetries they possess. The operations are selected
from the set of Euclidean plane isometries, which comprises (aside from translations) rotations, reflections
and glide reflections. If the restriction of demanding translational symmetry in two independent directions
is removed, one might construct tilings which present no periodicity at all, such as Penrose tilings. Yet
may still present rotational and reflection symmetries.
In nature, one finds that crystals can be classified according to a three-dimensional generalization of
the wallpaper groups – the space or Federov groups – of which 230 exist. This classification, of great
use to crystallographers, is possible since perfect crystals present the property of translational invariance
in three independent directions in space. These systems are thus the par excellence example of how
symmetries are present in the physical world: the properties of crystals signal the presence of regularity
at the most fundamental level of matter.
There is, once more, the possibility of constructing nonperiodic physical structures, known as qua-
sicrystals, for which the demand of translational invariance has been lifted. These objects can possess
rotational symmetries not allowed1 in periodic crystals, such as eightfold [6], tenfold (see Fig. 1.2), and
twelvefold [7] rotational symmetries. Examples of symmetry in biological systems proliferate: consider,
for instance, the structure of honeycombs or the fivefold rotational symmetry than can be seen in a
horizontally cut apple. Humans also tend to link symmetry with beauty, be it associated with works of
architecture or the bilateral symmetry of a person’s face [8].
So far we have only considered a geometrical notion of symmetry, associated with regularity. This
corresponds to the layman’s understanding of the word and one might wonder how one got to it.
1In the 2D wallpaper groups and the 3D Federov groups, the only possible rotational symmetries are twofold, threefold,fourfold, and sixfold rotational symmetries.
2
1.1 Evolution of the Concept of Symmetry
The term ‘symmetry’ has its origins in the Ancient Greek word συµµετρια, which results from the
fusion of συν (with) and µετρoν (measure) [9]. As suggested by its etymology, this term was used to
represent a notion of commensurability, the possibility of measuring using a common standard, which
translates into the presence of integer-based proportion relations.
In addition to the mathematical meaning of commensurability, where no subjectivity is implied, the
word συµµετρια became generally associated with beauty, harmony and unity. This is the meaning
which Plato ascribes to the word while referring to the human body in his dialogue Timaeus (c. 360 BC).
It is also in this dialogue that the Greek philosopher theorizes about the nature of the classical elements
– fire, water, air, and earth – which are, based on their properties, associated to four out of five possible
convex regular polyhedra. These are the so-called Platonic solids and Plato’s association is based on the
beauty he sees in them. Jumping forward to the sixteenth century, we see a revival of the attempt of
using platonic solids to describe nature in Johannes Kepler’s Mysterium Cosmographicum (1596). The
solids are now used to describe the geometry of planetary orbits, furnishing Kepler with a reasonable
approximation of the ratios between the radii of orbits, which were first taken to be circular. Despite
Kepler’s appeal to harmony, there is, however, no occurrence of ‘symmetry’ in his work.
Tracing the evolution of the meaning of ‘symmetry’ has proven to be a difficult endeavour to historians
since one is tempted to reinterpret scientific documents in light of a modern view of the concept. G.
Hon and B. Goldstein discriminate between two parallel paths in the evolution of the term, namely a
mathematical path vs. an aesthetic one [10]. In fact, unlike the Greek case in which the two meanings
of συµµετρια were present in a unique word, in Latin these were separated into commensurabiles and
symmetria. The former term was used exclusively in a scientific context, allowing us to follow the
mathematical path, while the latter pertains to a general, aesthetic notion of being well-proportioned,
corresponding to an altogether different evolution. In the remainder of this section, one follows Hon and
Goldstein’s historiographical work.
Concerning the mathematical path, one recognizes the usage of συµµετρια (and derived forms) as
commensurability in ancient works such as Plato’s Theaetetus (c. 369 BC), Aristotle’s Nicomachean
Ethics (c. 350 BC), and Euclid’s Elements (c. 300 BC). Indeed, in the Latin editions of these works,
the term συµµετρια would be translated to commensurabiles. Such is the case with Isaac Barrow’s
(1630-1677) edition of Euclid’s Elements, published in 1655. Barrow sets apart the two meanings of
συµµετρια, showing that they were understood at the time.
Looking into the aesthetic path, where judgement is key, one finds that συµµετρια was used to
describe ‘proper’ proportion, be it in Plato’s Timaeus, where ‘proper’ refers to beauty, in Aristotle’s
Nicomachean Ethics, where certain occurrences of συµµετρια are associated with moderation, or in
Ptolemy’s Almagest (c. 150), where it is used to convey the idea of suitability. It was Vitruvius who
coined the Latin term symmetria to characterize an entity, such as a building or a machine, whose parts
are joined gracefully by imposing well-chosen proportion relations between the parts themselves and
between the parts and the whole. This is how Vitruvius’ transliteration of the Greek word is understood
3
in the context of his theory of architecture, presented in his De architectura (c. 15 BC): a property of
a beautiful, well-coordinated unity. Claude Perrault (1613-1688), who translated the De architectura to
French in 1673, separated Vitruvius’ use of symmetry, which he takes to simply pertain to proportion,
from the common meaning of the word in seventeenth-century France. This common usage was associated
to a kind of geometrical correspondence, such as that which relates the disposition of windows between
the left and right sides of a building’s facade, and it represents an underdeveloped version of what we
today understand as mirror symmetry.
Further uses of symmetry in scientific literature are found to be technical extensions of the concept
of Vitruvius, such as Carl Linnaeus’ (1707-1778) use of the word connected to the idea of functionality
in his classification of plant species or Hauy’s formulation of a ‘law of symmetry’ in the context of
crystallography. The aesthetic path turns into a scientific one. Rene-Just Hauy (1743-1822), a French
mineralogist responsible for important steps in the mathematization of crystallography, initiated the
systematic use of the term ‘symmetry’ in that field. He gives, however, no definition of the term, which
is taken from context to refer to the geometry of the crystal, specifically to the relative disposition of its
facets. What he calls his ‘law of symmetry’ corresponds to the fact that certain rotations of a crystal
yield identical views. This hints towards the modern, geometrical notion of symmetry presented in the
beginning of the current chapter.
Other rare occurrences of the word symmetry in eighteenth-century documents have been recorded
in works of physics such as those by Henri-Louis Duhamel du Monceau (1700-1782) and Gaspard Monge
(1746-1818) regarding ship design and construction. Duhamel du Monceau and Monge appeal, respec-
tively, to equilibrium and to a bilateral equivalence between both sides of a vessel, but give no definition
of symmetry.
Hon and Goldstein argue that it was not until the work of Adrien-Marie Legendre (1752-1833) that
a turning point occurred. His studies of solid geometry in 1794 led him to explicitly define symmetry as
a relation – not a property of a whole – between two solids which are, in our modern terminology, each
other’s mirror image. Solids related in this way cannot be rotated in three-dimensional space such that
they are made to coincide. Even though Legendre gave no reason for his choice of the word symmetry,
one might conceive a link between his definition and Perrault’s geometrical correspondence.
It was after Legendre’s introduction of symmetry in solid geometry that adoption of term gradually
flourished in distinct domains of science. Andre-Marie Ampere (1775-1836) imported Legendre’s defini-
tion into chemistry, while other scientists, who were responsible for the introduction of the concept in
their respective fields, gave distinct, parallel meanings to symmetry. Pierre-Simon Laplace (1749-1827)
introduced it in probability to characterize a sequence of well-ordered events (symmetry as regularity),
while Sylvestre Lacroix (1765-1843) took symmetry in algebra to signal invariability of a function re-
garding permutation of its roots (symmetry as invariance). There is a close relation between Lacroix’s
definition and the way ‘symmetrical function’ is understood in current technical usage. This plurality of
meanings was often taken in stride. Such is the case with Augustin-Louis Cauchy (1789-1857), who used
the term in both algebra and geometry without commenting on the different meanings it was given.
4
A unification of different meanings was accomplished through the group-theoretical definition of sym-
metry. Group theory owes its development to the work of researchers such as Evariste Galois (1811-1832),
Marius Sophus Lie (1842-1899), and Christian Felix Klein (1849-1925). It is concerned with the study of
groups, an algebraic structure to be explored in the following section. The concept of group allows one to
tie togheter the various meanings of symmetry through the idea of invariance under specified operations.
The symmetries of a geometrical figure are thus identified with the possibility of invariance of said figure
under, say, reflections or rotations. The operations need not be geometrical in nature, and thus symmetry
is generalized, in modern science, beyond geometry.
1.2 Groups and Symmetry
In the context of abstract algebra, a group (G, ·) is defined by both an operation ‘·’, designated by
group multiplication or product, and a set2 G of elements g. The product, not necessarily commutative,
of two group elements g ∈ G and g′ ∈ G can thus be denoted by g · g′ or simply by g g′. If the set G is
finite, the group itself is said to be finite or discrete. For an uncountable number of elements the group is
said to be continuous. If, additionally, the group product is commutative, the group is said to be Abelian.
To complete this definition, the elements of G must satisfy the four following axioms:
• Closure: If g, g′ ∈ G and g′′ = g g′, then g′′ ∈ G.
• Identity: There exists an element e ∈ G such that g e = e g = g for every g ∈ G.
• Inverse: For every g ∈ G, there exists an element g−1 ∈ G, such that g g−1 = g−1g = e.
• Associativity: For every g, g′, g′′ ∈ G, the relationship (g g′)g′′ = g(g′g′′) holds.
To relate this abstract definition with geometrical symmetry it suffices to say that the elements of the
group are made to correspond to geometrical operations – such as rotations, reflections or translations
– and the group multiplication denotes composition of operations (associativity is guaranteed). There
should be no confusion between the operations which belong to G and the operation of group multipli-
cation, necessary for defining a group. The group property of closure implies that the composition of
two geometrical operations is itself a geometrical operation. The identity element e corresponds to per-
forming no operation whatsoever, while the existence of an inverse element tells us that for a geometrical
operation, there exists another which cancels its effect.
By looking at one of Escher’s symmetry drawings of Fig. 1.1, one can identify the operations which
leave it invariant (see caption) and thus determine to which wallpaper group it belongs3. The observed
properties of invariance of the system (the drawing), namely the geometrical operations which leave it
indistinguishable from its previous self, define the group.
The converse way of thinking is to specify the group and look for how the system must ‘respond’
to fulfil the conditions of invariance imposed by each group element. Take, for example, the group of
permutations of n elements, denoted Sn (non-Abelian for n > 2). Each of the group elements is a
permutation operation on some abstract space of n objects, such as a queue of n people, or a list of the n
2Often, this set-operation pair description is omitted and the group is identified with the set that defines it.3In the language of the wallpaper groups, Escher’s drawings correspond (if one again ignores colour) to the groups known
as: a) p4g, b) pg, and c) p2.
5
e (12) (13) (24) (234) (432) (132) (1234) (1324) (12)(34) (13)(24) (14)(23)
Table 1.1: Effect of some elements of the permutation group S4 on an array of different circles. Permu-
tations are written in the cycle notation. For example, (123) reads “1 goes to 2, 2 goes to 3, and 3 goes
to 1”, and (13)(24) corresponds to “1 goes to 3, 3 goes to 1, 2 goes to 4, and 4 goes to 2”.
arguments of a function. If one is given the function f(x, y, z) = a1 x2 +a2 y
2 +a3 z2 +b xy+c xyz, where
x, y, z are real variables and ai, b, c ∈ C are taken to be constant, imposing invariance under permutation
of the arguments of f , i.e. imposing invariance under S3, translates into demanding a1 = a2 = a3 and
b = 0 (no restriction arises for c). No exchange of arguments can change the value of the function.
To proceed, it is useful to define what is known as a representation of the group – a mapping between
each element ofG and ‘something’ that carries out the operation. For a drawing of Escher, this ‘something’
could be the command “Take the picture and carefully rotate it by 90 degrees!”. In a mathematical
language, group elements g are mapped into matrices Ug which act as linear transformations on some
vector space. In order to define a representation, the matrices must obey the relation Ug Ug′ = Ug g′ .
The vectors which belong to the vector space correspond to parts of the system under study.
To understand this, take the example of the group S4. Like all permutation groups, S4 is a discrete
group: it has a finite number of elements, #S4 = 4! = 24. Each element of S4 can be denoted, in what is
called the cycle notation, by a collection of number sequences separated by parenthesis. Each sequence
of numbers determines which objects to cycle. This notation can be clarified by looking at Table 1.1,
where some S4 permutations are applied to an array of four different circles. Since the notation relies on
cycles, one has, for example, the equivalence (432) = (324) = (243). The full set can be written as:
S4 = e, (12), (13), (14), (23), (24), (34), (12)(34), (13)(24), (14)(23), (234), (243), (134),
(143), (124), (142), (123), (132), (1234), (1243), (1324), (1342), (1423), (1432).(1.1)
A natural way of representing the group is to map each element g ∈ S4 into a 4× 4 matrix Ug in the
way suggested by the following examples (as one might guess, Ue = 14×4):
U(12)(34) =
0 1 0 0
1 0 0 0
0 0 0 1
0 0 1 0
, U(1234) =
0 0 0 1
1 0 0 0
0 1 0 0
0 0 1 0
, U(243) =
1 0 0 0
0 0 1 0
0 0 0 1
0 1 0 0
. (1.2)
These matrices form a four-dimensional representation of S4 and act on the space (C4) spanned by
the following vectors:
|1〉 ≡
1
0
0
0
, |2〉 ≡
0
1
0
0
, |3〉 ≡
0
0
1
0
, |4〉 ≡
0
0
0
1
. (1.3)
6
The choice of representation matrices of Eq. (1.2) becomes clear if one considers their action on the
vectors of (1.3). In looking for an object which is invariant under the group S4, one finds that multiples
of the vector |1〉+ |2〉+ |3〉+ |4〉 are clearly left unchanged after the action of any group element4, which
is to say after |i〉 → Ug |i〉, ∀ g ∈ S4. Other invariant expressions can be constructed, a simple example
being the scalar 〈y|x〉 = y∗i xi, where |x〉, |y〉 ∈ C4, i.e. |x〉 = xi|i〉, |y〉 = yi|i〉 and xi, yi ∈ C. This is
so since U†g = UTg = (Ug)
−1 = Ug−1 , and so 〈y|x〉 → 〈y|U†gUg|x〉 = 〈y|U(g−1g)|x〉 = 〈y|Ue|x〉 = 〈y|x〉.The operations act on parts of the system and, for a symmetric one, they cancel out. In fact, the above
example of a S3-symmetric function can be recast in this new language of representations. One rewrites
the function as f(x, y, z) = Aij rirj + Cijk rirjrk, with ~r = (x, y, z), the only nonzero entries of Aij and
Cijk being Aii = a for each i and C123 = c. Suppose now that ~r transforms according to5 the natural 3D
representation of S3, meaning that under the action of a certain g ∈ S3, ~r transforms as ri → (U′g)ij rj .
Here, the matrices U′g form a representation of S3 analogous to the one given for S4 in (1.2). One notices
that, given the structure of Aij and Cijk, the function remains unchanged under the action of any group
element, as expected.
A remark should be made regarding group construction. It is possible to take a group G and reduce the
set that defines it in such a way that the remaining structure H is still a group – H is said to be a subgroup
of G. S3 is a subgroup of S4 obtained by restricting ourselves to three objects instead of four, removing
all permutations which would affect that fourth object. Another subgroup of interest to us is A4, which is
obtained from S4 by keeping only elements, denoted even permutations, which can be decomposed into a
product of an even number of ‘pair switches’ – permutations which switch only two objects. Any element
of Sn can be decomposed into a product of ‘pair switches’: e.g. (1234) = (14) · (13) · (12), where we stress
that the present convention implies combining permutations from right to left (first we apply (12), then
(13), and finally (14)). The full set of even permutations of four elements is (#An = #Sn/2⇒ #A4 = 12):
A4 = e, (234), (243), (134), (143), (124), (142), (123), (132), (12)(34), (13)(24), (14)(23). (1.4)
To give an example concerning continuous groups, consider SO(n), the group of rotations in n dimen-
sions, of which SO(m) with m < n is trivially a subgroup (rotations in a plane can be seen as a subset
of rotations in 3D space). These groups are naturally represented by orthogonal rotation matrices.
One can also build groups by joining rather than reducing them. Starting from two groups G1 and
G2, one can construct a third group by considering their direct product, G3 = G1×G2. Here, the set G3
is indeed constructed through the Cartesian product of the others: each g3 can be identified by a pair
(g1, g2) and one has g3 · g′3 = (g1 · g′1, g2 · g′2). Demanding invariance of a system under G3 corresponds to
independently demanding its invariance under G1 and G2.
4This means that there is a subspace of C4 which is invariant under the action of these matrices, and thus there existsa basis in which all matrices Ug take the form:
Ug =
1 [Ug
] .
The matrices Ug correspond to a 3D representation in their own right, and one has thus decomposed (through a change ofbasis) our natural 4D representation into two of smaller dimension. When further reduction is impossible, one is left withso-called irreducible representations (or irreps.) along the diagonal.
5It is common, yet misleading, to phrase this as ~r ‘is in’ or ‘belongs to’ the representation under discussion.
7
To give an example of how the direct product works, consider the Abelian group Zn, which corresponds
to the set a, a2, . . . , an−1, an = e, where the relations between group elements are made explicit. The
direct product Z2 × Z2 produces a group with elements (e1, e2) ≡ e, (a, e2) ≡ a, (e1, b) ≡ b, (a, b) ≡ ab,where one has denoted the elements of the first Z2 by e1, a and the elements of the second Z2 by
e2, b. Although this group presents four elements it is not isomorphic6 to Z4, since a2 = b2 = ab2 = e,
corresponding to the only other possible group with four elements, the Klein group K ≡ Z2 × Z2.
1.3 Symmetry in Physics
1.3.1 From Classical to Quantum Mechanics
Symmetry, taken henceforth to signify invariance under specified operations, has been instrumental
to the development of modern physics. This group-theoretical notion was first imported from crys-
tallography into physics thanks to Pierre Curie who, citing the work of Auguste Bravais and Evgraf
Federov, investigated the connection between symmetries and physical phenomena in his 1894 paper Sur
la Symetrie dans les Phenomenes Physiques [11]. In this work, Curie formulated what is now known as
Curie’s Principle, which recognizes that the symmetries of a physical medium restrict which phenomena
can occur. Hence, to allow certain phenomena, the presence of certain asymmetries is required.
By looking at what one considers to be the laws of nature, one can identify symmetries of nature (or,
as Eugene Wigner called them, laws of invariance). According to Wigner [12], it was Einstein’s work on
special relativity in 1905 that brought the “reversal of a trend”: instead of reading off laws of invariance
from laws of nature, it became natural to establish laws of invariance and from there go on to derive
the laws of nature. Indeed, special relativity admits invariance under the Poincare group (also called
the inhomogeneous Lorentz group) which comprises Lorentz boosts, translations in space, translations in
time, and spatial rotations7. One can regard these invariances as a fundamental part of the structure of
the theory, the last three corresponding respectively to homogeneity, uniformity in time, and isotropy of
space (Wigner’s “older principles of invariance”).
Laws of invariance in theories described by the Lagrangian formalism can be associated with the
conservation of a physical quantity through Emmy Noether’s first theorem (proved in 1915 and published
in 1918 [13]). Invariance under translations in space is thus linked to the conservation of linear momen-
tum, invariance under translations in time to the conservation of energy, and invariance under spatial
rotations to the conservation of angular momentum. Additionally, an interesting connection between
symmetries and non-observable quantities is given by T.D. Lee [14]. Invariance under space translations,
time translations, or rotations implies that no absolute, preferred position, time, or spatial direction can
be observed, respectively.
6Two groups (G, ·) and (G′, ?) are said to be isomorphic, (G, ·) ∼= (G′, ?) if there exists a map ϕ : G→ G′ such that, forevery g1, g2 ∈ G, both ϕ(g1 · g2) = ϕ(g1) ? ϕ(g2) and ϕ(g1) = ϕ(g2) ⇒ g1 = g2 hold. An isomorphism denotes a strongequivalence which translates into saying that the two groups are essentially the same.
7Technically, we are restricting ourselves to transformations Λ which can be connected continuously to the group identity,ignoring time and space inversions for which det Λ = −1. Although these are isometries of Minkowski space, they mightnot be symmetries of a quantum field theory.
8
The advent of (non-relativistic) quantum mechanics brought with it the study of symmetries in a
novel context. Wigner pioneered this study, introducing his eponymous theorem (1931) [15] which states
that symmetry transformations U must be either unitary or antiunitary. This can be seen by taking
quantum states |φi〉, which transform as |φi〉 → |φ′i〉 = |Uφi〉, and noticing that by demanding |〈φ′i|φ′j〉| =|〈φi|φj〉| ∀ i, j one obtains that either 〈Uφi|Uφj〉 = 〈φi|φj〉 (U is a unitary operator) or 〈Uφi|Uφj〉 =
〈φj |φi〉 (U is an antiunitary, nonlinear operator). For infinitesimal transformations, parametrized as
U(ε) = 1 + i ε G (G is denoted the transformation generator), demanding unitarity – the majority
of operators of interest are unitary – implies that the generator is Hermitian, G† = G, allowing it
to be an observable. In the limit of a finite transformation, U(α) = limn→+∞ U(α/n)n, one obtains
U(α) = exp(i α G). In either case, the transformation operator U must commute with the system’s
Hamiltonian, [U , H] = 0, since a proper symmetry should not be spoilt by dynamics. This means that
G also commutes with H, which can be seen in Heisenberg’s representation to directly imply that G is a
conserved quantity, in line with Noether’s theorem.
1.3.2 New Kinds of Symmetry
A shift from spacetime to other kinds of symmetries eventually occurred. Such new symmetries
included permutation symmetry, which was introduced by Heisenberg and pertains to the indistinguisha-
bility of particles, the charge conjugation symmetry C (such discrete symmetries will be addressed in
the following section), and the so-called internal symmetries, such as isospin symmetry. Also due to
Heisenberg, isospin symmetry refers to the invariance of strong interactions when one transforms the
doublet (proton,neutron)T through an element of SU(2). U(n) is the (continuous) group of n×n unitary
matrices and SU(n) is the subgroup of U(n) obtained by keeping only matrices with unit determinant.
These transformations represent rotations in an abstract, ‘internal’ space. Disregarding electromagnetic
interactions and small mass differences, the validity of isospin symmetry allows one to consider protons
and neutrons as different states of the same particle – the nucleon. SU(2) rotations mix these two states
while leaving strong interactions unchanged. It is worth noting that isospin transformations are inherently
global: the same SU(2) rotation is applied in all points of space at the same time [16].
A generalization can be made from global to local symmetries, where the above abstract rotations can
be chosen to differ from point to point. Field theories which are based on invariance under internal, local
transformations are called gauge theories. In this context, the transformations themselves are denoted
gauge transformations. One of the simplest examples of a gauge theory is quantum electrodynamics
(QED), the theory which describes the interactions between light and electrically charged matter. The
corresponding gauge symmetry is U(1) or local phase invariance, which corresponds to admitting invari-
ance under transformations of the charged matter fields of the form ψ(x)→ ei q α(x) ψ(x), where q is the
electric charge associated with the field and α(x) is an arbitrary function of space and time. The matter
fields are said to belong to the defining representation of U(1). The introduction of a so-called gauge
boson with certain transformation properties, in this case the photon field, is crucial to ensure gauge
invariance of the QED Lagrangian. The idea of gauge invariance was first put forward by Hermann Weyl
9
in 1918 [17], as an (ultimately unfruitful) extension of Einstein’s work on general relativity8. It was only
in 1929 that Weyl established a connection between electromagnetism and local phase invariance [18].
Field theories based on local invariance under the non-Abelian gauge group SU(n) are known as
Yang-Mills theories and are at the heart of our current understanding of the subatomic world. They owe
their name to C. N. Yang and Robert Mills who, in 1954, explored the consequences of turning isospin
symmetry into a local symmetry [19]. In such a transition, several incompatibilities with experimental
evidence arise, namely the appearance of several gauge bosons which are massless, leading to forces with
an infinite range, and self-interacting, due to the non-Abelian character of the underlying theory [9, 16].
In 1961, S. Glashow put forward a proposal for the unification of the electromagnetic and weak
interactions based on the enlarged gauge group SU(2)×U(1) [20]. In this electroweak theory, whose
renormalizability (which translates into its calculability) was an issue, mass terms for the gauge bosons
had to be put by hand, explicitly invalidating the symmetry. Glashow’s theory was later independently
extended by S. Weinberg [21] and A. Salam [22] to include the Higgs mechanism (see Section 1.3.4).
Proof that this new theory was renormalizable was given by Gerard ’t Hooft in 1971 [23], while working
under Martinus Veltman. For their scientific contributions, Glashow, Weinberg and Salam were awarded
the 1979 Nobel Prize in Physics, while Veltman and ’t Hooft were similarly honoured in 1999.
The idea of invariance under a group of gauge transformations is a fruitful one. Quantum chromo-
dynamics, the standard theory of the strong interaction, is a gauge theory based on the group SU(3).
Electroweak and strong interactions are unified in the Standard Model of particle physics, which will be
presented in the following chapter. Additional examples of the use of symmetries in modern physics in-
clude grand unified theories (GUTs), based on groups which include the Standard Model as a subgroup,
supersymmetry (SUSY), where a symmetry between bosons and fermions exists, and discrete flavour
symmetries, to be explored in Chapter 3.
1.3.3 The Discrete Symmetries C, P and T
In quantum field theory (QFT), a special status is given to the discrete symmetries under the oper-
ations of parity (P), time reversal (T) and charge conjugation (C). The first two operations correspond
to spacetime symmetries already present in classical mechanics which were imported by Wigner into the
quantum context. The last operation, charge conjugation, presents no classical analogue. In light of
Wigner’s theorem, P and C correspond to unitary operators, while T can only be implemented as an
antiunitary operator [24].
A parity transformation corresponds to an inversion of all three spatial coordinates, (x, y, z) →(−x,−y,−z). This inversion can also be obtained by changing the sign of one coordinate and per-
forming a 180o rotation, hence the association of parity to reflections and mirror symmetry. A time
reversal transformation corresponds to a change in the sign of the time coordinate, t→ −t. Both parity
and time reversal correspond to improper Lorentz transformations (see Footnote 7).
8Weyl considered the possibility of arbitrary and local rescalings of the spacetime metric, which correspond to the changeof a local unit length or gauge, hence the current usage of the latter term [9].
10
Concerning parity, one might ask what goes on the other side of the mirror. As Lewis Carroll’s Alice
puts it, “things go the other way”: a right hand is converted into a left hand and vice-versa. One is
thus confronted with the concept of chirality, which corresponds to Legendre’s use of the word symmetry,
presented in Section 1.1. A object which differs from its mirror image is said to be chiral. The question is
whether or not nature cares about chirality at all. In fact, while neither humans nor cars are bilaterally
symmetric, there seems to be no reason to expect that their mirror images9, with inverted organs and
engines, represent an impossible biology or a faulty vehicle. The chirality of people and cars are considered
to be accidents of evolution and design [26] and parity is still expected to be a symmetry of microphysics.
However, reality defies intuition: parity is not a symmetry of weak interactions. The idea that this
might be so was put forward by T. D. Lee and C. N. Yang in 1956 [27], who proposed an experimental
test to parity by considering the beta decay of cobalt-60. This test, performed in 1957 by C. S. Wu et
al. [28], confirmed their hypothesis: there is a preferred direction for the emission of β radiation.
To clarify what is meant by saying that parity is not a symmetry of nature or, equivalently, that
parity symmetry is violated, consider the following dialogue:
Alice: Imagine I have a ball rotating ‘clockwise’ and invert the axes. I then get a ball rotating
‘counterclockwise’. Is this what parity violation means?
Bob: No, not at all. That’s just a matter of description.
Alice: How so?
Bob: Well, the ball is rotating in a certain way. Whether it is clockwise or counterclockwise
depends on your perspective, on how you define these words.
Alice: I see. Then what does parity violation mean?
Bob: Simple – in its extreme version it means that it is not possible to have the ball rotate in
the other direction. Nature just doesn’t allow it.
Alice: So in a frame where I say it rotates clockwise, no counterclockwise balls can be seen?
Bob: And vice-versa. Exactly.
Regarding time reversal, few observed macroscopic phenomena, like a performance of Bach’s crab
canon, exhibit a symmetry under such operation. However, the irreversibility of processes like burning a
piece of firewood answers to a thermodynamical arrow of time, which is dependent on macroscopically
probable configurations of a system and does not imply microscopical irreversability. In fact, the laws of
classical mechanics are T-symmetric. Nevertheless, nature is not classical, but quantum, and one begs the
question of whether time reversal is a symmetry of the laws of physics. Indeed, compelling experimental
evidence that time reversal symmetry is violated has been found in B-meson decay chains by the BaBar
collaboration [29]. As we shall briefly see, there is a fundamental link between T, P and C.
Charge conjugation is defined in the context of relativistic quantum mechanics and refers to the
exchange of particles and antiparticles. The latter correspond to the physically meaningful states resulting
from reinterpreting the negative energy solutions of the Dirac equation as positive energy solutions with
opposite U(1) charges. Like parity, charge conjugation is a symmetry of electromagnetic and strong
9To consider the mirror image of the world, one must go as far as the molecular level, since certain molecules, likesugars or gasoline, are chiral. The interesting phenomenon of tunnelling between molecular chiral states was investigatedby Friedrich Hund in 1927 [25].
11
interactions, but not of weak interactions. In general, both C-symmetry and P-symmetry are violated
by the same processes while the product CP of these operations seems to be a valid symmetry. This
suggests that it is CP and not C or P individually that constitutes a symmetry of the physical world.
Alas, this is not strictly true: in the neutral kaon system – the study of which owes much to the work
of Murray Gell-Mann and Abraham Pais [30] – a small but nonzero departure from CP-symmetry was
found in 1964 [31] and earned James Cronin and Val Fitch the 1980 Nobel Prize in Physics.
There is still hope for salvaging the fundamental role of the above discrete symmetries in physics if one
considers the CPT theorem, whose discovery and proof in the 1950s is credited to J. Schwinger [32], W.
Pauli [33], J. Bell [34], and G. Luders [35]. This theorem states that a combination of the three presented
discrete symmetries should be a symmetry of any reasonable quantum field theory, i.e. a theory which
has Lorentz invariance, positive energy and local causality [36]. Thus, since CPT is conserved10, CP
violation and T violation are fundamentally connected: one demands the other. So far, no deviation
from CPT-symmetry has been observed [29,37].
1.3.4 Symmetry Breaking
Symmetries can be defined as either exact, meaning unconditional validity, approximate, meaning valid
only under certain conditions, or broken [9]. Symmetry breaking refers to the destruction of symmetry,
the transition (even if abstract) between a situation where the invariance exists to one where it does not.
This ‘break’ can be said to occur explicitly or spontaneously. Let us first consider the former case.
Explicit symmetry breaking refers to a destruction of symmetry that can be traced to the physical
law, in particular, to the presence of Lagrangian terms which spoil the invariance. These terms can arise
either by construction, anomalously, or through higher-order effects. An example of symmetry-breaking
terms which arise by construction corresponds to the parity violating structure of weak interactions.
The transition from a classical description to a quantum one can be responsible for the appearance
of so-called anomalous terms in the Lagrangian which can arise, for example, from the regularization
procedure. Finally, symmetry breaking terms may arise due to non-renormalizable effects. To understand
this, consider a field theory which provides an effective description of reality, meaning it approximates
a broader theory at low energies: symmetries present in the effective theory might not be symmetries
of the high-energy theory. Symmetries which, although not postulated, are found to be present in the
theory and can be broken by quantum corrections or non-renormalizable effects are dubbed accidental.
Accidental symmetries of the SM include the global baryon number (B) and lepton flavour number (Li)
symmetries. These are anomalously broken (see Section 4.3), while B−L (L =∑i Li) is not. However, in
seesaw extensions of the SM (see Section 2.2), the accidental B−L is broken by the non-renormalizable
Majorana mass terms for neutrinos, which arise from the interactions of heavy states in the high-energy
theory.
Spontaneous symmetry breaking (SSB), in turn, corresponds not to an actual destruction of the sym-
metries of the laws but to their hiding, since it is the lowest energy state of the system (or vacuum state,
10This is a blatant abuse of language. To say, for example, that CP is (not) conserved means simply that nature is (not)CP-symmetric. There is no reference to a conserved quantity, but to a symmetry which is maintained.
12
|0〉) which presents an asymmetry11. Following M. Guidry [38], one can classify broken symmetry sys-
tems according to whether U |0〉 = |0〉 (Wigner mode), U |0〉 6= |0〉 and the symmetry is global (Goldstone
mode), or U |0〉 6= |0〉 and the symmetry is local (Higgs mode), U being the operator which realizes the
symmetry operation. The first case may correspond to the above explicit breaking examples while the
last two correspond to spontaneous symmetry breaking scenarios.
Classically, one can conceive of situations in which the choice of the lower energy state spoils existing
symmetries: the fall of a vertically held pole spontaneously breaks rotational symmetry as a particular
direction is chosen out of the existing infinite possibilities. One can see that the symmetry is hidden and
not destroyed because it is still present in the full set of solutions. Likewise, if the pole is constrained to
move in a plane, there are only two possible ground states and the choice of any of them spontaneously
breaks the symmetry. In quantum systems, however, spontaneous symmetry breaking will not occur
if the number of degenerate ground states is finite, since it is possible to construct a state from their
superposition [39]. The concept is only applicable to idealized infinite systems, such as a ferromagnet.
If a ferromagnetic material is heated above the Curie (critical) temperature, Tc, no preferred ori-
entation for the magnetic dipoles exists, resulting in zero net magnetization. However, as soon as the
temperature drops below Tc, the system transitions to a ground state where a net magnetization develops
in one of the infinite possible directions. A link to Curie’s principle is here readily found: it is only the
asymmetry of the situation that allows the phenomenon, i.e. the appearance of magnetization in the
absence of an applied magnetic field.
An important result which arises in the context of SSB is Goldstone’s Theorem [40], which refers to
the appearance of massless bosons – termed (Nambu-)Goldstone bosons – when a continuous symmetry
is spontaneously broken. The number of such bosons matches the number of generators12 of the broken
continuous group. Goldstone himself remarks that although “a method of losing symmetry is [...] highly
desirable in elementary particle theory”, there seems to be no way to avoid in this context the introduction
of “non-existent massless bosons”. A solution to this unphysical problem is found in the Higgs mechanism,
which owes its origins to the work of P. Higgs [43, 44], R. Brout and F. Englert [45], and G. Guralnik,
C. Hagen, and T. Kibble [46]. This mechanism consists in the appearance of gauge boson mass terms
when the gauge symmetry of the theory is spontaneously broken, thanks to the Goldstone bosons which
one expects to arise from the break. These unphysical massless bosons are ‘absorbed’ as mass degrees
of freedom by the previously massless gauge bosons. The Higgs mechanism is a crucial feature of the
Standard Model of particle physics, since it allows the generation of mass terms for fermions and gauge
bosons without explicitly compromising the underlying symmetry group. The recent discovery by the
ATLAS and CMS collaborations [47, 48] of a boson with a mass of 125.9 ± 0.4 GeV [37], compatible
with the Standard Model Higgs particle (which corresponds to an unabsorbed degree of freedom), will
certainly allow for a deeper look into the question of the origin of particle mass.
11Thus, unlike what one might suppose, the elliptical motion of planets as opposed to the isotropic character of Newton’slaw of universal gravitation does not constitute an example of spontaneous symmetry breaking.
12One might regard the group Zn = e, a, a2, . . . , an−1 as being generated by powers of one single element (to wit, a).Similarly, the elements of a continuous (Lie) group which lie infinitesimally close to the identity element form a vector space– tangent to the manifold which the group defines – spanned by basis vectors which are called the group generators [41].Exponentiation of these infinitesimal elements yields the remainder of the group (recall Section 1.3.1). In special cases,which will be of no concern to us, the number of Goldstone bosons might be less than the number of group generators [42].
13
1.4 A Philosophical Interlude
According to E. Castellani [49], symmetries in physical theories play four separate roles. A classifica-
tory role is easily identified in the crystallographic enterprise as well as in the classification of elementary
particles, an example being the work of Wigner on the representation theory of the Poincare group [50].
Symmetries also possess a normative role, in the sense that they regulate the form of the theory, as well
as a unifying role, present in the construction of theories, such as GUTs, which seek to join the funda-
mental interactions under a simple symmetry group. Lastly, symmetries can be attributed an explanatory
role, as they are taken to be fundamental principles which dictate how nature must behave. A notion
of simplification is transversal to the four roles. According to Curie’s principle, it is asymmetry which
allows a diversity of phenomena, which is to say, complexity.
The normative and explanatory roles lead to a methodological aspect: model building in modern
particle physics often relies on postulating the presence of certain symmetry properties, as well as the
breaking of these invariances, determining which Lagrangian terms are allowed. The increasingly central
role which has been given to symmetries is not without justification: both the prediction of the Ω−
particle, based on an incomplete classification of baryons and mesons (the Eightfold Way of Gell-Mann
and Ne’eman [51]), and its subsequent discovery, along with the prediction and discovery of the W and
Z bosons, mark extraordinary successes of the use of symmetry in physics. This predictive power seems
to imply that symmetries as a basis for the description of the physical world are ‘here to stay’13.
It is important to emphasize the epistemological character of symmetries: as Wigner points out [12],
the artificial (but fruitful) division of reality into initial conditions and laws of nature would not be
possible in the absence of invariace under spacetime displacements. Physics would then differ from place
to place and time to time, compromising the way scientific knowledge is obtained. There is also a close
connection between symmetries and objectivity. Lorentz invariance, for instance, establishes a physical
equivalence between different observers, with different perspectives. Equivalence renders conventions
irrelevant and only that which is invariant under the symmetry group is considered physical.
The effectiveness of symmetry in physics is analysed by P. Kosso [9], who makes a case against any
association of symmetry with design, stating that such a link is incompatible with the objective nature of
symmetries: these are introduced, in part, to remove dependences on the decisions of a conscient observer
from the physical laws. To end this interlude, one once again turns to Wigner, who ponders about the
unreasonable effectiveness of mathematics in the study of natural sciences [52]. In his words, “the miracle
of the appropriateness of [...] mathematics for the formulation of the laws of physics is a wonderful gift
which we neither understand nor deserve”.
13An interesting connection is established by Wigner [12] concerning the relationship between physical laws and eventsand that of symmetries and laws. Were we to know all future events, the correlations between them (the laws) wouldbecome unnecessary. In the same way, the full knowledge of the laws of nature (if such a thing is possible) would maketheir symmetry properties a mere curiosity.
14
1.5 The Asymmetry of Existence
We live in an apparently biased Universe in which there is, as far as one can see [53], a clear preference
for the presence of matter over antimatter. One of the greatest challenges of modern physics is to explain
the observed asymmetry (in the state of the system, not in the laws which govern it) in light of our present
knowledge of cosmology and particle physics. How could this imbalance come to be? One assumes that
it is either a consequence of the initial conditions of the Universe, i.e. an accident, or it is dynamically
generated as the Universe cools down. The latter hypothesis calls for a mechanism – baryogenesis – that
produces an excess of baryons over antibaryons at some point in the past and, therefore, allows for our
own existence.
1.5.1 Experimental Evidence
Although antimatter has been produced and studied, no primordial antimatter has been detected in
the observable Universe as of today. Local evidence for the baryon asymmetry of the Universe (BAU)
includes the scarcity of antimatter in the Earth, the absence of γ-ray production due to solar wind, and
the successful landing of planetary probes in the planets of our solar system.
Cosmic rays offer an additional probe into far away regions of the Universe: nonvanishing ratios
between the number of cosmic antiprotons and protons have been measured, the magnitude of which is
consistent with secondary proton production in collisions of cosmic rays with matter in the interstellar
medium, in processes such as p + p → 3p + p [54]. Additionally, no antinuclei such as antihelium (He)
have been found in cosmic radiation [55]. In short, the cosmic ray data is consistent with the absence of
large structures of antimatter in the Universe. An even more compelling evidence for the absence of such
structures comes from the missing γ-ray background one would expect from the collision of galaxies and
anti-galaxies, at the hypothetical boundaries of matter-antimatter domains.
1.5.2 The Tuning of Initial Conditions
One turns now to the problem of whether the observed BAU was generated dynamically or accidentally.
The reason why the accidental asymmetry scenario, based on proper initial conditions, is usually rejected
is twofold. On the one hand, generating the observed asymmetry would require one extra quark for every
∼ 107 antiquarks in the early Universe [56]. This presents itself as a problem of naturalness, and one
usually avoids this much fine-tuning. On the other hand, a constraint arises if the effects of inflation
are considered. Inflation corresponds to a rapid expansion of the scale factor in the early Universe (see
Section 4.1.1) and is predicted to exponentially dilute any initial asymmetry.
In the dynamical case, the symmetry violating processes should be weak enough in order to produce
the observed values of the BAU, which can be quantified in the present epoch through the parameter η,
defined as the ratio between the number density of baryonic number, nB, and that of photons, nγ :
η ≡ nB
nγ≡ nb − nb
nγ' nbnγ. (1.5)
15
The quantity nB is itself defined as the difference between the number densities of baryons (nb) and
antibaryons (nb).
Apart from the baryon-antibaryon imbalance, the value of η also signals the imbalance between matter
and radiation in the present-day Universe. The abundances of light elements and the anisotropies of the
cosmic microwave background (CMB) constrain the value of η independently. One verifies that both
conditions give compatible values for the BAU parameter, which when combined give [37]:
η = (6.19± 0.15)× 10−10. (1.6)
An equivalent description is given by the quantity YB, corresponding to the ratio between nB and the
current entropy density of the Universe. The approximate relationship YB ' η/7.04 holds.
1.5.3 The Possibility of a B-Symmetric Universe
The observation of matter dominance does not seem to exclude the existence of large quantities of
antimatter in far away regions of the Universe that would effectively make it baryon symmetric. Let
us assume a B-symmetric Universe, with any possible initial asymmetry being washed out by inflation.
One can then compute baryon and antibaryon densities at freeze-out – the time when the expansion rate
of the Universe, measured by the Hubble parameter H(t), surpasses the rate Γ of nucleon-antinucleon
annihilation (for details, see Section 4.1). These densities will coincide due to B-symmetry of the laws.
The temperature at freeze-out (H(T ) ∼ Γ) can be estimated as T ∼ 20 MeV [57], well below the nucleon
mass, mN ∼ 1 GeV, for which the following relation holds (cf. Eq. (4.17)):
nbnγ
=nbnγ∼(mN
T
)3/2
exp(−mN/T ). (1.7)
An unreasonable value of nb/nγ ∼ 10−19 is thus obtained at freeze-out. One concludes that some
mechanism must have segregated matter and antimatter when nb/nγ ∼ 10−10, which corresponds to
T ∼ 40 MeV, or earlier. However, such mechanism would need to operate at the scale of the observable
Universe and, at this time, the horizon (causally connected region) only contained a negligible amount
of matter (∼ 10−7M) [54]. Therefore one cannot avoid the conclusion that the Universe must have
possessed a baryon asymmetry already at early times.
To understand the creation of this asymmetry, one has to go beyond the realms of pure cosmology and
consider the role that particle physics may have played in its production. The following chapter is thus
dedicated to the Standard Model of particle physics and to extensions which can account for neutrino
masses, whose smallness will be a concern. Their presence allows for lepton mixing, which will be discussed
in Chapter 3. A compelling case was presented for a dynamically generated baryon asymmetry, which
implies the presence of an asymmetry within the physical laws (recall Curie’s principle). As previously
hinted, the construction of the SM relies heavily on the concepts of symmetry and symmetry breaking,
and one should investigate whether the asymmetry conditions for generating the BAU – the Sakharov
conditions – are satisfied in its context. This will be done in Chapter 4, where a popular baryogenesis
scenario accommodating naturally small neutrino masses – leptogenesis – is presented. In Chapter 5, the
viability of leptogenesis in a model which includes a discrete flavour symmetry and spontaneous breaking
of the CP symmetry is analysed.
16
The Standard Model of Particle
Physics and (slightly) Beyond 2Particle physics relies on the principle that the interactions between the constituents of the subatomic
world can be described, to a remarkable approximation, by the Standard Model. In spite of its successes
and repeated experimental verification, the SM cannot provide a satisfatory answer to some questions
which remain open. Among these is the fact that neutrino masses cannot be accounted for in the model,
as required by neutrino oscillations. In this chapter we review relevant aspects of the SM and consider
extensions in which small neutrino masses arise in a natural way.
2.1 Recap of the Electroweak Sector of the SM
The Standard Model is a relativistic quantum Yang-Mills theory built by postulating both an un-
derlying SU(3)c×SU(2)L×U(1)Y gauge symmetry group – the subscripts are labels which correspond,
respectively, to colour, left-handedness, and hypercharge – as well as the following field content:
φ ≡(φ+
φ0
)∼(2, 1/2
), `αL ≡
(ναLlαL
)∼(2,−1/2
), qαL ≡
(uαLdαL
)∼(2, 1/6
),
lαR ∼(1,−1
), uαR ∼
(1, 2/3
), dαR ∼
(1,−1/3
).
(2.1)
In the above, φ, `α and qα represent the Higgs, lepton, and quark doublets, respectively, while the fields
uα and dα correspond to up- and down-type quarks, and να and lα to neutrinos and charged leptons.
The index α runs over three generations (or families) of fermions.
Absent from the postulated field content are the twelve gauge bosons which appear in the theory
following the demand of local gauge invariance. One will henceforth focus on the electroweak sector,
which corresponds to the SU(2)L×U(1)Y gauge group, ignoring the (unbroken) strong SU(3)c symmetry.
The quantities given between parentheses in (2.1) refer to representation assigments under this electroweak
subgroup. The first one specifies how the field transforms under SU(2)L, to which three generators Ii
are associated. A value of ‘1’ signals that the field transforms as a singlet, Ii = 0, whereas ‘2’ indicates
that it transforms as a doublet, Ii = τ i/2 (τ i are the Pauli matrices). The second quantity refers to the
value taken by the hypercharge operator Y , the single generator of U(1)Y , for each (non-gauge) field.
The SM is also a chiral theory, meaning that its basic fermionic ingredients are eigenfunctions of the
chirality matrix γ5. In fact, it is possible to decompose a spinor field ψ into these right-handed and
17
left-handed chiral eigenfunctions (with eigenvalues +1 and −1 respectively):
ψ = ψR + ψL. (2.2)
One thus defines the operators PR,L which obey:
ψR =1 + γ5
2ψ ≡ PR ψ, ψL =
1− γ5
2ψ ≡ PL ψ. (2.3)
These operators are said to be chiral projectors since (PR,L)2 = PL,R, PR+PL = 1, and PRPL = PLPR =
0. The chiral fields ψR,L are two-component spinors and belong to the simplest nontrivial representations
of the Lorentz group [58], which grants them a fundamental role in the construction of the SM. The
dynamics of the fields are encoded in the Lagrangian (density) of the theory which reads1:
LSM = (Dµ φ)†(Dµ φ)− V (φ)− 1
4AiµνA
iµν − 1
4BµνB
µν
+ i `αL /D `αL + i qαL /D qαL + i lαR /D lαR + i uαR /D uαR + i dαR /D dαR
−(Ylαβ `αL φ lβR + H.c.
)−(Yuαβ qαL φ uβR + H.c.
)−(Ydαβ qαL φdβR + H.c.
).
(2.4)
One recognizes, in the above expression, the Klein-Gordon, Dirac and Proca kinetic Lagrangian terms
for scalar (spin 0), spinor (spin 1/2) and vector (spin 1) fields, respectively. However, the requirement of
gauge invariance under SU(2)L×U(1)Y implies a change to the usual kinetic terms of the fermions and
scalar Higgs doublet fields, namely the substitution of the ordinary derivative by a covariant one [41]:
∂µ → Dµ ≡ ∂µ − i g2Aiµ I
i − i gY Bµ Y, (2.5)
where g2 and gY are coupling constants associated with the gauge groups SU(2)L and U(1)Y , respectively.
This leads to the introduction of four real boson fields Aiµ and Bµ in the theory, whose transformation
properties are such that the kinetic terms are kept invariant under the action of the gauge group.
A modification is also in order for the Proca Lagrangian. Whereas the term for the Bµ field is a
typical one, with the usual definition Bµν ≡ ∂µBν − ∂νBµ, the term for the Aiµ gauge fields relies on the
following definition of Aiµν , to ensure invariance of the kinetic Proca term under the group SU(2)L:
Aiµν ≡ ∂µAiν − ∂νAiµ + g2 εijkAjµA
kν , (2.6)
where εijk is the totally antisymmetric (Levi-Civita) tensor of rank 3.
Further inspection of Eq. (2.4) reveals the absence of mass terms for all SM particles. Charged
fermion masses cannot be written down without explicitly breaking the symmetry group, since – as one
can see from Eq. (2.9) – there is no invariant way of combining an SU(2)L doublet with an SU(2)L
singlet, as required by the form of a Dirac mass term(PR,Lψ = ψPL,R
):
−mψψ = −m(ψR ψR + ψR ψL + ψL ψR + ψL ψL
)= −m
(ψR ψL + ψL ψR
). (2.7)
1 Aside from ignoring the strong sector, one omits gauge fixing and Faddeev-Popov terms in LSM. The Feynmanslash notation is also employed: /A ≡ γµAµ, where γµ are the Dirac matrices (µ = 0, 1, 2, 3) obeying γµ = γ0㵆γ0 and
γµ, γν = 2gµν . The chirality matrix is given by γ5 ≡ γ5 ≡ iγ0γ1γ2γ3 = γ5† = (γ5)2 and anticommutes with every γµ
(the number ‘5’ is a mere label, not a Dirac index). Additionally, one defines ψ ≡ ψ†γ0 and φ ≡ iτ2φ∗.
18
This clash with reality is resolved though the Higgs mechanism: after electroweak symmetry breaking
(EWSB), the Yukawa terms in the Lagrangian will give rise to the desired fermionic mass terms. The
former make up the last line of (2.4), where Yl, Yu and Yd are general complex matrices and ‘H.c.’
denotes Hermitian conjugation. Mass terms for the SM bosons will also arise after EWSB. In particular,
one might suppose that, due to the form of the scalar potential in LSM,
V (φ) ≡ µ2φ†φ+ λ(φ†φ
)2, (2.8)
there is a mass term for the scalar doublet components. However, this is not so straightforward since one
requires µ2 < 0 for EWSB to occur. Thus, this term is not a Klein-Gordon mass term.
Before proceeding, we present, for the sake of completeness, the explicit way in which the fields trans-
form under and element g of the gauge group, parametrized by the four local parameters(θi(x), η(x)
):
ψ → Ug ψ = exp(i θi(x) Ii
)︸ ︷︷ ︸
SU(2)L
exp(i η(x)Y
)︸ ︷︷ ︸
U(1)Y
ψ. (2.9)
2.1.1 Neutral and Charged Electroweak Currents
Expanding the covariant derivative (2.5) in LSM, one can extract the way in which the gauge bosons
interact with fermions. For the three generations of leptons,
L`Gauge =1
2`αL
(g2 /Ai τ i − gY /B
)`αL − gY lαR /B lαR
=1
2
(ναL lαL
) ( g2 /A3 − gY /B g2
(/A1 − i /A2
)
g2
(/A1 + i /A2
)−g2 /A3 − gY /B
) (ναLlαL
)− gY lαR /B lαR.
(2.10)
The diagonal interaction terms in weak isospin space – the space of SU(2)L rotations – are grouped
into the so-called neutral current (NC) Lagrangian, while the off-diagonal terms correspond to a charged
current (CC) Lagrangian. These terms determine the interactions of fermions with electroweak bosons:
L`NC =1
2ναL
[(g2 sW − gY cW
)γµAµ +
(g2 cW + gY sW
)γµZµ
]ναL
− 1
2lαL
[(g2 sW + gY cW
)γµAµ +
(g2 cW − gY sW
)γµZµ
]lαL
− gY lαR[cW γµAµ − sW γµZµ
]lαR,
(2.11)
L`CC =g2√
2
(ναL γ
µWµ lαL + lαL γµW †µ ναL
)≡ g2√
2j`,µCCWµ + H.c.. (2.12)
In the previous expressions one has employed the definitions
Wµ ≡A1µ + i A2
µ√2
, Zµ ≡ cos θW A3µ − sin θW Bµ , Aµ ≡ sin θW A3
µ + cos θW Bµ, (2.13)
where θW is the weak mixing or Weinberg angle, actually introduced by Glashow in 1961 [20], and the
shorthands cW ≡ cos θW and sW ≡ sin θW . The fields Wµ and Zµ correspond to the bosons responsible
for weak interactions while Aµ (no numerical index) corresponds to the photon.
By requiring that the neutrinos do not couple to the electromagnetic field, it follows that:
g2 sin θW = gY cos θW ⇒ gY /g2 = tan θW . (2.14)
19
Plugging this relation back into (2.11), one can extract the electromagnetic coupling of charged leptons:
L`A = −1
2lαL(g2 sW + gY cW
)γµAµ lαL − gY cW lαRγ
µAµ lαR
= −gY cW(lαL γ
µAµ lαL − lαR γµAµ lαR)
= −gY cW lL γµ lLAµ.
(2.15)
One recognizes, in the last line of this expression, the electromagnetic current and establishes the iden-
tification e = gY cos θW = g2 sin θW , where e represents the elementary charge. The procedure here
outlined can be similarly applied to extract the expressions for quark CCs and NCs. The electroweak
gauge interaction terms can thus be brought to the form:
LGauge = e jµAAµ +g
cos θWjµZ Zµ +
(g√2jµCCWµ + H.c.
). (2.16)
Here, jµA represents the electromagnetic NC for quarks and charged leptons, jµZ the weak NC for all quarks
and leptons, and jµCC the (weak) CCs, defined for the case of leptons in Eq. (2.12).
The presence of different conventions in the literature demands careful bookkeeping of signs entering
the definitions of couplings and fields – see Ref. [59] for a convention-independent notation. The choices
taken above coincide with those of Peskin and Schroeder [41].
2.1.2 The Higgs Mechanism
In the SM, the vacuum (recall Section 1.3.4) is identified by minimizing the scalar potential V (φ) of
(2.8) with respect to the fields φ. If µ2 > 0 and λ > 0, this is trivially satisfied at 〈φ+〉 = 〈φ0〉 = 0,
where the brackets denote the vacuum expectation value (VEV). In order to spontaneously break the
gauge symmetry, one takes µ2 < 0 and λ > 0. Now, some of the fields must possess a nonzero value in
the vacuum, where the potential is minimum. However, in order to preserve the charge neutrality and
isotropy of the vacuum, only neutral scalar fields can acquire a nonzero VEV, which is possible for the
lower isospin component of the Higgs doublet alone:
〈φ〉 =
(〈φ+〉〈φ0〉
)≡(
0
v
), v ∈ C, v 6= 0. (2.17)
The minimum of the potential will then be given by:
∂V (〈φ〉)∂〈φ〉 = 0 ⇒ |v|2 = −µ
2
2λ⇒ v = eiϑ
√−µ2
2λ. (2.18)
Choosing a particular minimum, hereafter ϑ = 0, realizes the spontaneous breaking of the symmetry.
There is still a residual, unbroken U(1) symmetry corresponding to electric charge conservation. The
electric charge operator Q is related to the weak isospin generator I3 (assumed to correspond to a
diagonal matrix) and to the hypercharge operator Y by the Gell-Mann–Nishijima relation, which reads:
Q = I3 + Y. (2.19)
To see which symmetry groups are preserved by the vacuum, one simply needs to check if the VEV of
the fields belongs to the kernel of the group generatorsG, since in this case ei αG〈φ〉 = (1+i αG+. . . )〈φ〉 =
20
〈φ〉 + i αG 〈φ〉 + · · · = 〈φ〉. This ceases to happen for the generators Ii and Y when the Higgs doublet
acquires a nonzero VEV, but not for the combination of (2.19), as expected:
Q 〈φ〉 =[I3 + Y
]〈φ〉 =
[1
2
(1 0
0 −1
)+
1
2
(1 0
0 1
)](0
v
)=
(1 0
0 0
)(0
v
)= 0. (2.20)
Since physical fields must have a zero value in the vacuum, one reparametrizes the Higgs doublet as:
φ(x) = exp
(iξi(x)
v
τ i
2
)
0
v +H(x)√
2
, (2.21)
with 〈H(x)〉 = 〈ξi(x)〉 = 0. H(x) and ξi(x) correspond to real fields, unlike φ0(x) and φ+(x) which are
complex, making transparent that there are four degrees of freedom in φ.
Prior to EWSB, one has the freedom to perform gauge transformations, which leave the Lagrangian
invariant. Choosing a particular gauge – the unitary gauge – one can bring φ to the following form by
cancelling the exponential of (2.21):
φ(x) =
0
v +H(x)√
2
. (2.22)
After EWSB, H(x) will thus correspond to the physical Higgs field and physics can be read off directly
from LSM by replacing φ with that of (2.22). In particular, one has:
Dµφ =
(0
∂µ(H/√
2))
+i g2
2
( √2W †µ
sec θW Zµ
)(v +H/
√2), (2.23)
V (φ) = λ
(H4
4+√
2 v H3 + 2 v2H2
)+ const., (2.24)
which imply, disregarding the constant as well as interaction terms:
∥∥Dµ φ∥∥2 − V (φ) =
1
2(∂µH)(∂µH) +
1
2(4v2λ)H2 +
g22 v
2
2W †µW
µ +1
2
g22 v
2
2 cos2 θWZµZ
µ + . . . (2.25)
From this last expression, one extracts the masses of the Higgs, W and Z bosons2:
mH = 2 v√λ =
√−2µ2, mZ =
g2 v√2 cos θW
, mW =g2 v√
2= cos θW mZ . (2.26)
Experiment yields the values mH = 125.9 ± 0.4 GeV, mZ = 91.1876 ± 0.0021 GeV, and mW =
80.385 ± 0.015 GeV [37] for the massive SM bosons. One sees that, in line with what was discussed
in Section 1.3.4, the ‘gauging-away’ of the three would-be Goldstone bosons ξi corresponds to their ab-
sorption as longitudinal/mass degrees of freedom of the gauge bosons, W± and Z. In the SM, by virtue
of (2.26), one expects the below defined ρ parameter to be unit at tree level:
ρ ≡ m2W
cos2 θW m2Z
. (2.27)
2The normalization of the kinetic Klein-Gordon term in (2.25) implies removing a factor of 1/2 in order to identify m2H .
For a Proca mass term, the factor to remove is also 1/2. While this works for mZ , the W mass term is an exception dueto the complex nature of the field, possessing an additional degree of freedom. Therefore, m2
W is simply the coefficient of
the W †µWµ term.
21
The inclusion of additional Higgs multiplets Φk in the standard recipe results in a deviation to this
formula, since all of the new VEVs will contribute to the masses of gauge bosons (Ik and I3k are the weak
isospin of the multiplet Φk and third isospin component of the multiplet component which acquires the
VEV, respectively) [60]:
ρ =
∑k
[Ik(1 + Ik)− (I3
k)2]v2k
2∑k (I3
k)2 v2k
. (2.28)
A surplus of Higgs singlets or doublets is seen to be inconsequential, whereas VEVs of higher isospin
multiplets are severely constrained by the experimental value ρexp = 1.0004+0.0003−0.0004 [37]. The closeness of
ρ to one is protected by the custodial symmetry [61,62], an accidental approximate symmetry of the SM
which becomes exact in the limit gY → 0, where θW = 0 by (2.14).
2.1.3 Fermion Masses and Mixing
Upon spontaneous breaking of the gauge symmetry, the Yukawa part of the Lagrangian (2.4) reads:
LYuk. = −(v +
H√2
)(Ylαβ lαL lβR + Yu
αβ uαL uβR + Ydαβ dαL dβR
)+ H.c.
= −Mlαβ lαL lβR −Mu
αβ uαL uβR −Mdαβ dαL dβR + H.c. + LHint.,
(2.29)
where Mψ ≡ vYψ is the mass matrix for the ψ fields and LHint. contains the interaction terms H ψLψR.
The fermion mass terms arising from EWSB are generally not diagonal, meaning one must ‘rotate’ the
fields to bring them to the physical basis. The mass eigenstates are obtained by diagonalizing3 Ml, Mu
and Md. This (bi-)diagonalization is achieved by unitary matrices V l,u,dL,R :
V lL†Ml V lR = diag(me,mµ,mτ ) ≡ Dl,
V uL†Mu V uR = diag(mu,mc,mt) ≡ Du,
V dL†Md V dR = diag(md,ms,mb) ≡ Dd,
(2.30)
where the mi are real and positive (for a proof, see Section 4.1 of [58]).
In light of (2.30), one sees that performing the following rotations4 corresponds to changing to a basis
where fermion fields have a definite mass:
lαL →(V lL)αβlβL, lαR →
(V lR)αβlβR,
uαL →(V uL)αβuβL, uαR →
(V uR)αβuβR,
dαL →(V dL)αβdβL, dαR →
(V dR)αβdβR.
(2.31)
Since, unlike what happens in a so-called weak-basis transformation, the previous transformations
distinguish SU(2)L doublet components, there will be a misalignment in the quark gauge interaction
terms. In particular, the quark charged-current Lagrangian terms, analogous to those of Eq. (2.12), now
3The word ‘diagonalizing’ might be misleading since one does not seek the eigenvalues of these matrices, but insteadwants to bring them to diagonal form through generally unrelated rotations (in flavour space) of the fields which make upthe mass term in the Lagrangian.
4Writing ψ → Uψ, for an invertible matrix U , is a quick way of saying that one can define a field ψ′ ≡ U−1ψ, then usethe relation ψ = Uψ′ to write the Lagrangian as a function of ψ′, and finally drop the primes.
22
Property # Parameters # Moduli # Phases
n×m General 2nm nm nm
n× n Symmetric n(n+ 1) n(n+ 1)/2 n(n+ 1)/2
n× n Unitary n2 n(n− 1)/2 n(n+ 1)/2
n× n Hermitian n2 n(n+ 1)/2 n(n− 1)/2
Table 2.1: Number of real parameters contained in complex matrices depending on their properties.
become (in terms of the rotated fields):
LqCC =g2√
2uβL
(V uL)∗αβγµ(V dL)αρdρLWµ + H.c.
=g2√
2uβL γ
µ(V uL† V dL
)βρdρLWµ + H.c.
≡ g2√2uαL γ
µ (VCKM)αβ dβLWµ + H.c.,
(2.32)
while the neutral current of Eq. (2.11) remains unchanged, implying that there are no flavour chang-
ing neutral currents at tree-level (GIM mechanism [63]). The matrix VCKM is known as the Cabibbo-
-Kobayashi-Maskawa matrix [64,65] or quark mixing matrix, and can be seen as a transformation which
relates massive down-type quarks with the linear combinations of them, dβ′L ≡ (VCKM)αβ dβL, that
interact with massive up-type quarks through the CC interaction.
The number of parameters for various classes of complex matrices can be found in Table 2.1. Being
unitary, the VCKM matrix can be described by n(n − 1)/2 moduli, which correspond to mixing angles,
since all entries of a unitary matrix have moduli ≤ 1, and n(n + 1)/2 phases, where n is the number
of generations. However, not all phases are physical, since some can be removed through the rephasing
of quark fields. This is possible because, aside from the CC terms, the Lagrangian possesses a U(1)2n
symmetry, i.e. is invariant under:
uαL,R → ei ϕuα uαL,R, dαL,R → ei ϕ
dα dαL,R (2.33)
Thus, one expects to identify 2n phases as unphysical. However, this is not the case since one can
decompose this transformation as a global rotation of, say, ei ϕu2 , and a subsequent rephasing where only
2n − 1 phases are available. The Lagrangian is unaffected by the first rotation, due to a global U(1)B
symmetry – corresponding to baryon number conservation – and so 2n − 1 phases are removable by
rephasing. For the case of interest, n = 3, one obtains three mixing angles θ12, θ13, θ23 ∈ [0, π/2], and a
single phase δ ∈ [0, 2π[. The standard parametrization of the VCKM is given by [66,67]:
VCKM =
c12 c13 s12 c13 s13 e−iδ
−s12 c23 − c12 s23 s13 eiδ c12 c23 − s12 s23 s13 e
iδ s23 c13
s12 s23 − c12 c23 s13 eiδ −c12 s23 − s12 c23 s13 e
iδ c23 c13
, (2.34)
where sij ≡ sin θij and cij ≡ cos θij . The δ phase is the only source of CP violation in the SM [26].
One now turns to the case of leptons. Since a right-handed component for the neutrino field is missing
from the SM field content of (2.1), one sees that the appearance of a Dirac mass term in LSM upon EWSB,
23
as in the case of other fermion fields, is not possible. Hence, neutrinos are strictly massless in the SM.
This implies that performing a rotation of the neutrino fields in flavour space is inconsequential as far as
LYuk. is concerned, since there is no mass term to spoil: neutrinos have definite (zero) mass in all bases.
This freedom can be used to cancel the effects that diagonalizing the charged lepton mass matrix has on
the lepton charged current Lagrangian, L`CC. By performing the transformation (recall (2.31)):
ναL →(V lL)αβνβL, (2.35)
the Lagrangian of (2.12) remains unchanged when expressed in terms of fields with definite mass. Lepton
mixing is thus absent from the SM.
Flavour neutrino fields νe,µ,τ are defined as the neutrino field combinations which are coupled by the
charged current to each massive charged lepton e, µ, τ . Since interactions do not mix lepton flavours,
lepton flavour numbers Li (i = e, µ, τ) – associated to the possibility of invariant rephasings of lepton
fields – and, by extension, total lepton number L =∑i Li, are conserved in the SM.
One is finally in the position to count the number of physical parameters in the SM. Aside from two
parameters which pertain to the strong sector, namely the strong coupling constant and a parameter
which quantifies the strong CP problem (see below), the SM depends on seventeen parameters, to be
taken as experimental input. These are the two electroweak couping constants gY and g2, six quark
masses mu,...,t and three charged lepton masses me,µ,τ , two scalar potential parameters, which can be
chosen to be mH and v, and the four VCKM mixing parameters.
Fermion masses comprise a large portion of this parameter list. Since the theory offers no prediction
for their values, their origin remains a mystery. One might also wonder why are there three generations
of fermions. The question of why the experimental values are what they are seems in general to be
metaphysical and unproductive. However, both the closeness of the VCKM to the identity matrix as well
as the presence of mass hierarchies between generations can be seen as suggestive of hidden relations
between parameters. Disparities between the orders of magnitude of fermion masses are made clear in
Fig. 2.1, where a summary of the fermionic content of the SM is given.
Other shortcomings of the theory include the absence of a description of gravity and of a viable dark
matter candidate, as well as the lack of solutions for the hierarchy and strong CP problems. The hierarchy
problem is related to the fine-tuning of parameters required to keep mH near the electroweak scale (∼ 102
GeV) as one demands validity of the model up to the Planck scale (∼ 1019 GeV). The strong CP problem
is in turn concerned with the absence of a CP violating term in the strong sector of the SM, despite there
being no symmetry which forbids it. Additionally, the SM cannot achieve gauge coupling unification at
high-energies, unlike what happens in supersymmetric models or GUTs, and, as we shall see in Section
4.3, is incapable of explaining the observed BAU.
These issues support the idea that the SM is an effective low-energy theory and not a final one.
Although some of these problems can be classified as theoretical prejudice, undeniable evidence for
physics beyond the Standard Model arises when one considers the experimental evidence for neutrino
oscillations [68]: neutrinos have small but nonzero masses. In the next section, simple extensions of the
SM will be considered in order to generate neutrino masses which are naturally small.
24
Figure 2.1: Mass hierarchy of the elementary fermions observed in Nature. Mass values and uncertainties
are obtained from [37] and references therein (light quarks present the highest relative mass uncertainties).
Charges are presented in units of elementary charge. Kinematical bounds from beta decay are considered
for neutrino masses.
2.2 Neutrinos Beyond the SM
2.2.1 The Neutrino Mass Term
A Dirac mass term for neutrinos can be defined consistently with the gauge symmetry if one considers a
straightforward extension obtained by adding three right-handed sterile neutrino fields ναR to the particle
content. These extra fields are two-component spinors and weak isospin singlets with null hypercharge
(hence ‘sterile’), and the extra Lagrangian terms read, before and after spontaneous symmetry breaking:
LνYuk. = −Yναβ `αL φ νβR + H.c.
EWSB−−−−−→ −(v +
H√2
)Yναβ ναL νβR + H.c. . (2.36)
Diagonalization of the neutrino mass matrix Mν = vYν is achieved by rotating the neutrino fields into a
basis of massive states ν1,2,3. The rotation (as was the case for quarks) spoils the CC alignment, making
lepton mixing possible. This mixing will be completely analogous to the quark case presented in the
previous section, with a mixing matrix parametrized as that of Eq. (2.34). However, by looking at Fig.
2.1, one sees that the entries of Yν must be unnaturally smaller than those of Yl,u,d in order to generate
neutrino masses . 1 eV. In fact, one might expect that by virtue of having the same origin, namely
electroweak symmetry breaking, the twelve elementary fermions would have comparable masses. Even
though this is not the case for charged fermions, which span over five orders of magnitude in mass, the
fact that neutrinos are such a strong outlier seems to signal the presence of an alternative mass generation
mechanism at work.
Since neutrinos are neutral fermions, it is possible to construct for them a Majorana mass term [69].
This is achieved by assuming that the chiral components of the neutrino field are not independent.
Consider the Dirac equation for a (free) fermion field with mass m, and the equations obtained by
25
applying to it the projectors PL and PR:
(i γµ∂µ −m
)ψ = 0 ⇒
i γµ∂µψR = mψL
i γµ∂µψL = mψR. (2.37)
If the chiral fields ψR and ψL are not independent, then one of these equations is redundant. Manipulating
the first one yields:
i γµ∂µψR = mψLH.c.−−−→ −i ∂µψ†R 㵆 = mψ†L
×γ0
−−−→ −i ∂µψR γµ = mψLtranspose−−−−−−−→ −i γµT ∂µψR
T= mψL
T.
(2.38)
Since the matrices −γµT also satisfy the anticommutation relation of Footnote 1 for the γµ matrices,
there is a unitary matrix C in Dirac space such that (see for instance section 5.2 of Ref. [70]):
γµT = −C−1 γµ C. (2.39)
This matrix, which also obeys CT = −C, is known as the charge conjugation matrix, as applying the
charge conjugation transformation (discussed in Section 1.3.3) to a field ψ results, with a standard phase
choice, in ψ → ψC ≡ C ψ T . Rewriting (2.38) and multiplying it by C to the left gives:
−i γµT ∂µψRT
= mψLT ⇔ i C−1 γµ ∂µ
(C ψR
T)
= mψLT
C×−−−→ i γµ ∂µ
(C ψR
T)
= m(C ψL
T)⇔ i γµ ∂µψ
CR = mψCL .
(2.40)
Performing the identification ψR = ψCL (⇒ ψL = ψCR), leads to a single, independent equation:
i γµ ∂µψL = mψCL , (2.41)
and the decomposition of Eq. (2.2) now reads ψ = ψL + ψCL , which implies
ψ = ψC , (2.42)
known as the Majorana condition. This condition entails that the particles associated with the field ψ,
now called a Majorana field, are their own antiparticles. Since charges of antiparticles are reversed, only
neutral fermions such as neutrinos can fulfil the said condition. Taking ψ = ν, one can build a neutrino
Majorana mass term, leading to a Majorana Lagrangian of the form (2.41) as follows:
LνMaj. =1
2ν(i /∂ −m
)ν . (2.43)
The 1/2 factor was introduced to prevent double counting of independent fields, guaranteeing consistency
in the normalization of QFT field operators [71]. The mass term alone reads:
Lν massMaj. = −1
2mν ν = −1
2m(νCL νL + H.c.
), (2.44)
where νCL can be rewritten by considering that νCL = −νTL C†. One notices that the above mass term is
allowed since νCL not only behaves as a right-handed field (PL νCL = 0, as expected), but also because it
transforms as νL under a Lorentz transformation and, thus, LνMaj. preserves Lorentz invariance. There
is, nonetheless, a symmetry that is clearly broken by the Majorana Lagrangian, namely lepton number L
26
– which automatically implies the breaking of B−L and B+L – as one has lost the liberty to rephase the
neutrino field. The rephasing ν → eiϕ ν would imply νC → eiϕ νC , since ν = νC , which is inconsistent
with the very definition of νC . If the mass m is null, however, one is still free to rephase the component
νL → eiϕ νL, since the kinetic term dissociates νL from νCL . The Majorana mass term can then be
understood, if m is small, as a perturbation to an effective Lagrangian which generates transitions ∆L = 2.
The fact that lepton number is broken implies that not as many phases can be removed from the lepton
mixing matrix as in the quark case. An overview of lepton mixing is postponed to Section 3.1.
One might ask if it is possible to introduce Majorana masses for neutrinos in the SM, given the available
field content. A mass term of the type (2.44) is forbidden since the νL are components of an SU(2)L
doublet and there is no field in the SM with the required electroweak assignments to aid in producing
an invariant term of that type which is also renormalizable. If one forgoes this last requirement, then
the lowest dimensional term which can be constructed using SM fields and induces Majorana neutrino
masses is the 5-dimensional Weinberg operator [72], which can be written as [73]:
LWein. = cαβ1
Λ
(`αCL φ∗)(φ† `βL
)+ H.c. = − cαβ
1
Λ
(φ† `αL
)TC†(φ† `βL
)+ H.c., (2.45)
where the cαβ are complex coefficients and Λ corresponds to an energy cutoff. The presence of nonrenor-
malizable terms in the Lagrangian implies that the SM is an effective theory, valid at low-energy, whereas
the ‘real’ theory becomes manifest only at energies of the order of a new, high-energy scale Λ.
It is worth noting that φ transforms under the electroweak group as a doublet with hypercharge
Y = −1/2, opposite to that of φ, and that after EWSB one gets5
φ(x) =
v +
H(x)√2
0
. (2.46)
As such, upon EWSB, one indeed obtains a Majorana neutrino mass term from (2.45):
LWein.EWSB−−−−−→ v2
Λ
(cαβ ναCL νβL + H.c.
)+ LHνint. ≡ −
1
2
(Mν
αβ ναCL νβL + H.c.
)+ LHνint. , (2.47)
where LHνint. contains interaction terms of neutrinos with the Higgs boson H(x) and Mν is the neutrino
(Majorana) mass matrix, for which flavour is taken into account.
The above considerations are independent of the form of the high-energy theory which gives rise to
the effective Weinberg operator LWein. at low-energy. In the next section, we consider seesaw extensions
of the SM in which Majorana neutrino masses arise. Their values turn out to be very small compared to
the rest of the fermions if a large enough scale Λ is considered, as suggested by (2.47).
2.2.2 The Seesaw Mechanism
If one regards the SM as an effective description of particle physics, one forcibly has to add new degrees
of freedom to the theory, taken to be heavy (with masses of order Λ), which are typically available at high-
-energies but decouple from the theory at low-energy. In seesaw extensions of the SM, the 5-dimensional
Weinberg operator arises after integrating out massive states from tree-level interactions in which they
5This fact was taken into account in (2.29) in order to obtain the mass terms of up-type quarks.
27
`L `L
φ φ
NR,ΣR
`L
`L φ
φ
∆YN,Σ YN,Σ Y∆ µ∗
Figure 2.2: Exchange interactions which in the effective theory give rise to the Weinberg operator of
(2.45). Seesaw types I and III correspond to the exchange of fermion fields NR and ΣR, respectively (left
diagram), while the type II seesaw mechanism is implemented through the exchange of scalar fields ∆
(right diagram).
are exchanged. Such an interaction reduces, at lower energy, to a four-point interaction of the form
φφ``, which produces neutrino mass terms following EWSB. Therefore, one must introduce fields ψ with
interactions terms of the form ψφ`, corresponding to type I or type III seesaw mechanisms, or both ψφφ
and ψ``, as is the case with the type II seesaw mechanism (see Fig. 2.2).
The Lagrangian terms which include the new fields must not spoil the SM gauge symmetry, which is
not accidental and must therefore be respected by the high-energy theory. Consequently, if one wishes to
include interaction terms of the form ψφ`, one immediately realizes that ψ must have zero hypercharge.
In order to preserve invariance under SU(2)L, one has only two possibilities: either ψ transforms as
a singlet (type I) or as a triplet (type III). That no other possibility exists follows from the fact that
the decomposition of the Kronecker product 2 ⊗ 2 ⊗ n of SU(2) representations into a direct sum (see
Footnote 4 of Chapter 1 and Ref. [74]) contains only a trivial representation, 1, for the cases n = 1,3.
Additionally, angular momentum conservation mandates these ψ to be fermionic (spin 1/2) fields. For
interaction terms of the form ψφφ and ψ``, one needs a field of hypercharge Y = 1 which could also be
an SU(2)L singlet or triplet. However, (2.19) dictates that such a singlet would be charged, making an
effective neutrino mass term impossible. Therefore, one is left with the triplet option, corresponding to
the type II mechanism. Each of these three standard extensions is now reviewed.
Type I Seesaw
In the type I seesaw paradigm [75–79], n′ sterile neutrino fields NiR ∼ (1, 0) are added to the SM
field content. Aside from a Yukawa term of the form (2.36), one is free to add a Majorana mass term as
that of (2.44) for the NiR. The extended Lagrangian of the theory will then read [73,80]:
Ltype I = LSM + iNiR /∂ NiR −[(
YN †)αi`αL φ NiR +
1
2MN
ij NiCRNjR + H.c.
], (2.48)
where greek indices run over SM generations while roman ones are tied to the n′ new fields. Thanks
to the sterility of the NiR, the covariant derivative in the kinetic term reduces to a regular one and the
presence of an L-violating Majorana mass term is not forbidden by gauge symmetry. While YN is a
general n′ × 3 complex matrix, MN is an n′ × n′ matrix which can be taken to be symmetric in flavour
28
space, since it is fully contracted with a symmetric quantity:
NiCRNjR = −NiTR C†NjR = −
(NiR
)aC†ab
(NjR
)b
=(NjR
)bC∗ba
(NiR
)a
= −NjTR C†NiR = NjCRNiR,
(2.49)
where the indices a, b refer to Dirac space and one has taken into account the anticommutation of fermion
fields. In order to write Feynman rules with which the amplitudes of physical processes are computed
one considers interacting fields of definite mass. Before EWSB, and neglecting finite temperature QFT
corrections, lepton and Higgs fields are massless and the focus is shifted to rotating the heavy mediators:
NiR →(V N)ijNjR, (2.50)
where V N is an n′ × n′ unitary matrix. By looking at Table 2.1 one verifies that V N contains enough
moduli and phases such that the above rotation brings MN to diagonal form, with real and positive
entries Mi. The mass eigenstates will then be the fields Ni ≡ NiR + NiCR, with NiR = PRNi. The
Lagrangian of (2.48) is henceforth assumed to be written in this mass basis.
Feynman rules are directly read off from the expanded interaction term in Ltype I:
Lint.type I = −
(YN †)
αi`αL φ NiR + H.c. = −YN∗
iα `αL ε φ∗ PRNiR + H.c.
= `kαL φj∗[YN∗iα εjk PR
]NiR +NiR
[YNiα ε
jk PL
]φj `kαL,
(2.51)
where superscript roman indices refer to weak isospin space and εij = i(τ2)ij
is the rank 2 antisymmetric
tensor. In these diagrams, no arrow is associated to Majorana fields. Rules for the proper treatment
of Feynman diagrams involving Majorana fermions are presented in Appendix A. Making use of the
φj
`kαL
NiRiYN∗
iα εjk PR
φj
`kαL
NiRiYN
iα εjk PL
Figure 2.3: Vertex contributions from the interaction Lagrangian (2.51) for type I seesaw. These contri-
butions do not depend on the choice of a fermion flow (see Appendix A).
rules presented in Fig. 2.3, one can write the amplitude for the type I exchange of the form `φ → `φ
corresponding to both the s-channel diagram of Fig. 2.2 and a t-channel contribution:
M(`iαL(p1)φj(p2)
NmR−−−→ φk(p3) `lβL(p4))
= u(p4)[iYN
mβ εkl PL
] i(/p1 + /p2 +Mm
)
(p1 + p2)2 −M2m + iε
[iYN
mα εji PL
]u(p1)
+ u(p4)[iYN
mβ εjl PL
] i(/p1 − /p3 +Mm
)
(p1 − p3)2 −M2m + iε
[iYN
mα εki PL
]u(p1).
(2.52)
In the above, no index sums are implied. In order to obtain a description at low-energy, the heavy
sterile fields must be integrated out from this amplitude, which can be achieved by considering the limit
29
M2m p2
n for all n = 1, . . . , 4 and m = 1, . . . , n′. One then obtains:
M(`iαL(p1)φj(p2)
NmR−−−→ φk(p3) `lβL(p4))' i u(p4) YN
mβ
1
MmYNmα
[εklεji + εkiεjl
]PL u(p1). (2.53)
Summing (2.53) over all possible NmR mediators yields:
M(`iαL(p1)φj(p2)→ φk(p3) `lβL(p4)
)' i u(p4)
(YNT 1
MNYN
)
αβ
[εklεji + εkiεjl
]PL u(p1), (2.54)
and matching the above to a four-point interaction results in the following effective Lagrangian term:
L4-pointtype I =
1
4`iαCL `
lβL
φk φj(
YNT 1
MNYN
)
αβ
[εklεji + εkiεjl
], (2.55)
where a 1/4 factor has been included since an identical particle factor of 4 arises when the vertex con-
tribution, present in (2.54), is extracted from (2.55). Upon closer inspection, one sees that this effective
Lagrangian term coincides6 with the Weinberg operator of (2.45) if one takes:
cαβ1
Λ=
1
2
(YNT 1
MNYN
)
αβ
. (2.56)
A direct consequence of this result is the relation between the high-energy matrices MN ,YN and the
low-energy neutrino mass matrix Mν (cf. (2.47)), which reads:
Mναβ = −2 v2 cαβ
1
Λ⇒ Mν = −YNT v2
MNYN . (2.57)
The number of extra parameters that the type I seesaw extension adds to the SM can be determined
by considering the basis where sterile fields have definite masses and lepton doublets and right-handed
charged-lepton fields have been rotated by V lL and V lR, respectively, so that Yl is diagonal. For n
SM lepton generations, the matrix YN contains 2n′n parameters (see Table 2.1) from which n can
be removed by rephasing the lepton doublets `αL (and singlets lαR). This amounts to n′ + 2n′n − nphysical parameters, of which n′(n + 1) are moduli and n(n′ − 1) are phases. For n = 3 and n′ = 3,
the high-energy theory relies on 7n′ − 3 = 18 parameters, to be contrasted with only 9 parameters at
low-energy: 3 masses, 3 mixing angles and 3 phases (see Section 3.1). An excess is present even when
one considers a minimal implementation of the type I mechanism, where n′ = 2. In such a case, only
two out of three SM neutrinos acquire nonzero masses and hence the low-energy description depends on
7 parameters (2 masses, 3 mixing angles, and 2 phases), while the high-energy theory is described by 11
physical quantities.
Type II Seesaw
The minimal type II seesaw scenario [81–85] requires the addition of a single scalar7 triplet ~∆ with
hypercharge Y = 1 to the SM content. One can choose a basis for the 3 × 3 SU(2)L generators Ii = T i
where the third generator T 3 is diagonal:
T 1 =1√2
0 1 0
1 0 1
0 1 0
, T 2 =
1√2
0 −i 0
i 0 −i0 i 0
, T 3 =
1 0 0
0 0 0
0 0 −1
. (2.58)
6The H.c. term of (2.45) follows from repeating the above procedure for the conjugated (reversed arrows) diagrams.7Renormalizability of the high-energy theory requires the new fields ψ to be bosonic. Although angular momentum
conservation forbids fields with spin higher than two, it does not in principle preclude the choice of a vector mediator.
The imposition of Lorentz invariance would in that case lead to a ψ`` term of the form ψµ `CL Γµ `L, where Γµ is a linearcombination of γµ and γ5γµ. However, due to the presence of left projectors PL, said term would automatically vanish.
30
The triplet components thus have a definite charge (see (2.19)) and one may write:
~∆ ≡
∆++
∆+
∆0
∼
(3, 1). (2.59)
In wanting to include a Lagrangian term involving two lepton doublets `L and one scalar triplet ~∆ one
must choose a combination of these fields which is invariant under SU(2)L. Consider the Clebsch-Gordan
decomposition of the Kronecker product of two doublets and one triplet of SU(2):
2⊗ 2⊗ 3 =(1⊕ 3
)⊗ 3 = 3⊗ 3⊕ 3 = 1⊕ 3⊕ 5⊕ 3. (2.60)
By analysing this decomposition, one notices that in order to extract an invariant term (associated with
the singlet representation, 1) one must obtain it from the product of two triplet representations, one of
which is obtained from the product of the two doublet representations 2 ⊗ 2. Looking at the Clebsch-
-Gordan coefficient table for SU(2) [86], one sees that for two doublets ~a = (a1, a2) and ~b = (b1, b2) the
combination which transforms as a triplet is:
a1 b11√2
(a1 b2 + a2 b1
)
a2 b2
∼ 3 , (2.61)
whereas the combination of the elements of two triplets ~c = (c1, c2, c3) and ~d = (d1, d2, d3) which trans-
forms as a singlet is given by:
1√3
(c1 d3 − c2 d2 + c3 d1
)∼ 1 . (2.62)
Overall factors, such as the 1/√
3 above can be ignored, since the combination is still invariant under the
gauge group. By taking ~d = ~∆ and ~c as the triplet found in (2.61), with ~a = ~b = `L, one finds that:
a1 b1 d3 −1√2
(a1 b2 + a2 b1
)d2 + a2 b2 d1 ∼ 1
⇒ νL νL ∆0 − 1√2νL lL ∆+ − 1√
2lL νL ∆+ + lL lL ∆++ ∼ 1
⇒(νL lL
)( ∆0 −∆+/√
2
−∆+/√
2 ∆++
)
︸ ︷︷ ︸≡∆
(νL
lL
)= `TL ∆ `L ∼ 1 ,
(2.63)
where in going from the second to the third line one has collected the components of ~∆ in the here
defined ∆ matrix. It is worth noting that the structure of this invariant term in isospin space has been
established independently of its structure in both Dirac and flavour spaces. In fact, to preserve Lorentz
invariance, `cL must appear to the left of `L, which is achieved by adding the matrix −C† to the term.
One thus concludes that the term −`TL C†∆ `L = `CL ∆ `L can be included in the type II Lagrangian, as
well as a similar term containing φ doublets (notice that φ∗ does not transform as an SU(2) doublet,
while φ = εφ∗ does). Under the SU(2)L×U(1)Y transformation of (2.9), one has ∆→ U∗g ∆ U†g.
Consider a straightforward generalization of the above which consists in adding not one but n′ scalar
triplets ∆i with Y = 1 to the field content. The above matrix notation allows one to write the extended
31
theory Lagrangian in the following compact form [80,87]:
Ltype II = LSM +(Dµ ~∆i
)†(Dµ
~∆i
)− V (φ,∆i)−
(Y∆i
αβ `αCL ∆i `βL + µi φ
T ∆i φ+ H.c.), (2.64)
where the remainder of the scalar potential reads (the Higgs-only term V (φ) is already included in LSM):
V (φ,∆i) =(M∆
ij
)2Tr[∆†i∆j
]+ λ1
ij φ†∆†i ∆j φ+ λ2
ij φ† φTr
[∆†i ∆j
]
+(λ3ijkl Tr
[∆†i ∆j
]Tr[∆†k ∆l
]+ λ4
ijkl Tr[∆†i ∆j ∆†k ∆l
]+ H.c.
).
(2.65)
While the Y∆i are n × n symmetric matrices, λ1,2 and M∆ are n′ × n′ Hermitian matrices (λ3,4 have
general complex-valued entries). A glance at Table 2.1 allows one to see that the unitary transformation
∆i → (V ∆)ij ∆j (2.66)
contains enough parameters to remove all phases from M∆ and all moduli except n′, which correspond
to triplet masses. Mimicking what was done in the type I case, one will henceforth work on this basis
where the ∆i have definite masses, i.e. M∆ij = Mi δij . The complex parameter µi has dimensions of mass
and one is now free to define the dimensionless parameter λi ≡ µi/Mi.
The Feynman rules can be read off8 from the Lagrangian, yielding the vertices of Fig. 2.4.
∆×i
`kβL
−i(−√2)1+δjk
Y∆i∗αβ PR
`jαL
∆×i
`kβL
−i(−√2)1+δjk
Y∆iαβ PL
`jαL
φj
φk
∆×i −i
(√2)1+δjk
µi
φj
φk
∆×i −i
(√2)1+δjk
µ∗i
Figure 2.4: Vertex contributions for type II interactions relevant for effective neutrino mass generation.
For each diagram, the appropriate ∆×i ∈ ∆0i ,∆
+i ,∆
++i is chosen such that electric charge is conserved.
The presented contributions are once again independent on the choice of a fermion flow.
The amplitude for the type II ``→ φφ exchange (rightmost diagram of Fig. 2.2) reads:
M(`iαL(p1) `jβL(p2)
∆×m−−→ φk(p3)φl(p4))
= v(p1)[− i(−√
2)1+δij
Y∆m
αβ PL
] i
(p1 + p2)2 −M2m + iε
[− i(√
2)1+δkl µ∗m
]u(p2),
(2.67)
8Technically one here applies functional derivatives with respect to the interacting fields to the related Lagrangian term.
32
where, once more, no index sums are implied. Integrating out of the heavy triplets is achieved by
considering the limit M2m p2
n for all n = 1, . . . , 4 and m = 1, . . . , n′:
M(`iαL(p1) `jβL(p2)
∆×m−−→ φk(p3)φl(p4))
' i v(p1)[(−√
2)1+δij
Y∆m
αβ
] 1
M2m
[(√2)1+δkl µ∗m
]PL u(p2)
= i v(p1)
(−(−√
2)δij(√
2)δkl 2µ∗m
M2m
Y∆m
αβ
)PL u(p2).
(2.68)
Taking into account that, for the reactions allowed by charge conservation, −(−√
2)δij(√
2)δkl = εklεji +
εkiεjl, one sees that this tree-level amplitude matches the following four-point Lagrangian:
L4-pointtype II =
1
4`iαCL `
lβL
φk φj(
2µ∗mM2m
Y∆m
αβ
)[εklεji + εkiεjl
], (2.69)
which is seen to correspond to the Weinberg operator for the choice:
cαβ1
Λ=
µ∗mM2m
Y∆m
αβ . (2.70)
The low-energy neutrino mass matrix is then given by (where a sum over the n′ triplets is considered):
Mν = −2λ∗i v2
MiY∆i ≡ 2ui Y
∆i . (2.71)
An alternative way of obtaining the above contribution for the neutrino mass matrix is by considering
that, thanks to EWSB, both φ0 and ∆0i acquire nonvanishing VEVs. In the approximation Mi v, the
Higgs VEV v remains unchanged, while the triplet VEV – which is said to be an induced VEV – is given
by [87]:
ui ≡ 〈∆0i 〉 = −λ
∗i v
2
Mi, (2.72)
and one has |ui| v for all i. In fact, due to the experimental constraint of a ρ parameter very close to
one, there is not much room available for the values of the ui. Considering Eq. (2.28), one sees that:
ρtype II =1 + 2
∑i u
2i /v
2
1 + 4∑i u
2i /v
2, (2.73)
and so the quantity√∑
i u2i is constrained by electroweak precision data to be of magnitude . 1 − 10
GeV [88,89]. However, for coupling constants of O(10−9) or higher, the most stringent constraint on the
scale of the Mi comes not from precision measurements but from the absolute scale of neutrino masses.
An unsatisfactory aspect of seesaw models pertains to a potential aggravation of the (Higgs) hierarchy
problem, which can be avoided by lowering the seesaw scale or considering supersymmetric versions of
the theory [90]. In the type II case, the presence of quartic couplings of the type φ2∆2, for instance,
would affect the running of the Higgs mass mH . Ignoring hereafter the scalar potential quartic couplings,
one sees that the type II extension introduces n′ triplet masses Mi, n′ Higgs-type couplings λi, and
12n′ − 3 Yukawa-type couplings (recall that the Y∆i are symmetric) to the parameter count, which in
the minimal case (n′ = 1) reduce to 11 undetermined high-energy quantities. A desirable feature of the
present model is the fact that it preserves the flavour structure, i.e. there is a direct correspondence
between high-energy and low-energy parameters in flavour space, as one can see from (2.71), as opposed
to the case of Eq. (2.57). Additionally, the presence of gauge interactions with triplet components opens
up new phenomenology, previously unavailable in the sterile type I scenario.
33
Type III Seesaw
One finally turns to the type III seesaw extension [91], in which n′ fermion triplets ~ΣiR with null
hypercharge are added to the SM. Taking the SU(2) generators to be those defined in (2.58), one has:
~ΣiR ≡
Σ+i R
Σ0i R
Σ−i R
∼
(3, 0). (2.74)
In order to see what extra Lagrangian terms are allowed by gauge symmetry, one refers to Eqs. (2.62)
and (2.63). From (2.62) one sees that invariantly combining two triplets ~ΣiR and ~ΣjR gives:
−MΣij
(1
2Σ0iC
R Σ0jR− Σ+
i
C
R Σ−j R
)+ H.c. ∼ 1, (2.75)
where MΣij is a symmetric n′ × n′ matrix. One can additionally extract from (2.63) an invariant combi-
nation of fields involving fermionic triplets and the doublets ˜L ≡ ε `∗L and φ = ε φ∗:
(νL lL
)(Σ0i R/√
2 −Σ+i R
Σ−i R −Σ0i R/√
2
)
︸ ︷︷ ︸≡ΣiR
(φ0∗
−φ−
)+ H.c. = `L ΣiR φ+ H.c. ∼ 1. (2.76)
In the above, one has employed φ− ≡ (φ+)∗ and a γ0 matrix has been included. Taking into account
flavour, the high-energy Lagrangian can then read [87,88]:
Ltype III = LSM + i ~ΣiR /D ~ΣiR −((
YΣ†)αi`αL ΣiR φ−+
1
2
(MΣ
ij Tr[ΣiCR ΣjR
]+ H.c.
), (2.77)
where one has made the redefinition Σ+i R → −Σ+
i R, making sense of (2.75) as a mass term: the fields
Σ0i = Σ0
iC
R + Σ0i R are given Majorana masses, whereas the Ψi ≡ Σ+
i
C
R + Σ−i R correspond to Dirac
fermions [92]. Feynman Rules can be straightforwardly obtained from the above Lagrangian. By following
an integrating-out procedure similar to that of the type I seesaw case, one arrives in this context at the
following effective neutrino mass matrix:
Mν = −YΣT v2
MΣYΣ . (2.78)
Turning to parameter counting one sees that, since MΣ is symmetric, there is enough freedom to diag-
onalize the triplet mass matrix, obtaining from this n′ parameters. Additionally, by rephasing lepton
doublets (and charged lepton singlets) one can remove n phases from the 2n′n parameters of MΣ. This
totals 7n′ − 3 undetermined high-energy quantities, as was the case for the type I seesaw mechanism.
The given canonical seesaw realizations of the Weinberg operator can be motivated by GUTs and
correspond only to a subset of possible SM extensions in which small neutrino masses are naturally
generated. Alternative mechanisms at tree-level include the inverse-seesaw model [93, 94], where the
conventional type I framework is extended through the addition of fermionic singlets whose nonzero
lepton number assignment is softly broken. Neutrino masses can also be generated radiatively, as is the
case of the Zee-Babu model [95,96], where two charged scalar singlets are added to the SM and Majorana
neutrino masses arise due to two-loop quantum corrections. Other exotic options for neutrino mass
generation include the breaking of R-parity without inducing proton decay in SUSY models [97, 98], as
well as theories with extra dimensions or expanded gauge symmetries (see [99] and references therein).
34
Lepton Mixing and
Discrete Family Symmetries 3In the previous chapter we have focused on the Standard Model of particle physics and seesaw ex-
tensions, in which naturally small neutrino masses are generated. As discussed of Section 2.1.3, lepton
mixing becomes non trivial if massive neutrinos are brought into the picture. At low-energies, significant
differences between the quark and lepton mixing patterns are experimentally observed. In particular, the
structure of lepton mixing hints towards the presence of discrete symmetries in the lepton sector, which
play a normative role in the physical theory (see Section 1.4). In the present chapter, lepton mixing
is briefly addressed and the role of family symmetries in the shaping of the mass matrices and mixing
pattern is explored. A model which generates near tribimaximal mixing, accommodating all available
neutrino data, is reviewed.
3.1 Lepton Mixing
As in the case of quarks, lepton mixing arises from the mismatch of lepton states which are massive
and those which participate in CC interactions. Assuming henceforth that neutrinos possess Majorana
masses, one sees that it is possible to rotate the fields to the mass basis through:
lαL →(V lL)αβlβL, lαR →
(V lR)αβlβR, ναL →
(V νL)αiνiL, (3.1)
since a unitary transformation V νL contains enough parameters to diagonalize the symmetric matrix
arising in the Majorana mass term (cf. Eq. (2.47) and Table 2.1). The neutrino fields with definite
masses are denoted by νi (i = 1, 2, 3). The neutrino mass matrix Mν then becomes:
V νLT Mν V νL = diag(m1,m2,m3) ≡ Dν . (3.2)
This rotation produces a misalignment in the Lagrangian term of Eq. (2.12):
L`CC →g2√
2νiL γ
µ(V νL† V lL
)iαlαLWµ + H.c. ≡ g2√
2νiL γ
µ (U†PMNS)iα lαLWµ + H.c., (3.3)
where the dagger in the definition of the lepton mixing matrix UPMNS, known as the Pontecorvo-Maki-
-Nakagawa-Sakata matrix [100–102], is purely conventional. The relation between states of definite flavour
(greek indices) and definite mass (roman indices) is given by:
ναL =(UPMNS
)αiνiL ⇔
νeLνµLντL
=
Ue1 Ue2 Ue3
Uµ1 Uµ2 Uµ3
Uτ1 Uτ2 Uτ3
ν1L
ν2L
ν3L
. (3.4)
35
ParameterResult of Global Fit with 1σ 3σ errors
Normal Ordering Inverted Ordering
∆m221
(10−5 eV2
)7.62 ± 0.19
+0.58−0.50
∆m231
(10−3 eV2
)2.53 +0.08
−0.10
+0.24−0.27
−(
2.40 +0.10−0.07
+0.28−0.25
)
sin2 θ12 0.320 +0.015−0.017
±0.050
sin2 θ23 0.49 +0.08−0.05
+0.15−0.10
0.53 +0.05
−0.07
+0.11−0.14
sin2 θ13 0.026 +0.003−0.004
+0.010−0.011
0.027 +0.003
−0.004
+0.010−0.011
δ(
0.83 +0.54−0.64
)π
any (
0.07 +1.93−0.07
)π
any
Table 3.1: Global fit results taken from Ref. [107] for the three-neutrino oscillation parameters (mass
differences, mixing angles and Dirac phase) and for both ordering possibilities (see text). At the 3σ level
there is no constraint on the value of δ (this is also true for the inverted ordering case already at 1σ).
The number of physical parameters contained in UPMNS can be determined as was done for the quark
case in Section 2.1.3 with one important caveat: due to the Majorana nature of neutrinos one has lost the
freedom to invariantly rephase the three corresponding fields. Therefore, one can only remove n phases
through the rotation of charged leptons instead of 2n − 1 from the unitary mixing matrix (the global
U(1)L is also broken), ending up, for n = 3 SM generations, with n(n − 1)/2 = 3 mixing angles and
n(n+ 1)/2− n = n(n− 1)/2 = 3 physical phases. A parametrization of the UPMNS is given by [37]:
UPMNS =
c12 c13 s12 c13 s13 e−iδ
−s12 c23 − c12 s23 s13 eiδ c12 c23 − s12 s23 s13 e
iδ s23 c13
s12 s23 − c12 c23 s13 eiδ −c12 s23 − s12 c23 s13 e
iδ c23 c13
︸ ︷︷ ︸≡VPMNS
1 0 0
0 eiα1 0
0 0 eiα2
︸ ︷︷ ︸≡KPMNS
, (3.5)
where sij ≡ sin θij and cij ≡ cos θij refer to the mixing angles θ12, θ13, θ23 ∈ [0, π/2] specific to the lepton
sector, not to be confused with those of (2.34). The δ ∈ [0, 2π[ phase is a Dirac-type phase, in an analogy
to quark mixing, while α1, α2 ∈ [0, 2π[ arise due to the Majorana character of neutrinos and are thus
called Majorana-type phases [58]. As mentioned in Section 2.2.1, lepton mixing breaks lepton flavour
numbers Li (and the total L in the Majorana case). This leads to the possibility of observing charged
lepton flavour violating processes which would arise radiatively, namely µ → e γ, µ − e conversion in
nuclei, and µ → 3 e [103]. So far, searches have yielded negative results, setting upper limits of order
10−12 on branching ratios and conversion rates [104–106].
As a consequence of lepton mixing, neutrinos produced with a definite flavour are allowed to oscillate
between flavours since they constitute quantum superpositions of massive states whose evolution in time
depends on the values of their (different) masses. Neutrino oscillations are only sensitive to the mass-
squared differences ∆m2ij ≡ m2
i −m2j , providing no information on the absolute neutrino mass scale. Since
the sign of ∆m231 is indeterminate, whereas ∆m2
21 is positive, there is room for two possible orderings of the
mass spectrum: normal ordering, with m1 < m2 < m3, and inverted ordering, for which m3 < m1 < m2.
If, furthermore, the absolute neutrino mass scale (corresponding to the mass of the lightest νi) is small
36
νe ν ντ
ν1
ν2
ν3
Figure 3.1: Depiction of lepton mixing for both a normally ordered and an inverted neutrino mass
spectrum, where the global fit data of Table 3.1 has been considered (left), to be compared with the
tribimaximal ansatz (right). The probability that a massive state νi is found as an α-flavour neutrino is
given by∣∣(UPMNS)αi
∣∣2.
enough, the spectrum can be regarded as hierarchical. One then has either a normal hierarchy, with
m1 < m2 m3, or an inverted hierarchy, for which m3 m1 < m2.
Results from the latest global fit to neutrino oscillation data [107] are summarized in Table 3.1. The
best fit values correspond to a solar neutrino mixing angle θ12 ' 34o, an atmospheric angle θ23 ' 44o
or 47o, depending on whether the mass spectrum displays normal or inverted ordering respectively, and
a reactor neutrino angle θ13 ' 9o, no longer consistent with zero at the 3σ level. The dimensionless
parameter r ≡ ∆m221 / |∆m2
31| ∼ 0.03 quantifies the hierarchy between the two mass-squared differences
∆m221 and ∆m2
31, which drive solar and atmospheric oscillations, respectively. The above results can be
given a visual interpretation through the diagram of Fig. 3.1.
Oscillation experiments are not sensitive to the Dirac or Majorana nature of neutrinos [108], and the
phases α1,2 cannot be measured through them. CP violation in the leptonic sector will depend on the
values of the three mixing phases. Thus, CP and T violations potentially measurable by neutrino oscilla-
tions will depend solely on the Dirac phase δ (Dirac-type CP violation). The experimental determination
of its value is sensitive to both the magnitude of the reactor angle and to the ordering of the neutrino
mass spectrum [80]. One gateway to the determination of the Majorana-type phases is neutrinoless dou-
ble beta decay [109], a rare process which is only possible if neutrinos are their own antiparticles, i.e. if
they are described by the Majorana formalism. The relevant model-dependent quantity for the rate of
said process is |mee|, defined as the absolute value of the element Mν11 computed in the basis where the
charged leptons have definite masses and the charged current is diagonal. It is useful, in this context
alone, to transform α2 → α2 + δ and regard the new α2 as the Majorana phase [110]. As such, one has:
∣∣mee
∣∣ =
∣∣∣∣∣∑
i
(UPMNS
)2eimi
∣∣∣∣∣ =∣∣∣(m1 c
212 +m2 s
212 e
2 i α1)c213 + m3 s
213 e
2 i α2
∣∣∣. (3.6)
37
A recent upper limit on |mee| corresponds to 0.2− 0.4 eV, given by the GERDA collaboration [111].
As was mentioned above, neutrino oscillations are not sensitive to the absolute neutrino mass scale,
but only to mass-squared differences. The value of this scale can in principle be directly obtained through
the observation of the beta decay spectrum end-point. In this case, one would measure an effective mass
mνe ≡√∑
i
∣∣UPMNS
∣∣2eim2i . Current tritium decay searches give an upper limit of mνe < 2 eV [37].
An additional (indirect) constraint on the absolute scale of neutrino masses is given by cosmological
considerations, which limit the sum mtotal =∑imi. Recent results from the Planck collaboration have
established an upper limit of mtotal < 0.66 eV (95% CL) [112] (where WMAP polarization data has
been considered). This value can however be seen to vary depending on the analysis performed. In
particular, adding baryon acoustic oscillation data to the analysis constrains the neutrino mass sum to
be mtotal < 0.23 eV (95% CL) [113].
From inspection of Fig. 3.1 it is apparent that the experimental mixing data bears a close resemblance
to the tribimaximal (TBM) pattern. The TBM hypothesis, put forward by Harrison, Perkins, and
Scott [114], corresponds to taking |Ve3|2 = 0, |Vµ3|2 = 1/2, and |Ve2|2 = 1/3, where V here denotes the
quark-like part VPMNS of the mixing matrix in (3.5). The matrix VPMNS is unitary and, under the TBM
ansatz, orthogonal, due to s13 = 0. The above assumptions therefore allow one to write:
V ′TBM =
√23
1√3
0
− 1√6
1√3
1√2
1√6− 1√
31√2
, (3.7)
where θ12 = arccos(√
2/3)' 35o, θ23 = 45o, said to be maximal1, and θ13 = 0o. Alternatively, by
making a redefinition of the Majorana phases αi → αi + π and by resorting to the rephasing of the
charged lepton fields (including a global phase eiϕ), one may write:
UTBM ≡
−eiϕ 0 0
0 −eiϕ 0
0 0 eiϕ
V ′TBM
1 0 0
0 −eiα1 0
0 0 −eiα2
= eiϕ
−√
23
1√3
0
1√6
1√3
1√2
1√6
1√3− 1√
2
︸ ︷︷ ︸≡VTBM
KPMNS, (3.8)
where KPMNS = diag(1, eiα1 , eiα2) is given in terms of the new redefined phases. The rephasing of
charged leptons, in conjunction with the Majorana phase redefinition, amounts to changing the mixing
angle ranges from [0, π/2] to [π/2, π]. The new mixing angles are then obtained through θij → π − θij .The possibility of a TBM mixing pattern motivated the study of discrete family symmetries (consid-
ered in the remainder of this chapter) which might govern the leptonic sector. However, exact TBM is
excluded by the aforementioned results on a nonzero reactor angle, given by the Daya-Bay, RENO and
Double Chooz collaborations [115–117] after hints from the T2K and MINOS collaborations [118, 119].
Nevertheless, one can still pursue the family symmetry approach, without having to resort to mass an-
archy [120]. Taking symmetry as a guiding principle, one may accommodate the above results by either
considering larger discrete groups in model building or regarding specific mixing patterns – like the TBM –
as a leading-order approximation, to be perturbed when higher-order corrections are taken into account.
1Since the νµ ↔ ντ oscillation probability is proportional, in the two-neutrino mixing approximation, to sin2(2 θ23), onesees that it is the choice θ23 = 45o which maximizes said probability.
38
3.2 Discrete Family Symmetries
Appealing to the normative role of symmetries, one sees that, in order to regulate the structure of
the leptonic mixing matrix, the symmetry operations must act on leptons of different flavours. Symme-
tries whose action is carried out in flavour space are known as flavour, family or horizontal symmetries,
to be contrasted with the case of GUT symmetries, which act on different members of the same fam-
ily/generation and are therefore vertical, in the sense of Fig. 2.1.
Family symmetries may be based on Abelian groups, such as the U(1)FN of the Froggatt-Nielsen
mechanism [121], which can account for hierarchies between masses and mixing matrix elements. For
non-hierarchical structures, however, one turns to non-Abelian symmetries, where different flavour fields
can be arranged into multiplets of the family symmetry group. One finally considers discrete (non-
-Abelian) family symmetry groups, which in general provide a simpler solution to the vacuum alignment
problem (to be defined in Section 3.2.2) than continuous groups [122]. For a review on lepton mixing and
family symmetries, we address the reader to Ref. [123].
Since family symmetries are to be imposed while respecting the underlying SM gauge group, one sees
that the fact that charged leptons and neutrinos belong to the same multiplet of SU(2)L forbids shaping
the charged lepton mass matrix Ml and the neutrino mass matrix Mν independently [124]. One therefore
resorts to a dynamical origin of the lepton Yukawa couplings adding new heavy (scalar) fields, dubbed
flavons, transforming non-trivially under the family symmetry, which is then spontaneously broken by
the flavon VEVs, at a scale typically much higher than that of EWSB.
3.2.1 Symmetries of the Mass Matrices
Due to the specific structure of the charged-lepton and neutrino mass terms, one can identify transfor-
mations of the fields in flavour space which leave those terms invariant, thus establishing the symmetries
of the mass matrices. Unless otherwise stated, we will work in a basis where both the charged-lepton
mass matrix and the CC Lagrangian are diagonal2. Thus, for charged leptons, one has in this basis:
Ml = Dl = diag(me,mµ,mτ ). (3.9)
By asking which are the symmetries of the mass matrix upon (unitary) rotations of the lepton fields in
flavour space, one sees that Ml is clearly invariant under the transformations lL → T lL and lR → T lR:
T †Ml T = Ml, (3.10)
where one has ommited flavour indices for simplicity and T corresponds to the three-dimensional matrix:
T =
1 0 0
0 ω 0
0 0 ω2
, (3.11)
where ω ≡ ei 2π/3. Hence, one identifies Z3 – the lowest-order group which constrains Ml to be diagonal
– with the symmetry of the charged lepton mass matrix3.
2This is often simply phrased as “working in the basis of diagonal charged leptons”.3In generalizing the present considerations to grand unified theories, it would be useful to consider a lepton mass matrix
only approximately diagonal, since in GUTs this matrix is related to the down-type quark matrix, which in light of quarkmixing is often taken to be non-diagonal.
39
Regarding the neutrino mass matrix, one sees that the Majorana mass term of Eq. (2.47) constrains
Mν to be symmetric, since, as was the case with (2.49), one can check that ναCL νβL is symmetric under
the exchange α ↔ β. The (bi)diagonalization of Mν is achieved through the UPMNS mixing matrix
(UPMNS = V lL†V νL = V νL in the basis we are working with):
Mν = U∗PMNS Dν U†PMNS = U∗PMNS diag(m1,m2,m3)U†PMNS. (3.12)
The matrix Mν will then be invariant under the (unitary) transformation νL → Ki νL, where
Ki = UPMNS ηi U†PMNS, η =
(13×3, diag(−,+,−), diag(+,+,−), diag(−,+,+)
), (3.13)
since:
KiT Mν Ki =(UPMNS η
i U†PMNS
)TMν
(UPMNS η
i U†PMNS
)
= U∗PMNS ηiT(UTPMNS U
∗PMNS
)Dν(U†PMNS UPMNS
)ηi U†PMNS
= U∗PMNS Dν U†PMNS = Mν .
(3.14)
Other four possible ηi matrices exist, obtained from the above by changing their sign, ηi → −ηi, but are
redundant in the description of the symmetry of Mν since a global sign cancels independently of its form.
By noticing that(η2)2
=(η3)2
= 1 and η4 = η2η3 = η3η2, one identifies the symmetry of the neutrino
mass matrix with the Klein group K = Z2 × Z2 of Section 1.2.
What was presented up to now regarding the symmetries of mass matrices is independent of the
form of UPMNS. For the particular TBM pattern of Eq. (3.8), one obtains the specific form of the Ki
transformation matrices from (3.13):
K1 = 1, K2 ≡ S =1
3
−1 2 2
2 −1 2
2 2 −1
, K3 ≡ U =
1 0 0
0 0 1
0 1 0
, K4 = S U = U S. (3.15)
Rephasing charged leptons and redefining Majorana phases would yield the same symmetry group but
different Ki, corresponding simply to a different choice of basis for the group representation.
3.2.2 Direct vs. Indirect Models
Invoking a discrete non-Abelian family symmetry, associated with a symmetry group G, one might
ask what are the available directions in which model building may proceed. The charged-lepton and
neutrino mass matrices exhibit the Z3 and K symmetries presented above, respectively, in the basis of
diagonal charged leptons. Focusing on a specific mixing pattern, one must consider matrices T , S and U
of a specific form (e.g. those of Eqs. (3.11) and (3.15)), which generate symmetry groups GT , GS and
GU , respectively [125]. One then wishes to obtain such symmetries at low-energy, after the spontaneous
breaking of the family symmetry by the VEVs of flavon fields. One obvious model building option is to
take GT , GS and GU as subgroups of G. Such class of models, where the desired symmetries of both
charged leptons and neutrinos are already present in the family group and are preserved by the flavon
VEVs, are called direct models [124].
In alternative to direct models, one may choose to work with a group which contains none of the
low-energy symmetries as subgroups, but nonetheless gives rise to them, accidentally, upon SSB. Such
40
models are termed indirect models, and the group G plays the indirect role of determining the allowed
directions for the flavon VEVs. This speaks directly to the so-called vacuum alignment problem, as the
model must account for both the GT symmetry in the charged-lepton sector, as well as the GS and GU
symmetries in the neutrino sector. These are here absent from the high-energy theory and must arise
due to a particular alignment of VEVs. Direct models are not free from the alignment problem since the
directions of the VEVs must preserve the relevant subgroups. Although the source of the flavon VEV
alignment is out of the scope of the present work, solutions of the alignment problem can be found, for
example, in the context of supersymmetric theories (see, for instance, [126]). Besides the two solutions
described above, an intermediate solution is found in semi-direct models, for which the family group G
contains some but not all of the desired subgroups generated by T , S and U .
Considering the particular case of TBM mixing, one sees that the generators take the forms given in
(3.11) and in (3.15). The simplest group which allows for exact TBM in the context of direct models
is G = S4 [127], presented in Section 1.2, as it is the smallest group which contains the TBM GT , GS
and GU as subgroups, i.e. can accomodate having the generators in the desired form in a common
basis. Regarding semi-direct models, a popular choice of G corresponds to the alternating group A4
[125,128–130], presented in the same section as above. This is not without justification, since A4 already
contains the TBM GT and GS as subgroups and is the smallest group which possesses a three-dimensional
irreducible representation. The three lepton flavours are thus naturally connected in A4-based models
as components of a single multiplet. The structure of the charged lepton mass matrix arises due to a
preserved GT ∼= Z3, while that of the neutrino mass matrix is established by both the presence of an
intact GS ∼= Z2 and of an accidental GU ∼= Z2.
In passing, we remark that extending A4-based models to the quark sector interestingly leads to
a unit mixing matrix at leading order, if one requires nondegenerate and nonvanishing quark masses
[131]. However, higher-order corrections seem to be too small to account for the observed VCKM mixing
angles [125], prompting the study of larger-order groups to achieve such unification (see, for instance,
Ref. [132]).
3.3 An A4 Model with Spontaneous CP Violation
Having established the philosophy by which one must abide, we turn, in the present section, to
the analysis of the model of Ref. [133], which implements an A4 family symmetry in a type II seesaw
framework with spontaneous CP violation. In this model, deviations to exact TBM are obtained by
perturbing the flavon VEV alignment. Aside from two real flavon fields, Φ and Ψ, which are singlets of
SU(2)L with null hypercharge, one adds two SU(2)L triplets ∆1,2 with Y = 1 and one complex scalar
field S, which is a singlet of the whole gauge group, to the particle content of the SM. Apart from the SM
gauge group and the A4 flavour symmetry, a Z4 symmetry which constrains the form of the Lagrangian
is also considered. The symmetry assignments of the fields are given in Table 3.2.
An interesting property of the present model is the absence of explicit CP violation at the Lagrangian
level, i.e. all parameters are real in the fundamental theory. The source of low and high-energy CP
41
Symmetry `L eR, µR, τR ∆1 ∆2 φ S Φ Ψ
A4 3 1,1′,1′′ 1 1 1 1 3 3
Z4 i −i 1 −1 i −1 i 1
SU(2)L × U(1)Y (2,−1/2) (1,−1) (3, 1) (3, 1) (2, 1/2) (1, 0) (1, 0) (1, 0)
Table 3.2: Representation assignments of the model fields under the action of the groups A4, Z4, and
gauge SU(2)L×U(1)Y . SM fields not covered in the present table transform trivially under A4 × Z4.
violation, namely that necessary to account for the BAU (see Chapter 4), is unique: the complex phase
of the VEV which S acquires. With the assignments of Table 3.2 in mind, one can find out which terms
are allowed in the Lagrangian of the model, the relevant part of which reads:
L = Ll + Lν − V CP×Z4 − V (Φ,Ψ), (3.16)
where Ll and Lν make up the Yukawa Lagrangian and V (Φ,Ψ) denotes the flavon part of the scalar
potential while V CP×Z4 corresponds to its remainder. The flavon-free potential is then given by:
V CP×Z4 = VS + Vφ + V∆ + VSφ + VS∆ + Vφ∆ + VSφ∆, (3.17)
which presents the following explicit form (one can straightforwardly check that these terms are the only
ones allowed by the imposed symmetries):
Vφ = µ2(φ) φ
†φ+ λ(φ)
(φ†φ
)2,
Vφ∆ =∑
i
(λ1i
(φ†∆†i∆iφ
)+ λ2
i
(φ†φ
)Tr[∆†i∆i
])+(λ2M2 φ
T ∆2 φ+ H.c.),
V∆ =∑
i
M2i Tr
[∆†i∆i
]+∑
i,j
[λ3ij Tr
[∆†i∆i
]Tr[∆†j∆j
]+ λ4
ij Tr[∆†i∆i∆
†j∆j
]],
VS = µ2S
(S2 + S∗2
)+m2
S S∗S + λS
(S∗S
)2+ λ′S
(S4 + S∗4
)+ λ′′S S
∗S(S2 + S∗2
),
VSφ = ηS(S∗S
) (φ†φ
)+ η′S
(S2 + S∗2
) (φ†φ
),
VS∆ =∑
i
Tr[∆†i∆i
](ξi(S2 + S∗2
)+ ξ′i S
∗S),
VSφ∆ = φT ∆1 φ(λ∆ S + λ′∆ S∗
)+ H.c..
(3.18)
As for the Yukawa Lagrangian, one sees that the usual SM charged lepton Yukawa term is forbidden
by the Z4 symmetry, as well as the standard type II seesaw coupling of triplets ∆1,2 to lepton doublets
(a coupling of the Higgs type, ∆φφ, is only allowed for ∆2). Instead, terms of these forms will arise as
effective operators – suppressed by a certain scale – where flavon fields and the singlet field S enter.
A detailed analysis of the structure of A4 is performed in Appendix B, where a specific basis for the
three-dimensional representation is chosen. This is the basis in which the representations of the group
elements s = (12)(34) and t = (123) (see Appendix B) simultaneously agree with the forms (3.11) and
(3.15) for the generators S and T : U3(s) = S, U3(t) = T .
In wanting to write A4-invariant terms involving lepton triplets, one must consider the Clebsch-Gordan
decomposition of the tensor product of two A4 triplet representations, 3⊗ 3 = 1⊕ 1′ ⊕ 1′′ ⊕ 3s ⊕ 3a.
42
The full computation of the (Clebsch-Gordan) coefficients which govern this decomposition is performed
in Appendix B. Due to the fact that the Majorana mass matrix is symmetric in flavour space, one restricts
oneself to products where the symmetric 3s arises. To obtain A4-invariant terms which include the barred
fields `L, one considers(`eL, `τL, `µL
), which transforms as an A4 triplet. For completeness, one presents
the invariant arising from the symmetric product of three triplets ~a, ~b and ~c, obtainable from (B.51) (a
global factor of 1/3 is included to keep in touch with the conventions of [133]):
1
3
(2a1b1c1 + 2a2b2c2 + 2a3b3c3 − a1b2c3 − a1b3c2 − a2b1c3 − a2b3c1 − a3b1c2 − a3b2c1
)∼ 1. (3.19)
Up to O(1/Λ, 1/Λ′), where Λ and Λ′ are taken to correspond to an assumed unique flavon scale and to
the scale of S decoupling, respectively, one obtains the following Yukawa Lagrangian, compatible4 with
the postulated symmetries (l1,2,3 = e, µ, τ):
Ll = −yle
Λ
(`L Φ
)1φ eR −
ylµΛ
(`L Φ
)1′′φµR −
ylτΛ
(`L Φ
)1′φ τR + H.c.,
Lν =1
Λ′∆1
(`TL C
† `L)1
(y1 S + y′1 S
∗)+y2
Λ∆2
(`TL C
† `L Ψ)1
+ H.c.,
(3.20)
where the bold superscripts indicate which field combination is chosen from the decomposition of the
triplet tensor product. Suppose now that the singlet S acquires a complex VEV 〈S〉 = vS eiα and the
flavon fields develop VEVs, which must be global minima of the scalar potential, in the generic directions
(ri, si ∈ R):
〈Φ〉 = (r1, r2, r3) , 〈Ψ〉 = (s1, s2, s3) . (3.21)
The (leptonic) Yukawa Lagrangian then becomes:
Ll = −Ylαβ `αL φ lβR + H.c. , Lν = −Y∆1
αβ `αCL ∆1 `βL −Y∆2
αβ `αCL ∆2 `βL + H.c., (3.22)
where one recognizes the familiar charged lepton Yukawa and type II seesaw terms, with:
Yl =1
Λ
r1 r3 r2
r2 r1 r3
r3 r2 r1
yle 0 0
0 ylµ 0
0 0 ylτ
, (3.23)
Y∆1 = y∆1
1 0 0
0 0 1
0 1 0
, y∆1
≡ vSΛ′(y1 e
iα + y′1 e−iα), (3.24)
Y∆2 =1
3
y2
Λ
2 s1 −s3 −s2
−s3 2 s2 −s1
−s2 −s1 2 s3
. (3.25)
Imposing the symmetries GT,S,U (with the desired generator representations T , S, and U) to be
manifest at low-energy constrains the direction of the flavon expectation values. Requiring Yl to be
symmetric under GT implies that it is diagonal and, as such, r1 ≡ r, while r2, r3 = 0. Demanding Y∆2
4This Lagrangian is not, however, the most general one allowed by the symmetries, up to the chosen order, as botha Weinberg operator term and the renormalizable ∆2
(`TL C
† `L)1
term can be included. Although the former might beremoved by a shaping symmetry, there is no possible additional symmetry which can forbid the latter without destroyingother desirable terms. Schematically, if (φφ)(∆1S), (`L`L)(∆1S), and (φφ)(∆2) are allowed, then so must (`L`L)(∆2).Such a term would modify Y∆2 by a term ∝ Y∆1 without affecting the mixing pattern, as one can see from (3.36).
43
to be symmetric under GU (often referred to as a ‘µ− τ symmetry’) yields s2 = s3 ≡ s. Finally, one can
check that for symmetry under the chosen representation of GS (commonly denoted as ‘magic symmetry’)
the constraint s1 = s arises. The VEV directions which allow for the TBM mixing pattern are thus:
〈Φ〉TBM = (r, 0, 0) , 〈Ψ〉TBM = (s, s, s) . (3.26)
One could equivalently have found this solution by requiring T 〈Φ〉 = 〈Φ〉 and S〈Ψ〉 = U〈Ψ〉 = 〈Ψ〉.
3.3.1 Spontaneous CP Violation
The spontaneous breaking of CP results from a non-trivial complex phase of the singlet VEV, 〈S〉 =
vS eiα. Assuming the scale of S decoupling to be much higher than both the electroweak and seesaw
scales, we can restrict ourselves to the VS term of the scalar potential (no induced VEVs arise) to verify
whether the potential minimization conditions allow for a non-trivial VEV S = 〈S〉 = vS eiα. For that,
we take
V0 ≡ VS∣∣S=〈S〉 = m2
S v2S + λ2
S v4S + 2
(µ2S + λ′′S v
2S
)v2S cos(2α) + 2λ′S v
4S cos(4α). (3.27)
The values of α and vS are obtained through the minimisation conditions:
∂V0
∂vS= 0
∂V0
∂α= 0
⇒
vS
[m2S + 2λS v
2S +
(2µ2
S + 4λ′′S v2S
)cos(2α) + 4λ′S v
2S cos(4α)
]= 0
v2S
[2(µ2S + λ′′S v
2S
)sin(2α) + 4λ′S v
2S sin(4α)
]= 0
. (3.28)
Since sin(4α) = 2 sin(2α) cos(2α), the lower condition defines three classes of solutions:
vS = 0 ∨ ( vS 6= 0 ∧ sin(2α) = 0 ) ∨ ( vS 6= 0 ∧ sin(2α) 6= 0 ) (3.29)
No CP violation arises from either the first (trivial) solution or from the second possibility (see Section
23.6 of Ref. [26]). One is then left with sin(2α) 6= 0, for which
m2S + 2λS v
2S + 4λ′S v
2S cos(4α)
+(2µ2
S + 4λ′′S v2S
)cos(2α) = 0
µ2S + λ′′S v
2S + 4λ′S v
2S cos(2α) = 0
⇒
vS =
√µ2S λ′′S − 2m2
S λ′S
4λS λ′S − 8λ′S2 − λ′′S
2
cos(2α) = −µ2S + λ′′S v
2S
4λ′S v2S
(3.30)
holds. Since the angle α is defined in the interval [0, 2π[, the above condition yields four possible solutions:
α, −α, π+α, and π−α (all to be taken mod 2π). One can see that the solution of Eq. (3.30) corresponds
to a global minimum in the region λS > 2λ′S > 0 ∧ λ′′S ' µS ' 0. In fact, for the CP-conserving solutions
which obey vS 6= 0 ∧ sin(2α) = 0, Eqs. (3.27) and (3.28) in this region give:
v '√
−m2S
2(λS + 2λ′S
) ⇒ V0 ' −m4S
4(λS + 2λ′S
) , (3.31)
since cos(4α) = 1−sin2(2α) = 1 for the considered class of solutions, while for the non-trivial CP-violating
solutions vS 6= 0 ∧ sin(2α) 6= 0 one gets:
vS '√
−m2S
2(λS − 2λ′S
) ⇒ V0 ' −m4S
4(λS − 2λ′S
) , (3.32)
since cos(2α) ' 0, corresponding to the four solutions: π/4, 3π/4, 5π/4, and 7π/4. Consistency mandates
that m2S < 0 for the given solutions to be viable. The assumption λS > 2λ′S > 0 selects the non-trivial
CP-breaking solution as that which minimizes the potential.
44
Parametrizing the complex singlet field as S = vS eiα + σ1 + i σ2, with 〈σi〉 = 0 ⇒ 〈S〉 = vS e
iα, one
can extract from the potential VS the mass eigenvalues for the two σi degrees of freedom. Collecting
the terms quadratic in the new fields – which amounts to taking the second derivatives of the potential
with respect to σi and σj – allows one to reach, for both the non-trivial minimization solution and the
particular parameter region considered above:
V σ1,σ2mass '
−m2S
λS − 2λ′S
[(λS + 2λ′S
)(σ2
1 + σ22
)+ 2(λS − 6λ′S
)σ1 σ2
]. (3.33)
Diagonalization of this quadratic form finally yields the positive squared masses M2σ1' −4m2
S and
M2σ2' −16m2
S λ′S/(λS − 2λ′S).
3.3.2 Neutrino Masses and Mixing
We have seen that, upon symmetry breaking, the Yukawa terms reduce to the leptonic part of the
type II seesaw Lagrangian. For the alignment condition of Eq. (3.26),
Yl =
ye 0 0
0 yµ 0
0 0 yτ
, Y∆1 = y∆1
1 0 0
0 0 1
0 1 0
, Y∆2 =
y∆2
3
2 −1 −1
−1 2 −1
−1 −1 2
, (3.34)
where ye,µ,τ ≡ r yle,µ,τ/Λ and y∆2≡ s y2/Λ. The coupling of heavy triplets to the Higgs bosons is also
available for both ∆i, since VSφ∆ produces such a term for ∆1 as soon as S acquires a VEV:
VSφ∆ = φT ∆1 φ(λ∆ S + λ′∆ S∗
)+ H.c.
SSB−−−→ λ1M1 φT ∆1 φ+ H.c., (3.35)
with λ1 ≡ vS(λ∆ eiα + λ′∆ e−iα
)/M1. Relying on the formalism already developed in Section 2.2.2, one
sees that the neutrino mass matrix reads (cf. Eq. (2.71)):
Mν =
2∑
i=1
−2λ∗i v2
MiY∆i = 2u1 Y∆1 + 2u2 Y∆2 , (3.36)
where the ui represent the triplet VEVs defined in (2.72). The differing sign of (3.36) with respect to [133]
is due to a different definition of the Yukawa couplings, y(′)1,2 → − y
(′)1,2 .
Neutrino masses can be obtained, in light of (3.2), by taking the square root of the eigenvalues of
Mν†Mν , with UPMNS as the change of basis matrix. It is useful to define beforehand the quantities zi
(> 0) and β through 2u1 y∆1≡ z1 e
iβ and 2u2 y∆2≡ z2. All information regarding CP violation is then
encoded5 in the phase β. The diagonalization procedure yields:
Dν = diag(m1,m2,m3) = diag(∣∣z1 e
iβ + z2
∣∣, z1,∣∣z1 e
iβ − z2
∣∣), (3.37)
and one finds that, as expected, the mixing matrix is of TBM form: UPMNS = UTBM, with indeterminate
ϕ and KPMNS. To obtain them, one notices that, since KPMNS is diagonal,
e−iϕK∗PMNS = ±√V TTBM Mν VTBM (Dν)−1. (3.38)
Recalling that KPMNS = diag(1, eiα1 , eiα2), and defining σ± ≡ arg(z2 ± z1 eiβ), one arrives at:
ϕ = −σ+/2 , α1 = (σ+ − β)/2 , α2 = (σ+ − σ−)/2, (3.39)
where we have restricted ourselves to the positive sign solution.
5In fact, as the mixing pattern is TBM, there will be no Dirac-type CP violation and β enters solely in Majorana phases.
45
mtotal
m1
m2
m3
Π 9 Π
8
5 Π
4
11 Π
8
3 Π
2
1
10-1
10-2
mi and mtotal HeVL
Β
Planck
Exclusion
maximum mtotal allowed by cosmology
Figure 3.2: Predicted values for neutrino masses as a function of the CP-violating angle β. The plot is
restricted to the region [π, 3π/2] since the functions mi(β) are symmetric around β = π. Due to Eqs.
(3.41) and (3.42), the absolute neutrino mass scale is bounded from below: m1 & 1.6 × 10−2 eV. The
spectrum is the most hierarchical for β = π, while it presents the highest degeneracy near β = π/2, 3π/2.
The Planck upper limit of mtotal < 0.66 eV has been considered (see Section 3.1).
As for the mass spectrum, one sees that m2 > m1 implies z2 + 2 z1 cosβ < 0, while z2 − 2 z1 cosβ is
positive for normal ordering (m3 > m2) and negative for inverted ordering (m3 < m2). Normal ordering
(considered herafter) then entails cosβ < 0 ⇒ β ∈ [π/2, 3π/2], while it is impossible to implement
inverted ordering since the above conditions demand that both cosβ and − cosβ be negative (z1,2 > 0).
By manipulating (3.37), one can express z1 and z2 as a function of β and mass-squared differences:
z2 =
√m2
1 +m23 − 2m2
2
2=
√∆m2
31 − 2 ∆m221
2
z1 =m2
1 −m23
4 z2 cosβ= − 1
2 cosβ
∆m231√
2(∆m2
31 − 2 ∆m221
). (3.40)
Taking into account the large hierarchy bewteen solar and atmospheric squared-mass differences, one
obtains the approximate solution (using the best fit data of Table 3.1):
z2 '√
∆m231
2' 3.56× 10−2 eV ⇒ z1 ' −
1
2 cosβ
√∆m2
31
2=−z2
2 cosβ' −1.78× 10−2 eV
cosβ. (3.41)
Also, since m2 = z1, one is able to express neutrino masses in terms of z1(β) and ∆m221,31:
m1 =√m2
2 −∆m221 =
√z2
1 −∆m221 , m3 =
√m2
2 −∆m223 =
√z2
1 −∆m221 + ∆m2
31, (3.42)
a result which is shown in Fig. 3.2.
In the approximation of Eq. (3.41), the Majorana phases α1,2 are given by:
σ1 ' −β
σ2 ' arctan
(1
3tanβ
) ⇒
α1 ' −β
α2 ' −1
2
[β + arctan
(1
3tanβ
)] . (3.43)
46
3.3.3 Nonzero Reactor Neutrino Mixing Angle
In order to attain an acceptable agreement with experimental data within the presented framework,
one must account for the deviation θ13 6= 0 from the TBM pattern. This is achieved by considering
perturbations to the flavon alignment of Eq. (3.26), which might arise due to non-renormalizable correc-
tions to the potential V (Φ,Ψ). In particular, one focuses one the case where only the flavon VEV 〈Ψ〉is modified, with small ε1,2, to:
〈Ψ〉 = s(1, 1 + ε1, 1 + ε2). (3.44)
This case, which is the one considered in Chapter 5 for the analysis of the viability of leptogenesis in
the present model, allows for Dirac-type CP violation, in principle observable via neutrino oscillations,
unlike what would happen if 〈Φ〉 alone were perturbed6. Perturbing the 〈Ψ〉 VEV does not affect the
charged lepton Yukawa matrix – which conveniently remains diagonal – nor Y∆1 , but modifies Y∆2 ,
yielding, according to Eq. (3.25):
Y∆2 =y∆2
3
2 −1− ε2 −1− ε1
−1− ε2 2(1 + ε1
)−1
−1− ε1 −1 2(1 + ε2
)
. (3.45)
Through the same diagonalization procedure described in the previous section, keeping terms up to
O(ε1, ε2), one arrives at the perturbed neutrino mass eigenvalues7:
m21,3 ' z2
1 +1
3
(z2
2 (3 + 2ε1 + 2ε2)± 2 z1 z2 (3 + ε1 + ε2) cosβ)
, m22 ' z2
1 . (3.46)
For small enough perturbations, |ε1,2| 1, only normal ordering is once again allowed, and Fig. 3.2
remains qualitatively unaltered. As for the mixing pattern, one obtains the following deviations to TBM:
sin2 θ12 '1
3
[1 +
2
3(ε1 + ε2)
], sin2 θ23 '
1
2
[1 +
1
3(ε1 − ε2)
], sin2 θ13 '
1
2
(ε1 − ε2
6 cosβ
)2
. (3.47)
The above result can be straightforwardly obtained in the context of nondegenerate perturbation theory
(see, for instance, section 6.1 of [86]) and by considering the approximation ∆m231 ' ∆m2
32 ∆m221.
As far as mixing phases are concerned, one can measure Dirac-type CP violation through the following
invariant [80,134], which depends on the Dirac phase δ:
JCP ≡ Im(U11 U22 U
∗12 U
∗21
)=
1
8sin(2 θ12) sin(2 θ13) sin(2 θ23) cos θ13 sin δ, (3.48)
where one has used U as short for UPMNS. For ∆m231 ∆m2
21 and at first order in ε1,2, one obtains the
result JCP ' tanβ (ε2− ε1)/36⇒ sin δ ' ± sinβ (the sign depends on the direction of the perturbation).
One is now in a position to consider experimental constraints on the perturbed version of model.
This is done in Fig. 3.3: the left part illustrates the allowed regions of the (ε1, ε2) plane, while the right
part shows the corresponding possible values of JCP. One obtains a limit 0.02 . |JCP| . 0.05 (at 3σ)
associated with the ranges π/2 . β . 5π/8 and 11π/8 . β . 3π/2, which imply 3π/8 . δ . 5π/8 or
11π/8 . δ . 13π/8.
6Such a case is explored in detail in Ref. [133], where it is found to fail in accomodating the recent experimental results.A mixed option, where both 〈Ψ〉 and 〈Φ〉 are perturbed is also possible.
7This result revises that of Eq. (40) in [133], in which a factor of 2 is missing.
47
Figure 3.3: Scatter plot of the experimentally allowed regions in the(ε1, ε2
)plane (left), where exact TBM
is seen to be excluded, and corresponding regions of the(JCP, β
)plane (right). There is no constraint
on the sign of either JCP or sinβ. Red bands denote the Planck collaboration exclusion. Perturbations
were varied in the range [−0.2, 0.2]. Uncertainty intervals were obtained from Table 3.1.
Π 9 Π
8
5 Π
411 Π
83 Π
2
1
10-1
10-2
10-3
ÈmeeÈ
Β
TBM case
Perturbed case
ÈmeeÈ upper limit from GERDA
Figure 3.4: Values for the neutrinoless double beta decay parameter∣∣mee
∣∣, in the exact TBM and
perturbed cases, as a function of β (the region [π/2, π] is redundant). Both ε1 and ε2 were varied in
the range [−0.2, 0.2]. Red bands correspond to Planck and GERDA exclusions, while the green band
indicates the (aproximate) region of survival of the model, obtained from Fig. 3.3.
Finally, one focuses on the Majorana-type phases, which impact neutrinoless double beta decay.
Since the perturbations do not affect the Mν11 matrix element at first order (cf. (3.45)), the quantity
∣∣mee
∣∣ as a function of z1, z2 and β coincides, in the perturbed model, with that of the TBM case:∣∣mee
∣∣TBM
=∣∣z1 e
iβ + 2 z2/3∣∣. However, the relationship (3.41) between the mi and the zi is modified8:
z1 ' −1
2 cosβ
√∆m2
31
2, z2 '
[1− 1
3
(ε1 + ε2
)]√
∆m231
2, (3.49)
which reduces to the TBM case when ε1,2 → 0, as expected. The allowed values for the parameter∣∣mee
∣∣
as a function of the model phase β are presented in Fig. 3.4.
This perturbed version of the model will be reconsidered later on, in Chapter 5, where the viability
of implementing leptogenesis within its context is analysed.
8This once more revises a result given in [133].
48
Baryogenesis through
Leptogenesis 4We have so far considered how to extend our current knowledge of particle physics to accomodate the
experimental observation of nonzero neutrino masses. One might now wonder if the theory is sufficient to
explain the observed imbalance between matter and antimatter. In this chapter, the ingredients needed to
produce a nonzero baryon asymmetry are presented. We will recall why the SM is insufficient to account
for the observed BAU, and how seesaw extensions can provide a natural solution to this problem through
thermal leptogenesis. With this purpose, we start by briefly surveying some topics from the domains of
cosmology and statistical physics, which present themselves as relevant to the remainder of this work.
4.1 Topics of Cosmology and Thermodynamics
4.1.1 Cosmological Inflation
In the Standard Model of Cosmology (SMC), space-time is described by the Friedmann-Robertson-
-Walker metric (FRW) [135]:
ds2 = dt2 −R(t)2
[dr2
1− k r2+ r2
(dθ2 + sin2 θ dθ dφ2
)], (4.1)
which describes an isotropic space of constant (normalized) curvature k, and where (t, r, θ, φ) are dimen-
sionless comoving coordinates. The present cosmological data is consistent with a flat Universe [136],
corresponding to k = 0. The variable R(t) denotes the cosmic scale factor and its evolution is determined
by the Friedmann equation, where H(t) ≡ R(t)/R(t) is the Hubble parameter:
H(t)2 =8πG
3ρ(t)− k
R(t)2+
Λ
3. (4.2)
In this equation, Λ represents the cosmological constant, G Newton’s gravitational constant and ρ(t) the
total energy density of matter and radiation in the Universe at cosmic time t.
Cosmological inflation [137,138] is invoked in order to explain the flatness of the present-day Universe,
as well as to solve the horizon1 and GUT monopole problems. According to the inflationary scenario, the
scale factor of the Universe underwent an exponential increase in the early Universe. During this period,
the energy density ρ(t) remained approximately constant so that H ' const., implying R(t) ∝ exp(H t).
1The horizon problem pertains to the observation of a homogeneous and isotropic Universe, in line with the cosmologicalprinciple but in contradiction with what is expected without inflation. Inflation allows for the expansion of a region whichis causally connected (and thus allowed to thermalize in the early Universe) to the size of the observable Universe.
49
Furthermore, the density of any conserved charge scaled with the inverse of the volume of the Universe,
leading to:
nB = nb − nb ∼ R(t)−3 , (4.3)
for the density of baryonic charge nB defined in Section 1.5.2. Since successful inflation demands a number
H t & 70 of exponential folds [139], primordial asymmetries are extremely diluted. One should thus have
a great imbalance in baryon number from the start in order to survive this dilution, in contradiction with
the assumption of a constant matter energy density during inflation. As this is an unfruitful solution,
one turns to mechanisms which dynamically generate the observed baryon asymmetry.
4.1.2 Equilibrium Thermodynamics
In order to describe the evolution of early Universe particle content after inflation has taken place (but
prior to EWSB), one turns to the domain of statistical physics. To each particle species i one associates a
phase-space distribution fi(~p, ~x, t), which is taken to depend solely on energy E and time t, fi ≡ fi(E, t),as a consequence of the assumed homogeneity (no dependence on position) and isotropy (no dependence
on the direction of linear momentum) of the early – hot and dense – Universe plasma. The internal
degrees of freedom gi for a certain particle species (of mass mi) appear explicitly in its expressions for the
number density ni, energy density ρi and pressure pi, which are given as a function of the phase-space
distribution by:
ni ≡gi
(2π)3
∫d3p fi
(Ei(~p)
), (4.4)
ρi ≡gi
(2π)3
∫d3pEi(~p) fi
(Ei(~p)
), (4.5)
pi ≡gi
(2π)3
∫d3p
|~p |23Ei(~p)
fi(Ei(~p)
), (4.6)
where energy depends only on the magnitude of the momentum, Ei(~p) = Ei(|~p|) =√|~p|2 +m2
i (we have
omitted time dependence). The non-trivial formula for pressure has been derived in the context of kinetic
theory (see, for instance, section 4.2 of Ref. [140]). For a species of spin si, one has gi = 2si+1 if mi > 0,
gi = 2 if mi = 0 and si > 0, and gi = 1 if2 both mi = 0 and si = 0 [141].
A situation of kinetic equilibrium, which is typically enforced through scatterings in the plasma, is
defined by the possibility of having the different fi cast into the form (units of kB = 1 are considered):
fi(E(~p)
)=
1
exp[(Ei(~p)− µi) /Ti
]± 1
, (4.7)
corresponding to Fermi-Dirac (‘+’ sign) and Bose-Einstein (‘−’ sign) quantum distribution functions,
associated with fermions and bosons, respectively. In the cases for which the exponential in the denom-
inator overshadows the ±1, the above distribution reduces to the known (classical) Maxwell-Boltzmann
one. The uncertainty in the fi(E(~p)
)is thus encoded into a single time-dependent parameter, the chem-
ical potential µi, whose value is fixed by the interactions if the system is in chemical equilibrium [142].
In such a case, the interaction i + j → k + l implies that the relation µi + µj = µk + µl holds. We will
2In the context of a chiral description, each chiral fermionic component ψR,L has gi = 1.
50
henceforth use the superscript ‘eq’ to refer to situations of (local3) thermal equilibrium, achieved when
both kinetic and chemical equilibria conditions are verified.
Neglecting quantum aspects of the distribution, the kinetic equilibrium density ni reduces to:
ni = eµi/Tigi
(2π)3
∫d3p e−Ei/Ti =
gi eµi/Ti
(2π)3
∫∫dΩ
∫e−Ei/Ti |~p|2 d|~p|, (4.8)
where∫∫
dΩ = 4π since there are no angular dependences. Omitting the index i and defining z ≡ E/T and
x ≡ m/T allows one to perform a change of integration variables, yielding (notice that |~p| d|~p| = E dE):
n e−µ/T =g
2π2
∫ ∞
m
e−E/T E |~p| dE =g T 2
2π2
∫ ∞
x
e−z z |~p| dz =g T 3
2π2
∫ ∞
x
e−z z√z2 − x2 dz. (4.9)
The above can be expressed in terms of a second-kind modified Bessel function of order 2. The modified
Bessel function of the second kind of order ν, Kν , can be found on page 376, section 9.6.23 of the
Abramowitz and Stegun Handbook of Mathematical Functions [144] and is given by (for Re (ν) > −1/2
and for |arg(y) | < π/2):
Kν(y) ≡√π(y/2)ν
Γ(ν + 1/2)
∫ ∞
1
e−yt(t2 − 1)ν−1/2 dt, (4.10)
where Γ represents the gamma function, which extends the factorial to complex arguments. We will be
interested in cases for which y ∈ R and ν ∈ N, in particular ν = 1, 2(Γ(3/2) =
√π/2, Γ(5/2) = 3
√π/4
).
For the case ν = 2, one then has:
K2(y) =y2
3
∫ ∞
1
e−yt(t2 − 1)3/2 dt =1
y2
∫ ∞
y
e−t t√t2 − y2 dt. (4.11)
This last result has been obtained through the change of variables y t→ t and integration by parts. One
then concludes that a classical equilibrium distribution for a particle of mass mi is given by:
ni =gi T
3i
2π2eµi/Ti x2
i K2(xi)mi→ 0−−−−−→ ni =
gi T3i
π2eµi/Ti , (4.12)
where the massless (or relativistic) limit, Ti mi implies ximi/Ti → 0, has been also considered.
The opposite, non-relativistic limit Ti mi for the number density can be obtained by looking at the
behaviour the Bessel function as x→∞:
K2(x)x→∞−−−−−→ e−x
[√π
2
√1
x+O
(1
x3/2
)], (4.13)
leading to, for Ti mi:
ni = gi
(mi Ti
2π
)3/2
exp[−(mi − µi
)/Ti
], (4.14)
from which one can see that the density for a non-relativistic species is Boltzmann suppressed.
Considering quantum corrections to the relativistic species density of (4.12) results in:
ni =
ζ(3)gi T
3i
π2(Bose-Einstein)
3
4ζ(3)
gi T3i
π2(Fermi-Dirac)
, (4.15)
3Thermal equilibrium will never be attained globally but only locally since there is no timelike spatially constant Killingvector in the FRW metric [143]. For a slow enough expansion, one can ignore this technicality.
51
where one has also considered the limit µi Ti (see, for example, Appendix C of Ref. [141]), valid as
long as one ignores scenarios with degenerate fermions or Bose condensation. The number density for
photons, for which gγ = 2 (two polarizations) and µ(eq)γ = 0 (photon number is not conserved), is thus
given by:
nγ =2 ζ(3)
π2T 3, (4.16)
where T denotes the photon temperature, often simply refered to as the temperature of the Universe,
since it describes the thermal equilibrium state of the early Universe plasma.
For non-relativistic particles, on the other hand, polylogarithmic corrections introduced by quantum
corrections can be ignored, as the quantum expression for the number density matches the classical
expression (4.14), at first order in mi/Ti.
We can now make Eq. (1.7) sensible, as it results from the ratio between the (thermal, Ti = T )
equilibrium densities (4.14) and nγ given in Eq. (4.16):
neqi
nγ=gi√π
25/2
(mi
T
)3/2
exp[−(mi − µeq
i )/T]' gi
√π
25/2
(mi
T
)3/2
exp(−mi/T
), (4.17)
where the approximation is valid in the limit µeqi /T 1.
4.1.3 Expansion, Entropy and Degrees of Freedom
In what follows, we will be interested in the radiation dominated era of the thermal history of the
Universe (T & 1 eV), for which the total pressure and energy density are related by the equation of state
p = ρ/3. To determine the dependence of the Hubble parameter on the temperature T , one turns to the
Friedmann equation. For a flat metric, with no cosmological constant, Eq. (4.2) reduces to:
H(T )2 =8πG
3ρ(T ) ⇒ H(T ) =
√8πG
3
√ρ(T ) . (4.18)
The dominant contribution for the total energy density comes from relativistic species, for which:
ρi(T ) =
π2
30gi T
4i (Bose-Einstein)
7
8
π2
30gi T
4i (Fermi-Dirac)
. (4.19)
Hence, the total energy density ρ(t) is given by:
ρ(T ) =∑
i
ρi(T ) =
[ ∑
bosons
gi
(TiT
)4
+7
8
∑
fermions
gi
(TiT
)4]π2
30T 4 ≡ g∗(T )
π2
30T 4. (4.20)
The dependence of the total number of relativistic degrees of freedom g∗ on temperature may be ignored
and, therefore, g∗ is taken to be constant at high enough energies. Summing over SM species, which
are all relativistic for temperatures T & 100 GeV (Ti ' T ), one obtains g∗ = 106.75. By considering√G = m−1
Pl , where mPl ' 1.22× 1019 GeV is the Planck mass, one then has:
H(T ) =
√8π3
90g
1/2∗
T 2
mPl' 1.66 g
1/2∗
T 2
mPl. (4.21)
52
As a consequence of the first law of thermodynamics and (4.18), the equation of state p = ω ρ implies
ρ ∝ R−3(1+ω) and R ∝ t2(1+ω)/3, and so, in the radiation dominated era, R ∝√t. This allows one to write
the Hubble parameter as a function of time, as well as to relate cosmic time with photon temperature:
H(t) =1
2t⇒ t(T ) ' 0.301 g
−1/2∗
mPl
T 2. (4.22)
It is useful to introduce the entropy density s, defined as the ratio between the entropy per comoving
volume S and the physical volume V = R3, i.e.
s ≡ S
R3=∑
i
pi + ρiTi
=2π2
45
[ ∑
bosons
gi
(TiT
)3
+7
8
∑
fermions
gi
(TiT
)3]T 3 ≡ 2π2
45g∗S T
3, (4.23)
where the equation of state p = ρ/3 and Eq. (4.19) have been considered. Together with Eq. (4.16), the
above relation establishes the relation s =(π4/(45 ζ(3))
)g∗S nγ ' 1.8 g∗S nγ . For the early periods of
the history of the Universe we are interested in, g∗S can be replaced by g∗. In the absence of entropy
production (S = const.), s(t) ∝ R(t)−3, which implies that the ratio Yi ≡ ni/s remains constant in
the absence of production or destruction of i-species particles, as the number of particles in a comoving
volume element, Ni ≡ niR3, is then proportional to ni/s. The ratio of number density per entropy
density, Yi, will then be a useful quantity to track the evolution of a particle species. One is free to write
ni/nγ ' 1.8 g∗S Yi. At present times, the seas of decoupled photons and neutrinos determine the value
g∗S = 3.9, implying ηi ≡ ni/nγ ' 7.04Yi. Constant entropy per comoving volume additionally implies
that g∗S T3R3 remains constant during the expansion of the Universe, and so T ∝ g
−1/3∗S R−1. In the
case of constant g∗S , the relation T ∼ R−1 follows.
We will henceforth consider that the assumption of kinetic equilibrium entails the equality of all Ti
governing the distributions (4.7) of particles in the thermal bath which, in particular, will match the
photon temperature T .
4.1.4 Brief Thermal History of the Universe
Significant events in the evolution of the Universe have been driven by departures from thermal equi-
librium. As a rule of thumb, one may consider an interaction to be out of equilibrium if the corresponding
rate Γ is not fast enough to accompany the expansion of the Universe4, i.e. Γ H(T ). If this is the case,
interactions freeze-out and the particle species decouples from the plasma. On the contrary, if interactions
are fast, Γ H(T ), one takes the corresponding particle species to be in (thermal) equilibrium. The
non-trivial case is located in between, with Γ ∼ H(T ), and demands a more careful and quantitative
treatment, which relies on the Boltzmann transport equation (to be presented in Section 4.5).
The central events in the thermal evolution of the Universe are summarized in Fig. 4.1. Following the
inflationary era, a baryon asymmetry must have developed prior to (e.g. GUT baryogenesis or leptogenesis
scenarios) or in coincidence with (electroweak baryogenesis) the electroweak phase transition (EWPT),
which occurred for temperatures of the order of T ∼ 100 GeV. As the temperature decreased, more and
more SM species became non-relativistic, contributing to the decrease of both g∗ and g∗S . After the
4The case of a massless species which has decoupled from the plasma presents an exception, as such a species will retainan equilibrium distribution (with a naturally different temperature from that of the plasma) even after the decoupling.
53
Figure 4.1: Brief thermal history of the Universe. The family symmetry breaking scale is often taken to
be of the order of the GUT scale. The location of the seesaw scale, which coincides with that of thermal
leptogenesis (Section 4.4), can vary greatly depending on the model implementation.
quantum chromodynamics (QCD) phase transition, which is believed to occur for temperatures around
200−400 MeV [145], the existing baryon asymmetry was fed into the process of Big Bang (or primordial)
nucleosynthesis (BBN), corresponding to the formation of light nuclei for 10 MeV & T & 100 keV. All
excess nucleons and antinucleons would have annihilated for temperatures below T ∼ mπ ∼ 100 MeV,
while electrons and positrons did so at a lower scale, T ∼ 2me ∼ 1 MeV. At even lower temperatures,
T ∼ 0.1 eV, nuclei and free electrons were brought together to form neutral atoms, in what is misleadingly
referred to as recombination. Atom formation was accompanied by the lowering of the number of free
electrons, allowing for the decoupling of photons, which made up the cosmic microwave background,
characterized at present times by a temperature T 0γ ' 2.73 K. One might wonder about the origin of the
excess of electrons over positrons if a mechanism focused solely on the generation of a baryon asymmetry
is considered. Such excess may arise through weak interactions (fast at high-energies), without affecting
total lepton number – part of which may be stored in yet unobserved neutrino seas – or the apparent
electric charge neutrality of the Universe.
4.2 The Sakharov Conditions
From the discussion of Sections 1.5 and 4.1.1 one concludes that, in order to explain the present
BAU, a baryon number asymmetry must be dynamically generated after inflation has taken place. In
fact, taking an initial state of zero baryon number, B = 0, a baryon asymmetry can be generated if the
following sufficient conditions, presented by Andrei Sakharov in 1967 [146], hold:
• B symmetry is violated.
• C and CP symmetries are violated.
• Some departure from thermal equilibrium occurs.
54
Although the Sakharov conditions are not necessary conditions for the BAU generation [147], trying to
get around any one of the three is generally not easy and might require troublesome assumptions, such
as CPT violation, which is associated to the breaking of Lorentz invariance (see Section 1.3.3).
B Violation
The baryon number operator is given, as a function of time, by [57]:
B(t) =1
3
∑
quarks q
∫d3x :q†(~x, t) q(~x, t) : , (4.24)
where q(~x, t) corresponds to the quark field operator and colons refer to Wick ordering. From the
discussion of Section 1.5.3, it is clear why B-violating interactions are needed for the generation of any
baryon asymmetry. Although no direct experimental proof for non-conservation of B exists yet, grand
unified theories and standard electroweak theory demand it at sufficiently high energies.
C and CP Violation
In order to understand the need for both C and CP violation, one must consider the action of these
transformations on the B operator. Adopting standard conventions for the phases, one has:
P q(~x, t) P−1 = γ0 q(−~x, t), C q(~x, t) C−1 = i γ2 q†(~x, t), T q(~x, t) T−1 = −i q(~x,−t) γ5 γ0γ2, (4.25)
where P , C and T are the operators which carry out the parity, charge conjugation and time reversal
operations, respectively. Taking into account the unitarity of P and C and the anti-unitarity of T , one
easily obtains the transformations for q†:
P q†(~x, t) P−1 = q†(−~x, t) γ0, C q†(~x, t) C−1 = i q(~x, t) γ2, T q†(~x, t) T−1 = −i γ2γ0 γ5 q†(~x,−t).(4.26)
Considering that, since quarks are fermions, :q q† : = − :q† q : holds, one obtains:
P B P−1 = B , C B C−1 = −B , CP B (CP )−1 = −B, (4.27)
CP T B(0) (CP T )−1 = −B(0). (4.28)
This shows that the B is odd under C and CP transformations and, thus, a nonzero expectation value of
B, demands the presence of both C and CP violation.
Departure from Equilibrium
The expectation value of the B(t) operator is given by:
⟨B(t)
⟩= Tr
[ρD(t) B(t)
], (4.29)
where ρD(t) denotes the density operator. In the Heisenberg representation, one has:
B(t) = eiHt B(0) e−iHt. (4.30)
55
The third Sakharov condition can be understood by considering a scenario of thermal equilibrium at a
temperature T , for which ρD = exp(−H/T )/Z, with a partition function Z = Tr[exp(−H/T )
]. Using
the above relations and considering an equilibrium scenario, one obtains:
⟨B(t)
⟩T
=1
ZTr[e−H/T B(t)
]=
1
ZTr[e−H/T eiHt B(0) e−iHt
]
=1
ZTr[e−H/T B(0)
]=⟨B(0)
⟩T,
(4.31)
where the cyclic nature of the trace and the possibility of commuting the exponential operators have been
used. Assuming a CPT invariant Hamiltonian, CP T H (CP T )−1 = H, and taking into account (4.28)
and (4.31), one sees that:
⟨B(t)
⟩T
=⟨B(0)
⟩T
=1
ZTr[e−H/T B(0)
]
=1
ZTr[(CP T )−1(CP T ) e−H/T (CP T )−1(CP T ) B(0)
]
=1
ZTr[(CP T ) e−H/T (CP T )−1(CP T ) B(0) (CP T )−1
]
=1
ZTr[e−H/T
(−B(0)
)]= −
⟨B(0)
⟩T.
(4.32)
Hence, the expectation value of the baryon number operator vanishes in thermal equilibrium and so, even
in the presence of B, C and CP violating interactions, no asymmetries can be generated.
Naıvely, one might expect that the reason the two first Sakharov conditions do not suffice is because
inverse decays would wash out the asymmetries generated by decays if thermal equilibrium is imposed.
However, this is not the case. In fact, decays and inverse decays both push the asymmetry in the same
direction. Other scattering processes must then be responsible for the washout, which is to be expected
from the unitarity of the QFT scattering matrix. A departure from equilibrium conditions is then needed,
and corresponds to the third and final requirement considered for the BAU generation.
4.3 Is the SM Enough?
The successful generation of the observed BAU depends on the nature of the model under considera-
tion. Despite its great successes, the Standard Model of particle physics seems to fall short in providing
viable conditions for baryogenesis. In fact, as we shall now verify, all necessary Sakharov ingredients are
avaliable in the SM, but cannot satisfactorily account for the value of η ∼ 10−10 given in Eq. (1.6).
B + L Violation
The SM Lagrangian is invariant under global rephasing of both quark and lepton fields. At the classical
level, one can associate to these transformations the currents JBµ (x) and JL
µ (x), obeying ∂µJB,Lµ (x) = 0.
This guarantees the conservation of the quantum numbers (baryon and lepton numbers) associated to
the charge operators B =∫d3xJB
0 (x) and L =∫d3xJL
0 (x). However, these accidental SM symmetries
are broken beyond the classical field aproximation as the JB,Lµ are no longer conserved due to the Adler-
-Bell-Jackiw chiral anomaly [148,149]. In the present case, one obtains the anomalous divergences [57]:
∂µJBµ = ∂µJL
µ = n
(g2
2
32π2Ai
µνAiµν −
g2Y
32π2Bµν Bµν
), (4.33)
56
≠`ieL
`iµL`iτL
qj1L
qk1L
ql1L
qj2L
qk2L
ql2L
qj3L
qk3L
ql3L
Figure 4.2: Schematic representation of the vacuum structure of the electroweak theory (left) and effective
diagram for the transition between vacua (right). Esphaleron corresponds to the height of the energy
barrier. Vacuum states are assigned an integer topological charge NCS. In the 12-fermion diagram,
numbers denote generations, quark line colours refer to the SU(3)c group, and roman indices pertain to
isospin (not all combinations are possible, as the amplitude is proportional to εij εkl).
where n = 3 is the number of SM fermion generations and the tilde represents the dual of a field strength
tensor, Fµν ≡ εµναβFαβ/2 (the convention ε0123 = +1 is considered). The above implies that B and
L (and consequently B + L) are violated – the SM satisfies the first Sakharov condition – whereas the
combination B − L is conserved. This is clear from (4.33) since one can construct a conserved current
JB−Lµ ≡ JB
µ − JLµ with ∂µJB−L
µ = 0. Due to cancellations, the gauge symmetry group of the SM is not
anomalously broken and the theory remains renormalizable [150].
The non-Abelian character of the gauge theory translates into a non-trivial topological structure,
which includes an infinite number of ground states to which a topological charge, the Chern-Simmons
winding number NCS ∈ Z [151], is associated. The non-conservation of baryon and lepton numbers may
be visualized as transitions between vacua (leftmost diagram of Fig.4.2), where the meaningful quantity
is the difference between charges, ∆NCS. A selection rule applies:
∆B = ∆L = n∆NCS = 3 ∆NCS . (4.34)
Such transitions may occur via the instanton field, which is associated to a tunneling scenario between
adjacent vacua: ∆B = ∆L = ±3. At present day collision energies, however, these processes are expo-
nentially suppressed by a factor of e−16π2/g22 ∼ 10−160 [152].
An alternative way of switching between vacua is due to the sphaleron field [153, 154] (see rightmost
diagram of Fig. 4.2): if the temperature of the thermal bath to which the SM fields are coupled is
high enough (T > TEW ∼ 100 GeV), (B + L)-violating processes may occur as the barrier Esphaleron
of Fig. 4.2 can be surpassed by thermal fluctuations. Comparing the rate of these (B + L)-violating
reactions, Γsphalerons, with the expansion rate of the Universe, one concludes that they are in thermal
equilibrium for temperatures up to ∼ 1012 GeV [155]. Thus, the SM presents a mechanism for baryon
number violation which in the early Universe is fast and unsuppressed.
57
≠
»
Figure 4.3: Diagram for the expansion of a ‘true vacuum’ bubble. The imbalances produced in CP
violating interactions near the wall will be converted into a baryon asymmetry by sphaleron processes
(still thermal ouside).
The Electroweak Phase Transition
Skipping ahead to the third Sakharov condition, one can make two key points regarding the departure
from thermal equilibrium in a SM baryogenesis scenario. First, that such a departure must occur during
the electroweak phase transition (T ∼ TEW ), and second that a first-order phase transition is required
for sucessful baryogenesis. The claim that baryogenesis within the SM corresponds to a particular case
of electroweak baryogenesis can be understood by considering that for T > TEW the reaction rates
for SM interactions are much larger than H(T ), and so no deviations from thermal equilibrium occur.
Additionally, since the particle content of the model is massless above the EWPT, no CP-violation effects
are observed for such temperatures (CP is conserved due to quark mass degeneracy [26]).
A first-order EWPT proceeds as follows [156]: as the Universe cools, it reaches a critical temperature
Tc at which both the formation and expansion of ‘true vacuum’ bubbles with a nonzero Higgs VEV
take place. Particles in the Universe plasma interact with the bubble walls, as illustrated in Fig. 4.3,
generating an asymmetry in some quantum number, which is then carried back to the unbroken space
(where the Higgs VEV is still zero). In this space, sphalerons will be responsible for converting the
produced asymmetry into a baryon number asymmetry. As the bubble expands, it sweeps regions where
a baryon number imbalance is present and freezes it, since the rate of sphaleron reactions inside the bubble
is highly suppressed5. In the above description, the Higgs VEV plays the role of an order parameter.
Were the EWPT second-order (or not strong enough) and sphaleron processes would still be active after
the transition, washing away any asymmetry that was potentially generated. The order of the EWPT
is highly dependent on the physical Higgs mass, mH . In particular, for mH & 80 GeV [157] (as is the
case) the phase transition is already second-order. This analysis thus rules out the SM ‘as is’ as a viable
candidate in explaining the observed BAU, independently of any CP violation mechanism which may be
present within the model.
5The condition of high suppression of sphaleron effects corresponds to a so-called strong first-order phase transition.
58
Not Enough CP
We now address the remaining Sakharov condition. Setting aside the problem of whether or not the
EWPT is first-order, one can estimate the strength of CP violation in the SM through the measure dCP ,
constructed from rendering the invariant (m2t−m2
c)(m2t−m2
u)(m2c−m2
u)(m2b−m2
s)(m2b−m2
d)(m2s−m2
d)JCP
dimensionless by dividing it by the 12th power of TEW . One obtains dCP ∼ 10−19, which should also
be an estimate for the matter-antimatter asymmetry generated in the interactions between SM particles
and the expanding bubble wall. The fact that this value is orders of magnitude away from η ∼ 10−10
constitutes an argument for why the amount of CP violation present in the SM is insufficient for successful
baryogenesis. Although there has been some debate over the validity and possible enhancement of the
presented estimate [158, 159], it is a general consensus that one has to search for extra, non-standard
sources of CP violation. Nevertheless, since one has established that the phase transition is not first-
-order, this remains a moot point in the context of electroweak baryogenesis.
4.4 Thermal Leptogenesis
The Standard Model appears to be insufficient in explaining the presently observed BAU in light of
the Sakharov conditions. However, one may consider extensions to the SM, such as supersymmetry or
the addition of a second Higgs doublet, which provide additional sources of CP violation and guarantee
that the electroweak phase transition is strongly first-order. In particular, two Higgs doublet models
are found to provide a sufficiently enlarged parameter space to accomodate the desired phase transition
strength [160]. For the case of supersymmetric models, a large enough mass splitting between the two
stops is required for the EWPT to be first-order.
Aside from electroweak baryogenesis and other exotic scenarios [147] such as the Affleck-Dine mech-
anism [161], associated with the rolling of a scalar field (possibly a superpartner of an SM fermion), an
alternative class of baryogenesis scenarios presents itself if one considers the out-of-equilibrium decays of
considerably massive particles in the very early Universe, T TEW . One such particularly interesting
scenario is that of leptogenesis [162], in which the decays of heavy particles – such as the seesaw medi-
ators of Section 2.2.2 – violate lepton number L. Lepton-number asymmetries would then be converted
into baryon-number asymmetries by electroweak sphaleron processes, whereas the required CP violation
would arise from complex couplings in the heavy particle interaction Lagrangian. We will henceforth
focus on the case of thermal leptogenesis, meaning that the heavy particles are produced thermally, fol-
lowing reheating, by scattering processes in the plasma. The additional assumption of a hierarchical mass
spectrum for the added heavy species is taken6, implying that the significant contribution to the BAU
comes from the decays of the lightest of such particles: asymmetries produced by the heavier ones are
subject to erasure through interactions involving the lightest.
Having accounted for possible deviations from thermal equilibrium due to the extreme mass of the new
states and the expansion of the Universe, one turns to the remaining Sakharov conditions. If the decays of
6Such an assumption might conflict with the requirement of low reheating temperatures included in the solution of the so--called SUSY gravitino problem. One way to avoid this incompatibility is to turn to resonant leptogenesis scenarios [163,164],which can accomodate low-scale leptogenesis with small mass splitings between the new particles.
59
Figure 4.4: Effect of electroweak sphalerons on the quantum numbers B and L. For a certain amount of
existing B−L, unaffected by sphalerons, the plasma evolves along one of the thin diagonal lines until the
condition of (4.35) is met.
the new species violate L alone, then B−L is also broken. This is a crucial aspect of baryogenesis models
operating at high scales T TEW , as electroweak sphalerons (which tend to wash out all asymmetries)
cannot erase B−L, thus classified as a non-thermalizing mode (see Fig. 4.4). For a second-order EWPT,
the following relation between lepton and baryon numbers at the electroweak scale is imposed by sphaleron
interactions in equilibrium [165,166]:
YB =12
37YB−L ⇒ YB = −12
25YL = −0.48YL . (4.35)
Next, we discuss the remaining Sakharov condition, namely CP violation.
4.4.1 CPT, Unitarity and CP Asymmetries
In the context of QFT, to the transition between asymptotic states (or sets of states) i and j, one
associates a quantum amplitude M(i → j), to be calculated following the Feynman rules of the theory.
Denoting the CP-conjugates of i and j by i and j, respectively, CPT invariance implies [141]:
CPT invariance: M(i→ j) =M(j → i), (4.36)
whereas the imposition of CP invariance translates simply to:
CP invariance: M(i→ j) =M(i→ j). (4.37)
By joining the above conditions, one obtains a relation valid under T invariance:
T invariance: M(i→ j) =M(j → i). (4.38)
Finally, the unitarity of the scattering matrix in QFT (also known as the S-matrix) implies7:
∑
j
|M(i→ j)|2 =∑
j
|M(j → i)|2 CPT−−−−→∑
j
|M(i→ j)|2 =∑
j
∣∣M(i→ j)∣∣2 , (4.39)
7 One is presently ignoring the effects of quantum statistics. A more general treatment is presented, for instance, inAppendix A.2 of [141], where Pauli blocking and Bose enhancing factors enter the above expression.
60
`
`
Mone-loop
X = X
`
`
Mtree
X
Figure 4.5: Tree-level and one-loop diagrams for the process X → ` ` whose interference generates a CP
asymmetry when compared to the conjugate process. The higher-order diagram is illustrative and may
represent either a wave or vertex correction.
where a sum over all possible states or sets of states j has been considered (such sum includes any particle j
as well as its CP-conjugate j). The above relations allow one to parametrize the CP asymmetries between
different decay modes of a heavy state, X, with mass M . Suppose X = X and X have two distinct decay
modes, namely X → ` ` and X → ` `. One can then write:
|M(X → ` `)|2 = |M(` `→ X)|2 = (1 + ε/2) |M0|2,
|M(X → ` `)|2 = |M(` `→ X)|2 = (1− ε/2) |M0|2,(4.40)
where |M0|2 is the value |M|2 would take in the absence of a CP asymmetry ε. From Eq. (4.40) and
from the fact that the two-body decay rate Γ of a particle obeys8 Γ ∝ |M|2, with the proportionality
constant depending solely on particle masses, one obtains:
ε = 2Γ(X → ` `)− Γ(X → ` `)
Γ(X → ` `) + Γ(X → ` `). (4.41)
The expressions given above can be straightforwardly generalized. An additional consequence from the
unitarity of the S-matrix is that squared absolute values of amplitudes for CP-conjugate processes may
only differ beyond the tree-level computation. In fact, the first-order contribution towards a nonzero CP
asymmetry is found in the interference between the tree-level and the one-loop correction diagrams (see
Fig. 4.5). One has, for coupling constants of order y:
Γ ∝ |Mtree +Mloop +O(y5)|2 = |Mtree|2 +M∗treeMloop +MtreeM∗loop +O(y6). (4.42)
Thus, in the numerator of Eq. (4.41) one must expand decay rates up to the order of the interference,
while for the denominator one can take the approximation Γ ' |Mtree|2. By extracting coupling constants
yi from amplitudes in the following schematic manner:
Mtree ≡ y1Atree , Mloop ≡ y2 y∗3 y4Aloop , (4.43)
8The square modulus of the amplitude is here assumed to be averaged over initial and summed over final degrees offreedom of the interacting particles.
61
one can, in practice, write the above numerator as:
Γ(X → ` `)− Γ(X → ` `) =(M`` ∗
treeM``loop +M``
treeM`` ∗loop
)−(M` ` ∗
tree M` `loop +M` `
treeM` ` ∗loop
)
= 4 Im(y1 y
∗2 y3 y
∗4
)Im(A∗treeAloop
).
(4.44)
This expression clearly shows that, in order to have a nonvanishing CP asymmetry, one must allow
complex coupling constants and Im(A∗treeAloop
)6= 0. Verifying these conditions demands for the presence
of at least twoX-type particles whose masses are such that the particles running along the loop are allowed
to be on-shell.
4.5 Boltzmann Equation(s)
The Boltzmann transport equation (BE) allows one to quantitatively describe the evolution in time
of the (non-equilibrium) phase-space distribution fψ(Eψ, t) of a particle species ψ influenced both by
interactions – which can be responsible for the creation or destruction of such particles – as well as by
the expansion of the Universe. The BE can be summarily written as:
L fψ = C fψ , (4.45)
where L corresponds to the Liouvillian operator [167]:
L fψ ≡∂fψ∂t−H(t)
|~pψ|2Eψ
∂fψ∂Eψ
, (4.46)
which codifies the effects of the expansion of the Universe on the evolution of particle distributions, while
C denotes the collision operator [140]:
C fψ ≡ −∑
ψ,a,...↔ i,j,...
1
2Eψ
∫dΠ′ (2π)4 δ4(pout − pin)
[|M|2dir fψfa . . . − |M|2inv fifj . . .
], (4.47)
which accounts for the effect of particle interactions. In the above expression, one sums over all possible
reactions involving the ψ species. The shorthand dΠ′ = dΠa . . . dΠi dΠj . . . is considered, where one has
made use of the definition:
dΠi ≡gi
(2π)3
d3pi2Ei
, (4.48)
where the 2Ei factors arise from a delta function imposing the energy-momentum relations E2i = |~pi|2 +
m2i . The argument of the delta function of (4.47) contains both the initial and final four-momenta,
defined as pin ≡ pψ + pa + . . . and pout ≡ pi + pj + . . .. The amplitudes |M|2dir and |M|2inv correspond to
the reactions ψ, a, . . . → i, j, . . . and i, j, . . . → ψ, a, . . ., respectively, and are averaged over both initial
and final spin degrees of freedom. The (global) signs with which these amplitudes enter the BE reflect
whether the process is responsible for the creation or destruction of ψ particles. Both a symmetry factor
and a multiplicity factor will have to be included in the above expression to account for identical particles
in the interaction and the possibility of producing/destroying more than one ψ-type particle, respectively.
Quantum effects have been ignored in writing (4.47), and can be included through the substitutions:
fψfa → fψfa (1± fi) (1± fj), fifj → fifj (1± fψ) (1± fa), (4.49)
62
where the ‘+’ and ‘-’ signs correspond to Bose enhancing and Pauli blocking factors, respectively (see
Footnote 7), which we henceforth neglect. To proceed, one considers the evolution of particle densities
ni, obtained from fi through Eq. (4.4). One thus integrates both sides of the BE, obtaining, for the
left-hand side (and using integration by parts):
gψ(2π)3
∫L fψ d
3p = nψ −∫∫
dΩ
∫ | ~pψ|4Eψ
∂fψ∂Eψ
d| ~pψ| = nψ + 3H nψ , (4.50)
while the right-hand side yields:
gψ(2π)3
∫C fψ d
3p = −∑
ψ,...↔...
∫dΠ (2π)4 δ4(pout − pin)
[|M|2dir fψfa . . . − |M|2inv fifj . . .
], (4.51)
where dΠ ≡ dΠψ dΠ. One can further manipulate the BE by noticing that:
ni + 3Hni =1
R3
d
dt
(niR
3)
=S
R3Yi = s Yi . (4.52)
By defining the variable z ≡ M/T , where M represents a relevant mass scale (chosen in the case of
thermal hierarchical leptogenesis to coincide with that of the lightest decaying particle) one sees that:
Yi =dYidz
dz
dt=
dYidz
Md(1/T )
dt=
dYidz
M
(− 1
T 2
)(dH
dt
/dH
dT
)= z H
dYidz
, (4.53)
where Eq. (4.22) for H(t) in the radiation dominated era has been used. The left-hand side of the BE
now reads:
ni + 3Hni = s z HdYidz
. (4.54)
Putting together what we have so far explicitly gives:
sHzdYψdz
=−∑
ψ,...↔...
(#ψ)
∫gψ d
3pψ(2π)32Eψ
ga d3pa
(2π)32Ea, . . .
gi d3pi
(2π)32Ei
gj d3pj
(2π)32Ej. . .
×(2π)4 δ4(pi + pj + . . .− pψ − pa − . . .)1
Πk(#k)!
[|M|2dir fψfa . . . − |M|2inv fifj . . .
],
(4.55)
where (#ψ) counts the number of destroyed ψ particles in the interaction ψ + a+ . . .→ i+ j + . . ., and(Πk(#k)!
)−1is the product of symmetry factors 1/(#k)! for all particles k involved in the reaction, each
with a multiplicity of (#k). The gi factors, present in the definition (4.48), are henceforth absorbed by
the amplitudes, which are no longer averaged but both summed over the degrees of freedom of the initial
and final states. dΠi will then simply read dΠi = d3pi/((2π)32Ei
).
All that remains is to simplify the right-hand side of the BE. We will be interested in doing so for
two particular cases, namely those of two-body decays and 2↔ 2 scatterings.
4.5.1 Two-body Decays and Inverse Decays
Consider the two-body decays of a particle species ψ through the reactions ψ → i+ j and i+ j → ψ,
respectively. We denote a parametrization of the CP asymmetry in amplitudes by:
|M(ψ → i+ j)|2 = α(ε)|M0|2 , |M(i+ j → ψ)|2 = β(ε)|M0|2 , (4.56)
where α(ε) and β(ε) contain the chosen parametrization (such as that of Eq. (4.40)).
63
The assumption of kinetic (but not chemical equilibrium) allows one to write phase-space distributions
in terms of number densities and equilibrium distributions/densities (cf. Eqs. (4.4) and (4.7)) as:
fψ = nψf eqψ
neqψ
=nψ
n′eqψ
f ′eqψ , (4.57)
where the primes denote that both f ′eqψ and n′eq
ψ correspond to a zero equilibrium chemical potential, µeqψ .
The equality is valid as the term eµeqψ /T cancels in the above expression. From now on, primes are dropped
and we work with zero chemical potential equilibrium densities and distributions (f ′eqi = e−Ei/T ).
We now split the BE (4.55) for the decaying particle and considered reactions into the direct (decay)
and indirect (inverse decay) parts. The direct part reads (symmetry factors are denoted by 1/S):
−∫dΠψ dΠi dΠj(2π)4δ4(pψ − pi − pj)
1
S |M(ψ → i+ j)|2 fψ
=− nψneqψ
∫d3pψ(2π)3
e−Eψ/T1
S
∫dΠi dΠj(2π)4δ4(pψ − pi − pj) |M|2dir
=− nψneqψ
∫gψ d
3pψ(2π)3
1
S1
2Eψ
∫dΠi dΠj(2π)4δ4(pψ − pi − pj)
1
gψ|M|2dir
e−Eψ/T
=− nψneqψ
∫gψ d
3pψ(2π)3
Γ(ψ → i+ j) e−Eψ/T ,
(4.58)
where in the last line we recognize the expression for the decay rate of ψ in an unspecified frame of
reference. The factor gψ has been explicitly and exceptionally included, since decay rates are typically
given in terms of an amplitude averaged over initial spins and summed over final ones. This rate can be
related to the decay rate in the rest frame ΓR by (Mψ is the mass of ψ particles):
Γ(ψ → i+ j) =Mψ
EψΓR(ψ → i+ j) . (4.59)
Expression (4.58) then reads:
− nψneqψ
∫gψ d
3pψ(2π)3
Mψ
EψΓR(ψ → i+ j) e−Eψ/T ≡ −nψ
⟨Mψ
EψΓR⟩eq
ψ
. (4.60)
In the last line we have defined what is usually called a thermal average [167]. Since in the rest frame
pψ = (Mψ,~0), one has that ΓR is independent of ~pψ and, therefore, it can be extracted from the thermal
average (we choose not to extract Mψ). On the other hand, Eψ is a function of the momentum, as it
does not necessarily refer to the rest frame. One explicitly has:
⟨Mψ
Eψ
⟩eq
ψ
=1
neqψ
∫gψ d
3pψ(2π)3
Mψ
Eψe−Eψ/T =
gψ(2π)3
∫ (Mψ
Eψ
)e−Eψ/T d3pψ
gψ(2π)3
∫e−Eψ/T d3pψ
. (4.61)
Taking into account the definition of the modified Bessel function given in (4.10) , one sees that:
K1(y) = y
∫ ∞
1
e−yt√t2 − 1 dt =
1
y
∫ ∞
y
e−t t√t2 − y2 dt, (4.62)
and thus the numerator may be written as (z ≡ Eψ/T , x ≡Mψ/T ):
gψ(2π)3
∫ (Mψ
Eψ
)e−Eψ/T d3pψ =
gψT3
2π2
∫ ∞
x
Mψ
Eψz√z2 − x2 e−z dz =
gψT3
2π2x2K1(x). (4.63)
64
In turn, the denominator neqψ is given in the left-hand side of Eq. (4.12). At the end, the thermal average
reduces to:
⟨Mψ
Eψ
⟩eq
ψ
=K1(x)
K2(x)=K1(Mψ/T )
K2(Mψ/T ). (4.64)
In conclusion, the direct part of the BE right-hand side, given in (4.58), simply reads:
−nψK1(Mψ/T )
K2(Mψ/T )α(ε) ΓR0 ≡ −
nψneqψ
α(ε) γD , γD = neqψ ΓR0
K1(Mψ/T )
K2(Mψ/T ), (4.65)
where ΓR0 corresponds to ΓR with |M0|2 instead of |M(ψ → i+j)|2, and γD is the decay reaction density.
For the inverse part of the BE, Eq. (4.58) is modified by changing the global sign, the amplitude and
fψ → fi fj . Since we are under an integral, where the delta function imposes conservation of energy, the
relation f eqi f eq
j = f eqψ is valid and the computations follow as in the direct case. A caveat arises regarding
(4.60), in which an neqψ is absorbed. To keep the calculation unchanged, one multiplies and divides the
expression by neqψ , resulting finally in
+ni njneqi n
eqj
neqψ
K1(Mψ/T )
K2(Mψ/T )β(ε) ΓR0 = +
ni njneqi n
eqj
β(ε) γD , (4.66)
for the inverse part. To summarize, the BE for decays and inverse decays reads:
sHzdYψdz
= −[nψneqψ
α(ε)− ni njneqi n
eqj
β(ε)
]γD . (4.67)
If one is interested in the evolution of Yi, the above is modified to dYi/dz = −(#i) dYψ/dz, where
(#i) = 2 if i = j and unity otherwise. Assuming no CP violation and that ni,j = neqi,j , one can write:
z
Yψ
dYψdz∼ −
(nψneqψ
− 1
)ΓR0H
, (4.68)
which means that the number of ψ particles in a comoving volume is not significantly altered for ΓR0 H
and interactions have thus frozen-out [54], making the rule of thumb presented in Section 4.1.4 sensible.
4.5.2 2↔ 2 Scatterings
Concerning 2 ↔ 2 scatterings ψ + a ↔ i + j, we consider the relation (4.57) once more, and define
the parametrization
|M(ψ + a→ i+ j)|2 = µ(ε)|M0|2 , |M(i+ j → ψ + a)|2 = ν(ε)|M0|2 . (4.69)
The BE (4.55) is taken as a starting point. Considering both the direct (ψ + a → i + j) and inverse
(i+ j → ψ + a) parts of the equation:
− (#ψ)
∫dΠψ dΠa dΠi dΠj(2π)4δ4(pψ + pa − pi − pj)
1
S
[|M|2dir fψfa − |M|2inv fifj
]
=− (#ψ)
∫dΠψ . . . dΠj(2π)4δ4(. . .)
1
S |M|20
[µ(ε)
nψ naneqψ n
eqaf eqψ f
eqa − ν(ε)
ni njneqi n
eqj
f eqi f
eqj
]
=− (#ψ)
[µ(ε)
nψ naneqψ n
eqa− ν(ε)
ni njneqi n
eqj
] ∫dΠψdΠa f
eqψ f
eqa
∫dΠidΠj(2π)4δ4(. . .)
1
S |M|20
=− (#ψ)
[. . .
] ∫d3pψ(2π)3
d3pa(2π)3
f eqψ f
eqa
F
EψEa
1
4F
∫dΠidΠj(2π)4δ4(. . .)
1
S |M|20
,
(4.70)
65
where one has used (4.57) and the relation f eqi f eq
j = f eqψ f eq
a . Inside the curly brackets one recognizes
the usual formula for a 2 ↔ 2 scattering cross section σ. We have also introduced the quantity F ≡√(pψ · pa)2 −M2
ψm2a and defined the Møller velocity in terms of particle velocities ~vi = ~pi /Ei [168]:
vMøl ≡F
EψEa=[|~vψ − ~va |2 − |~vψ × ~va |2
]1/2, (4.71)
which reduces to the relative velocity between particles when these are parallel. Its importance was first
recognized by Gondolo and Gelmini [167] for the computation of cosmological relic densities. One can
now rewrite Eq. (4.70) in the form:
−(#ψ)
[µ(ε)
nψ naneqψ n
eqa− ν(ε)
ni njneqi n
eqj
]neqψ n
eqa
1
neqψ n
eqa
∫d3pψ(2π)3
d3pa(2π)3
(σ vMøl
)f eqψ f
eqa
︸ ︷︷ ︸≡〈σvMøl〉eq
ψ,a
, (4.72)
where the thermal average 〈σvMøl〉eqψ,a has been set. Defining the reaction density γ2 ≡ neq
ψ neqa 〈σvMøl〉eq
ψ,a,
one can write the BE for the considered scatterings in a manner formally similar to that of Eq. (4.73):
sHzdYψdz
= −(#ψ)
[nψ naneqψ n
eqaµ(ε)− ni nj
neqi n
eqj
ν(ε)
]γ2 . (4.73)
To conclude, we now turn to the determination of a clearer formula for the scattering reaction density
γ2:
γ2 =
∫dp3ψ
(2π)3
dp3a
(2π)3σ(s)
F (s)
EψEaf eqψ f
eqa
=
∫dp3ψ
(2π)3
dp3a
(2π)3
[ ∫d4P δ(P0 − Eψ − Ea) δ3(~P − ~pψ − ~pa)
]F (s)σ(s)
EψEae−(Eψ+Ea)/T
=
∫d4P e−P0/T
∫dp3ψ
(2π)3Eψ
dp3a
(2π)3Eaδ4(P − pψ − pa)F (s)σ(s)
︸ ︷︷ ︸explicitly Lorentz invariant
,
(4.74)
where an integration term in P = (P 0, ~P ) has been introduced (the quantity between square brackets is
unity) and the dependency of F and σ on the square of the center-of-mass energy s = (pψ + pa)2 (not
to be mistaken with the entropy density) has been specified. Since the singled out quantity is Lorentz
invariant, we can compute it in any frame (but not the whole integral). In particular, one chooses the
frame in which pψ + pa = (√s,~0). This quantity reads:
∫dp3ψ
(2π)3Eψ
dp3a
(2π)3Eaδ(√s− Eψ − Ea) δ3(~pψ + ~pa)F (s)σ(s)
=4π
(2π)6
∫d|~pψ| |~pψ|
δ(|~pψ| − |~pψ|CM)
Eψ + EaF (s)σ(s) =
4π
(2π)6|~pψ|CM
F (s)σ(s)√s
.
(4.75)
In the center-of-mass frame, |~pψ|CM is given by:
|~pψ|CM =
√(s−M2
ψ −m2a
)2 − 4M2ψm
2a
2√s
≡
√λ(s,M2
ψ,m2a
)
2√s
=
√s
2
√λ(1,M2
ψ/s,m2a/s)
=F (s)√s, (4.76)
where one has defined the function λ(x, y, z) as in [169]. One can now write the reaction density as the
simple integral:
γ2 =
∫d4Pe−P0/T
1
4(2π)52 s λ
(1,M2
ψ/s,m2a/s)σ(s)
︸ ︷︷ ︸≡ σ(s)
=1
8π
∫d4P
(2π)4e−P0/T σ(s) , (4.77)
66
where one has also defined the so-called reduced (or dimensionless) cross section σ(s). Carrying out the
integration gives (P 2 = s):
γ2 =1
4(2π)5
∫4π dP0 d|~P ||~P |2 e−P0/T σ(s) =
1
4(2π)4
∫dP0
∫ds |~P | e−P0/T σ(s)
=1
64π4
∫ ∞
smin
ds σ(s)
[ ∫ ∞√s
dP0
√P 2
0 − s e−P0/T
],
(4.78)
where smin = max(Mψ + ma
)2,(mi + mj
)2. The term in square brackets can be evaluated in terms
of a modified Bessel function, yielding:
∫ ∞√s
dP0
√P 2
0 − s e−P0/T = T√sK1
(√s
T
). (4.79)
One then obtains the expression for the reaction density:
γ2 =T
64π4
∫ ∞
smin
ds√s σ(s) K1
(√s
T
). (4.80)
In this equation, the reduced cross section can also be given a clearer expression in terms of the Mandel-
stam variable t ≡ (pψ − pi)2. Since σ(s) is a Lorentz invariant quantity, it too can be computed in the
center-of-mass frame, namely
σ = 2 s λ(1,M2
ψ/s,m2a/s)σ(s) ≡ 2 s λ(s)σ(s)
= 2 s λ(s)
1
4F (s)
∫dp3i
(2π)3 2Ei
dp3j
(2π)3 2Ej(2π)4δ4(pψ + pa − pi − pj)
1
S |M(s, t)|20
= 2 s λ(s)1
2s√λ(s)
1
4(2π)2
∫dp3i dp
3j
EiEjδ3( . . . )δ
(√s− Ei − Ej
) 1
S |M(s, t)|20
=
√λ(s)
16π2
∫dp3i
EiEjδ(√s− Ei − Ej
) 1
S |M(s, t)|20.
(4.81)
Defining angular coordinates with respect to a particular z axis, chosen to coincide with ~pi, we get:
d3pi = sin θ dθ dϕ d|~pi| |~pi|2 = d(cos θ) dϕ dEiEi |~pi|, d(cos θ) =1
2|~pψ||~pi|dt , (4.82)
which allows us to write:
σ =
√λ(s)
16π2
∫dp3i
EiEjδ(√s− Ei − Ej
) 1
S |M(s, t)|20 (4.83)
=
√λ(s)
16π2
∫d(cos θ) dϕ dEiEi |~pi|
EiEjδ(√s− Ei − Ej
) 1
S |M(s, t)|20 (4.84)
=
√λ(s)
16π2
∫dϕ
∫dEi
∫d(cos θ)
|~pi|Ej
δ
(√s− Ei −
√m2j + |~pj |2
)1
S |M(s, t)|20 . (4.85)
After expanding the delta function and performing some straightforward manipulations, one finally arrives
at:
σ(s) =1
8πs
∫dt
1
S |M(s, t)|20 . (4.86)
The reaction dentity γ2 of Eq. (4.80), to be included in (4.73), is then given as an integral of this reduced
cross section, for which the amplitude is summed over both initial and final state degrees of freedom, as
previously mentioned.
67
In the context of leptogenesis, Boltzmann equations can be used to quantify the evolution of an
asymmetry in lepton number, generated via the out-of-equilibrium decay of heavy seesaw mediators. In
general, the result of computing the BE solution can be encoded into a single parameter 0 < η < 1, which
describes the efficiency in producing a net asymmery. In view of this, one usually writes:
YB = C YL = C ε η YX∣∣TM , (4.87)
where ε is the CP asymmetry in leptonic decays and C is the sphaleron conversion factor between baryon
and lepton numbers, which, for a second-order phase transition, is given in Eq. (4.35). The equilibrium
number density divided by the entropy density at high temperatures is given by:
YX∣∣TM =
3
4ζ(3)
gXπ2
(2π2
45g∗S
)−1
=135 ζ(3) gX
8π4 g∗S' 1.95× 10−3 gX . (4.88)
We conclude this chapter by summarizing the main approximations which have been considered so
far in the derivations presented above. These are:
• Absence of entropy production [167].
• Taking the available degrees of freedom to be independent of temperature in the early radiation
dominated era, g∗(T ) ' g∗S(T ) ' const. = 106.75.
• Neglecting quantum statistics factors in general, namely Pauli blocking and Bose enhancing
corrections to the collision operator.
• Working in zero temperature field theory, as opposed to considering finite temperature correc-
tions, belonging to the domain of thermal field theory. These effects have been considered by
Guidice et al. [169] for type I seesaw leptogenesis, resulting in corrections of order ∼ 10%, at most.
We also ignore the effect of Sommerfeld corrections to cross sections (see, for instance, [170]).
• The classical kinetic theory assumption of a weakly interacting, dilute gas of particles, which allows
one to consider only binary collisions.
• The assumption of molecular chaos (stosszahlansatz), which corresponds to saying that the mo-
menta of particles in a volume element are uncorrelated and thus simple products of phase-space
distribution functions arise in the BE [140].
• Insisting on the kinetic equlibrium parametrization of (4.7), with Ti = T and all the uncertainty
in the distributions contained in the chemical potential [142]. This is taken by Luty [171] to be
viewed as an ansatz which dictates the form of relevant out-of-equilibrium effects. Computations
where this assumption has been relaxed have been carried out for type I seesaw leptogenesis [172].
• Taking the quantum amplitude, both for scatterings or decays, to be independent of thermal
motion relative to the plasma (see appendix A of [169]).
Some of the above approximations are commonly employed to render the problem treatable, without
significantly hindering the quality of results.
Having obtained the general form of the kinetic equations which govern the number densities in a
thermal plasma, we now apply them to the case where the decaying particles generating the lepton
asymmetries are the heavy scalar triplets of the type II seesaw mechanism discussed in Section 2.2.2.
68
Type II Seesaw Leptogenesis 5In this chapter we will apply the formalism developed in Chapter 4 for type II seesaw leptogenesis,
namely that of CP asymmetries, lepton to baryon number conversion, and Boltzmann equations, including
the computation of decay rates, reduced cross sections and reaction densities. We follow the most general
approach where flavour effects are considered, taking into account processes which change the number
densities in each flavour. After obtaining the full set of flavoured Boltzmann equations, these are solved
numerically in the context of the A4 model with spontaneous CP violation discussed in Chapter 3.
5.1 Flavoured CP Asymmetries from Triplet Decays
We start by parametrizing the branching ratios for each scalar triplet decay channel beyond tree-
level, i.e. taking into account the CP asymmetries arising from the interference between tree-level and
higher-order diagrams, in a manner consistent with CPT invariance and unitarity (cf. Section 4.4.1).
Namely,
|M(∆i → `α `β)|2 = |M(`α `β → ∆∗i )|2 =(Bαβi,L − εαβi /2
)|M0|2,
|M(∆∗i → `α `β)|2 = |M(`α `β → ∆i)|2 =(Bαβi,L + εαβi /2
)|M0|2,
|M(∆i → φφ)|2 = |M(φ∗ φ∗ → ∆∗i )|2 =(Bi,H + εi/2
)|M0|2,
|M(∆∗i → φ∗ φ∗)|2 = |M(φφ→ ∆i)|2 =(Bi,H − εi/2
)|M0|2.
(5.1)
Here, (4.40) has been generalized to include the tree-level branching ratios Bαβi,L and Bi,H , as the scalar
triplets possess both a lepton doublet (for clarity, one omits the SU(2) L subscript) and a Higgs doublet
decay mode1. The CP asymmetries and branching ratios defined above obey:
∑
α≥ β
Bαβi,L ≡ Bi,L ,∑
α≥ β
εαβi ≡ εi , Bi,L +Bi,H = 1 . (5.2)
As a function of decay rates (recall (4.41)), the asymmetries are given by:
εαβi = 2Γ(∆∗i → `α `β)− Γ(∆i → `α `β)
Γ∆i + Γ∆∗i
=Γ(∆∗i → `α `β)− Γ(∆i → `α `β)
Γ∆i
, (5.3)
1 Doublet components are indistinguishable above the electroweak phase transition and one can then regard a lepton orHiggs doublet as a single species with g` = g` = 2 and gφ = gφ∗ = 2. The same applies to triplets and triplet components,and g∆i = g∆∗i
= 3.
69
∆i
`β
`α
∆i
φ
φ
Tree-Level
∆i ∆j
φ
φ
`α
`β
∆i ∆j
`α
`β
`µ
`ν
One-Loop
Figure 5.1: Tree-level diagrams for the decays of type II seesaw scalar triplets and one-loop diagrams
contributing to the decay process ∆i → `α `β . The one-loop diagrams, which correspond to the rightmost
diagram of Fig. 4.5, represent a wave-function correction to the tree-level amplitude (unlike the type I
case, where a vertex correction is also present).
where Γ∆iand Γ∆∗i
denote the total decay rates of the triplet and its conjugate, respectively, which
are equal as a consequence of CPT invariance. The interfering diagrams which are relevant for the
computation of the CP asymmetry are presented in Fig. 5.1.
Resorting to the Feynman rules presented in Fig. 2.4, one can write the amplitudes for the diagrams of
Fig. 5.1 and their CP conjugates. Since all triplet components possess the same dynamics at high-energy,
we consider amplitudes and decay rates computed for a single component, chosen for simplicity to be ∆0i
(this clearly has no effect on the asymmetry of (5.3)). Amplitudes are denoted by the superscripts [+2]
and [−2], according to the lepton number L produced in each reaction. Amplitudes for the Higgs channel
are obtained directly from Fig. 2.4. Above the EWPT, fermions are taken to be massless, while Higgs
doublets can be given a nonvanishing thermal mass mφ. For the tree-level amplitudes one has:
M[+2]tree = −2iY∆i∗
αβ uα PR vβ ,
M[−2]tree = −2iY∆i
αβ uα PL vβ .(5.4)
The one-loop amplitudes Mloop are given by the sum of contributions from the two wave-correction
diagrams of Fig. 5.1 (a topological symmetry factor of 1/(1 + δµν) is introduced for the loop):
M[+2]loop = − i
4π2
∑
j 6=i
uα PR vβM2i −M2
j
[µ∗i µj Y
∆j∗αδ R+ I+ Y
∆j∗αδ Y∆j
µν Y∆i∗µν
R′ + I ′1 + δµν
],
M[−2]loop = − i
4π2
∑
j 6=i
uα PL vβM2i −M2
j
[µi µ
∗j Y
∆j
αδ R+ I+ Y∆j
αδ Y∆j∗µν Y∆i
µν
R′ + I ′1 + δµν
],
(5.5)
where R + I and R′ + I ′ contain both a real and an imaginary part (hence the designations R and
I):
R+ I = ∆ε − ln(M2i /µ
2)
+ 2 ln 2 + iπr − ln(1− r2)− r ln
(1 + r
1− r
),
R′ + I ′ = M2i
(∆ε − ln
(M2i /µ
2)
+ 2 + iπ),
(5.6)
where r =√
1− 4m2φ /M
2i and ∆ε ≡ 2/ε− γ + ln(4π) contains the one-loop divergence (γ is the Euler-
-Mascheroni constant) which has been isolated through dimensional regularization (4D ⇔ ε → 0, and µ
represents the typical dimensionful auxiliary parameter).
70
The rate of a two-body decay is explicitly given in terms of the corresponding amplitude by:
Γ =1
S|~q|
8πM2i
|M|2 , (5.7)
where |~q| is the solution to Mi =√m2
1 + |~q|2 +√m2
2 + |~q|2 (mi denote masses of the decay products).
For leptons in the final state, |~q| = Mi/2, while for the Higgs channel |~q| =(Mi/2
)r. In the above
equation, a sum over final spin states,∑sα,sβ
|uα PL,R vβ |2 = 2 pα · pβ = M2i , is implicit in |M|2. One
therefore has, for the total tree-level decay rate:
Γ∆i=
1
16πMi
[ ∑
α≥ β
1
1 + δαβ|M(∆i → `α `β)|2 +
1
2|M(∆i → φφ)|2 r
]
=1
16πMi
[1
2
∑
α,β
(4|Y∆i
αβ |2M2i
)+
1
2
(4|µi|2
)r
]mφMi−−−−−−→ Mi
8π
(Tr[Y∆i
†Y∆i
]+ |λi|2
),
(5.8)
where λi ≡ µi/Mi, and one has taken the limit mφ Mi ⇒ r → 1. At this point, we can also give
expressions for the tree-level branching ratios:
Bαβi,L =2
1 + δαβ
∣∣Y∆i
αβ
∣∣2
Tr[Y∆i
†Y∆i
]+ |λi|2
, Bi,H =|λi|2
Tr[Y∆i
†Y∆i
]+ |λi|2
. (5.9)
By expanding the (beyond tree-level) decay rate as in (4.42), one then obtains the expression for the CP
asymmetry:
εαβi =1
1 + δαβ
1
16πMi
∑
µ,ν
8π
Mi
2 Re[M[+2]∗
tree M[+2]loop −M
[−2]∗tree M[−2]
loop
]
Tr[Y∆i
†Y∆i
]+ |λi|2
. (5.10)
Inserting now the above amplitudes into this expression and considering Mj Mi one finally arrives at:
εαβi = − 2
1 + δαβ
1
2π
∑
j 6=i
Mi
Mj
Im(λ∗i λjY
∆i
αβY∆j∗αβ + (Mi/Mj) Tr
[Y∆i
†Y∆j
]Y∆i
αβY∆j∗αβ
)
Tr[Y∆i
†Y∆i
]+ |λi|2
, (5.11)
which is split into two contributions, the second of which clearly reduces to zero in the unflavoured case,
i.e. upon summing over the flavour indices, α ≥ β. Although we could write (5.11) as a function of the
mass matrices Mν , as is done for instance in Ref. [133], we will not do so since the couplings are defined
at high scales. The connection between the low-energy Mν and the high-energy Y∆ would require a
proper treatment relying on renormalization-group effects.
5.2 Boltzmann Equations for Type II Seesaw
In order to track the evolution of a lepton asymmetry generated by the out-of-equilibrium decays of
the lightest Higgs triplet, we consider the network of Boltzmann equations which account for the relevant
out-of-equilibrium reactions, namely: decays, inverse decays, ∆L = 2 s-channel and t-channel scatterings,
and gauge scatterings, which will tend to keep triplets close to thermal equilibrium (absent in the type I
thermal leptogenesis scenario). The diagrams for these processes are depicted in Fig. 5.2.
In the following, we will be interested in the BEs which govern the evolution of the number densities
(normalized to the entropy density) of both the sum, ΣY∆i ≡ Y∆i + Y∆∗i, and the difference, δY∆i ≡
71
`β
`α
Decays and Inverse Decays
∆i ∆i
φ
φ
`α
∆i
φ
φ`β
s-Channel Scatterings
Gauge Scatterings
Fermions and Higgs
∆i
∆i
f, φ
f, φ
Ai, B
`α
∆i
φ
φ`β
t-Channel Scatterings
t-channel
∆i
∆i
∆i
Aa, B
Ab, B
u-channel
∆i
∆i
∆i
Aa, B
Ab, B
cubic
Aa, B
∆i
∆i
quartic
Ab, B
Ac, B
∆i
∆i
Aa, B
Ab, B
Figure 5.2: Scalar triplet interactions relevant to the BE out-of-equilibrium analysis, where one considers
the diagrams presented for decays, inverse decays and s- and t-channel scatterings and their charge con-
jugates (reverse all arrows), as well as gauge scattering reactions (f denotes any SM fermion). Reactions
can proceed in both the direct and inverse direction.
Y∆i− Y∆∗i
, between densities of triplets and their conjugates, and the number densities of the three
differences B/3 − Lα, denoted by YBLα. We choose to work with these three differences between (a
third of) baryon number and each lepton flavour number, as they correspond to non-thermalizing modes,
meaning they are unaffected by rapid sphaleron processes. The total B − L can be obtained from the
sum of these quantities and is directly related to the density of baryon number through (4.35). The
evolution of the YBLα will then only be influenced by the out-of-equilibrium reactions involving heavy
triplets, which break B− L. The BE for YBLα is obtained from that of δY`α ≡ Y`α − Y`α by [173]:
sHzd δY`αdz
= f(δY`α , δYφ
)⇒ sHz
dYBLα
dz= −f
((CL)αβ YBLβ , (CH)β YBLβ
), (5.12)
meaning that, aside from a global sign change, one introduces a matrix CL and a vector CH which
encode the effects of equilibrium interactions in the plasma, relating each YBLα with all three δY`α and
with δYφ ≡ Yφ − Yφ∗ , respectively. The entries of CL and CH will then depend on interactions which
are fast (Γ > H), such as gauge, heavy fermion Yukawa, and sphaleron interactions – collectively known
as spectator processes – and thus impose relations between the various chemical potentials. Solving the
linear system which contains these relations for different temperature regimes yields [174]:
1012 GeV . T . 1013 GeV : CL =
(−1 0 00 −1 0
−1/16 −1/16 −3/4
), CH = − 1
8
(3 3 4
),
1011 GeV . T . 1012 GeV : CL = 1460
(−196 34 24
34 −196 249 9 −156
), CH = − 1
115
(41 41 56
),
108 GeV . T . 1011 GeV : CL = 11074
(−906 120 120
75 −688 2875 28 −688
), CH = − 1
179
(37 52 52
),
T . 108 GeV : CL = 2711
(−221 16 16
16 −221 1616 16 −221
), CH = − 16
79
(1 1 1
).
(5.13)
The entries may vary slightly depending on the quark Yukawa interactions which are taken to be in
equilibrium [175]. Even though in some regimes some lepton flavours are indistinguishable, matrices are
72
kept 3 × 3 by convenience, as is done in [173]. Above T ∼ 1013 GeV, one takes lepton flavours to be
indistinguishable, since all lepton Yukawa interactions are out-of-equilibrium.
As seen in Section 4.5, the BE for a particle species ψ affected by the reaction ψ(+a)→ i+ j is (cf.
(4.67) and (4.73)):
sHzdYψdz
= −(#ψ)
[nψ(na)
neqψ (na
a)α(ε) +
ni njneqi n
eqj
β(ε)
]γ . (5.14)
If more than one interaction comes into play, the right-hand side is augmented by other terms of the same
kind. In order to obtain the BE for both the triplet sum ΣY∆iand difference δY∆i
, one first considers
the equations for Y∆iand Y∆∗i
individually. We work with an equation for the whole triplet population
and not a specific component, which allows a straightforward inclusion of coannihilations in the BE [176].
Expressions for α(ε) and β(ε) are to be extracted from the parametrization of (5.1). An exact result for
ΣY∆iis given, considering only triplet decays (and inverse decays), by:
sHzdΣY∆i
dz= −
ΣY∆i
Y eq∆i
−∑
α≥ β
[Y`αY`β
Y eq`
2
(Bαβi,L + εαβi /2
)+Y`αY`β
Y eq`
2
(Bαβi,L − εαβi /2
)]
−[Y 2φ
Y eqφ
2
(Bi,H − εi/2
)+
Y 2φ∗
Y eqφ
2
(Bi,H + εi/2
)]γD ,
(5.15)
where γD is given in (4.65) (with ΓR0 = Γ∆i). In order to write the BEs in a condensed form, some
useful approximations are considered. In particular, the expansion Yi∗Yj∗ ' Yi Yj − Yi δYj − Yj δYi, for
particle species i and j, presents itself as very useful. Additional approximations include neglecting terms
of the type δY ε, making Y ε ' Y eq ε and, in some cases, taking YiYj + Yi∗Yj∗ ' Y eqi Y eq
j . One also has
ΣY eq∆i
= 2Y eq∆i
and defines γΣD ≡ 2γD, thus arriving at the partial result (no gauge scatterings yet):
sHzdΣY∆i
dz= −
(ΣY∆i
Y eq∆i
− 2
)γD = −
(ΣY∆i
ΣY eq∆i
− 1
)γΣD . (5.16)
Proceeding in a similar manner for gauge triplet interactions as well as for the remaining quantities of
interest, namely δY∆i and YBLα, the full system of Boltzmann equations finally reads:
sHzdΣY∆i
dz= −
(ΣY∆i
ΣY eq∆i
− 1
)γΣD − 2
[(ΣY∆i
ΣY eq∆i
)2
− 1
]γA , (5.17)
sHzd δY∆i
dz= −
[δY∆i
ΣY eq∆i
+1
2
∑
α≥ β
Bαβi,L(CL)αµ YBLµ + (CL)βν YBLν
Y eq`
−Bi,H(CH)µ YBLµ
Y eqφ
]γΣD , (5.18)
sHzdYBLα
dz= −
∑
β
(1 + δαβ)γD
[εαβi
(ΣY∆i
ΣY eq∆i
+ 1− 2Bi,H
)
− 2Bαβi,L
(εi +
δY∆i
ΣY eq∆i
+1
2
(CL)αµ YBLµ + (CL)βν YBLν
Y eq`
)](5.19)
− 2[(1 + δαβ)(γ′s)αβ + (γt)αβ
][1
2
(CL)αµ YBLµ + (CL)βν YBLν
Y eq`
+(CH)µ YBLµ
Y eqφ
],
where a sum over µ and ν (but not over α) is implied, and equilibrium distributions are given by (nγ is
found in (4.16)):
neq∆i
=g∆i
2π2TM2
i K2
(Mi
T
), neq
` =3
4ζ(3)
g`π2
T 3, neqφ = ζ(3)
gφπ2
T 3. (5.20)
73
Here, the degrees of freedom gi are those indicated in Footnote 1 of the present chapter. Notice that
equilibrium distributions are the same for particles and antiparticles. In the above, γA and γt denote the
reaction densities for gauge and t-channel scatterings, while γ′s is defined as:
(γ′s)αβ = (γs)αβ −Bαβi,LBi,H γD , (5.21)
with γs denoting the s-channel reaction density. In fact, when one includes the BE right-hand side term for
the evolution of YBL corresponding to the s-channel process, one must use a so-called subtracted s-channel
reaction density γsubs , instead of the full γs since the latter includes a resonance – real intermediate state
(RIS) – which has already been taken into account by decays and inverse decays. The subtraction is done
in the narrow width approximation (see, for instance, [169]) and amounts to having:
(γsubs )αβ = (γs)αβ −
(Bαβi,L ±
εαβi2
)(Bi,H ±
εi2
)γD = (γ′s)αβ ∓
[εi2Bαβi,L +
εαβi2Bi,H
]γD +O(ε2) ,
(5.22)
where the ± corresponds to reactions which create/destroy lepton number, respectively. The term orig-
inating from the content of the square brackets has been moved, in the final BE expression (5.19), next
to the decay term, while γ′s appears alongside γt. The inclusion of the (subtracted) s-channel scattering
term is crucial2 for the consistency of the BEs: without it, we are limited to processes which create an
asymmetry even in thermal equilibrium [141] (cf. (4.40) and discussion at the end of Section 4.2). The
computation of the scattering reaction densities γA,s,t is addressed in the following section.
5.3 Scattering Reaction Densities
The general expression for scattering reaction densities as a function of the respective reduced cross
sections σ is given in Eq. (4.80), where – for massless fermions (above the EWPT) and under the
assumption of a negligible Higgs thermal mass – smin = 4M2i for gauge scatterings (two triplets in
the initial state), and smin = 0 for s- and t-channel processes. The reduced cross sections which enter the
above expressions are obtained from the amplitudes through Eq. (4.86). For the case of gauge scatterings3,
the amplitudes themselves are computed by considering the interactions of triplets with gauge bosons,
which can be read off from the expansion of the covariant derivative, given by (2.5) (with the generators
of (2.58)), in the seesaw Lagrangian (2.64).
A non-trivial step arises during the computation of diagrams with massless gauge bosons in the
final state (SSB is yet to happen), for which one must forcibly consider the following expression for the
polarization sums [178] (with boson four momenta p3µ and p4µ):
Pµν(pi) =∑
σ
εµ(pi, σ) ε∗ν(pi, σ) = −gµν +piµ ην + piν ηµ
pi · η− η2
piµ piν(pi · η)2
, (5.23)
where εµ(pi) are polarization four-vectors, the σ denote possible polarizations, and ηµ is a four-vector
which verifies η ·ε(pi, σ) = 0 and η ·pi 6= 0. The valid choice η = p3 +p4 is considered in our computation.
The integration limits for the Mandelstam variable t arising in (4.86) are, for a generic scattering i+ j →2Notice that what is typically ignored in the literature – due to its generally small magnitude – is γ′s, and not γsub
s .3A helpful reference concerning gauge interaction amplitudes for charged scalars is Ref. [177].
74
k + l, given in terms of masses and s (the Mandelstam variable, not the entropy density) by:
m2i +m2
k −1
2s
(s+m2
i −m2j
)(s+m2
k −m2l
)± 1
2s
√λ(s,m2
i ,m2j
)λ(s,m2
k,m2l
), (5.24)
where the definition of λ(x, y, z) has been given in Eq. (4.76).
Integrating over t results in the following reduced cross sections for the gauge scatterings (the sum of
which is denoted by σA)4:
σ(∆i∆∗i → ff , φφ∗) =
1
16π
(50 g4
2 + 41 g4Y
)r3i ,
σ(∆i∆∗i → AaAb) =
g42
π
[(5 + 34
M2i
s
)ri − 24
(M2i
s
)2(s
M2i
− 1
)ln
(1 + ri1− ri
)], (5.25)
σ(∆i∆∗i → AaB,BB) =
3
2π
(4g2
2g2Y + g4
Y
)[(1 + 4
M2i
s
)ri − 4
(M2i
s
)2(s
M2i
− 2
)ln
(1 + ri1− ri
)],
where ri ≡√
1− 4M2i /s. Similarly, for s- and t-channel scatterings, one obtains (notice that this cross
section corresponds to the full γs):
σαβs (s) =3|µi|2
∣∣Y ∆i
αβ
∣∣2
πM2i
1
1 + δαβ
[x
(x− 1)2 +(Γ∆i
/Mi
)2], (5.26)
σαβt (s) =6|µi|2
∣∣Y ∆i
αβ
∣∣2
πM2i
[− 1
1 + x+
ln(x+ 1)
x
], (5.27)
where xi ≡ s/M2i and the notation σs,t is self-evident. In the unflavoured limit, i.e. after a sum in flavour
indices (∑α≥β for σs and
∑α,β for σt), these reaction densities agree with those5 of Ref. [179] in the
absence of four-point ∆L = 2 interactions.
5.4 Leptogenesis in an A4 Model
We now turn to a particular framework – the A4-based model presented in Chapter 3 – in the context
of which leptogenesis is, in principle, possible through the out-of-equilibrium decays of type II seesaw
mediators. It is here assumed that the masses of the triplets are hierarchical M2 M1, meaning that
any asymmetry generated due to the decoupling of the heaviest triplet is erased by interactions involving
the lightest, which is in thermal equilibrium at that time. We will thus be interested in tracking the
evolution of Y∆1 . The CP asymmetries for both triplets read:
εαβ1 =1
3π
z1z2 |u1|2M21
4|u1|4M21 + 3z2
1v4
sinβ
︸ ︷︷ ︸≡ε01
−1 0 0
0 0 1
0 1 0
, (5.28)
εαβ2 =1
3π
z1z2 |u2|2M22
4|u2|4M22 + 2z2
2v4
[1 +
2
9
z22 v
4(ε1 + ε2)
2M22 |u2|4 + z2
2v4
]sinβ
︸ ︷︷ ︸≡ε02
−1 0 0
0 0 1
0 1 0
, (5.29)
where (5.11) (M2 M1) was used in deriving the expression for εαβ1 , and εαβ2 is obtained from the same
expression with a global minus sign since now Mi Mj (cf. squared-mass difference denominators in
4These results are in agreement with those of Ref. [179], with gauge reduced cross sections behaving as σ(∆i∆∗i →
ff , φφ∗) ∼ r3 and σ(∆i∆∗i → AaAb, AaB,BB) ∼ r in the high-energy limit (not low-energy, as is mistakingly remarked).
5In both eqs. (17a) and (17b) of Ref. [179] every “MT ” arising within square brackets should be replaced by an “mT ”.
75
Figure 5.3: Contours of the (magnitude of the) maximum CP asymmetries in the decays of hierarchical
scalar triplets ∆1,2. For temperatures above T ∼ 1013 GeV, lepton flavours are no longer distinguishable
and one enters the unflavoured regime.
(5.5)) and the second term in Eq. (5.11) vanishes. Notice that the total CP asymmetries vanish in this
model (purely flavoured leptogenesis) and that diagrams with leptons running in the loop produce no
contribution to the asymmetries. The correcting factor arising in square brackets for the ε2 asymmetry
contains perturbations (not to be confused with total CP asymmetries) which enter the expression solely
through the trace in the denominator of (5.11). Maximization of these asymmetries with respect to the
|ui| (the zi are fixed as functions of neutrino masses and perturbations by (3.49)) gives:
ε01,max =
z2
12√
3πv2M1 sinβ , for |u1|2 =
√3
2
z1v2
M1, (5.30)
ε02,max '
z1
12√
2πv2
[1 +
1
9(ε1 + ε2)
]M2 sinβ , for |u2|2 '
1√2
[1− 1
9(ε1 + ε2)
]z2v
2
M2. (5.31)
Expressing this in terms of the angle β and squared mass differences yields:
ε01,max '
√∆m2
31
12√
6πv2
[1− 1
3(ε1 + ε2)
]M1 sinβ , (5.32)
ε02,max ' −
√∆m2
31
48πv2
[1− 1
9(ε1 + ε2)
]M2 tanβ . (5.33)
The magnitude of the maximum CP asymmetry ε0i,max is shown as a contour plot in the (β,Mi)-plane in
Fig. 5.3.
To solve the BE network for the case at hand, a Fortran-based programme was constructed. In its
context, all quantities are made dimensionless and reaction densities are obtained from the numerical
integration of reduced cross sections. To numerically integrate the BEs – and thus arrive at entropy-
-normalized number densities YX – we resort to the Runge-Kutta-Fehlberg method.
76
90 95 100 105 110 115
10-8
10-9
10-10
10-11
10-12
10-13
10-14
245 250 255 260 265 270
10-8
10-9
10-10
10-11
10-12
10-13
10-14
M1 = 1´1012
GeV
M1 = 5´1012
GeV
YB
Β HºLFigure 5.4: Scatter plot of the baryon asymmetry generated in randomly-chosen perturbed versions of the
model, for M1 = 1012 GeV (black) and M1 = 5× 1012 GeV (cyan), which are characterized by different
values of β. The green horizontal bar spans a 30% deviation for YB from the central value of Eq. (1.6).
A set of working assumptions is considered, namely M2 = 10M1 and |u1| = u2 (equality of VEVs),
which automatically imply λ2 = 10 |λ1| through (2.72). The mass of the lightest triplet, M1, is chosen
such that the (maximum) CP asymmetry is large but still in the flavoured regime (see Fig. 5.3). The
value of |λ1| (or, equivalently, that of the VEV u1) is in turn fixed by the best-fit value of the solar
squared-mass difference ∆m221, as:
|λ1|2 =M2
1 ∆m221
4v4(eig(2)− eig(1)
) , (5.34)
where eig(i) are the numerically-determined (ordered) eigenvalues of the (perturbed) matrix (Y∆1 +
Y∆2)† (Y∆1 + Y∆2), which are proportional to neutrino squared masses.
Both z1 and z2 are randomly generated in the range [10−3, 2], while the perturbations ε1 and ε2
are varied in [−0.2, 0.2]. In Fig. 5.4 we show a scatter plot of YB for two different values of M1, for
randomly-chosen perturbed versions of the model. One expects that if the choice of parameters – which
the programme selects randomly – maximizes the CP asymmetry, more points should be accessible above
the experimental value of YB.
We now restrict ourselves to a particular point, chosen to lie in the region above the green band,
namely that which corresponds to z1 ' 0.2595, z2 ' 0.0656, ε1 ' −0.1668, ε2 ' 0.1682, and β ' 255.2o.
Reaction densities and the evolution of asymmetries for this particular perturbed version of the model
are given in Fig. 5.5. A baryon asymmetry YB ' 5.93 × 10−10 is obtained, following the decoupling of
the lightest triplet, by summing over all three YBLα and converting the resulting YB−L asymmetry into
a purely baryonic one through (4.35). For such a case, one additionally has ε01 ' −2.02 × 10−5 and the
77
Figure 5.5: Reaction densities normalized to the product H(T )nγ(T ) (left) and evolution of the various
densities considered in the BE network (right). RIS subtraction has been considered, and the given γ′s is
equal to (γ′s)ee + (γ′s)µτ , with (γ′s)µτ = 2(γ′s)ee, while γt is equal to (γt)ee + (γt)µτ + (γt)τµ, with all these
terms identical between them. Gauge scatterings keep triplets close to equilibrium.
branching ratios obey Bµτ1,L = 2Bee1,L ' 0.654.
The efficiency parameter defined in (4.87) can be generalized, in the case of flavour, by considering
flavoured efficiencies ηα which translate the strenght with which an asymmetry in each lepton flavour is
generated:
YB = C
(∑
α,β
ηα εαβ
)YX∣∣TM . (5.35)
Notice that, for scalar triplets, YX = ΣY∆i. For the case under consideration, one obtains the flavoured
efficiencies ηe ' 0.686, and ηµ ' ητ ' 0.346.
We have seen that the model is sufficient to account for the observed baryon asymmetry of the
Universe. This behaviour may appear borderline. However, it must be emphasized that some regions
of parameter space remain to be explored. One has assumed that triplet masses are hierarchical, taken
to imply that asymmetries produced prior to the decay of the lightest triplet have been washed out.
The fact that CP asymmetries in the out-of-equilibrium decays of ∆2 can overshadow those of ∆1 by as
much as an order of magnitude (see Fig. 5.3) challenges this assumption. Flavour considerations may
nevertheless come to our rescue, as a sufficiently massive ∆2 may decay during the unflavoured regime,
for which all different lepton flavours are out-of-equilibrium and indistinguishable. This fact enters in
the BEs through CL and CH and the structure of the model prevents, in such a case, the generation of
asymmetries.
78
Concluding Remarks
Particle physics operates on the most fundamental level of science: the subatomic world. At present,
our understanding of fundamental particles and their interactions relies on a powerful guiding principle:
the principle of symmetry. Its validity spans from quantum to macroscopic scales and, therefore, it is
only natural to adopt it as a guide in searching for the explanation of the unsolved mysteries of our
Universe. Driven by the observed relations between quantities, physical theories ascribe a central role
to symmetries. These arise in modern physics through the abstract language of group theory, which
generalizes spacetime symmetries to local internal ones. The latter are at the heart of the SM and, thus,
of our current understanding of elementary particles.
In the last couple of years we have witnessed important advances in particle physics. The LHC
ATLAS and CMS discovery of the long-awaited Higgs boson at CERN is gradually clarifying the structure
of electroweak symmetry breaking and providing ultimate proofs of the SM. Still, and in spite of its
numerous successes and repeated confirmations beyond the classical level, there is a plethora of questions
for which the SM offers no answer. Perhaps the best example of the standard theory limitations is the
phenomenon of neutrino oscillations, a purely quantum mechanical process which implies nonvanishing
neutrino masses and mixing. Since neutrinos are strictly massless in the SM, the observation of neutrino
oscillations provides a solid evidence for new physics. From the theoretical viewpoint, the longstanding
problem of fermion mass origin is now augmented by a bizarre fact: neutrino masses are very small when
compared with those of other fermions, and the neutrino mixing pattern is completely different from that
observed in the quark sector.
The most popular SM extensions accommodating massive neutrinos in a natural way are those in
which neutrino masses are generated through the tree-level exchange of new heavy particles (see Chapter
2). Interactions (and masses) of these extra states with the SM degrees of freedom determine the flavour
structure of the effective Majorana neutrino mass resulting from the decoupling of the heavy particles and
from electroweak symmetry breaking. Those interactions and masses are, in general, free parameters of
the theory which are consistent with the gauge and Lorentz symmetries. As it is well known, this leaves
too much room for arbitrariness. Therefore, inspired by their normative role, one often considers new
symmetries to shape the fermion mass and mixing pattern. In Chapter 3 we have identified the symmetries
of the effective (Majorana) neutrino mass matrix and analysed the phenomenological viability of a specific
model based on an A4 non-Abelian discrete symmetry. In this scenario, CP-violating effects stem from
a single complex phase associated to the VEV of a scalar singlet. The numerical analysis presented in
Section 3.3 shows that the model is compatible with all data provided by neutrino oscillation experiments
79
and predicts large CP-violating effects, which could be detected in future reactor, long-baseline and
accelerator neutrino experiments.
In our constant search for a deeper understanding of the Universe, we are often faced with the problem
of comparing different scales and distances. Intuitively, phenomena which occur at the subatomic level
are, in general, uncorrelated with those happening at cosmological distances: how can the structure of
the Universe at large scales have anything to do with interactions among fundamental particles? The
key to this enigma relies on considering the present Universe as the result of an evolutionary, dynamic
process.
Although symmetries are crucial for the explanation of particle interactions, the truth is that we owe
our existence to symmetry breaking, in line with Curie’s Principle. In the SM, the Higgs mechanism is
responsible for giving mass to the gauge bosons and (some) fermions, keeping photons massless. However,
it remains to be explained why there is more matter than antimatter in the Universe since, after all, this
is a requirement for our existence. Once more, this fact does not find a satisfactory explanation in
the SM framework. A remarkable feature of the seesaw paradigm discussed in Chapter 2 is that the
presence of heavy neutrino mass mediators provides a natural explanation for the fact that we live in a
matter-dominated Universe. The connection between neutrino phenomenology and cosmology may thus
be established within the leptogenesis scenario, where the excess of baryons in the Universe relies on a
lepton asymmetry, as discussed in Chapter 4. For the specific case of type II seesaw, the starting point to
generate that lepton asymmetry is the out-of-equilibrium decay of the heavy-triplet scalars. In Section 5.1
we have obtained the general expressions for the CP asymmetries generated by the interference between
one-loop and tree-level decay diagrams, while in Section 5.2 the relevant set of Boltzmann equations which
govern the evolution of lepton number densities in each flavour have been obtained. The viability of the
leptogenesis scenario in the framework of the A4 model analysed in Section 3.3 has been also explored
in Section 5.4. We have concluded that the model not only allows to reproduce the current neutrino
mass and mixing pattern, but also generates a sufficiently large baryon asymmetry of the Universe, in
accordance with the experimental result.
Our conclusions justify the purpose of this thesis: to relate neutrino masses and mixing, symmetries
and the origin of matter. What we have presented is just an example (among many others) of how
complementary studies may shed some light on answering open questions in fundamental physics, the
future of which hinges on the combined exploration of physical phenomena at the energy, intensity
and cosmic frontiers. Neutrino oscillations represent the most fruitful example of physics at the intensity
frontier and the next years will surely reveal something more about the elusive neutrinos, like the neutrino
mass hierarchy and leptonic CP violation. On the other hand, the synergies between observations at
the energy frontier provided by accelerators like the LHC and those at the intensity frontier can be
complemented with data from cosmological experiments in the search of a more complete description of
Nature.
Quaerendo Invenietis (“By seeking, you will discover”).
80
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88
Computing Diagrams
with Majorana Fermions ASuppose one wants to study a given scattering (or decay) process. One starts by drawing all possible
Feynman diagrams corresponding to that process, up to a certain order. Consider now one of such
diagrams. The philosophy for computing its amplitude M without resorting explicitly to the charge
conjugation matrix C is, according to Ref. [180], as follows:
• Assign to each fermion chain in the diagram an arbitrary orientation (fermion flow).
• Proceeding along each fermion flow chain, starting at its termination (for closed loops, at an arbi-
trary propagator), write down the appropriate spinors, vertex terms (see below) and propagators
in accordance with the rules of Fig. A.1.
• Include a factor of (−1) for every closed loop.
• Choose a reference order for the order of the spinors in the expression of the amplitude (the ordering
refers to the particle label of each spinor).
• After exhausting every fermion chain, include a factor of (−1)p in the amplitude, where p denotes
the permutation parity of the spinor order in this diagram with respect to the previously chosen
reference order.
• Include a combinatorial factor, related to the topology of the diagram [181,182]. Majorana fermions
are to be treated here in the same way as real scalar or vector fields.
• For diagrams involving vector bosons, include the appropriate propagators and polarization vectors.
Vertex contributions are extracted from the Lagrangian density by reading off a quantity hΓ, where Γ
includes all nontrivial matrices in Dirac space and h corresponds to a numerical factor. The vertex term
will then be either i η hΓ or i η hΓ′ depending on the fermion flow choice (see Fig. A.1), where η denotes
an identical particle factor – one has η =∏i ni! for a vertex involving ni (indistinguishable) particles of
each type i – and Γ′ is defined as Γ′ ≡ C ΓT C−1. In most cases of interest, Γ = PR,L = C PTR,L C−1 = Γ′,
and so vertex contributions will be independent from the choice of a fermion flow.
89
u(p, s)
v(p, s)
v(p, s)
u(p, s)
i η hΓ i η hΓ′
ψc
ψ
ψc
ψ
S(p) ≡ 1
/p−m + iε
i S(+p)
i S(−p)
i S(+p)
Figure A.1: Rules for writing propagator, vertex and spinor contributions obtained from Ref. [180]. In
our Feynman diagrams, time is taken to flow from the left to the right.
90
Clebsch-Gordan Coefficients for A4 BB.1 General Description of the Group
The A4 group, introduced in Section 1.2, corresponds to the group of even permutations of four
elements. The group contains nG ≡ #A4 = 12 elements, explicitly given in (1.4), and four conjugacy
classes which partition the whole set:
C1 = e, n1 = 1,
C2 = (12)(34), (13)(24), (14)(23), n2 = 3,
C3 = (123), (142), (134), (243), n3 = 4,
C4 = (132), (124), (143), (234), n4 = 4,
(B.1)
where ni is the number of elements of each class Ci. The results presented in this appendix extensively
rely on group theory definitions and results, most of which can be found in Ref. [183].
The number of irreducible representations of the group (irreps., see Footnote 4 of Chapter 1) equals
the number of its conjugacy classes, nC = 4. Since the 1D (ir)reps. of A4 coincide with those of
A4/[A4, A4], and since the commutator [A4, A4] is isomorphic to the Klein group K = Z2 × Z2, one has
#(A4/[A4, A4]) = 12/4 = 3. Hence, A4/[A4, A4] ∼= Z3 and so A4 has 3 (degenerate) 1D representations.
The dimension n of the remaining irreducible representation of A4 is given by:
∑
irreps. µ
nµ2 = nG ⇒ 12 + 12 + 12 + n2 = 12 ⇒ n = 3, (B.2)
where nµ represents the dimension of the irrep. labeled by µ.
The character (χµi ) table for this group is given in Table B.1. It can be constructed from the known 1D
irreps of Z3 and by considering the following relations of completeness and orthonormality, respectively:
∑
µ
ninG
(χµi )∗χµj = δij . (B.3)
∑
i
ninG
(χµi )∗χνi = δµν , (B.4)
From relation (B.3) one obtains the value of χ33 :
∑
µ
n3
nG(χµ3 )
∗χµ3 = 1 ⇒
∑
µ
|χµ3 |2
=12
4⇒ |1|2 + |ω|2 +
∣∣ω2∣∣2 +
∣∣χ33
∣∣2 = 3 ⇒ χ33 = 0. (B.5)
91
χ1 χ2 χ3 χ4
1 1 1 1 1
1′ 1 1 ω ω2
1′′ 1 1 ω2 ω
3 3 χ32 = −1 χ3
3 = 0 χ34 = 0
Table B.1: Character table for the group A4. The entries correspond to the value of the character
χµi ≡ Tr Uµ(g), g ∈ Ci for each of the nC irreducible representations. As in Section 1.2, UR(g) ≡ URg
corresponds, for a given group representation R, to the matrix onto which g is mapped. The definition
ω ≡ exp(i 2π/3) for one of the cube roots of unity is also considered.
e s2 = t3 (123) t (132) t2
(12)(34) s (142) sts (124) tst = st2s
(13)(24) t2st (134) ts (143) st2
(14)(23) tst2 (243) st (234) t2s
Table B.2: Explicit decomposition of the elements of A4 in terms of s and t.
By the same argument, one shows that χ34 = 0. Finally, for χ3
2 and from (B.4), one has:
∑
i
ninG
(χ1i
)∗χ3i = 0 ⇒
∑
i
ni χ3i = 0 ⇒ 1× 3 + 3χ3
2 = 0 ⇒ χ32 = −1. (B.6)
The whole group can be obtained from two generators, s ≡ (12)(34) and t ≡ (123). The explicit
decomposition of group elements in terms of products of these generators is presented in Table B.2.
B.2 Choice of an Explicit 3D Representation
For 1D representations, the character coincides with the representation. For the remaining 3D irrep.,
however, one must make a choice and select which of the non-trivial group elements corresponds to a
diagonal matrix. Here, the choice of Altarelli and Feruglio [125] is considered. In particular, U3(t), the
3× 3 matrix representing t in the 3D irrep. of A4 is chosen to be diagonal.
It is important to mention that the choice of basis for the 3D representation affects the value of the
Clebsch-Gordan coefficients (CGCs) and thus consistency is required. Although our choice of U3(t) will
differ slightly from that of Altarelli and Feruglio, the CGCs obtained here are the same up to meaningless
normalization. An alternative option for the shape of the 3D irrep. matrices comes from considering
U3(s) diagonal instead, as is done by Ma and Rajasekaran [128].
While working with discrete groups one can always (and will here) choose a basis for which the
representation matrices are unitary. Since t3 = e and U3(e) = 13×3, the elements on the diagonal of
U3(t) will be cube roots of unity. In order to obtain the right character, χ33 = 0 (t ∈ C3), these roots
92
must all be different and one may choose:
U3(t) =
1 0 0
0 ω 0
0 0 ω2
. (B.7)
Constructing the explicit form of U3(s) automatically defines all matrices U3(g) with g ∈ A4, thanks
to the possibility of decomposing every group element into products of t and s. Since s2 = e, U3(s)2 =
13×3 = U3(s)−1U3(s), and so the matrix U3(s) is its own inverse. Due to it is unitarity, U3(s) is also
Hermitian. Imposing the condition χ32 = −1, one has:
U3(s) =
a b c
b∗ d e
c∗ e∗ −1− a− d
, (B.8)
with a and d constrained to be real. Since ts ∈ C3, one must have Tr[U3(ts)] = χ33 = 0, and st2 ∈ C4, so
Tr[U3(st2)] = χ34 = 0. Thus:
Tr[U3(ts)
]= 0
Tr[U3(st2)
]= 0
⇒
a+ ωd− ω2(1 + a+ d) = 0
a+ ω2d− ω(1 + a+ d) = 0⇒
a = −1/3
d = −1/3. (B.9)
The requirement that U3(s) is unitary corresponds to having orthonormal vectors in the matrix lines:
1/9 + |b|2 + |c|2 = 1/9 + |b|2 + |e|2 = 1/9 + |c|2 + |e|2 = 1 ⇒ |b|2 = |c|2 = |e|2 = 4/9. This condition
can be fulfilled with b, c and e real and positive: b = c = e = 2/3. We have thus explicitly built the 3D
irreducible representation of A4, in a basis corresponding to the matrices:
U3(e) =
1 0 0
0 1 0
0 0 1
, U3(t) =
1 0 0
0 ω 0
0 0 ω2
, U3(s) =
1
3
−1 2 2
2 −1 2
2 2 −1
. (B.10)
B.3 The Tensor Product Representation
Every group representation R can be decomposed into a direct sum of irreducible representations
(chosen unitary). In the vector space basis where this decomposition is possible one has:
UR(g) =⊕
irreps. µ
aµ Uµ(g), ∀ g ∈ A4, (B.11)
where aµ denotes the multiplicity of the irrep. µ.
Consider in particular the representation obtained from the tensor product of two 3D irreps. of A4,
denoted 3 ⊗ 3. To obtain the multiplicities a1, a1′ , a1′′ and a3, one can use the following expression,
which is a consequence of (B.4) and (B.11):
aµ = (χµ)† · χR, where χR ≡
(——–
√ninG
χRi ——–
). (B.12)
93
Taking into account the property χµ⊗νi = χµi × χνi , one obtains, for the multiplicities:
χ1 =1√12
(1,√
3, 2, 2)
χ1′ =1√12
(1,√
3, 2ω, 2ω2)
χ1′′ =1√12
(1,√
3, 2ω2, 2ω)
χ3 =1√12
(3, −√
3, 0, 0)
χ3⊗3 =1√12
(9,√
3, 0, 0)
⇒
a1 =1× 9 +
√3×√
3
12= 1
a1′ =1× 9 +
√3×√
3
12= 1
a1′′ =1× 9 +
√3×√
3
12= 1
a3 =3× 9−
√3×√
3
12= 2
. (B.13)
As a consequence of (B.11) and (B.13), one concludes that 3⊗ 3 = 1⊕ 1′ ⊕ 1′′ ⊕ 3⊕ 3. This means
that there is a certain basis for which every U3⊗3(g) matrix will be unitary and of the form:
U3⊗3(g) =
U1(g)
U1′(g)
U1′′(g) U3(g)
U3(g)
. (B.14)
As we calculate the CGCs in the next section we will see that there is a possibility for certain symmetric
and antisymmetric choices which will allow to make sense out of writing 3⊗3 = 1⊕1′⊕1′′⊕3s⊕3a.
B.4 Computing the CGCs
We will denote the basis in which U3⊗3(g) has the form (B.14) by a tilde. Consider two A4 triplets,
~a = (a1, a2, a3) and ~b = (b1, b2, b3), with ai, bi ∈ C. Under the action of an element of the group they
will transform according to some 3D representation matrices. Assuming henceforth that they transform
according to the choice of (B.10), one has:
~at−→ ~a ′ = U3(t)~a =
1 0 0
0 ω 0
0 0 ω2
a1
a2
a3
=
a1
ωa2
ω2a3
=
a′1a′2a′3
, (B.15)
~as−→ ~a ′′ = U3(s)~a =
1
3
−1 2 2
2 −1 2
2 2 −1
a1
a2
a3
=
1
3
−a1 + 2a2 + 2a3
2a1 − a2 + 2a3
2a1 + 2a2 − a3
=
a′′1a′′2a′′3
. (B.16)
We now need to see how the tensor product(~a⊗~b
)transforms:
(~a⊗~b
)t−→(~a⊗~b
)′= U3⊗3(t)
(~a⊗~b
), (B.17)
(~a⊗~b
)≡ (a1b1, a1b2, a1b3, a2b1, a2b2, a2b3, a3b1, a3b2, a3b3), (B.18)
(~a⊗~b
)′≡ (a1b1
′, a1b2′, a1b3
′, a2b1′, a2b2
′, a2b3′, a3b1
′, a3b2′, a3b3
′). (B.19)
94
To determine the shape of U3⊗3(t), one simply considers what happens to the products aibj when
both ~a and ~b are transformed simultaneously under U3(t). Using (B.15):
(~a⊗~b
)′= U3⊗3(t)
(~a⊗~b
)=((
U3(t)~a)⊗(U3(t)~b
) )=
a1
ωa2
ω2a3
⊗
b1
ωb2
ω2b3
= (a1b1, ω a1b2, ω2 a1b3, ω a2b1, ω
2 a2b2, a2b3, ω2 a3b1, a3b2, ω a3b3).
(B.20)
One thus concludes that U3⊗3(t) = diag(1, ω, ω2, ω, ω2, 1, ω2, 1, ω). In the same manner, one can
resort to the transformation rule of (B.16) in order to obtain the explicit form of U3⊗3(s):
U3⊗3(s) =1
3
−U3
s 2 U3s 2 U3
s
2 U3s −U3
s 2 U3s
2 U3s 2 U3
s −U3s
=
1
9
1 −2 −2 −2 4 4 −2 4 4
−2 1 −2 4 −2 4 4 −2 4
−2 −2 1 4 4 −2 4 4 −2
−2 4 4 1 −2 −2 −2 4 4
4 −2 4 −2 1 −2 4 −2 4
4 4 −2 −2 −2 1 4 4 −2
−2 4 4 −2 4 4 1 −2 −2
4 −2 4 4 −2 4 −2 1 −2
4 4 −2 4 4 −2 −2 −2 1
. (B.21)
We now need to find the vectors vi that make the change of basis which leaves U3⊗3(t) and U3⊗3(s)
in the form of (B.14), i.e. transforms these matrices into U3⊗3(t) and U3⊗3(s). We denote by S the
change of basis matrix, whose columns are the vectors vi as written in the current basis (no tilde):
S =
| | |v1 v2 · · · v9
| | |
, (B.22)
such that U3⊗3(g) = S−1 U3⊗3(g)S ⇔ U3⊗3(g) = S U3⊗3(g)S−1.
In the tilde basis, the change of basis vectors, vi = S−1vi, correspond (by definition of a change of
basis) to vectors ei with zeros in every entry except for a one in the j-th position. We construct the vi
vectors by requiring that they be orthonormal, which implies the unitarity of the change of basis matrix,
S† = S−1. Before proceeding, one presents the explicit matrix representations of the generators in the
tilde basis. For t, the matrix is U3⊗3(t) = diag(1, ω, ω2, 1, ω, ω2, 1, ω, ω2), while for s, one has:
U3⊗3(s) =
13×3
1
3
−1 2 2
2 −1 2
2 2 −1
1
3
−1 2 2
2 −1 2
2 2 −1
. (B.23)
One may first determine vi with i = 1, 2, 3 by noticing that these vectors must be eigenvectors of both
U3⊗3(t) and U3⊗3(s) with eigenvalues λ′1 = 1, λ′2 = ω, λ′3 = ω2 and λ′′1 = λ′′2 = λ′′3 = 1, respectively.
Let us determine v1. One sees that the eigenvectors of U3⊗3(t) with eigenvalue 1 are clearly linear
combinations of e1, e6 and e8. At the same time, v1 must be an eigenvector of U3⊗3(s) with the same
95
eigenvalue. The space of such eigenvectors is spanned by:
p1 =(0, 1, 0, 1, 0, 0, 0, 0, 1
),
p2 =(0, 0, 1, 1, 0, 0, 0, 1, 0
),
p3 =(− 1, 1, 1, 1, 0, 0, 1, 0, 0
),
p4 =(1, 0,−1,−1, 0, 1, 0, 0, 0
),
p5 =(1,−1, 0,−1, 1, 0, 0, 0, 0
).
(B.24)
Combining the constraints, one sees that there can be no p1, p3 or p5 in v1 since these vectors can
never be obtained by linear combinations of e1, e6 and e8. The vectors p2 and p4 remain. From the
fact that their linear combination must cancel the third component, one concludes that v1 ∝ p2 + p4.
Normalizing the sum yields:
v1 =1√3
(1, 0, 0, 0, 0, 1, 0, 1, 0
). (B.25)
The vector v2 will also be a linear combination of the pi, and simultaneously, by looking at U3⊗3(t),
a combination of e2, e4 and e9 – the eigenvectors of U3⊗3(t) with eigenvalue ω. Now, p2 through p5 are
excluded, leading to the conclusion that v2 ∝ p1, and so:
v2 =1√3
(0, 1, 0, 1, 0, 0, 0, 0, 1
). (B.26)
Finally, by looking at U3⊗3(t), one sees that v3 will be a linear combination of e3, e5 and e7, which
excludes p1, p2 and p4. Once again, one must consider v1 ∝ p3 + p5 to get rid of any components aside
from the third, fifth and seventh, concluding that:
v3 =1√3
(0, 0, 1, 0, 1, 0, 1, 0, 0
). (B.27)
To find out the remaining vi, we restrict ourselves to the space spanned by the orthogonal1 vectors:
q1 = (−1, 0, 0, 0, 0, 1, 0, 0, 0),
q2 = (1, 0, 0, 0, 0, 1, 0,−2, 0),
q3 = (0,−1, 0, 1, 0, 0, 0, 0, 0),
q4 = (0, 1, 0, 1, 0, 0, 0, 0,−2),
q5 = (0, 0,−1, 0, 1, 0, 0, 0, 0),
q6 = (0, 0, 1, 0, 1, 0,−2, 0, 0).
(B.28)
Consider now the effect of U3⊗3(t) on v4. One sees that v4 and v7 are eigenvectors of U3⊗3(t) with
eigenvalue 1. Thus, v4 and v7 will be linear combinations of e1, e6 and e8 orthogonal to v1. Looking at
the available qi vectors, one realizes that v4 and v7 must be given by combinations of q1 and q2:
v4 = x q1 + y q2 =(y − x, 0, 0, 0, 0, x+ y, 0,−2y, 0
),
v7 = x q1 + y q2 =(y − x, 0, 0, 0, 0, x+ y, 0,−2y, 0
).
(B.29)
1The qi vectors are not only orthogonal between themselves but also span the space orthogonal to the one of v1, v2 andv3. Normalization of these vectors is avoided as it is cumbersome to work with.
96
The values of x, y, x and y are chosen such that v4 and v7 are orthogonal and normalized, since S is
taken to be unitary. A general vector(~a⊗~b
)will transform to the tilde basis as:
˜(~a⊗~b
)= S†
(~a⊗~b
)⇔
a1b1
a1b2...
a3b3
=
———– v∗1 ———–
———– v∗2 ———–...
———– v∗9 ———–
a1b1
a1b2...
a3b3
. (B.30)
Looking at (B.30) and the previous expressions (B.29) for v4 and v7, one sees that:
a2b1 = (y − x)∗a1b1 + (x+ y)∗a2b3 + (−2y)∗a3b2,
a3b1 = (y − x)∗a1b1 + (x+ y)∗a2b3 + (−2y)∗a3b2.(B.31)
In the tilde basis, (a2b1, a2b2, a2b3) and (a3b1, a3b2, a3b3) transform in the same way under the action
of any group element. It is therefore safe to symmetrize and antisymmetrize the sum a2b1 + a3b1 in the
indexes i, j of the aibj . One then obtains constraints on x, y, x and y by assuming that v4 leads to a
symmetric a2b1 and v7 to an antisymmetric a3b1.
Once one has obtained v4 and v7, the remaining vectors vi will follow since, in the tilde basis, the
transformations of v4, v5 and v6 are related (the same happens for v7, v8 and v9). One will later see that
the (anti)symmetrization choice will naturally extend from v4 (v7) to v5 and v6 (v8 and v9).
Consider the following definitions for the coefficients Sij and Aij :
a2b1 ≡∑
i,j
Sij aibj , a3b1 ≡∑
i,j
Aij aibj . (B.32)
One starts by expanding the sum of a2b1 and a3b1 in terms of the aibj , as implied by (B.31):
∑i,j σij aibj ≡
∑i,j Sij aibj +
∑i,j Aij aibj = a2b1 + a3b1
= (y − x)∗a1b1 + (x+ y)∗a2b3 + (−2y)∗a3b2
+ (y − x)∗a1b1 + (x+ y)∗a2b3 + (−2y)∗a3b2
= (y + y − x− x)∗a1b1 + (x+ x+ y + y)∗a2b3 + (−2y − 2y)∗a3b2.
(B.33)
To constrain v4 and v7, (anti)symmetrization conditions are imposed:
S11 = σ11 ∧ A11 = 0
S23 = S32 =σ23 + σ32
2
A23 = −A32 =σ23 − σ32
2
⇒
(y − x)∗ = (y + y − x− x)∗ ∧ (y − x)∗ = 0
(x+ y)∗ = (−2y)∗ =(x+ x− y − y)∗
2
(x+ y)∗ = (2y)∗ =(x+ x+ 3y + 3y)∗
2
. (B.34)
Solving the previous system yields x = y and x = −3y. One can then write v4 and v7, unnormalized:
v4 = x(− 4/3, 0, 0, 0, 0, 2/3, 0, 2/3, 0
), (B.35)
v7 = x(0, 0, 0, 0, 0, 2, 0,−2, 0
). (B.36)
These vectors are clearly orthogonal, as one could have anticipated from the definitions of (B.32).
The orthornormality of the remaining vi will not be spoilt by the (anti)symmetrization choices.
97
Normalization implies |x|2 = 9/24 and |x|2 = 1/8. Choosing x = −(1/2)√
3/2 and x = 1/(2√
2):
v4 =1√6
(2, 0, 0, 0, 0,−1, 0,−1, 0
), (B.37)
v7 =1√2
(0, 0, 0, 0, 0, 1, 0,−1, 0
). (B.38)
Regarding the remaining vi vectors, one sees that v5 and v8 are eigenvectors of U3⊗3(t) with eigenvalue
ω and that v6 and v9 are eigenvectors of U3⊗3(t) with eigenvalue ω2. Thus, v5 and v8 will be linear
combinations of e2, e4, and e9, orthogonal to v2, whereas v6 and v9 will be linear combinations of e3, e5,
and e7, orthogonal to v3. Scanning the available qi vectors, one realizes that v5 and v8 must be given by
combinations of q3 and q4, and that v6 and v9 must be given by combinations of q5 and q6:
v5 = γ q3 + δ q4, v6 = ε q5 + η q6, v8 = γ q3 + δ q4, v9 = ε q5 + η q6. (B.39)
In the tilde basis, by looking at the explicit form of U3⊗3(s), one obtains the following relation:
U3⊗3(s) v4 = −1
3v4 +
2
3v5 +
2
3v6 ⇒ S U3⊗3(s)S−1S v4 = −1
3S v4 +
2
3S v5 +
2
3S v6
⇒ U3⊗3(s) v4 = −1
3v4 +
2
3v5 +
2
3v6 .
(B.40)
Then, by (B.29):
3xU3⊗3(s) q1 + 3 yU3⊗3(s) q2 = −v4 + 2v5 + 2v6
⇒ 3x q′′1 + 3 y q′′2 = −x q1 − y q2 + 2 γ q3 + 2 δ q4 + 2 ε q5 + 2 η q6.(B.41)
From the explicit form of U3⊗3(s) in Eq. (B.21) one reads off q′′1 and q′′2 , which can be decomposed
as a linear combination of qi vectors as follows2:
q′′1 = −1
3
(q1 + q3 + q4 + q5 + q6
), q′′2 = −1
3
(q2 + 3q3 + q4 − 3q5 + q6
). (B.42)
Inserting this result in (B.41), and using the fact that the qi are linearly independent, gives:
−x− 3y = 2γ
x− y = 2δ
−x+ 3y = 2ε
−x− y = 2η
⇒
γ = − 12x− 3
2y
δ = 12x− 1
2y
ε = − 12x+ 3
2y
η = − 12x− 1
2y
. (B.43)
By noticing that the same procedure can be applied to v7, v8 and v9 to obtain the barred variables,
and by considering (B.34) as well as the given choices for x and x, it follows that:
γ = 0
δ = 23x = − 1√
6
ε = −x = 32
1√6
η = − 13x = 1
21√6
∧
γ = − 12 x− 3
2 y = −2x = − 1√2
δ = 12 x− 1
2 y = 0
ε = − 12 x+ 3
2 y = x = 12√
2
η = − 12 x− 1
2 y = −x = − 12√
2
. (B.44)
2Notice that the subspace spanned by v1, v2 and v3 has already been dealt with.
98
One can now write vi for i = 5, 6, 8, 9:
v5 =1√6
(0,−1, 0,−1, 0, 0, 0, 0, 2
), (B.45)
v6 =1√6
(0, 0,−1, 0, 2, 0,−1, 0, 0
), (B.46)
v8 =1√2
(0, 1, 0,−1, 0, 0, 0, 0, 0
), (B.47)
v9 =1√2
(0, 0,−1, 0, 0, 0, 1, 0, 0
). (B.48)
Using the above Eqs. (B.25)-(B.27), (B.37)-(B.38), and (B.45)-(B.48) for the change of basis vectors
vi one can determine the (unitary) change of basis matrix. It’s inverse is explicitly given by:
S−1 = S† =1√6
√2 0 0 0 0
√2 0
√2 0
0√
2 0√
2 0 0 0 0√
2
0 0√
2 0√
2 0√
2 0 0
2 0 0 0 0 −1 0 −1 0
0 −1 0 −1 0 0 0 0 2
0 0 −1 0 2 0 −1 0 0
0 0 0 0 0√
3 0 −√
3 0
0√
3 0 −√
3 0 0 0 0 0
0 0 −√
3 0 0 0√
3 0 0
. (B.49)
To obtain the CGCs, consider a general vector(~a⊗~b
)in the non-tilde basis, where ~a and ~b transform
according to (B.15) and (B.16). Describing this vector in the tilde basis produces:
˜(~a⊗~b
)= S†
(~a⊗~b
)⇒
a1b1
a1b2
a1b3
a2b1
a2b2
a2b3
a3b1
a3b2
a3b3
=
1√3(a1b1 + a2b3 + a3b2)
1√3(a1b2 + a2b1 + a3b3)
1√3(a1b3 + a2b2 + a3b1)
1√6(2a1b1 − a2b3 − a3b2)
1√6(−a1b2 − a2b1 + 2a3b3)
1√6(−a1b3 + 2a2b2 − a3b1)
1√2(a2b3 − a3b2)
1√2(a1b2 − a2b1)
1√2(−a1b3 + a3b1)
. (B.50)
This, for example, means that when one applies any group transformation, a1b1 remains invari-
ant, which means 1√3(a1b1 + a2b3 + a3b2) transforms trivially (common factors like 1/
√3 can be ig-
nored). Thus, considering (B.14), one finally extracts the Clebsch-Gordan decomposition for the 3⊗ 3 =
1⊕ 1′ ⊕ 1′′ ⊕ 3s ⊕ 3a tensor product representation from (B.50):
a1b1 + a2b3 + a3b2 ∼ 1,
a1b2 + a2b1 + a3b3 ∼ 1′,
a1b3 + a2b2 + a3b1 ∼ 1′′,
2a1b1 − a2b3 − a3b2
−a1b2 − a2b1 + 2a3b3
−a1b3 + 2a2b2 − a3b1
∼ 3s,
a2b3 − a3b2
a1b2 − a2b1
−a1b3 + a3b1
∼ 3a. (B.51)
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