Neutrinos, Symmetries and the Origin of Matter Engineering Physics

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

i ii 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 Funda¸cãopara a Ciênciae Tecnologia (FCT) and Centro de F´ısica Teóricade Part´ıculas(CFTP), thanking, in particular, the kindness of CláudiaRomão,who has lent a helping hand whenever needed, and the support of Professor Jorge Romão,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 AntónioCoutinho, to whom I thank for our spontaneous (and crucial) discussions about physics, Copernicus and chameleons. I end by expressing my thanks to JoãoLoureiro, who was there even if two thousand kilometers away, and to S´ılvia Conde, for a superposition of all possible reasons.

iii iv Este trabalho foi financiado pela Funda¸cãopara a Ciênciae Tecnologia, sob o projecto PTDC/FIS/102120/2008.

This work was supported by Funda¸c˜aopara a Ciˆenciae Tecnologia, under the grant PTDC/FIS/102120/2008.

v vi Resumo

As simetrias como leis de invariânciadesempenham um papel fundamental na constru¸cãode teo- rias f´ısicas. Em particular, as simetrias de gauge estãona base do presente conhecimento do mundo subatómico,que assenta no Modelo Padrãoda f´ısicade part´ıculas. Apesar de repetido sucesso, este modelo tem que ser necessariamente expandido àluz da existênciade massas e mistura de neutrinos. Na presente disserta¸cãosãoexploradas extensõesdo Modelo Padrãobaseadas no mecanismo seesaw onde a supressãoda massa dos neutrinos énaturalmente explicada. Massas de neutrinos nãonulas conduzem a mistura leptónica,cuja estrutura se aproxima a um padrão tribimaximal, apontando para a poss´ıvel presen¸cade simetrias discretas na teoria a altas energias – como a invariânciasob transforma¸cões do grupo A4, considerado neste trabalho. O Modelo Padrãorevela-se igualmente insuficiente na explica¸cãoda assimetria bariónicado Universo. Nos modelos seesaw époss´ıvel gerar dinamicamente essa assimetria atravésdos decaimentos dos novos estados pesados (fora de equil´ıbriotérmico)mediante o mecanismo de leptogénese,cuja eficiênciaéde- terminada numericamente resolvendo o sistema de equa¸cõesde Boltzmann adequado. Nesta disserta¸cão, apresenta-se a análisede um modelo particular para viola¸cãoespontâneada simetria CP onde se explicam as massas e mistura de neutrinos impondo uma simetria discreta A4. A implementa¸cãodo mecanismo de leptogéneseneste contexto édiscutida em detalhe.

Palavras-chave: Assimetria bariónicado Universo; Leptogénese;Massa e mistura de neutrinos; Mecanismo seesaw; Simetrias; Viola¸cãode CP

vii viii 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 necessarily 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

ix x 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

xi 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

xii 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): electron 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 uncertainties 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

xiii 5.1 Tree-level diagrams for the decays of type II seesaw scalar triplets and one-loop diagrams contributing to the decay process ∆ ` ` ...... 70 i → α β 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 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 12 12 the model, for M1 = 10 GeV (black) and M1 = 5 10 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

xiv 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) U(1) ...... 42 L× Y

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

xv xvi 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

xvii xviii Symmetries and Asymmetries in Nature 1

It 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.

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 P´olya [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 quasicrystals, 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 µετρ´ oν (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 fa¸cade,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 Haüy’sformulation of a ‘law of symmetry’ in the context of crystallography. The aesthetic path turns into a scientific one. René-JustHaüy(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, respectively, 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. André-MarieAmpère(1775-1836) imported Legendre’s definition 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 regarding 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 symmetry. 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 g0 G can thus be denoted by g g0 or simply by g g0. 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, g0 G and g00 = g g0, then g00 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, g0, g00 G, the relationship (g g0)g00 = g(g0g00) 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 multiplication, 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 performing 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”.

2 2 2 arguments of a function. If one is given the function f(x, y, z) = a1 x +a2 y +a3 z +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 of G 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 Ug0 = Ug g0 . 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), { (1.1) (143), (124), (142), (123), (132), (1234), (1243), (1324), (1342), (1423), (1432) . } A natural way of representing the group is to map each element g S4 into a 4 4 matrix U in the ∈ × g way suggested by the following examples (as one might guess, Ue = 14×4):

0 1 0 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 U(12)(34) =   , U(1234) =   , U(243) =   . (1.2) 0 0 0 1 0 1 0 0 0 0 0 1       0 0 1 0 0 0 1 0 0 1 0 0             4 These matrices form a four-dimensional representation of S4 and act on the space (C ) spanned by the following vectors:

1 0 0 0 0 1 0 0 1   , 2   , 3   , 4   . (1.3) | i ≡ 0 | i ≡ 0 | i ≡ 1 | i ≡ 0         0 0 0 1                

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 | i | i | i | i is to say after i U i , g S4. Other invariant expressions can be constructed, a simple example | i → g | i ∀ ∈ ∗ 4 being the scalar y x = y xi, where x , y C , i.e. x = xi i , y = yi i and xi, yi C. This is h | i i | i | i ∈ | i | i | i | i ∈ † T −1 † so since U = U = (U ) = U −1 , and so y x y U U x = y U −1 x = y U x = y x . g g g g h | i → h | g g| i h | (g g)| i h | e| i h | i 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 5 Cijk being Aii = a for each i and C123 = c. Suppose now that ~r transforms according to the natural 3D 0 representation of S3, meaning that under the action of a certain g S3, ~r transforms as r (U ) r . ∈ i → g ij j 0 Here, the matrices Ug 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 S can be decomposed into a product of ‘pair switches’: e.g. (1234) = (14) (13) (12), where we stress n · · 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 (#A = #S /2 #A4 = 12): n n ⇒

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 dimensions, 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 0 0 0 (g1, g2) and one has g3 g = (g1 g , g2 g ). Demanding invariance of a system under G3 corresponds to · 3 · 1 · 2 independently demanding its invariance under G1 and G2.

4 4 This means that there is a subspace of C which is invariant under the action of these matrices, and thus there exists a basis in which all matrices Ug take the form: 1  " # Ug =   .  Ue g 

The matrices Ue g correspond to a 3D representation in their own right, and one has thus decomposed (through a change of basis) our natural 4D representation into two of smaller dimension. When further reduction is impossible, one is left with so-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 { } 6 2 2 2 e2, b . Although this group presents four elements it is not isomorphic to Z4, since a = b = ab = e, { } corresponding to the only other possible group with four elements, the Klein group Z2 Z2. K ≡ ×

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 crystallography 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 Symétrie dans les PhénomènesPhysiques [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 Poincarégroup (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 momentum, 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 (G0,?) are said to be isomorphic, (G, ·) =∼ (G0,?) if there exists a map ϕ : G → G0 such that, for every g1, g2 ∈ G, both ϕ(g1 · g2) = ϕ(g1) ? ϕ(g2) and ϕ(g1) = ϕ(g2) ⇒ g1 = g2 hold. An isomorphism denotes a strong equivalence 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 might not 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 Uˆ must be either unitary or antiunitary. This can be seen by taking quantum states φ , which transform as φ φ0 = Uφˆ , and noticing that by demanding φ0 φ0 = | ii | ii → | ii | ii |h i| ji| φ φ i, j one obtains that either Uφˆ Uφˆ = φ φ (Uˆ is a unitary operator) or Uφˆ Uφˆ = |h i| ji| ∀ h i| ji h i| ji h i| ji φ φ (Uˆ is an antiunitary, nonlinear operator). For inﬁnitesimal transformations, parametrized as h j| ii Uˆ(ε) = 1 + i ε Gˆ (Gˆ is denoted the transformation generator), demanding unitarity – the majority of operators of interest are unitary – implies that the generator is Hermitian, Gˆ† = Gˆ, allowing it n to be an observable. In the limit of a ﬁnite transformation, Uˆ(α) = limn→+∞ Uˆ(α/n) , one obtains Uˆ(α) = exp(i α Gˆ). In either case, the transformation operator Uˆ must commute with the system’s Hamiltonian, [U,ˆ Hˆ ] = 0, since a proper symmetry should not be spoilt by dynamics. This means that Gˆ also commutes with Hˆ , which can be seen in Heisenberg’s representation to directly imply that Gˆ 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 invariance 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 chromodynamics, 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 include 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 operations 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 change of 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, like sugars or gasoline, are chiral. The interesting phenomenon of tunnelling between molecular chiral states was investigated by 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. L¨uders[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 = L ) is not. However, in − i i seesaw extensions of the SM (see Section 2.2), the accidental B L is broken byP 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 symmetries 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- | i tems according to whether Uˆ 0 = 0 (Wigner mode), Uˆ 0 = 0 and the symmetry is global (Goldstone | i | i | i 6 | i mode), or Uˆ 0 = 0 and the symmetry is local (Higgs mode), Uˆ being the operator which realizes the | i 6 | i 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 orientation 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’s law of universal gravitation does not constitute an example of spontaneous symmetry breaking. 12 2 n−1 One might regard the group Zn = {e, a, a , . . . , a } 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 Poincarégroup [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 fundamental 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 events and that of symmetries and laws. Were we to know all future events, the correlations between them (the laws) would become unnecessary. In the same way, the full knowledge of the laws of nature (if such a thing is possible) would make their 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 :

n n n n η B b − b b . (1.5) ≡ nγ ≡ nγ ' nγ

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, m 1 GeV, for which the following relation holds (cf. Eq. (4.17)): N ∼ 3/2 nb nb mN = exp( mN /T ). (1.7) nγ nγ ∼ T − An unreasonable value of n /n 10−19 is thus obtained at freeze-out. One concludes that some b γ ∼ mechanism must have segregated matter and antimatter when n /n 10−10, which corresponds to b γ ∼ 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 −7 of matter ( 10 M ) [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 2

Particle 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 underlying SU(3) SU(2) U(1) gauge symmetry group – the subscripts are labels which correspond, c× L× Y respectively, to colour, left-handedness, and hypercharge – as well as the following ﬁeld content:

+ φ ναL uαL φ 0 2, 1/2 , `αL 2, 1/2 , qαL 2, 1/6 , ≡ φ ! ∼ ≡ lαL ! ∼ − ≡ dαL! ∼ (2.1) l 1, 1 , u 1, 2/3 , d 1, 1/3 . αR ∼ − αR ∼ αR ∼ − 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) U(1) gauge group, ignoring the (unbroken) strong SU(3) symmetry. L× Y c The quantities given between parentheses in (2.1) refer to representation assigments under this electroweak i subgroup. The first one specifies how the field transforms under SU(2)L, to which three generators I 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:

1 + γ5 1 γ5 ψ = ψ P ψ, ψ = − ψ P ψ. (2.3) R 2 ≡ R L 2 ≡ L

2 These operators are said to be chiral projectors since (PR,L) = 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:

µ † 1 i iµν 1 µν SM = (D φ) (D φ) V (φ) A A B B L µ − − 4 µν − 4 µν (2.4) + i `αL D/ `αL + i qαL D/ qαL + i lαR D/ lαR + i uαR D/ uαR + i dαR D/ dαR

Yl ` φ l + H.c. Yu q φ u + H.c. Yd q φ d + H.c. . − αβ αL βR − αβ αL βR − αβ αL βR One recognizes, in the above expression, the Klein-Gordon,e 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) U(1) implies a change to the usual kinetic terms of the fermions and L× Y scalar Higgs doublet ﬁelds, namely the substitution of the ordinary derivative by a covariant one [41]:

i i ∂ D ∂ i g2 A I i g B Y, (2.5) µ → µ ≡ µ − µ − Y µ where g2 and gY are coupling constants associated with the gauge groups SU(2)L and U(1)Y , respectively. i This leads to the introduction of four real boson ﬁelds Aµ 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 B ∂ B ∂ B , the term for the Ai gauge fields relies on the µν ≡ µ ν − ν µ µ i following definition of Aµν , to ensure invariance of the kinetic Proca term under the group SU(2)L:

i i i ijk j k A ∂ A ∂ A + g2 ε A A , (2.6) µν ≡ µ ν − ν µ µ ν 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 : m ψ ψ = m ψ ψ + ψ ψ + ψ ψ + ψ ψ = m ψ ψ + ψ ψ . (2.7) − − R R R L L R L L − R L L R 1 Aside from ignoring the strong sector, one omits gauge ﬁxing and Faddeev-Popov terms in LSM. The Feynman µ µ µ 0 µ† 0 slash notation is also employed: A/ ≡ γ Aµ, where γ are the Dirac matrices (µ = 0, 1, 2, 3) obeying γ = γ γ γ and µ ν µν 5 0 1 2 3 5† 5 2 µ {γ , γ } = 2g . The chirality matrix is given by γ ≡ γ5 ≡ iγ γ γ γ = γ = (γ ) and anticommutes with every γ (the number ‘5’ is a mere label, not a Dirac index). Additionally, one deﬁnes ψ ≡ ψ†γ0 and φe ≡ iτ 2φ∗.

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 SM, L 2 V (φ) µ2φ†φ + λ φ†φ , (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 transform under and element g of the gauge group, parametrized by the four local parameters θi(x), η(x) : ψ U ψ = exp i θi(x) Ii exp i η(x) Y ψ. (2.9) → g

SU(2)L U(1)Y | {z } | {z } 2.1.1 Neutral and Charged Electroweak Currents

Expanding the covariant derivative (2.5) in SM, one can extract the way in which the gauge bosons L interact with fermions. For the three generations of leptons,

` 1 i i = ` g2 A/ τ g B/ ` g l B/ l LGauge 2 αL − Y αL − Y αR αR (2.10) 1 g A/3 g B/ g A/1 i A/2 ν 2 Y 2 αL / = ναL lαL 1 − 2 3− gY lαR B lαR. 2 g2 A/ + i A/ g2 A/ gY B/ ! lαL ! − − − 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:

` 1 µ µ = ν g2 s g c γ A + g2 c + g s γ Z ν LNC 2 αL W − Y W µ W Y W µ αL 1 µ µ l g2 s + g c γ A + g2 c g s γ Z l (2.11) − 2 αL W Y W µ W − Y W µ αL g l c γµA s γµZ l , − Y αR W µ − W µ αR

` g2 µ µ † g2 `,µ CC = ναL γ Wµ lαL + lαL γ W ναL j Wµ + H.c.. (2.12) L √2 µ ≡ √2 CC In the previous expressions one has employed the definitions 1 2 Aµ + i Aµ 3 3 Wµ ,Zµ cos θW A sin θW Bµ ,Aµ sin θW A + cos θW Bµ, (2.13) ≡ √2 ≡ µ − ≡ µ where θW is the weak mixing or Weinberg angle, actually introduced by Glashow in 1961 [20], and the shorthands c cos θ and s sin θ . The fields W and Z correspond to the bosons responsible W ≡ W W ≡ W µ µ 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 θ = g cos θ g /g2 = tan θ . (2.14) W Y W ⇒ Y W

19 Plugging this relation back into (2.11), one can extract the electromagnetic coupling of charged leptons:

` 1 µ µ A = lαL g2 sW + gY cW γ Aµ lαL gY cW lαRγ Aµ lαR L −2 − (2.15) = g c l γµA l l γµA l = g c l γµ l A . − Y W αL µ αL − αR µ αR − Y W L L µ 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:

µ g µ g µ Gauge = e jA Aµ + jZ Zµ + jCC Wµ + H.c. . (2.16) L cos θW √2 µ µ Here, jA represents the electromagnetic NC for quarks and charged leptons, jZ the weak NC for all quarks µ and leptons, and jCC 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, h i h i 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+ 0 φ = h i , v C, v = 0. (2.17) h i φ0 ≡ v ∈ 6 h i! ! The minimum of the potential will then be given by:

∂V ( φ ) µ2 µ2 h i = 0 v 2 = v = eiϑ − . (2.18) ∂ φ ⇒ | | −2λ ⇒ 2λ h i r 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 generators G, since in this case ei α G φ = (1+i α G+... ) φ = h i h i

20 φ + i α G φ + = φ . This ceases to happen for the generators Ii and Y when the Higgs doublet h i h i ··· h i acquires a nonzero VEV, but not for the combination of (2.19), as expected:

1 1 0 1 1 0 0 1 0 0 Q φ = I3 + Y φ = + = = 0. (2.20) h i h i 2 0 1 2 0 1 v 0 0 v ! ! ! ! ! − Since physical ﬁelds must have a zero value in the vacuum, one reparametrizes the Higgs doublet as:

0 ξi(x) τ i φ(x) = exp i H(x) , (2.21) v 2 v +  √2     with H(x) = ξi(x) = 0. H(x) and ξi(x) correspond to real ﬁelds, unlike φ0(x) and φ+(x) which are h i h i 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):

0 φ(x) = H(x) . (2.22) v +  √2     After EWSB, H(x) will thus correspond to the physical Higgs ﬁeld and physics can be read oﬀ directly from SM by replacing φ with that of (2.22). In particular, one has: L † 0 i g2 √2 Wµ Dµφ = + v + H/√2 , (2.23) ∂µ H/√2 ! 2 sec θW Zµ! H 4 V (φ) = λ + √2 v H3 + 2 v2 H2 + const., (2.24) 4 which imply, disregarding the constant as well as interaction terms:

2 2 2 2 2 1 µ 1 2 2 g2 v † µ 1 g2 v µ Dµ φ V (φ) = (∂ H)(∂µH) + (4v λ)H + WµW + 2 ZµZ + ... (2.25) − 2 2 2 2 2 cos θW

From this last expression, one extracts the masses of the Higgs, W and Z bosons2:

2 g2 v g2 v mH = 2 v √λ = 2 µ , mZ = , mW = = cos θW mZ . (2.26) − √2 cos θW √2 p Experiment yields the values m = 125.9 0.4 GeV, m = 91.1876 0.0021 GeV, and m = H ± Z ± W 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:

2 mW ρ 2 2 . (2.27) ≡ cos θW mZ

2 2 The normalization of the kinetic Klein-Gordon term in (2.25) implies removing a factor of 1/2 in order to identify mH . 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 due 2 to the complex nature of the ﬁeld, possessing an additional degree of freedom. Therefore, mW is simply the coeﬃcient of † µ the WµW term.

21 The inclusion of additional Higgs multiplets Φk in the standard recipe results in a deviation to this 3 formula, since all of the new VEVs will contribute to the masses of gauge bosons (Ik and Ik are the weak isospin of the multiplet Φk and third isospin component of the multiplet component which acquires the VEV, respectively) [60]:

3 2 2 Ik(1 + Ik) (I ) v ρ = k − k k . (2.28) 2 (I3)2 v2 P k k k A surplus of Higgs singlets or doublets is seen toP be inconsequential, whereas VEVs of higher isospin +0.0003 multiplets are severely constrained by the experimental value ρexp = 1.0004−0.0004 [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 g 0, where θ = 0 by (2.14). Y → W

2.1.3 Fermion Masses and Mixing

Upon spontaneous breaking of the gauge symmetry, the Yukawa part of the Lagrangian (2.4) reads:

H l u d Yuk. = v + Y lαL lβ + Y uαL uβ + Y dαL dβ + H.c. L − √2 αβ R αβ R αβ R (2.29) = Ml l l Mu u u Md d d + H.c. + H , − αβ αL βR − αβ αL βR − αβ αL βR Lint. where Mψ vYψ is the mass matrix for the ψ ﬁelds and H contains the interaction terms H ψ ψ . ≡ Lint. L R 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 d l,u,d and M . This (bi-)diagonalization is achieved by unitary matrices VL,R :

† V l Ml V l = diag(m , m , m ) Dl, L R e µ τ ≡ V u† Mu V u = diag(m , m , m ) Du, (2.30) L R u c t ≡ † V d Md V d = diag(m , m , m ) Dd, L R d s b ≡ 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:

l V l l , l V l l , αL → L αβ βL αR → R αβ βR u V u u , u V u u , (2.31) αL → L αβ βL αR → R αβ βR d V d d , d V d d . αL → L αβ βL αR → R αβ βR 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 instead wants to bring them to diagonal form through generally unrelated rotations (in flavour space) of the fields which make up the 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 ψ0 ≡ U −1ψ, then use the relation ψ = Uψ0 to write the Lagrangian as a function of ψ0, 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):

q g2 u ∗ µ d = uβ V γ V dρ Wµ + H.c. LCC √2 L L αβ L αρ L g2 µ u† d = uβ γ V V dρ Wµ + H.c. (2.32) √2 L L L βρ L g2 µ uαL γ (VCKM) dβ Wµ + H.c., ≡ √2 αβ L while the neutral current of Eq. (2.11) remains unchanged, implying that there are no ﬂavour changing 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 0 relates massive down-type quarks with the linear combinations of them, d (VCKM) d , that βL ≡ αβ βL 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:

u d u ei ϕα u , d ei ϕα d (2.33) αL,R → αL,R αL,R → αL,R

Thus, one expects to identify 2n phases as unphysical. However, this is not the case since one can u decompose this transformation as a global rotation of, say, ei ϕ2 , 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]: ∈ −iδ c12 c13 s12 c13 s13 e iδ iδ VCKM =  s12 c23 c12 s23 s13 e c12 c23 s12 s23 s13 e s23 c13  , (2.34) − − iδ − iδ s12 s23 c12 c23 s13 e c12 s23 s12 c23 s13 e c23 c13  − − −    where s sin θ and c cos θ . The δ phase is the only source of CP violation in the SM [26]. ij ≡ ij ij ≡ ij 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 SM upon EWSB, L

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

Yuk. is concerned, since there is no mass term to spoil: neutrinos have deﬁnite (zero) mass in all bases. L This freedom can be used to cancel the eﬀects that diagonalizing the charged lepton mass matrix has on the lepton charged current Lagrangian, ` . By performing the transformation (recall (2.31)): LCC

ν V l ν , (2.35) αL → L αβ βL 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 numberP 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 m near the electroweak scale ( 102 H ∼ 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:

ν ν EWSB H ν Yuk. = Y `αL φ νβ + H.c. v + Y ναL νβ + H.c. . (2.36) L − αβ R −−−−−→ − √2 αβ R e 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:

µ i γ ∂µψR = m ψL µ i γ ∂µ m ψ = 0 . (2.37) − ⇒  µ i γ ∂µψL = m ψR 

If the chiral ﬁelds ψR and ψL are not independent, then one of these equations is redundant. Manipulating the ﬁrst one yields:

µ H.c. † µ† † i γ ∂µψR = m ψL i ∂µψR γ = m ψL −−−→ − (2.38) ×γ0 transpose T T i ∂ ψ γµ = m ψ i γµT ∂ ψ = m ψ . −−−→ − µ R L −−−−−−−→ − µ R L 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 T choice, in ψ ψC C ψ . Rewriting (2.38) and multiplying it by C to the left gives: → ≡

µT T T −1 µ T T i γ ∂µψR = m ψL i C γ ∂µ C ψR = m ψL − ⇔ (2.40) × T T C i γµ ∂ C ψ = m C ψ i γµ ∂ ψC = m ψC . −−−→ µ R L ⇔ µ R L Performing the identiﬁcation ψ = ψC ( ψ = ψC ), leads to a single, independent equation: R L ⇒ L R

µ C i γ ∂µψL = m ψL , (2.41)

C and the decomposition of Eq. (2.2) now reads ψ = ψL + ψL , 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:

1 ν = ν i ∂/ m ν . (2.43) LMaj. 2 − 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:

1 1 ν mass = m ν ν = m νC ν + H.c. , (2.44) LMaj. −2 −2 L L where νC can be rewritten by considering that νC = νT C†. One notices that the above mass term is L L − L C C allowed since νL not only behaves as a right-handed ﬁeld (PL νL = 0, as expected), but also because it transforms as ν under a Lorentz transformation and, thus, ν preserves Lorentz invariance. There L LMaj. 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 ν eiϕ ν , since the kinetic term dissociates ν from νC . The Majorana mass term can then be L → L L L 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]:

T 1 C ∗ † 1 † † † Wein. = c ` φ φ ` + H.c. = c φ ` C φ ` + H.c., (2.45) L αβ Λ αL βL − αβ Λ αL βL e e e e 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 − e H(x) v + φ(x) = √2 . (2.46)   0 e   As such, upon EWSB, one indeed obtains a Majorana neutrino mass term from (2.45):

2 EWSB v C Hν 1 ν C Hν Wein. c ν ν + H.c. + M ν ν + H.c. + , (2.47) L −−−−−→ Λ αβ αL βL Lint. ≡ −2 αβ αL βL Lint. where Hν contains interaction terms of neutrinos with the Higgs boson H(x) and Mν is the neutrino Lint. (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 Wein. at low-energy. In the next section, we consider seesaw extensions L 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 `L φ

NR, ΣR ∆ N,Σ YN,Σ Y Y∆ µ∗

φ φ `L φ

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], n0 sterile neutrino ﬁelds N (1, 0) are added to the SM iR ∼ 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]:

N † 1 N C type I = SM + i N ∂/ N Y ` φ N + M N N + H.c. , (2.48) L L iR iR − αi αL iR 2 ij iR jR where greek indices run over SM generations while romane ones are tied to the n0 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 n0 3 complex matrix, MN is an n0 n0 matrix which can be taken to be symmetric in ﬂavour × ×

28 space, since it is fully contracted with a symmetric quantity:

C T † † NiR NjR = NiR C NjR = NiR Cab NjR − − a b (2.49) = N C∗ N = N T C† N = N C N , jR b ba iR a − jR iR jR iR 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:

N V N N , (2.50) iR → ij jR where V N is an n0 n0 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 M . The mass eigenstates will then be the ﬁelds N N + N C , with N = P N . The i i ≡ iR iR iR R i 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 type I: L † int. YN YN∗ ∗ type I = αi `αL φ NiR + H.c. = iα `αL ε φ PR NiR + H.c. L − − (2.51) ∗ k j N∗ jk N jk j k = `αL φ Yiα εe PR NiR + NiR Yiα ε PL φ `αL,

h i h i ij where superscript roman indices refer to weak isospin space and εij = i τ 2 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 φj

NiR N jk NiR N jk i Yiα∗ ε PR i Yiα ε PL

k k `αL `αL

Figure 2.3: Vertex contributions from the interaction Lagrangian (2.51) for type I seesaw. These contributions 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:

i j NmR k l ` (p1) φ (p2) φ (p3) ` (p4) M αL −−−→ βL N kl i p/1 + p/2 + Mm N ji = u(p4) i Ymβ ε PL 2 2 i Ymα ε PL u(p1) (2.52) (p1 + p2) Mm + i h i − h i N jl i p/1 p/3 + Mm N ki + u(p4) i Ymβ ε PL − 2 2 i Ymα ε PL u(p1). (p1 p3) Mm + i h i − − h i 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 M 2 p2 for all n = 1,..., 4 and m = 1, . . . , n0. One then obtains: m n Nm 1 `i (p ) φj(p ) R φk(p ) `l (p ) i u(p ) YN YN εklεji + εkiεjl P u(p ). (2.53) αL 1 2 3 βL 4 4 mβ mα L 1 M −−−→ ' Mm h i Summing (2.53) over all possible NmR mediators yields:

T 1 `i (p ) φj(p ) φk(p ) `l (p ) i u(p ) YN YN εklεji + εkiεjl P u(p ), (2.54) αL 1 2 3 βL 4 4 N L 1 M → ' M αβ h i and matching the above to a four-point interaction results in the following eﬀective Lagrangian term:

1 T 1 4-point = `i C `l φk φj YN YN εklεji + εkiεjl , (2.55) type I αL βL N L 4 M αβ h i where a 1/4 factor has been included since an identical particle factor of 4 arises when the vertex contribution, 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:

1 1 T 1 c = YN YN . (2.56) αβ Λ 2 MN αβ 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: 2 1 T v Mν = 2 v2 c Mν = YN YN . (2.57) αβ − αβ Λ ⇒ − MN 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 l l l charged-lepton fields have been rotated by VL and VR, respectively, so that Y is diagonal. For n SM lepton generations, the matrix YN contains 2 n0n parameters (see Table 2.1) from which n can be removed by rephasing the lepton doublets ` (and singlets l ). This amounts to n0 + 2 n0n n αL αR − physical parameters, of which n0(n + 1) are moduli and n(n0 1) are phases. For n = 3 and n0 = 3, − the high-energy theory relies on 7n0 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 n0 = 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) generators Ii = T i × L where the third generator T 3 is diagonal: 0 1 0 0 i 0 1 0 0 1 1 − T 1 = 1 0 1 ,T 2 = i 0 i ,T 3 = 0 0 0 . (2.58) √2   √2  −    0 1 0 0 i 0 0 0 1      −        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. C µ µ The imposition of Lorentz invariance would in that case lead to a ψ`` term of the form ψµ `L Γ `L, where Γ is a linear µ 5 µ combination of γ and γ γ . 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:

∆++ ∆~ ∆+ 3, 1 . (2.59) ≡   ∼ ∆0     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 b1 1  a1 b2 + a2 b1  3 , (2.61) √2 ∼  a b   2 2    whereas the combination of the elements of two triplets ~c = (c1, c2, c3) and d~ = (d1, d2, d3) which transforms as a singlet is given by: 1 c1 d3 c2 d2 + c3 d1 1 . (2.62) √3 − ∼ 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: 1 a1 b1 d3 a1 b2 + a2 b1 d2 + a2 b2 d1 1 − √2 ∼ 0 1 + 1 + ++ νL νL ∆ νL lL ∆ lL νL ∆ + lL lL ∆ 1 ⇒ − √2 − √2 ∼ (2.63) 0 + ∆ ∆ /√2 νL T νL lL + − ++ = `L ∆ `L 1 , ⇒ ∆ /√2 ∆ ! lL ! ∼ − ≡∆ where in going from the second to| the third{z 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, `c must appear to the left of ` , which is achieved by adding the matrix C† to the term. L L − One thus concludes that the term `T C† ∆ ` = `C ∆ ` can be included in the type II Lagrangian, as − L L L L 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) U(1) transformation of (2.9), one has ∆ U∗ ∆ U† . e L× Y → g g Consider a straightforward generalization of the above which consists in adding not one but n0 scalar e 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]:

µ † ∆i C T type II = SM + D ∆~ D ∆~ V (φ, ∆ ) Y ` ∆ ` + µ φ ∆ φ + H.c. , (2.64) L L i µ i − i − αβ αL i βL i i e e where the remainder of the scalar potential reads (the Higgs-only term V (φ) is already included in SM): L

∆ 2 † 1 † † 2 † † V (φ, ∆i) = Mij Tr ∆i ∆j + λij φ ∆i ∆j φ + λij φ φ Tr ∆i ∆j (2.65) 3 † † 4 † † + λijkl Tr ∆i ∆j Tr ∆k ∆l + λijkl Tr ∆i ∆j ∆k ∆l + H.c. . While the Y∆i are n n symmetric matrices, λ1,2 and M∆ are n0 n0 Hermitian matrices (λ3,4 have × × general complex-valued entries). A glance at Table 2.1 allows one to see that the unitary transformation

∆ (V ∆) ∆ (2.66) i → ij j contains enough parameters to remove all phases from M∆ and all moduli except n0, 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. Mij = Mi δij. The complex parameter µi has dimensions of mass and one is now free to define the dimensionless parameter λ µ /M . i ≡ i i The Feynman rules can be read off8 from the Lagrangian, yielding the vertices of Fig. 2.4.

j j `αL `αL

∆i× ∆i× 1+δjk ∆i 1+δjk ∆i i √2 Yαβ∗ PR i √2 Yαβ PL − − − −

k k `βL `βL

φj φj

∆i× 1+δjk ∆i× 1+δjk i √2 µi i √2 µi∗ − −

φk φk

Figure 2.4: Vertex contributions for type II interactions relevant for eﬀective neutrino mass generation. For each diagram, the appropriate ∆× ∆0, ∆+, ∆++ is chosen such that electric charge is conserved. i ∈ { i i i } 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: →

× i j ∆m k l ` (p1) ` (p2) φ (p3) φ (p4) αL βL M −−→ (2.67) 1+δ i 1+δ √ ij ∆m √ kl ∗ = v(p1) i 2 Yαβ PL 2 2 i 2 µm u(p2), − − (p1 + p2) M + i − − m h i h i 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 M 2 p2 for all n = 1,..., 4 and m = 1, . . . , n0: m n × i j ∆m k l ` (p1) ` (p2) φ (p3) φ (p4) M αL βL −−→ 1+δ 1 1+δ √ ij ∆m √ kl ∗ i v(p1) 2 Yαβ 2 2 µm PL u(p2) (2.68) ' − Mm h i ∗h i δij δkl 2 µm ∆m = i v(p1) √2 √2 Y P u(p2). − − M 2 αβ L m δ δ Taking into account that, for the reactions allowed by charge conservation, √2 ij √2 kl = εklεji + − − εkiεjl, one sees that this tree-level amplitude matches the following four-point Lagrangian: 1 2 µ∗ 4-point = ì C `l φk φj m Y∆m εklεji + εkiεjl , (2.69) type II αL βL 2 αβ L 4 Mm h i which is seen to correspond to the Weinberg operator for the choice: ∗ 1 µm ∆m cαβ = 2 Yαβ . (2.70) Λ Mm The low-energy neutrino mass matrix is then given by (where a sum over the n0 triplets is considered): ∗ 2 ν 2 λi v ∆i ∆i M = Y 2 ui Y . (2.71) − Mi ≡ An alternative way of obtaining the above contribution for the neutrino mass matrix is by considering that, thanks to EWSB, both φ0 and ∆0 acquire nonvanishing VEVs. In the approximation M v, the i i Higgs VEV v remains unchanged, while the triplet VEV – which is said to be an induced VEV – is given by [87]: ∗ 2 0 λi v ui ∆i = , (2.72) ≡ h i − Mi and one has u v for all i. In fact, due to the experimental constraint of a ρ parameter very close to | i| one, there is not much room available for the values of the ui. Considering Eq. (2.28), one sees that: 2 2 1 + 2 i ui /v ρtype II = , (2.73) 1 + 4 u2/v2 Pi i and so the quantity u2 is constrained by electroweakP precision data to be of magnitude 1 10 i i . − GeV [88,89]. However,pP for coupling constants of (10−9) or higher, the most stringent constraint on the O 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, 0 0 one sees that the type II extension introduces n triplet masses Mi, n Higgs-type couplings λi, and 12n0 3 Yukawa-type couplings (recall that the Y∆i are symmetric) to the parameter count, which in − the minimal case (n0 = 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

0 ~ 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:

+ Σi R Σ~ Σ0 3, 0 . (2.74) iR ≡  i R  ∼ Σ−  i R   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:

1 C MΣ Σ0C Σ0 Σ+ Σ− + H.c. 1, (2.75) − ij 2 i R j R − i R j R ∼ where MΣ is a symmetric n0 n0 matrix. One can additionally extract from (2.63) an invariant combi- ij × nation of ﬁelds involving fermionic triplets and the doublets ` ε `∗ and φ = ε φ∗: L ≡ L 0 + 0∗ Σ /√2 Σ φ f e ν l i R − i R + H.c. = ` Σ φ + H.c. 1. (2.76) L L − 0 √ − L iR Σi Σi R/ 2! φ ! ∼ R − −

≡ΣiR e In the above, one has employed| φ−{z (φ+)∗ and} a γ0 matrix has been included. Taking into account ≡ ﬂavour, the high-energy Lagrangian can then read [87, 88]:

Σ† 1 Σ C type III = SM + i Σ~ D/ Σ~ Y ` Σ φ + M Tr Σ Σ + H.c. , (2.77) L L iR iR − αi αL iR − 2 ij iR jR h i where one has made the redefinition Σ+ Σ+ , makinge sense of (2.75) as a mass term: the fields i R → − i R C Σ0 = Σ0C + Σ0 are given Majorana masses, whereas the Ψ Σ+ + Σ− correspond to Dirac i i R i R i ≡ i R i R 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:

2 T v Mν = YΣ YΣ . (2.78) − MΣ Turning to parameter counting one sees that, since MΣ is symmetric, there is enough freedom to diagonalize the triplet mass matrix, obtaining from this n0 parameters. Additionally, by rephasing lepton doublets (and charged lepton singlets) one can remove n phases from the 2 n0n parameters of MΣ. This totals 7n0 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 3

In the previous chapter we have focused on the Standard Model of particle physics and seesaw extensions, 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:

l V l l , l V l l , ν V ν ν , (3.1) αL → L αβ βL αR → R αβ βR αL → L αi iL ν since a unitary transformation VL 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 ν masses are denoted by νi (i = 1, 2, 3). The neutrino mass matrix M then becomes:

ν T ν ν ν V M V = diag(m1, m2, m3) D . (3.2) L L ≡ This rotation produces a misalignment in the Lagrangian term of Eq. (2.12):

` g2 µ ν † l g2 µ † CC νiL γ V V lαL Wµ + H.c. νiL γ (U ) lαL Wµ + H.c., (3.3) L → √2 L L iα ≡ √2 PMNS iα 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:

νeL Ue1 Ue2 Ue3 ν1L ν = UPMNS ν ν = U U U ν . (3.4) αL αi iL ⇔  µL  µ1 µ2 µ3  2L ν U U U ν  τ L  τ1 τ2 τ3  3L      

35 Result of Global Fit with 1σ {3σ} errors Parameter Normal Ordering Inverted Ordering ∆m2 10−5 eV2 7.62 0.19 +0.58 21 ± −0.50 ∆m2 10−3 eV2 2.53 +0.08 +0.24 2.40 +0.10 +0.28 31 −0.10 −0.27 − −0.07 −0.25 2 +0.015 sin θ12 0.320 0.050 −0.017 ± 2 +0.08 +0.15 +0 .05 +0.11 sin θ23 0.49 −0.05 −0.10 0.53 −0.07 −0.14

2 +0.003 +0.010 +0.003 +0.010 sin θ13 0.026 −0.004 −0.011 0.027 −0.004 −0.011

+0.54 +1.93 δ 0.83 −0.64 π any 0.07 −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]: − − −iδ c12 c13 s12 c13 s13 e 1 0 0 iδ iδ iα1 UPMNS = s12 c23 c12 s23 s13 e c12 c23 s12 s23 s13 e s23 c13 0 e 0 , (3.5) − − −    iδ iδ iα2 s12 s23 c12 c23 s13 e c12 s23 s12 c23 s13 e c23 c13 0 0 e  − − −        ≡VPMNS ≡KPMNS where s sin| θ and c cos θ refer to the{z mixing angles θ12, θ13, θ23 [0},| π/2] specific{z to the} lepton ij ≡ ij ij ≡ ij ∈ 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 ∆m2 m2 m2, providing no information on the absolute neutrino mass scale. Since ij ≡ i − j 2 2 the sign of ∆m31 is indeterminate, whereas ∆m21 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 ν1

ν2

ν3

νe ν ντ

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 2 given by (UPMNS)αi .

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 o o best fit values correspond to a solar neutrino mixing angle θ12 34 , an atmospheric angle θ23 44 ' ' or 47o, depending on whether the mass spectrum displays normal or inverted ordering respectively, and o a reactor neutrino angle θ13 9 , no longer consistent with zero at the 3σ level. The dimensionless ' parameter r ∆m2 / ∆m2 0.03 quantifies the hierarchy between the two mass-squared differences ≡ 21 | 31| ∼ 2 2 ∆m21 and ∆m31, 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 oscillations 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 double 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 m , defined as the absolute value of the element Mν computed in the basis where the | ee| 11 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: →

2 2 2 2 i α1 2 2 2 i α2 mee = UPMNS mi = m1 c12 + m2 s12 e c13 + m3 s13 e . (3.6) ei i X

37 A recent upper limit on m corresponds to 0.2 0.4 eV, given by the GERDA collaboration [111]. | ee| − 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 2 2 m UPMNS m . Current tritium decay searches give an upper limit of m < 2 eV [37]. νe ≡ i ei i νe An additionalq (indirect) constraint on the absolute scale of neutrino masses is given by cosmological P considerations, which limit the sum mtotal = i mi. Recent results from the Planck collaboration have established an upper limit of mtotal < 0.66 eVP (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 2 2 2 Scott [114], corresponds to taking V 3 = 0, V 3 = 1/2, and V 2 = 1/3, where V here denotes the | e | | µ | | e | 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:

2 √1 3 3 0 0 1 1 1 VTBM =  q√ √ √  , (3.7) − 6 3 2  √1 √1 √1   6 − 3 2    o o 1 o where θ12 = arccos 2/3 35 , θ23 = 45 , said to be maximal , and θ13 = 0 . Alternatively, by ' making a redeﬁnitionp 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:

iϕ 2 √1 0 e 0 0 1 0 0 − 3 3 − iϕ 0 iα iϕ 1 1 1 UTBM 0 e 0 V 0 e 1 0 = e  √q √ √  KPMNS, (3.8) ≡  −  TBM  −  6 3 2 iϕ iα2 0 0 e 0 0 e  √1 √1 √1     −   6 3 − 2        ≡VTBM

iα1 iα2 where KPMNS = diag(1, e , e ) is given in terms of the new| redefined{z 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 θ π θ . ij → − ij The possibility of a TBM mixing pattern motivated the study of discrete family symmetries (considered 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.

1 2 Since the νµ ↔ ντ oscillation probability is proportional, in the two-neutrino mixing approximation, to sin (2 θ23), one o sees that it is the choice θ23 = 45 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 family/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 transformations 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:

l l M = D = 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 l T l and l T l : L → L R → R T † Ml T = Ml, (3.10) where one has ommited flavour indices for simplicity and T corresponds to the three-dimensional matrix: 1 0 0 T = 0 ω 0  , (3.11) 0 0 ω2   i 2π/3   l where ω e . Hence, one identifies Z3 – the lowest-order group which constrains M 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 quark mixing 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 ν C M to be symmetric, since, as was the case with (2.49), one can check that ναL νβL is symmetric under ν the exchange α β. The (bi)diagonalization of M is achieved through the UPMNS mixing matrix ↔ l † ν ν (UPMNS = VL VL = VL in the basis we are working with):

ν ∗ ν † ∗ † M = UPMNS D UPMNS = UPMNS diag(m1, m2, m3) UPMNS. (3.12)

The matrix Mν will then be invariant under the (unitary) transformation ν Ki ν , where L → L

i i † K = UPMNS η U , η = 13×3, diag( , +, ), diag(+, +, ), diag( , +, +) , (3.13) PMNS − − − − since: T iT ν i i † ν i † K M K = UPMNS η UPMNS M UPMNS η UPMNS ∗ iT T ∗ ν † i † (3.14) = UPMNS η UPMNS UPMNS D UPMNS UPMNS η UPMNS

∗ ν † ν = UPMNS D UPMNS = M . 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. 2 2 By noticing that η2 = η3 = 1 and η4 = η2η3 = η3η2, one identiﬁes the symmetry of the neutrino mass matrix with the Klein group = Z2 Z2 of Section 1.2. K × What was presented up to now regarding the symmetries of mass matrices is independent of the i form of UPMNS. For the particular TBM pattern of Eq. (3.8), one obtains the speciﬁc form of the K transformation matrices from (3.13):

1 2 2 1 0 0 1 − K1 = 1,K2 S = 2 1 2 ,K3 U = 0 0 1 ,K4 = SU = U S. (3.15) ≡ 3  −  ≡   2 2 1 0 1 0  −        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 symmetries presented above, respectively, in the basis of K 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 ΦΨ 0 00 A4 3 1, 1 , 1 1 1 1 1 3 3

Z4 i i 1 1 i 1 i 1 − − − SU(2) U(1) (2, 1/2) (1, 1) (3, 1) (3, 1) (2, 1/2) (1, 0) (1, 0) (1, 0) L × Y − −

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 + ν V CP×Z4 V (Φ, Ψ), (3.16) L L L − − where l and ν make up the Yukawa Lagrangian and V (Φ, Ψ) denotes the ﬂavon part of the scalar L L potential while V CP×Z4 corresponds to its remainder. The ﬂavon-free potential is then given by:

CP×Z4 V = 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):

2 † † 2 Vφ = µ(φ) φ φ + λ(φ) φ φ , 1 † † 2 † † T Vφ∆ = λi φ ∆i ∆iφ + λi φ φ Tr ∆i ∆i + λ2 M2 φ ∆2 φ + H.c. , i X 2 † 3 † † e4 e† † V∆ = Mi Tr ∆i ∆i + λij Tr ∆i ∆i Tr ∆j∆j + λij Tr ∆i ∆i∆j∆j , i i,j X X h i 2 2 ∗2 2 ∗ ∗ 2 0 4 ∗4 00 ∗ 2 ∗2 (3.18) VS = µS S + S + mS S S + λS S S + λS S + S + λS S S S + S , ∗ † 0 2 ∗2 † VSφ = ηS S S φ φ + ηS S + S φ φ ,

† 2 2 0 ∗ V ∆ = Tr ∆ ∆ ξ S + S + ξ S S , S i i i ∗ i i X T 0 ∗ VSφ∆ = φ ∆1 φ λ∆ S + λ∆ S + H.c.. As for the Yukawae e 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 0 00 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, `τ L, `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 2a1b1c1 + 2a2b2c2 + 2a3b3c3 a1b2c3 a1b3c2 a2b1c3 a2b3c1 a3b1c2 a3b2c1 1. (3.19) 3 − − − − − − ∼ Up to (1/Λ, 1/Λ0), where Λ and Λ0 are taken to correspond to an assumed unique ﬂavon scale and to O the scale of S decoupling, respectively, one obtains the following Yukawa Lagrangian, compatible4 with the postulated symmetries (l1,2,3 = e, µ, τ):

l l l l ye yµ yτ = `L Φ 1 φ eR `L Φ 100 φ µR `L Φ 10 φ τR + H.c., L − Λ − Λ − Λ (3.20) ν 1 T † 0 ∗ y2 T † = ∆1 ` C ` y1 S + y S + ∆2 ` C ` Ψ + H.c., L Λ0 L L 1 1 Λ L L 1 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 = v eiα and the h i S 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) h i h i The (leptonic) Yukawa Lagrangian then becomes:

l l ν ∆1 C ∆2 C = Y ` φ l + H.c. , = Y ` ∆1 ` Y ` ∆2 ` + H.c., (3.22) L − αβ αL βR L − αβ αL βL − αβ αL βL where one recognizes the familiar charged lepton Yukawa and type II seesaw terms, with:

l r1 r3 r2 y 0 0 1 e Yl = r r r 0 yl 0 , (3.23) Λ  2 1 3  µ  r r r 0 0 yl  3 2 1  τ      1 0 0 ∆1 vS iα 0 −iα Y = y∆ 0 0 1 , y∆ y1 e + y e , (3.24) 1   1 ≡ Λ0 1 0 1 0     2 s1 s3 s2 ∆2 1 y2 − − Y = s3 2 s2 s1 . (3.25) 3 Λ − −  s2 s1 2 s3 − −    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

∆2 symmetric under G implies that it is diagonal and, as such, r1 r, while r2, r3 = 0. Demanding Y T ≡ 4This Lagrangian is not, however, the most general one allowed by the symmetries, up to the chosen order, as both T † a Weinberg operator term and the renormalizable ∆2 `L C `L 1 term can be included. Although the former might be removed by a shaping symmetry, there is no possible additional symmetry which can forbid the latter without destroying other 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 G (often referred to as a ‘µ τ symmetry’) yields s2 = s3 s. Finally, one can U − ≡ 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) h i h i One could equivalently have found this solution by requiring T Φ = Φ and S Ψ = U Ψ = Ψ . h i h i h i h i h i

3.3.1 Spontaneous CP Violation

The spontaneous breaking of CP results from a non-trivial complex phase of the singlet VEV, S = h i iα vS e . 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 = v eiα. For that, h i S we take

2 2 2 4 2 00 2 2 0 4 V0 V = m v + λ v + 2 µ + λ v v cos(2α) + 2 λ v cos(4α). (3.27) ≡ S S=hSi S S S S S S S S S S The values of α and vS are obtained through the minimisation conditions:

∂V0 = 0 v m2 + 2 λ v2 + 2 µ2 + 4 λ00 v2 cos(2α) + 4 λ0 v2 cos(4α) = 0 ∂v S S S S S S S S S S . (3.28)   h i  ∂V0 ⇒ 2 2 00 2 0 2  = 0  vS 2 µS + λS vS sin(2α) + 4 λS vS sin(4α) = 0 ∂α   h i Since sin(4α) = 2 sin(2α) cos(2α), the lower condition deﬁnes three classes of solutions:

v = 0 ( v = 0 sin(2α) = 0 ) ( v = 0 sin(2α) = 0 ) (3.29) S ∨ S 6 ∧ ∨ S 6 ∧ 6 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) = 0, for which 6 2 00 2 0 2 2 0 2 µS λS 2 mS λS mS + 2 λS vS + 4 λS vS cos(4α) v = S 0 − 0 2 00 2  2 00 2  s4 λS λS 8 λS λS + 2 µS + 4 λS vS cos(2α) = 0 − − (3.30)  ⇒  µ2 + λ00 v2  2 00 2 0 2  cos(2α) = S S S µS + λSvS + 4 λS vS cos(2 α) = 0 0 2 − 4 λS vS   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 λ > 2λ0 > 0 λ00 µ 0. In fact, for the CP-conserving solutions S S ∧ S ' S ' which obey v = 0 sin(2α) = 0, Eqs. (3.27) and (3.28) in this region give: S 6 ∧ 2 4 mS mS v − 0 V0 0 , (3.31) ' s2 λS + 2 λS ⇒ ' −4 λS + 2 λS since cos(4α) = 1 sin2(2α) = 1 for the considered class of solutions, while for the non-trivial CP-violating − solutions v = 0 sin(2α) = 0 one gets: S 6 ∧ 6 2 4 mS mS vS − V0 , (3.32) ' 2 λ 2 λ0 ⇒ ' −4 λ 2 λ0 s S − S S − S since cos(2α) 0, corresponding to the four solutions: π/4, 3π/4, 5π/4, and 7π/4. Consistency mandates ' 2 0 that mS < 0 for the given solutions to be viable. The assumption λS > 2λS > 0 selects the non-trivial CP-breaking solution as that which minimizes the potential.

44 iα iα Parametrizing the complex singlet field as S = v e + σ1 + i σ2, with σ = 0 S = v e , one S h ii ⇒ h i S 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: 2 σ1,σ2 mS 0 2 2 0 V − λ + 2 λ σ + σ + 2 λ 6 λ σ1 σ2 . (3.33) mass ' λ 2λ0 S S 1 2 S − S S − S h i Diagonalization of this quadratic form finally yields the positive squared masses M 2 4 m2 and σ1 ' − S M 2 16 m2 λ0 /(λ 2 λ0 ). σ2 ' − S S S − 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),

ye 0 0 1 0 0 2 1 1 l ∆1 ∆2 y∆2 − − Y = 0 y 0 , Y = y∆ 0 0 1 , Y = 1 2 1 , (3.34)  µ  1   3 − −  0 0 y 0 1 0 1 1 2  τ    − −   l      where y r y /Λ and y∆ s y2/Λ. The coupling of heavy triplets to the Higgs bosons is also e,µ,τ ≡ e,µ,τ 2 ≡ available for both ∆i, since VSφ∆ produces such a term for ∆1 as soon as S acquires a VEV:

T 0 ∗ SSB T V ∆ = φ ∆1 φ λ∆ S + λ S + H.c. λ1 M1 φ ∆1 φ + H.c., (3.35) Sφ ∆ −−−→ iα 0 −iα with λ1 v λ∆ e + λe e e/M1. Relying on the formalism already developede e in Section 2.2.2, one ≡ S ∆ sees that the neutrino mass matrix reads (cf. Eq. (2.71)):

2 ∗ 2 ν 2 λi v ∆i ∆1 ∆2 M = Y = 2 u1 Y + 2 u2 Y , (3.36) − M i=1 i X 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(0) y(0) . 1,2 → − 1,2 Neutrino masses can be obtained, in light of (3.2), by taking the square root of the eigenvalues of ν † ν M M , with UPMNS as the change of basis matrix. It is useful to define beforehand the quantities zi iβ (> 0) and β through 2 u1 y∆ z1 e and 2 u2 y∆ z2. All information regarding CP violation is then 1 ≡ 2 ≡ encoded5 in the phase β. The diagonalization procedure yields:

ν iβ iβ D = diag(m1, m2, m3) = diag z1 e + z2 , z1, z1 e z2 , (3.37) − and one ﬁnds that, as expected, the mixing matrix is of TBM form: U PMNS = UTBM , with indeterminate

ϕ and KPMNS. To obtain them, one notices that, since KPMNS is diagonal,

−iϕ ∗ T ν ν −1 e K = V M VTBM (D ) . (3.38) PMNS ± TBM q iα1 iα2 iβ Recalling that KPMNS = diag(1, e , e ), and deﬁning σ± arg(z2 z1 e ), 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 mi and mtotal eV

1 H L maximum mtotal allowed by cosmology

mtotal 10-1

m3 Planck m2 Exclusion m1 10-2 Π 9 Π 5 Π 11 Π 3 Π Β 8 4 8 2

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. −2 (3.41) and (3.42), the absolute neutrino mass scale is bounded from below: m1 1.6 10 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 cos β < 0, while z2 2 z1 cos β is − positive for normal ordering (m3 > m2) and negative for inverted ordering (m3 < m2). Normal ordering (considered herafter) then entails cos β < 0 β [π/2, 3π/2], while it is impossible to implement ⇒ ∈ inverted ordering since the above conditions demand that both cos β and cos β be negative (z1 2 > 0). − , By manipulating (3.37), one can express z1 and z2 as a function of β and mass-squared diﬀerences:

2 2 2 2 2 m1 + m3 2 m2 ∆m31 2 ∆m21 z2 = − = − 2 2  r r  2 2 2 . (3.40)  m1 m3 1 ∆m31  z1 = − = 4 z2 cos β −2 cos β 2 2 2 ∆m31 2 ∆m21  −  q 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):

2 2 −2 ∆m31 −2 1 ∆m31 z2 1.78 10 eV z2 3.56 10 eV z1 = − − × . (3.41) ' r 2 ' × ⇒ ' −2 cos β r 2 2 cos β ' cos β 2 Also, since m2 = z1, one is able to express neutrino masses in terms of z1(β) and ∆m21,31:

2 2 2 2 2 2 2 2 2 m1 = m ∆m = z ∆m , m3 = m ∆m = z ∆m + ∆m , (3.42) 2 − 21 1 − 21 2 − 23 1 − 21 31 q q q q 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 α1 β ' − ' −   . (3.43)  1 ⇒  1 1  σ2 arctan tan β  α2 β + arctan tan β  ' 3  ' −2 3     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 = 0 from the TBM pattern. This is achieved by considering 6 perturbations to the flavon alignment of Eq. (3.26), which might arise due to non-renormalizable corrections to the potential V (Φ, Ψ). In particular, one focuses one the case where only the flavon VEV Ψ h i is modified, with small ε1,2, to:

Ψ = s(1, 1 + ε1, 1 + ε2). (3.44) h i 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 h i h i charged lepton Yukawa matrix – which conveniently remains diagonal – nor Y∆1 , but modiﬁes Y∆2 , yielding, according to Eq. (3.25):

2 1 ε2 1 ε1 ∆2 y∆2 − − − − Y = 1 ε2 2 1 + ε1 1 . (3.45) 3 − − −  1 ε1 1 2 1 + ε2 − − −    Through the same diagonalization procedure described in the previous section, keeping terms up to 7 (ε1, ε2), one arrives at the perturbed neutrino mass eigenvalues : O

2 2 1 2 2 2 m z + z (3 + 2ε1 + 2ε2) 2 z1 z2 (3 + ε1 + ε2) cos β , m z . (3.46) 1,3 ' 1 3 2 ± 2 ' 1 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:

2 2 1 2 2 1 1 2 1 ε1 ε2 sin θ12 1 + (ε1 + ε2) , sin θ23 1 + (ε1 ε2) , sin θ13 − . (3.47) ' 3 3 ' 2 3 − ' 2 6 cos β 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 ∆m2 ∆m2 ∆m2 . 31 ' 32 21 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:

∗ ∗ 1 JCP Im U11 U22 U U = sin(2 θ12) sin(2 θ13) sin(2 θ23) cos θ13 sin δ, (3.48) ≡ 12 21 8 2 2 where one has used U as short for UPMNS. For ∆m ∆m and at ﬁrst order in ε1 2, one obtains the 31 21 , result JCP tan β (ε2 ε1)/36 sin δ sin β (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 hΨi and hΦi 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 sin β. 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. − 1

mee upper limit from GERDA

-1 10 È È

mee

10-2 È È TBM case Perturbed case

10-3 Π 9 Π 5 Π 11 Π 3 Π Β 8 4 8 2

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. ν Since the perturbations do not aﬀect the M11 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: iβ 8 m = z1 e + 2 z2/3 . However, the relationship (3.41) between the m and the z is modiﬁed : ee TBM i i 2 2 1 ∆m31 1 ∆m31 z1 , z2 1 ε1 + ε2 , (3.49) ' −2 cos β 2 ' − 3 2 r r which reduces to the TBM case when ε1 2 0, as expected. The allowed values for the parameter m , → ee 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 4

We 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]:

dr2 ds2 = dt2 R(t)2 + r2 dθ2 + sin2 θ dθ dφ2 , (4.1) − 1 k r2 − which describes an isotropic space of constant (normalized) curvature k, and where (t, r, θ, φ) are dimensionless 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: ≡ 8πG k Λ H(t)2 = ρ(t) + . (4.2) 3 − R(t)2 3 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 cosmological principle but in contradiction with what is expected without inflation. Inflation allows for the expansion of a region which is 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:

−3 nB = n n R(t) , (4.3) b − b ∼ 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 f (~p,~x,t), which is taken to depend solely on energy E and time t, f f (E, t), i i ≡ i 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:

g n i d3p f E (~p) , (4.4) i ≡ (2π)3 i i Z g ρ i d3p E (~p) f E (~p) , (4.5) i ≡ (2π)3 i i i Z g ~p 2 p i d3p | | f E (~p) , (4.6) i ≡ (2π)3 3 E (~p) i i Z i where energy depends only on the magnitude of the momentum, E (~p) = E ( ~p ) = ~p 2 + m2 (we have i i | | | | i omitted time dependence). The non-trivial formula for pressure has been derived inp 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, 2 gi = 2 if mi = 0 and si > 0, and gi = 1 if 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): 1 fi E(~p) = , (4.7) exp (Ei(~p) µi) /Ti 1 − ± 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 denominator 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 chemical 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 µ + µ = µ + µ holds. We will → i j k l 2 In 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:

g g e µi/Ti n = e µi/Ti i d3p e −Ei/Ti = i dΩ e−Ei/Ti ~p 2 d ~p , (4.8) i (2π)3 (2π)3 | | | | Z ZZ Z where dΩ = 4π since there are no angular dependences. Omitting the index i and defining z E/T and ≡ x m/TRR allows one to perform a change of integration variables, yielding (notice that ~p d ~p = E dE): ≡ | | | | ∞ 2 ∞ 3 ∞ −µ/T g −E/T g T −z g T −z 2 2 n e = 2 e E ~p dE = 2 e z ~p dz = 2 e z z x dz. (4.9) 2π m | | 2π x | | 2π x − Z Z Z p 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): | | √π(y/2)ν ∞ K (y) e−yt(t2 1)ν−1/2 dt, (4.10) ν ≡ Γ(ν + 1/2) − Z1 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:

2 ∞ ∞ y −yt 2 3/2 1 −t 2 2 K2(y) = e (t 1) dt = 2 e t t y dt. (4.11) 3 1 − y y − Z Z p 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:

3 3 gi Ti µi/Ti 2 mi → 0 gi Ti µi/Ti n = e x K2(x ) n = e , (4.12) i 2π2 i i −−−−−→ i π2 where the massless (or relativistic) limit, T m implies x m /T 0, has been also considered. i i i i i → The opposite, non-relativistic limit T m for the number density can be obtained by looking at the i i behaviour the Bessel function as x : → ∞

x → ∞ −x π 1 1 K2(x) e + , (4.13) −−−−−→ 2 x O x3/2 r r leading to, for T m : i i m T 3/2 n = g i i exp m µ /T , (4.14) i i 2π − i − i i h i 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:

g T 3 ζ(3) i i (Bose-Einstein) π2 n = , (4.15) i  3  3 gi T  ζ(3) i (Fermi-Dirac) 4 π2  3Thermal equilibrium will never be attained globally but only locally since there is no timelike spatially constant Killing vector in the FRW metric [143]. For a slow enough expansion, one can ignore this technicality.

51 where one has also considered the limit µ T (see, for example, Appendix C of Ref. [141]), valid as i i long as one ignores scenarios with degenerate fermions or Bose condensation. The number density for (eq) photons, for which gγ = 2 (two polarizations) and µγ = 0 (photon number is not conserved), is thus given by:

2 ζ(3) n = T 3, (4.16) γ π2 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):

neq g √π m 3/2 g √π m 3/2 i = i i exp (m µeq)/T i i exp m /T , (4.17) n 25/2 T − i − i ' 25/2 T − i γ h i where the approximation is valid in the limit µeq/T 1. i

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:

8πG 8πG H(T )2 = ρ(T ) H(T ) = ρ(T ) . (4.18) 3 ⇒ r 3 p The dominant contribution for the total energy density comes from relativistic species, for which:

π2 g T 4 (Bose-Einstein) 30 i i ρi(T ) =  . (4.19)  7 π2  g T 4 (Fermi-Dirac) 8 30 i i  Hence, the total energy density ρ(t) is given by:

4 4 2 2 Ti 7 Ti π 4 π 4 ρ(T ) = ρ (T ) = g + g T g∗(T ) T . (4.20) i i T 8 i T 30 ≡ 30 i " bosons fermions # X X X

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 (T T ), one obtains g∗ = 106.75. By considering & i ' −1 19 √G = m , where mPl 1.22 10 GeV is the Planck mass, one then has: Pl ' × 3 2 2 8π 1/2 T 1/2 T H(T ) = g∗ 1.66 g∗ . (4.21) r 90 mPl ' mPl

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:

1 −1/2 mPl H(t) = t(T ) 0.301 g∗ . (4.22) 2t ⇒ ' T 2

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.

2 3 3 2 S pi + ρi 2π Ti 7 Ti 3 2π 3 s = = g + g T g∗ T , (4.23) ≡ R3 T 45 i T 8 i T ≡ 45 S i i " bosons fermions # X X X where the equation of state p = ρ/3 and Eq. (4.19) have been considered. Together with Eq. (4.16), the 4 above relation establishes the relation s = π /(45 ζ(3)) g∗ n 1.8 g∗ n . For the early periods of S γ ' S γ 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 Y n /s remains constant in ∝ i ≡ i the absence of production or destruction of i-species particles, as the number of particles in a comoving volume element, N n R3, is then proportional to n /s. The ratio of number density per entropy i ≡ i i density, Yi, will then be a useful quantity to track the evolution of a particle species. One is free to write n /n 1.8 g∗ Y . At present times, the seas of decoupled photons and neutrinos determine the value i γ ' S i g∗ = 3.9, implying η n /n 7.04 Y . Constant entropy per comoving volume additionally implies S i ≡ i γ ' i 3 3 −1/3 −1 that g∗ T R remains constant during the expansion of the Universe, and so T g R . In the S ∝ ∗S −1 case of constant g∗ , the relation T R follows. S ∼ 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 equilibrium. 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 retain an 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 m 100 MeV, ∼ π ∼ while electrons and positrons did so at a lower scale, T 2m 1 MeV. At even lower temperatures, ∼ e ∼ 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 0 2.73 K. One might wonder about the origin of the γ ' 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]:

1 B(ˆ t) = d3x :q†(~x,t) q(~x,t): , (4.24) 3 quarks X q Z 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 Bˆ operator. Adopting standard conventions for the phases, one has:

Pˆ q(~x,t) Pˆ−1 = γ0 q( ~x,t), Cˆ q(~x,t) Cˆ−1 = i γ2 q†(~x,t), Tˆ q(~x,t) Tˆ−1 = i q(~x, t) γ5 γ0γ2, (4.25) − − − where Pˆ, Cˆ and Tˆ are the operators which carry out the parity, charge conjugation and time reversal operations, respectively. Taking into account the unitarity of Pˆ and Cˆ and the anti-unitarity of Tˆ, one easily obtains the transformations for q†:

Pˆ q†(~x,t) Pˆ−1 = q†( ~x,t) γ0, Cˆ q†(~x,t) Cˆ−1 = i q(~x,t) γ2, Tˆ q†(~x,t) Tˆ−1 = i γ2γ0 γ5 q†(~x, t). − − − (4.26) Considering that, since quarks are fermions, :q q† : = :q† q : holds, one obtains: −

Pˆ Bˆ Pˆ−1 = Bˆ , Cˆ Bˆ Cˆ−1 = Bˆ , CˆPˆ B(ˆ CˆPˆ)−1 = Bˆ, (4.27) − − CˆPˆTˆ B(0)ˆ (CˆPˆTˆ)−1 = B(0)ˆ . (4.28) −

This shows that the Bˆ is odd under C and CP transformations and, thus, a nonzero expectation value of B,ˆ demands the presence of both C and CP violation.

Departure from Equilibrium

The expectation value of the B(ˆ t) operator is given by:

B(ˆ t) = Tr ρD(t) B(ˆ t) , (4.29) where ρD(t) denotes the density operator. In the Heisenberg representation, one has:

ˆ ˆ B(ˆ t) = eiHt B(0)ˆ 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 ρ = exp( H/Tˆ )/Z, with a partition function Z = Tr exp( H/Tˆ ) . Using D − − the above relations and considering an equilibrium scenario, one obtains: 1 ˆ 1 ˆ ˆ ˆ B(ˆ t) = Tr e−H/T B(ˆ t) = Tr e−H/T eiHt B(0)ˆ e−iHt T Z Z (4.31) 1 ˆ = Tr e−H/T B(0)ˆ = B(0)ˆ , Z T where the cyclic nature of the trace and the possibility of commuting the exponential operators have been used. Assuming a CPT invariant Hamiltonian, CˆPˆTˆ Hˆ (CˆPˆTˆ)−1 = Hˆ , and taking into account (4.28) and (4.31), one sees that:

1 ˆ B(ˆ t) = B(0)ˆ = Tr e−H/T B(0)ˆ T T Z 1 ˆ = Tr (CˆPˆTˆ)−1(CˆPˆTˆ) e−H/T (CˆPˆTˆ)−1(CˆPˆTˆ) B(0)ˆ Z h i (4.32) 1 ˆ = Tr (CˆPˆTˆ) e−H/T (CˆPˆTˆ)−1(CˆPˆTˆ) B(0)ˆ (CˆPˆTˆ)−1 Z h i 1 ˆ = Tr e−H/T B(0)ˆ = B(0)ˆ . Z − − T Hence, the expectation value of theh baryon number operatori vanishes in thermal equilibrium and so, even in the presence of B, C and CP violating interactions, no asymmetries can be generated. Na¨ıvely, 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 consideration. 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 B L µ B,L level, one can associate to these transformations the currents Jµ (x) and Jµ (x), obeying ∂ Jµ (x) = 0. This guarantees the conservation of the quantum numbers (baryon and lepton numbers) associated to ˆ 3 B ˆ 3 L the charge operators B = d x J0 (x) and L = d x J0 (x). However, these accidental SM symmetries B,L are broken beyond the classicalR field aproximationR as the Jµ are no longer conserved due to the Adler- -Bell-Jackiw chiral anomaly [148, 149]. In the present case, one obtains the anomalous divergences [57]: 2 2 g µν g ∂µJ B = ∂µJ L = n 2 Ai Ai Y Bµν B , (4.33) µ µ 32π2 µν − 32π2 µν f e 56 i `µL ì i ≠ eL `τ L qj l 1L q3L

k k q1 L q3 L

l j q1L q3L

j l q2L q2L k q2 L

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, F µν εµναβF /2 (the convention ε0123 = +1 is considered). The above implies that B and ≡ αβ L (and consequently B + L) are violated – the SM satisfies the first Sakharov condition – whereas the e combination B L is conserved. This is clear from (4.33) since one can construct a conserved current − J B−L J B J L with ∂µJ B−L = 0. Due to cancellations, the gauge symmetry group of the SM is not µ ≡ µ − µ µ 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- ± −16 2 2 −160 nentially suppressed by a factor of e π /g2 10 [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 > T 100 GeV), (B + L)-violating processes may occur as the barrier Esphaleron EW ∼ 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 T ), and second that a ﬁrst-order phase transition is required ∼ EW 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 (m2 m2)(m2 m2 )(m2 m2 )(m2 m2)(m2 m2)(m2 m2)J t − c t − u c − u b − s b − d s − d CP dimensionless by dividing it by the 12th power of T . One obtains d 10−19, which should also EW CP ∼ 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 mechanism [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 T . One such particularly interesting EW scenario is that of leptogenesis [162], in which the decays of heavy particles – such as the seesaw mediators 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, following 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 T , as electroweak sphalerons (which tend to wash out all asymmetries) EW 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]: 12 12 YB = YB−L YB = YL = 0.48 YL . (4.35) 37 ⇒ −25 − 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 (i j), to be calculated following the Feynman rules of the theory. M → Denoting the CP-conjugates of i and j by i and j, respectively, CPT invariance implies [141]:

CPT invariance: (i j) = (j i), (4.36) M → M → whereas the imposition of CP invariance translates simply to:

CP invariance: (i j) = (i j). (4.37) M → M → By joining the above conditions, one obtains a relation valid under T invariance:

T invariance: (i j) = (j i). (4.38) M → M → Finally, the unitarity of the scattering matrix in QFT (also known as the S-matrix) implies7:

CPT 2 (i j) 2 = (j i) 2 (i j) 2 = (i j) , (4.39) |M → | |M → | −−−−→ |M → | M → j j j j X X X X 7 One is presently ignoring the eﬀects of quantum statistics. A more general treatment is presented, for instance, in Appendix A.2 of [141], where Pauli blocking and Bose enhancing factors enter the above expression.

60 one-loop Mtree M ` `

X = X 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: → →

2 2 2 (X ` `) = (` ` X) = (1 + ε/2) 0 , |M → | |M → | |M | (4.40) 2 2 2 (X ` `) = (` ` X) = (1 ε/2) 0 , |M → | |M → | − |M |

2 2 where 0 is the value would take in the absence of a CP asymmetry ε. From Eq. (4.40) and |M | |M| from the fact that the two-body decay rate Γ of a particle obeys8 Γ 2, with the proportionality ∝ |M| constant depending solely on particle masses, one obtains:

Γ(X ` `) Γ(X ` `) ε = 2 → − → . (4.41) Γ(X ` `) + Γ(X ` `) → → 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:

5 2 2 ∗ ∗ 6 Γ tree + loop + (y ) = tree + loop + tree + (y ). (4.42) ∝ |M M O | |M | Mtree M M Mloop O

Thus, in the numerator of Eq. (4.41) one must expand decay rates up to the order of the interference, 2 while for the denominator one can take the approximation Γ tree . By extracting coupling constants ' |M | yi from amplitudes in the following schematic manner:

∗ tree y1 tree , loop y2 y y4 loop , (4.43) M ≡ A M ≡ 3 A

8The square modulus of the amplitude is here assumed to be averaged over initial and summed over ﬁnal degrees of freedom of the interacting particles.

61 one can, in practice, write the above numerator as:

`` ∗ `` `` `` ∗ ` ` ∗ ` ` ` ` ` ` ∗ Γ(X ` `) Γ(X ` `) = tree loop + tree loop tree loop + tree loop → − → M M M M − M M M M (4.44) ∗ ∗ ∗ = 4 Im y1 y y3 y Im loop . 2 4 Atree A This expression clearly shows that, in order to have a nonvanishing CP asymmetry, one must allow ∗ complex coupling constants and Im loop = 0. Verifying these conditions demands for the presence Atree A 6 of at least two X-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]:

2 ∂fψ ~pψ ∂fψ L fψ H(t)| | , (4.46) ≡ ∂t − Eψ ∂Eψ which codiﬁes the eﬀects of the expansion of the Universe on the evolution of particle distributions, while C denotes the collision operator [140]:

1 0 4 4 2 2 C fψ dΠ (2π) δ (pout pin) dir fψfa ... inv fifj ... , (4.47) ≡ − 2Eψ − |M| − |M| {ψ,a,...X↔ i,j,...} Z h i which accounts for the eﬀect of particle interactions. In the above expression, one sums over all possible 0 reactions involving the ψ species. The shorthand dΠ = dΠa . . . dΠi dΠj ... is considered, where one has made use of the deﬁnition:

3 gi d pi dΠi 3 , (4.48) ≡ (2π) 2Ei where the 2E factors arise from a delta function imposing the energy-momentum relations E2 = ~p 2 + i i | i| 2 mi . The argument of the delta function of (4.47) contains both the initial and final four-momenta, 2 2 defined as pin p + p + ... and pout p + p + .... The amplitudes and correspond to ≡ ψ a ≡ i j |M|dir |M|inv 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:

f f f f (1 f ) (1 f ), f f f f (1 f ) (1 f ), (4.49) ψ a → ψ a ± i ± j i j → i j ± ψ ± a

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):

g ~p 4 ∂f ψ L f d3p =n ˙ dΩ | ψ| ψ d ~p =n ˙ + 3 H n , (4.50) (2π)3 ψ ψ − E ∂E | ψ| ψ ψ Z ZZ Z ψ ψ while the right-hand side yields:

gψ 3 4 4 2 2 C f d p = dΠ (2π) δ (pout pin) f f ... f f ... , (4.51) (2π)3 ψ − − |M|dir ψ a − |M|inv i j Z {ψ,...X↔...} Z h i where dΠ dΠ dΠ. One can further manipulate the BE by noticing that: ≡ ψ 1 d S n˙ + 3Hn = n R3 = Y˙ = s Y˙ . (4.52) i i R3 dt i R3 i i 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:

dY dz dY d(1/T ) dY 1 dH dH dY Y˙ = i = i M = i M = z H i , (4.53) i dz dt dz dt dz −T 2 dt dT dz where Eq. (4.22) for H(t) in the radiation dominated era has been used. The left-hand side of the BE now reads:

dY n˙ + 3Hn = s z H i . (4.54) i i dz

Putting together what we have so far explicitly gives:

3 3 3 3 dYψ gψ d pψ ga d pa gi d pi gj d pj sHz = (#ψ) 3 3 ,... 3 3 ... dz − (2π) 2Eψ (2π) 2Ea (2π) 2Ei (2π) 2Ej {ψ,...X↔...} Z (4.55) 4 4 1 2 2 (2π) δ (pi + pj + ... pψ pa ...) dir fψfa ... inv fifj ... , × − − − Πk(#k)! |M| − |M| h i where (#ψ) counts the number of destroyed ψ particles in the interaction ψ + a + ... i + j + ..., and → −1 Πk(#k)! is 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 3 3 and ﬁnal states. dΠi will then simply read dΠi = d pi/ (2π) 2Ei . 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:

2 2 2 2 (ψ i + j) = α(ε) 0 , (i + j ψ) = β(ε) 0 , (4.56) |M → | |M | |M → | |M | 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:

eq fψ nψ 0eq fψ = nψ eq = 0eq fψ , (4.57) nψ nψ

0eq 0eq eq where the primes denote that both fψ and nψ correspond to a zero equilibrium chemical potential, µψ . µeq/T The equality is valid as the term e ψ cancels in the above expression. From now on, primes are dropped

0eq −Ei/T and we work with zero chemical potential equilibrium densities and distributions (fi = e ). 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 1 dΠ dΠ dΠ (2π)4δ4(p p p ) (ψ i + j) 2 f − ψ i j ψ − i − j |M → | ψ Z 3 S nψ d pψ −Eψ /T 1 4 4 2 = eq 3 e dΠi dΠj(2π) δ (pψ pi pj) dir − nψ (2π) − − |M| Z S Z (4.58) n g d3p 1 1 1 = ψ ψ ψ dΠ dΠ (2π)4δ4(p p p ) 2 e−Eψ /T − neq (2π)3 2E i j ψ − i − j g |M|dir ψ Z S ψ Z ψ n g d3p = ψ ψ ψ Γ(ψ i + j) e−Eψ /T , − neq (2π)3 → ψ Z 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 R related to the decay rate in the rest frame Γ by (Mψ is the mass of ψ particles):

M Γ(ψ i + j) = ψ ΓR(ψ i + j) . (4.59) → Eψ →

Expression (4.58) then reads:

n g d3p M M eq ψ ψ ψ ψ ΓR(ψ i + j) e−Eψ /T n ψ ΓR . (4.60) −neq (2π)3 E → ≡ − ψ E ψ Z ψ ψ ψ In the last line we have deﬁned what is usually called a thermal average [167]. Since in the rest frame R pψ = (Mψ,~0), one has that Γ 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:

g M ψ ψ e−Eψ /T d3p M eq 1 g d3p M (2π)3 E ψ ψ = ψ ψ ψ e−Eψ /T = Z ψ . (4.61) eq 3 g Eψ ψ nψ (2π) Eψ ψ −Eψ /T 3 Z e d pψ (2π)3 Z Taking into account the deﬁnition of the modiﬁed Bessel function given in (4.10) , one sees that:

∞ ∞ −yt 2 1 −t 2 2 K1(y) = y e t 1 dt = e t t y dt, (4.62) 1 − y y − Z p Z p and thus the numerator may be written as (z E /T , x M /T ): ≡ ψ ≡ ψ 3 ∞ 3 gψ Mψ −Eψ /T 3 gψT Mψ 2 2 −z gψT 2 3 e d pψ = 2 z z x e dz = 2 x K1(x). (4.63) (2π) Eψ 2π x Eψ − 2π Z Z p

64 eq In turn, the denominator nψ is given in the left-hand side of Eq. (4.12). At the end, the thermal average reduces to:

M eq K (x) K (M /T ) ψ = 1 = 1 ψ . (4.64) E K (x) K (M /T ) ψ ψ 2 2 ψ In conclusion, the direct part of the BE right-hand side, given in (4.58), simply reads:

K1(Mψ/T ) R nψ eq R K1(Mψ/T ) nψ α(ε)Γ0 eq α(ε) γD , γD = nψ Γ0 , (4.65) − K2(Mψ/T ) ≡ − nψ K2(Mψ/T )

R R 2 2 where Γ corresponds to Γ with 0 instead of (ψ i+j) , and γ is the decay reaction density. 0 |M | |M → | D For the inverse part of the BE, Eq. (4.58) is modiﬁed by changing the global sign, the amplitude and f f f . Since we are under an integral, where the delta function imposes conservation of energy, the ψ → i j eq eq eq relation fi fj = fψ is valid and the computations follow as in the direct case. A caveat arises regarding eq (4.60), in which an nψ is absorbed. To keep the calculation unchanged, one multiplies and divides the eq expression by nψ , resulting ﬁnally in

ni nj eq K1(Mψ/T ) R ni nj + eq eq nψ β(ε)Γ0 = + eq eq β(ε) γD , (4.66) ni nj K2(Mψ/T ) ni nj for the inverse part. To summarize, the BE for decays and inverse decays reads:

dY n n n sHz ψ = ψ α(ε) i j β(ε) γ . (4.67) dz − neq − neqneq D ψ i j If one is interested in the evolution of Y , the above is modiﬁed to dY /dz = (#i) dY /dz, where i i − ψ eq (#i) = 2 if i = j and unity otherwise. Assuming no CP violation and that ni,j = ni,j, one can write:

z dY n ΓR ψ ψ 1 0 , (4.68) Y dz ∼ − neq − H ψ ψ which means that the number of ψ particles in a comoving volume is not signiﬁcantly altered for ΓR H 0 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

2 2 2 2 (ψ + a i + j) = µ(ε) 0 , (i + j ψ + a) = ν(ε) 0 . (4.69) |M → | |M | |M → | |M | 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: → 1 (#ψ) dΠ dΠ dΠ dΠ (2π)4δ4(p + p p p ) 2 f f 2 f f − ψ a i j ψ a − i − j |M|dir ψ a − |M|inv i j Z S 4 4 1 2 nψ na eq eq ni nj eq eq = (#ψ) dΠψ . . . dΠj(2π) δ (...) 0 µ(ε) eq eq fψ fa ν(ε) eq eq fi fj − |M| nψ na − ni nj Z S (4.70) nψ na ni nj eq eq 4 4 1 2 = (#ψ) µ(ε) eq eq ν(ε) eq eq dΠψdΠa fψ fa dΠidΠj(2π) δ (...) 0 − n na − n n |M| ψ i j Z Z S d3p d3p F 1 1 = (#ψ) ... ψ a f eqf eq dΠ dΠ (2π)4δ4(...) 2 , − (2π)3 (2π)3 ψ a E E 4F i j |M|0 Z ψ a Z S

65 eq eq eq eq where one has used (4.57) and the relation fi fj = fψ fa . Inside the curly brackets one recognizes the usual formula for a 2 2 scattering cross section σ. We have also introduced the quantity F ↔ ≡ (p p )2 M 2 m2 and defined the Møller velocity in terms of particle velocities ~v = ~p /E [168]: ψ · a − ψ a i i i q 1/2 F 2 2 vMøl = ~vψ ~va ~vψ ~va , (4.71) ≡ EψEa | − | − | × | h i 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: 3 3 nψ na ni nj eq eq 1 d pψ d pa eq eq (#ψ) µ(ε) ν(ε) n n σ vMøl f f , (4.72) − neqneq − neqneq ψ a neqneq (2π)3 (2π)3 ψ a ψ a i j ψ a Z eq ≡hσvMøliψ,a eq | {z eq} eq eq where the thermal average σvMøl has been set. Defining the reaction density γ2 n n σvMøl , h iψ,a ≡ ψ a h iψ,a one can write the BE for the considered scatterings in a manner formally similar to that of Eq. (4.73):

dYψ nψ na ni nj sHz = (#ψ) µ(ε) ν(ε) γ2 . (4.73) dz − neqneq − neqneq ψ a i j To conclude, we now turn to the determination of a clearer formula for the scattering reaction density

γ2:

3 3 dpψ dpa F (s) eq eq γ2 = σ(s) f f (2π)3 (2π)3 E E ψ a Z ψ a dp3 3 ψ dpa 4 3 F (s) σ(s) −(Eψ +Ea)/T = d P δ(P0 Eψ Ea) δ (P~ ~pψ ~pa) e (2π)3 (2π)3 − − − − E E (4.74) Z Z ψ a dp3 dp3 = d4P e−P0/T ψ a δ4(P p p ) F (s) σ(s) , (2π)3E (2π)3E − ψ − a Z Z ψ a explicitly Lorentz invariant where an integration term| in P = (P 0, P~ ) has been{z introduced (the quantity} between square brackets is 2 unity) and the dependency of F and σ on the square of the center-of-mass energy s = (pψ + pa) (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: dp3 dp3 ψ a δ(√s E E ) δ3(~p + ~p ) F (s) σ(s) (2π)3E (2π)3E − ψ − a ψ a Z ψ a (4.75) 4π δ( ~pψ ~pψ CM) 4π F (s) σ(s) = d ~p ~p | | − | | F (s) σ(s) = ~p CM . (2π)6 | ψ| | ψ| E + E (2π)6 | ψ| √s Z ψ a In the center-of-mass frame, ~p CM is given by: | ψ| 2 2 2 2 2 2 2 s Mψ ma 4 Mψ ma λ s, Mψ, ma √s F (s) − − − 2 2 ~pψ CM = = λ 1,Mψ/s, ma/s = , (4.76) | | q 2√s ≡ q 2√s 2 √s q where one has defined the function λ(x, y, z) as in [169]. One can now write the reaction density as the simple integral: 4 4 −P0/T 1 2 2 1 d P −P0/T γ2 = d P e 2 s λ 1,M /s, m /s σ(s) = e σˆ(s) , (4.77) 4(2π)5 ψ a 8π (2π)4 Z Z ≡ σˆ(s) | {z } 66 where one has also defined the so-called reduced (or dimensionless) cross sectionσ ˆ(s). Carrying out the integration gives (P 2 = s):

1 2 −P0/T 1 −P0/T γ2 = 4π dP0 d P~ P~ e σˆ(s) = dP0 ds P~ e σˆ(s) 4(2π)5 | || | 4(2π)4 | | Z∞ ∞ Z Z (4.78) 1 2 −P0/T = 4 ds σˆ(s) √ dP0 P0 s e , 64π s s − Z min Z q 2 2 where smin = max Mψ + ma , mi + mj . The term in square brackets can be evaluated in terms of a modiﬁed Besseln function, yielding: o

∞ √s 2 −P0/T √ √ dP0 P0 s e = T s K1 . (4.79) s − T Z q One then obtains the expression for the reaction density:

T ∞ √s γ2 = ds √s σˆ(s) K1 . (4.80) 64π4 T Zsmin In this equation, the reduced cross section can also be given a clearer expression in terms of the Mandel- stam variable t (p p )2. Sinceσ ˆ(s) is a Lorentz invariant quantity, it too can be computed in the ≡ ψ − i center-of-mass frame, namely

σˆ = 2 s λ 1,M 2 /s, m2/s σ(s) 2 s λ(s) σ(s) ψ a ≡ 1 dp3 dp3 1 = 2 s λ(s) i j (2π)4δ4(p + p p p ) (s, t) 2 4F (s) (2π)3 2E (2π)3 2E ψ a − i − j |M |0 Z i j S 3 3 (4.81) 1 1 dpi dpj 3 1 2 = 2 s λ(s) 2 δ ( ... )δ √s Ei Ej (s, t) 0 2s λ(s) 4(2π) EiEj − − |M | Z S λ(s) dp3 1 = p i δ √s E E (s, t) 2. 16π2 E E − i − j |M |0 p Z i j S Deﬁning angular coordinates with respect to a particular z axis, chosen to coincide with ~pi, we get:

1 d3p = sin θ dθ dϕ d ~p ~p 2 = d(cos θ) dϕ dE E ~p , d(cos θ) = dt , (4.82) i | i| | i| i i | i| 2 ~p ~p | ψ|| i| which allows us to write:

1 1 σˆ(s) = dt (s, t) 2 . (4.86) 8πs |M |0 Z S

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 = CYL = C ε η YX T M , (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:

2 −1 3 gX 2π 135 ζ(3) gX −3 Y = ζ(3) g∗ = 1.95 10 g . (4.88) X T M 4 π2 45 S 8π4 g ' × X ∗S

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∗ (T ) const. = 106.75. ' S ' 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 T = T and all the uncertainty • i 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 5

In 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, 2 ∗ 2 αβ αβ 2 (∆ ` ` ) = (` ` ∆ ) = B ε /2 0 , |M i → α β | |M α β → i | i,L − i |M | ∗ 2 2 αβ αβ 2 (∆i `α `β) = (`α `β ∆i) = Bi,L + εi /2 0 , |M → | |M → | |M | (5.1) 2 ∗ ∗ ∗ 2 2 (∆ φ φ) = (φ φ ∆ ) = B + ε /2 0 , |M i → | |M → i | i,H i |M | ∗ ∗ ∗ 2 2 2 (∆ φ φ ) = (φ φ ∆ ) = B ε /2 0 . |M i → | |M → i | i,H − i |M | αβ Here, (4.40) has been generalized to include the tree-level branching ratios Bi,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:

Bαβ B , εαβ ε ,B + B = 1 . (5.2) i,L ≡ i,L i ≡ i i,L i,H ≥ ≥ αXβ αXβ As a function of decay rates (recall (4.41)), the asymmetries are given by:

Γ(∆∗ ` ` ) Γ(∆ ` ` ) Γ(∆∗ ` ` ) Γ(∆ ` ` ) εαβ = 2 i → α β − i → α β = i → α β − i → α β , (5.3) i ∗ Γ∆i + Γ∆i Γ∆i

1 Doublet components are indistinguishable above the electroweak phase transition and one can then regard a lepton or Higgs doublet as a single species with g` = g` = 2 and gφ = gφ∗ = 2. The same applies to triplets and triplet components, and g∆ = g ∗ = 3. i ∆i

69 Tree-Level One-Loop

`α φ φ `α `µ `α

∆i ∆i ∆i ∆j ∆i ∆j

` φ `β `β β φ `ν

Figure 5.1: Tree-level diagrams for the decays of type II seesaw scalar triplets and one-loop diagrams contributing to the decay process ∆ ` ` . The one-loop diagrams, which correspond to the rightmost i → α β 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∆i and 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, 0 we consider amplitudes and decay rates computed for a single component, chosen for simplicity to be ∆i (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:

[+2] ∆i∗ tree = 2i Yαβ u¯α PR vβ , M − (5.4) [−2] = 2i Y∆i u¯ P v . Mtree − αβ α L β

The one-loop amplitudes loop are given by the sum of contributions from the two wave-correction M diagrams of Fig. 5.1 (a topological symmetry factor of 1/(1 + δµν ) is introduced for the loop):

0 0 [+2] i u¯α PR vβ ∗ ∆j ∗ ∆j ∗ ∆j ∆i∗ R + I loop = 2 2 2 µi µj Yαδ R + I + Yαδ Yµν Yµν { } , M −4π M M { } 1 + δµν j6=i i − j X (5.5) 0 0 [−2] i u¯ P v ∆ ∆ R + I = α L β µ µ∗ Y j R + I + Y j Y∆j ∗ Y∆i { } , Mloop −4π2 M 2 M 2 i j αδ { } αδ µν µν 1 + δ 6= i j µν Xj i − where R + I and R0 + I0 contain both a real and an imaginary part (hence the designations R and { } { } I): 1 + r R + I = ∆ ln M 2/µ2 + 2 ln 2 + iπr ln(1 r2) r ln , { } − i − − − 1 r − (5.6) R0 + I0 = M 2 ∆ ln M 2/µ2 + 2 + iπ , { } i − i where r = 1 4 m2 /M 2 and ∆ 2/ γ + ln(4π) contains the one-loop divergence (γ is the Euler- − φ i ≡ − -Mascheroniq constant) which has been isolated through dimensional regularization (4D 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:

1 ~q Γ = | | 2 , (5.7) 8πM 2 |M| S i where ~q is the solution to M = m2 + ~q 2 + m2 + ~q 2 (m denote masses of the decay products). | | i 1 | | 2 | | i For leptons in the ﬁnal state, ~q p= M /2, whilep for the Higgs channel ~q = M /2 r. In the above | | i | | i 2 2 2 equation, a sum over ﬁnal spin states, u¯α PL,R vβ = 2 pα pβ = M , is implicit in . One sα,sβ | | · i |M| therefore has, for the total tree-level decayP rate:

1 1 2 1 2 Γ∆i = (∆i `α `β) + (∆i φφ) r 16π Mi 1 + δαβ |M → | 2|M → | α ≥ β X (5.8) mφMi † 1 1 ∆i 2 2 1 2 Mi ∆i ∆i 2 = 4 Yαβ Mi + 4 µi r Tr Y Y + λi , 16π Mi 2 | | 2 | | −−−−−−→ 8π | | α,β X where λ µ /M , and one has taken the limit m M r 1. At this point, we can also give i ≡ i i φ i ⇒ → expressions for the tree-level branching ratios:

2 ∆i 2 αβ 2 Yαβ λi B = ,Bi,H = | | . (5.9) i,L ∆ † ∆ 2 ∆ † ∆ 2 1 + δαβ Tr Y i Y i + λi Tr Y i Y i + λi | | | | By expanding the (beyond tree-level) decay rate as in (4.42), one then obtains the expression for the CP asymmetry:

[+2]∗ [+2] [−2]∗ [−2] 2 Re tree tree αβ 1 1 8π M Mloop − M Mloop εi = † . (5.10) 1 + δ 16πM M h ∆i ∆i 2 i αβ i µ,ν i Tr Y Y + λi X | | Inserting now the above amplitudes into this expression and considering M M one finally arrives at: j i † ∗ ∆i ∆j ∗ ∆i ∆j ∆i ∆j ∗ Im λ λjY Y + (Mi/Mj) Tr Y Y Y Y αβ 2 1 Mi i αβ αβ αβ αβ εi = † , (5.11) − 1 + δ 2π M ∆i ∆i 2 αβ 6= j Tr Y Y + λi Xj i | | 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∆ Y∆ + Y∆∗ , and the diﬀerence, δY∆ i ≡ i i i ≡

71 Decays and Inverse Decays s-Channel Scatterings t-Channel Scatterings

`α φ `α φ `α φ ∆i ∆i ∆i ∆i

`β φ `β φ `β φ

Gauge Scatterings ∆ f,φ ∆ Aa,B ∆ ∆ Ab,B ∆ Aa,B i i i Aa,B i i Ai, B Aa, B

∆i b ∆i ∆i A ,B ∆i b A ,B c b ∆i f,φ ∆i A ,B ∆i A ,B Fermions and Higgs t-channel u-channel cubic quartic

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 conjugates (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∆ Y∆∗ , between densities of triplets and their conjugates, and the number densities of the three i − i differences B/3 L , denoted by YBL . We choose to work with these three differences between (a − α α 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` Y` Y by [173]: − α ≡ α − `α

d δY`α dYBLα sHz = f δY , δY sHz = f (C ) YBL , (C ) YBL , (5.12) dz `α φ ⇒ dz − L αβ β H β β 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 δY Y Y ∗ , respectively. The entries of C and C will then depend on interactions which φ ≡ φ − φ L H 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]:

−1 0 0 12 13 1 10 GeV . T . 10 GeV : CL = 0 −1 0 ,CH = 3 3 4 , −1/16 −1/16 −3/4 − 8 −196 34 24 11 12 1 1 10 GeV . T . 10 GeV : CL = 34 −196 24 ,CH = 41 41 56 , 460 9 9 −156 − 115 (5.13) −906 120 120 8 11 1 1 10 GeV . T . 10 GeV : CL = 75 −688 28 ,CH = 37 52 52 , 1074 75 28 −688 − 179 −221 16 16 8 2 16 T . 10 GeV : CL = 16 −221 16 ,CH = 1 1 1 . 711 16 16 −221 − 79 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)): dY n (n ) n n sHz ψ = (#ψ) ψ a α(ε) + i j β(ε) γ . (5.14) dz − neq(na) neqneq ψ a i j 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∆i and difference δY∆i , one first considers

∗ the equations for Y∆i and 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∆i is given, considering only triplet decays (and inverse decays), by:

d ΣY ΣY Y` Y` Y` Y` sHz ∆i = ∆i α β Bαβ + εαβ/2 + α β Bαβ εαβ/2 dz − Y eq − eq2 i,L i eq2 i,L − i ∆i ≥ Y` Y` αXβ 2 2 (5.15) Y Y ∗ φ B ε /2 + φ B + ε /2 γ , − eq2 i,H − i eq2 i,H i D Yφ Yφ R where γD is given in (4.65) (with Γ0 = Γ∆i ). In order to write the BEs in a condensed form, some useful approximations are considered. In particular, the expansion Y ∗ Y ∗ Y Y Y δY Y δY , for i j ' i j − i j − j i particle species i and j, presents itself as very useful. Additional approximations include neglecting terms eq eq eq of the type δY ε, making Y ε Y ε and, in some cases, taking Y Y + Y ∗ Y ∗ Y Y . One also has ' i j i j ' i j eq eq Σ ΣY = 2 Y and deﬁnes γ 2γD, thus arriving at the partial result (no gauge scatterings yet): ∆i ∆i D ≡ d ΣY ΣY ΣY sHz ∆i = ∆i 2 γ = ∆i 1 γΣ . (5.16) dz − Y eq − D − ΣY eq − D ∆i ∆i 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:

d ΣY ΣY ΣY 2 sHz ∆i = ∆i 1 γΣ 2 ∆i 1 γ , (5.17) dz − ΣY eq − D − ΣY eq − A ∆i ∆i d δY δY 1 (C ) Y + (C ) Y (C ) Y sHz ∆i = ∆i + Bαβ L αµ BLµ L βν BLν B H µ BLµ γΣ , (5.18) dz − ΣY eq 2 i,L Y eq − i,H Y eq D ∆i ≥ ` φ αXβ d Y ΣY sHz BLα = (1 + δ )γ εαβ ∆i + 1 2B dz − αβ D i ΣY eq − i,H ( ∆i Xβ δY 1 (C ) Y + (C ) Y 2Bαβ ε + ∆i + L αµ BLµ L βν BLν (5.19) − i,L i ΣY eq 2 Y eq ∆i `

0 1 (CL)αµ YBLµ + (CL)βν YBLν (CH )µ YBLµ 2 (1 + δαβ)(γs)αβ + (γt)αβ eq + eq , − 2 Y` Yφ ) h i 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)):

eq g∆i 2 Mi eq 3 g` 3 eq gφ 3 n = TM K2 , n = ζ(3) T , n = ζ(3) T . (5.20) ∆i 2π2 i T ` 4 π2 φ π2

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 0 reaction densities for gauge and t-channel scatterings, while γs is deﬁned as:

(γ0 ) = (γ ) Bαβ B γ , (5.21) s αβ s αβ − i,L i,H D 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 sub reaction density γs , 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:

εαβ ε ε εαβ (γsub) = (γ ) Bαβ i B i γ = (γ0 ) i Bαβ + i B γ + (ε2) , s αβ s αβ − i,L ± 2 i,H ± 2 D s αβ ∓ 2 i,L 2 i,H D O (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 0 to the decay term, while γs 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 sectionsσ ˆ is given in Eq. (4.80), where – for massless fermions (above the EWPT) and under the 2 assumption of a negligible Higgs thermal mass – smin = 4Mi 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):

pi ην + pi ηµ pi pi P (p ) = ε (p , σ) ε∗(p , σ) = g + µ ν η2 µ ν , (5.23) µν i µ i ν i − µν p η − (p η)2 σ i i X · · where εµ(pi) are polarization four-vectors, the σ denote possible polarizations, and ηµ is a four-vector which veriﬁes η ε(p , σ) = 0 and η p = 0. The valid choice η = p3 +p4 is considered in our computation. · i · i 6 The integration limits for the Mandelstam variable t arising in (4.86) are, for a generic scattering i + j → 2 0 sub Notice that what is typically ignored in the literature – due to its generally small magnitude – is γs, and not γs . 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: 1 1 m2 + m2 s + m2 m2 s + m2 m2 λ s, m2, m2 λ s, m2 , m2 , (5.24) i k − 2s i − j k − l ± 2s i j k l q 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 4 which is denoted byσ ˆA) : 1 σˆ(∆ ∆∗ ff,¯ φφ∗) = 50 g4 + 41 g4 r3 , i i → 16π 2 Y i g4 M 2 M 2 2 s 1 + r σˆ(∆ ∆∗ AaAb) = 2 5 + 34 i r 24 i 1 ln i , (5.25) i i → π s i − s M 2 − 1 r i − i 3 M 2 M 2 2 s 1 + r σˆ(∆ ∆∗ AaB,BB) = 4g2g2 + g4 1 + 4 i r 4 i 2 ln i , i i → 2π 2 Y Y s i − s M 2 − 1 r i − i where r 1 4M 2/s. Similarly, for s- and t-channel scatterings, one obtains (notice that this cross i ≡ − i section correspondsp to the full γs):

2 ∆i 2 αβ 3 µi Yαβ 1 x σˆs (s) = | | , (5.26) 2 2 2 πM i 1 + δαβ (x 1) + Γ∆ /Mi − i 2 ∆i 2 αβ 6 µi Yαβ 1 ln(x + 1) σˆt (s) = | | 2 + , (5.27) πM i − 1 + x x where x s/M 2 and the notationσ ˆ is self-evident. In the unﬂavoured limit, i.e. after a sum in ﬂavour i ≡ i s,t 5 indices ( α≥β forσ ˆs and α,β forσ ˆt), these reaction densities agree with those of Ref. [179] in the absence ofP four-point ∆L =P 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:

2 2 1 0 0 αβ 1 z1z2 u1 M1 − ε1 = 4 | 2 | 2 4 sin β 0 0 1 , (5.28) 3π 4 u1 M + 3z v   | | 1 1 0 1 0 ≡ 0   ε1  

| {z2 2 } 2 4 1 0 0 αβ 1 z1z2 u2 M2 2 z2 v (ε1 + ε2) − ε2 = 4 | 2 | 2 4 1 + 2 4 2 4 sin β 0 0 1 , (5.29) 3π 4 u2 M + 2z v 9 2M u2 + z v   | | 2 2 2 | | 2 0 1 0 ≡ 0   ε2   | {z αβ } αβ where (5.11) (M2 M1) was used in deriving the expression for ε , and ε is obtained from the same 1 2 expression with a global minus sign since now M M (cf. squared-mass diﬀerence denominators in i j 4 ∗ These results are in agreement with those of Ref. [179], with gauge reduced cross sections behaving asσ ˆ(∆i∆i → ¯ ∗ 3 ∗ a b a ff, φφ ) ∼ r andσ ˆ(∆i∆i → A A ,A B,BB) ∼ r in the high-energy limit (not low-energy, as is mistakingly remarked). 5 In both eqs. (17a) and (17b) of Ref. [179] every “MT ” arising within square brackets should be replaced by an “me T ”.

75 Figure 5.3: Contours of the (magnitude of the) maximum CP asymmetries in the decays of hierarchical 13 scalar triplets ∆1 2. For temperatures above T 10 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 u (the z are fixed as functions of neutrino masses and perturbations by (3.49)) gives: | i| i 2 0 z2 2 √3 z1v ε1,max = M1 sin β , for u1 = , (5.30) 12√3πv2 | | 2 M1 2 0 z1 1 2 1 1 z2v ε2,max 1 + (ε1 + ε2) M2 sin β , for u2 1 (ε1 + ε2) . (5.31) ' 12√2πv2 9 | | ' √2 − 9 M2 Expressing this in terms of the angle β and squared mass differences yields:

2 0 ∆m31 1 ε1 max 1 (ε1 + ε2) M1 sin β , (5.32) , ' 12√6πv2 − 3 p 2 0 ∆m31 1 ε 1 (ε1 + ε2) M2 tan β . (5.33) 2,max ' − 48πv2 − 9 p 0 The magnitude of the maximum CP asymmetry εi,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 YB 10-8 10-8

10-9 10-9

10-10 10-10

10-11 10-11

10-12 10-12

-13 12 -13 10 M1 = 1´10 GeV 10 12 M1 = 5´10 GeV

10-14 10-14 90 95 100 105 110 115 245 250 255 260 265 270 Β º Figure 5.4: Scatter plot of the baryon asymmetry generated in randomly-chosen perturbed versions of the 12 H12L model, for M1 = 10 GeV (black) and M1 = 5 10 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 = 10 M1 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 | | 2 squared-mass difference ∆m21, as: 2 2 2 M1 ∆m21 λ1 = , (5.34) | | 4v4 eig(2) eig(1) − 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. −3 Both z1 and z2 are randomly generated in the range [10 , 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, o namely that which corresponds to z1 0.2595, z2 0.0656, ε1 0.1668, ε2 0.1682, and β 255.2 . ' ' ' − ' ' Reaction densities and the evolution of asymmetries for this particular perturbed version of the model −10 are given in Fig. 5.5. A baryon asymmetry YB 5.93 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 ε0 2.02 10−5 and the 1 ' − ×

77 Figure 5.5: Reaction densities normalized to the product H(T ) nγ (T ) (left) and evolution of the various 0 densities considered in the BE network (right). RIS subtraction has been considered, and the given γs is 0 0 0 0 equal to (γs)ee + (γs)µτ , with (γs)µτ = 2(γs)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 Bµτ = 2 Bee 0.654. 1,L 1,L ' 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 ηα εαβ YX T M . (5.35) α,β ! X

Notice that, for scalar triplets, YX = ΣY∆i . For the case under consideration, one obtains the flavoured efficiencies η 0.686, and η η 0.346. e ' µ ' τ ' 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 Bibliography

[1] M. C. Escher and J. E. Brigham, The graphic work of M.C. Escher, (2001) Taschen.

[2] J. Bosdriesz (Director) and M. Plukker (Producer), Metamorphose: M.C. Escher, 1898-1972 [Doc- umentary], (1999) Netherlands: Radio Netherlands Television.

[3] G. P´olya, Z. Kristallogr. 60 (1924) 278.

[4] A. L. Mackay, Physica A 114 (1982) 609.

[5] D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, Phys. Rev. Lett. 53 (1984) 1951.

[6] N. Wang, H. Chen, and K. H. Kuo, Phys. Rev. Lett. 59 (1987) 1010.

[7] T. Ishimasa, H.-U. Nissen, and Y. Fukano, Phys. Rev. Lett. 55 (1985) 511.

[8] G. Rhodes, Annu. Rev. Psychol. 57 (2006) 199.

[9] K. Brading and E. Castellani, Symmetries in Physics: Philosophical Reﬂections, (2003) Cambridge University Press.

[10] G. Hon and B. Goldstein, From Summetria to Symmetry: The Making of a Revolutionary Scientiﬁc Concept, (2008) Springer.

[11] P. Curie, J. Phys. Theor. Appl. 3 (1894) 393.

[12] E. Wigner, Symmetries and Reﬂections: Scientiﬁc Essays of Eugene P. Wigner, (1967) Indiana University Press.

[13] E. Noether, G¨ott.Nachr. (1918) 235.

[14] T. D. Lee, Particle Physics and Introduction to Field Theory, (1981) Harwood Academic Publishers.

[15] E. Wigner, Group Theory and its Application to the Quantum Mechanics of Atomic Spectra, (1959) Academic Press.

[16] G. ’t Hooft, Sci. Am. 242 (1980) 104.

[17] H. Weyl, Sitzber. Preuss. Akad. (1918) 465.

[18] H. Weyl, Z. Phys. 56 (1929) 330.

81 [19] C. N. Yang and R. Mills, Phys. Rev. 96 (1954) 191.

[20] S. Glashow, Nucl. Phys. 22 (1961) 579.

[21] S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264.

[22] A. Salam, Conf. Proc. C680519 (1968) 367.

[23] G. ’t Hooft, Nucl. Phys. B 35 (1971) 167.

[24] J. Sakurai, Modern Quantum Mechanics, (1994) Addison-Wesley Publishing Company.

[25] F. Hund, Z. Phys. 43 (1927) 805.

[26] G. C. Branco, L. Lavoura, and J. P. Silva, CP Violation, (1999) Oxford University Press.

[27] T. D. Lee and C. N. Yang, Phys. Rev. 104 (1956) 254.

[28] C. S. Wu et al., Phys. Rev. 105 (1957) 1413.

[29] J. P. Lees et al. (BaBar Collaboration), Phys. Rev. Lett. 109 (2012) 211801.

[30] M. Gell-Mann and A. Pais, Phys. Rev. 97 (1955) 1387.

[31] J. Christenson, J. Cronin, V. Fitch, and R. Turlay, Phys. Rev. Lett. 13 (1964) 138.

[32] J. Schwinger, Phys. Rev. 82 (1951) 914.

[33] W. Pauli, Niels Bohr and the Development of Physics: Essays Dedicated to Niels Bohr on the Occasion of His Seventieth Birthday, (1955) McGraw-Hill.

[34] J. S. Bell, Proc. Roy. Soc. A 231 (1955) 79.

[35] G. L¨uders,Ann. Phys. 2 (1957) 1.

[36] S. Hawking, Phys. Rev. D 32 (1985) 2489.

[37] J. Beringer et al. (Particle Data Group), Phys. Rev. D 86 (2012) 010001 and 2013 partial update for the 2014 edition (URL: http://pdg.lbl.gov).

[38] M. Guidry, Gauge Field Theories: An Introduction with Applications, (1991) John Wiley & Sons.

[39] S. Weinberg, The Quantum Theory of Fields, Vol II. Modern Applications, (1996) Cambridge Uni- versity Press.

[40] J. Goldstone, Nuovo Cimento 19 (1961) 154.

[41] M. Peskin and D. Schroeder, An Introduction to Quantum Field Theory, (1995) Westview Press.

[42] J. Goldstone, A. Salam, and S. Weinberg, Phys. Rev. 127 (1962) 965.

[43] P. Higgs, Phys. Rev. Lett. 13 (1964) 508.

82 [44] P. Higgs, Phys. Rev. 145 (1966) 1156.

[45] F. Englert and R. Brout, Phys. Rev. Lett. 13 (1964) 321.

[46] G. Guralnik, C. Hagen, and T. Kibble, Phys. Rev. Lett. 13 (1964) 585.

[47] G. Aad et al. (ATLAS Collaboration), Phys. Lett. B 716 (2012) 30.

[48] S. Chatrchyan et al. (CMS Collaboration), Phys. Lett. B 716 (2012) 1.

[49] E. Castellani, Found. Sci. 7 (2002) 181.

[50] E. Wigner, Ann. Math. 40 (1939) 149.

[51] M. Gell-Mann and Y. Ne’eman, The Eightfold Way, (1964) W. A. Benjamin.

[52] E. Wigner, Commun. Pur. Appl. Math 13 (1960) 1.

[53] A. Cohen, A. De R´ujula,and S. Glashow, Astrophys. J. 495 (1998) 539.

[54] E. Kolb and M. Turner, The Early Universe, (1990) Westview Press.

[55] K. Abe et al. (BESS Collaboration), Phys. Rev. Lett. 108 (2012) 131301.

[56] S. Davidson, E. Nardi, and Y. Nir, Phys. Rep. 466 (2008) 105.

[57] W. Bernreuther, Lect. Notes Phys. 591 (2002) 237.

[58] C. Giunti and C. W. Kim, Fundamentals of Neutrino Physics and Astrophysics, (2007) Oxford University Press.

[59] J. C. Rom˜aoand J. P. Silva, Int. J. Mod. Phys. A 27 (2012) 1230025.

[60] P. Langacker, Phys. Rept. 72 (1981) 185.

[61] S. Weinberg, Phys. Rev. D 19 (1979) 1277.

[62] L. Susskind, Phys. Rev. D 20 (1979) 2619.

[63] S. Glashow, J. Iliopoulos, and L. Maiani, Phys. Rev. D 2 (1970) 1285.

[64] N. Cabibbo, Phys. Rev. Lett. 10 (1963) 531.

[65] M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49 (1973) 652.

[66] L. Maiani, Phys. Lett. B 62 (1976) 183.

[67] L.-L. Chau and W.-Y Keung, Phys. Rev. Lett. 53 (1984) 1802.

[68] Y. Fukuda et al. (Super-Kamiokande Collaboration), Phys. Rev. Lett. 81 (1998) 1562.

[69] E. Majorana, Nuovo Cimento 14 (1937) 171.

[70] P. West, Introduction to Strings and Branes, (2012) Cambridge University Press.

83 [71] P. Pal, Am. J. Phys. 79 (2011) 485.

[72] S. Weinberg, Phys. Rev. Lett. 43 (1979) 1566.

[73] A. Broncano, M. B. Gavela, and E. Jenkins, Phys. Lett. B 552 (2003) 177.

[74] C. Itzykson and M. Nauenberg, Rev. Mod. Phys. 38 (1966) 95.

[75] P. Minkowski, Phys. Lett. B 67 (1977) 421.

[76] M. Gell-Mann, P. Ramond, and R. Slansky, in Supergravity (eds. P. van Nieuwenhuizen and D. Freedman), (1979) North-Holland Publishing Company.

[77] T. Yanagida, Conf. Proc. C7902131 (1979) 95.

[78] S. Glashow, in Quarks and Leptons (eds. M. L´evyet al.), (1980) Plenum Publishing.

[79] R. Mohapatra and G. Senjanovic, Phys. Rev. Lett. 44 (1980) 912.

[80] G. C. Branco, R. Gonz´alezFelipe, and F. R. Joaquim, Rev. Mod. Phys. 84 (2012) 515.

[81] W. Konetschny, W. Kummer, Phys. Lett. B 70 (1977) 433.

[82] T. P. Cheng and L.-F. Li, Phys. Rev. D 22 (1980) 2860.

[83] J. Schechter and J. Valle, Phys. Rev. D 22 (1980) 2227.

[84] G. Lazarides, Q. Shaﬁ, and C. Wetterich, Nucl. Phys. B 181 (1981) 287.

[85] R. Mohapatra and G. Senjanovic, Phys. Rev. D 23 (1981) 165.

[86] D. Griﬃths, Introduction to Quantum Mechanics, (1995) Prentice Hall.

[87] A. Abada et al., J. High Energy Phys. 0712 (2007) 061.

[88] F. del Aguila and J.A. Aguilar-Saavedra, Nucl. Phys. B 813 (2009) 22.

[89] S. Kanemura and K. Yagyu, Phys. Rev. D 85 (2012) 115009.

[90] Z. Xing and S. Zhou, Neutrinos in Particle Physics, Astronomy and Cosmology, (2011) Springer.

[91] R. Foot, H. Lew, X. G. He, and G. C. Joshi, Z. Phys. C 44 (1989) 441.

[92] A. Abada et al., Phys. Rev. D 78 (2008) 033007.

[93] R. Mohapatra, Phys. Rev. Lett. 56 (1986) 561.

[94] R. Mohapatra and J. Valle, Phys. Rev. D 34 (1986) 1642.

[95] A. Zee, Nucl. Phys. B 246 (1986) 99.

[96] K. S. Babu, Phys. Lett. B 203 (1988) 132.

[97] F. de Campos et al., J. High Energy Phys. 0805 (2008) 048.

84 [98] X. Ji, R. Mohapatra, S. Nussinov, and Y. Zhang, Phys. Rev. D 78 (2008) 075032.

[99] M.-C. Chen and J. Huang, Mod. Phys. Lett. A 26 (2011) 1147.

[100] B. Pontecorvo, Zh. Eksp. Teor. Fiz. 33 (1957) 549.

[101] Z. Maki, M. Nakagawa, and S. Sakata, Prog. Theor. Phys. 28 (1962) 870.

[102] B. Pontecorvo, Zh. Eksp. Teor. Fiz. 53 (1967) 1717.

[103] D. N. Dinh, A. Ibarra, E. Molinaro, and S. T. Petcov, J. High Energy Phys. 1208 (2012) 125.

[104] J. Adam et al. (MEG Collaboration), Phys. Rev. Lett. 110 (2013) 201801.

[105] J. Kaulard et al. (SINDRUM II Collaboration), Phys. Lett. B 422 (1998) 334.

[106] U. Bellgardt et al. (SINDRUM Collaboration), Nucl. Phys. B 299 (1988) 1.

[107] M. T´ortola,Fortschr. Phys. 61 (2013) 427.

[108] S. M. Bilenky, J. Hosek, and S. T. Petcov, Phys. Lett. B 94 (1980) 495.

[109] W. H. Furry, Phys. Rev. 56 (1939) 1184.

[110] W. Rodejohann, Int. J. Mod. Phys. E 20 (2011) 1833.

[111] M. Agostini et al. (GERDA Collaboration), Phys. Rev. Lett. 111 (2013) 122503.

[112] P. A. R. Ade et al. (Planck Collaboration), arXiv:1303.5062 [astro-ph.CO].

[113] P. A. R. Ade et al. (Planck Collaboration), arXiv:1303.5076 [astro-ph.CO].

[114] P. F. Harrison, D. H. Perkins, and W. G. Scott, Phys. Lett. B 530 (2002) 167.

[115] F. P. An et al. (DAYA-BAY Collaboration), Phys. Rev. Lett. 108 (2012) 171803.

[116] J. K. Ahn et al. (RENO Collaboration), Phys. Rev. Lett. 108 (2012) 191802.

[117] Y. Abe et al. (Double Chooz Collaboration), Phys. Rev. D 86 (2012) 052008.

[118] K. Abe et al. (T2K Collaboration), Phys. Rev. Lett. 107 (2011) 041801.

[119] K. Abe et al. (MINOS Collaboration), Phys. Rev. Lett. 107 (2011) 041801.

[120] L. J. Hall, H. Murayama, and N. Weiner, Phys. Rev. Lett. 84 (2000) 2572.

[121] C. D. Froggatt and H. B. Nielsen, Nucl. Phys. B 147 (1979) 277.

[122] G. Altarelli and F. Feruglio, Nucl. Phys. B 720 (2005) 64.

[123] S. F. King and C. Luhn, Rept. Prog. Phys. 76 (2013) 056201.

[124] S. F. King and C. Luhn, J. High Energy Phys. 0910 (2009) 093.

[125] G. Altarelli and F. Feruglio, Nucl. Phys. B 741 (2006) 215.

85 [126] I. de M. Varzielas, S. F. King, and G. G. Ross, Phys. Lett. B 648 (2007) 201.

[127] C. S. Lam, Phys. Rev. Lett. 101 (2008) 121602.

[128] E. Ma and G. Rajasekaran, Phys. Rev. D 64 (2001) 113012.

[129] K. S. Babu, E. Ma, and J. W. F. Valle, Phys. Lett. B 552 (2003) 207.

[130] M. Hirsch et al., Phys.Rev. D 69 (2004) 093006.

[131] R. Gonz´alezFelipe, H. Serˆodio,and J. P. Silva, Phys. Rev. D 87 (2013) 055010.

[132] S. Bhattacharya, E. Ma, A. Natale, and D. Wegman, Phys. Rev. D 87 (2013) 013006.

[133] G. C. Branco, R. Gonz´alezFelipe, F. R. Joaquim, and H. Serˆodio,Phys. Rev. D 86 (2012) 076008.

[134] C. Jarlskog, Phys. Rev. Lett. 55 (1985) 1039.

[135] R. D’Inverno, Introducing Einstein’s Relativity, (1992) Clarendon Press.

[136] C. L. Bennett et al. (WMAP Collaboration), Astrophys. J. Suppl. S. 208 (2013) 20.

[137] A. H. Guth, Phys. Rev. D 23 (1981) 347.

[138] A. L. Linde, Phys. Lett. B 108 (1982) 389.

[139] A. D. Dolgov, Phys. Atom. Nucl. 73 (2010) 815.

[140] K. Huang, Statistical Mechanics (2nd ed.), (1987) John Wiley & Sons.

[141] E. Kolb and S. Wolfram, Nucl. Phys. B 172 (1980) 224.

[142] S. Dodelson, Modern Cosmology, (2003) Academic Press.

[143] J. Bernstein, Kinetic Theory in the Expanding Universe, (2004) Cambridge University Press.

[144] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, (1964) Dover Publications.

[145] A. Bazavov et al., Phys. Rev. D 80 (2009) 014504.

[146] A. Sakharov, Pisma Zh. Eksp. Teor. Fiz. 5 (1967) 24.

[147] A. D. Dolgov, Phys. Rept. 222 (1992) 309.

[148] S. L. Adler, Phys. Rev. 177 (1969) 2426.

[149] J. S. Bell and R. Jackiw, Nuovo Cim. A 60 (1969) 47.

[150] C. Bouchiat, J. Iliopoulos, and P. Meyer, Phys. Lett. B 38 (1972) 519.

[151] S.-S. Chern and J. Simons, Ann. Math. 99 (1974) 48.

[152] G. ’t Hooft, Phys. Rev. Lett. 37 (1976) 8.

[153] F. R. Klinkhamer and N.S. Manton, Phys. Rev. D 30 (1984) 2212.

86 [154] V. A. Kuzmin, V. A. Rubakov, and M.E. Shaposhnikov, Phys. Lett. B 155 (1985) 36.

[155] D. Bodeker, G. D. Moore, and K. Rummukainen, Phys. Rev. D 61 (2000) 056003.

[156] D. E. Morrissey and M. J. Ramsey-Musolf, New J. Phys. 14 (2012) 125003.

[157] K. Kajantie, M. Laine, K. Rummukainen, and M. Shaposhnikov, Phys. Rev. Lett. 77 (1996) 2887.

[158] G. R. Farrar and M. E. Shaposhnikov, Phys. Rev. D 50 (1994) 774.

[159] M. B. Gavela et al., Nucl. Phys. B 430 (1994) 382.

[160] J. Cline and P.-A. Lemieux, Phys. Rev. D 55 (1997) 3873.

[161] I. Aﬄeck and M. Dine, Nucl. Phys. B 249 (1985) 361.

[162] M. Fukugita and T. Yanagida, Phys. Lett. B 174 (1986) 45.

[163] A. Pilaftsis, Phys. Rev. D 56 (1997) 5431.

[164] A. Pilaftsis and T. Underwood, Phys. Rev. D 72 (2005) 113001.

[165] J. A. Harvey and M. S. Turner, Phys. Rev. D 42 (1990) 3344.

[166] M. Laine and M. Shaposhnikov, Phys. Rev. D 61 (2000) 117302.

[167] P. Gondolo and G. Gelmini, Nucl. Phys. B 360 (1991) 145.

[168] S. R. de Groot, W. A. van Leeuwen, and C. G. van Weert, Relativistic kinetic theory, (1980) North-Holland Publishing Company.

[169] G. F. Giudice et al., Nucl. Phys. B 685 (2004) 89.

[170] A. Strumia, Nucl. Phys. B 809 (2009) 308.

[171] M. A. Luty, Phys. Rev. D 45 (1992) 455.

[172] A. Basbol and S. Hannestad, J. Cosmol. Astropart. P. 0701 (2007) 003.

[173] R. Barbieri, P. Creminelli, A. Strumia, and N. Tetradis, Nucl. Phys. B 575 (2000) 61.

[174] E. Nardi, Y. Nir, E. Roulet, and J. Racker, J. High Energy Phys. 0601 (2006) 164.

[175] A. Abada et al., J. High Energy Phys. 0609 (2006) 010.

[176] T. Hambye, New J. Phys. 14 (2012) 125014.

[177] C. F. Berger, L. Covi, S. Kraml, and F. Palorini, J. Cosmol. Astropart. P. 0810 (2008) 005.

[178] T. P. Cheng and L.-F. Li, Gauge Theory of Elementary Particle Physics, (1984) Clarendon Press.

[179] T. Hambye, M. Raidal, and A. Strumia, Phys. Lett. B 632 (2006) 667.

[180] A. Denner, H. Eck, O. Hahn, and J. K¨ublbeck, Nucl. Phys. B 387 (1992) 467.

87 [181] G. Goldberg, Phys. Rev. D. 32 (1985) 3331.

[182] M. Veltman, Diagrammatica: The Path to Feynman Diagrams, (1994) Cambridge University Press.

[183] W.-K. Tung, Group Theory in Physics, (1985) World Scientiﬁc Publishing Company.

88 Computing Diagrams with Majorana Fermions A

Suppose 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 without resorting explicitly to the charge M 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 h Γ, 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 h Γ or i η h Γ0 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 0 0 T −1 T −1 0 each type i – and Γ is defined as Γ C ΓQC . In most cases of interest, Γ = PR,L = CP C = Γ , ≡ R,L and so vertex contributions will be independent from the choice of a fermion flow.

89 iηh Γ iηh Γ0

iS(+p)

1 iS( p) S(p) − ≡ /p m + i − iS(+p)

u(p, s)

v(p, s)

ψc ψc u(p, s)

ψ ψ v(p, s)

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 B B.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 n #A4 = 12 elements, explicitly given in (1.4), and four conjugacy G ≡ classes which partition the whole set:

C1 = e , n1 = 1, { }

C2 = (12)(34), (13)(24), (14)(23) , n2 = 3, { } (B.1) C3 = (123), (142), (134), (243) , n3 = 4, { }

C4 = (132), (124), (143), (234) , n4 = 4, { } 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 = Z2 Z2, one has K × #(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:

n 2 = n 12 + 12 + 12 + n2 = 12 n = 3, (B.2) µ G ⇒ ⇒ irreps. X µ where nµ represents the dimension of the irrep. labeled by µ. µ The character (χi ) 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: n i (χµ)∗ χµ = δij . (B.3) n i j µ G X n i (χµ)∗ χν = δµν , (B.4) n i i i G X 3 From relation (B.3) one obtains the value of χ3 :

n 12 2 2 3 (χµ)∗ χµ = 1 χµ 2 = 1 2 + ω 2 + ω2 + χ3 = 3 χ3 = 0. (B.5) n 3 3 ⇒ | 3 | 4 ⇒ | | | | 3 ⇒ 3 µ G µ X X

91 χ1 χ2 χ3 χ4 1 1 1 1 1 10 1 1 ω ω2 100 1 1 ω2 ω 3 3 χ3 = 1 χ3 = 0 χ3 = 0 2 − 3 4

Table B.1: Character table for the group A4. The entries correspond to the value of the character χµ Tr Uµ(g), g C for each of the n irreducible representations. As in Section 1.2, UR(g) UR i ≡ ∈ i C ≡ g 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.

3 3 By the same argument, one shows that χ4 = 0. Finally, for χ2 and from (B.4), one has:

n ∗ i χ1 χ3 = 0 n χ3 = 0 1 3 + 3 χ3 = 0 χ3 = 1. (B.6) n i i ⇒ i i ⇒ × 2 ⇒ 2 − i G i X X 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 3 3 representation matrices are unitary. Since t = e and U (e) = 13×3, the elements on the diagonal of 3 3 U (t) will be cube roots of unity. In order to obtain the right character, χ = 0 (t C3), these roots 3 ∈

92 must all be diﬀerent and one may choose:

1 0 0 3 U (t) = 0 ω 0  . (B.7) 0 0 ω2     3 3 Constructing the explicit form of U (s) automatically deﬁnes all matrices U (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 = 3 −1 3 3 3 13×3 = U (s) U (s), and so the matrix U (s) is its own inverse. Due to it is unitarity, U (s) is also Hermitian. Imposing the condition χ3 = 1, one has: 2 −

a b c 3 ∗ U (s) = b d e  , (B.8) c∗ e∗ 1 a d  − − −    3 3 2 with a and d constrained to be real. Since ts C3, one must have Tr[U (ts)] = χ = 0, and st C4, so ∈ 3 ∈ 3 2 3 Tr[U (st )] = χ4 = 0. Thus:

Tr U3(ts) = 0 a + ωd ω2(1 + a + d) = 0 a = 1/3 − − . (B.9) 3 2 2 ( TrU (st ) = 0 ⇒ ( a + ω d ω(1 + a + d) = 0 ⇒ ( d = 1/3 − − 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:

1 0 0 1 0 0 1 2 2 1 − U3(e) = 0 1 0 , U3(t) = 0 ω 0 , U3(s) = 2 1 2 . (B.10)     3  −  0 0 1 0 0 ω2 2 2 1      −       

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:

R µ U (g) = a U (g), g A4, (B.11) µ ∀ ∈ irreps. M µ 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, a10 , a100 and a3, one can use the following expression, ⊗ which is a consequence of (B.4) and (B.11):

n a = (˜χµ)† χ˜R, whereχ ˜R ——– i χR ——– . (B.12) µ · ≡ n i r G

93 Taking into account the property χµ⊗ν = χµ χν , one obtains, for the multiplicities: i i × i 1 χ˜1 = 1, √3, 2, 2 √12 1 9 + √3 √3  10 1 2 a1 = × × = 1  χ˜ = 1, √3, 2ω, 2ω 12  √12    1 9 + √3 √3  00  a 0 = × × = 1  1 1 2  1  χ˜ = 1, √3, 2ω , 2ω  12 . (B.13)  √12 ⇒  √ √   1 9 + 3 3  1  a100 = × × = 1 χ˜3 = 3, √3, 0, 0 12 √ −  12  3 9 √3 √3   a3 = × − × = 2  3⊗3 1  12  χ˜ = 9, √3, 0, 0   √12     0 00 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:

U1(g) 0 U1 (g)  00  U1 (g)       3⊗3  3  U (g) =   U (g)   . (B.14)                3   U (g)            As we calculate the CGCs in the next section we will see that there is a possibility for certain symmetric 0 00 and antisymmetric choices which will allow to make sense out of writing 3 3 = 1 1 1 3s 3a. ⊗ ⊕ ⊕ ⊕ ⊕

B.4 Computing the CGCs

3⊗3 We will denote the basis in which U (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:

0 1 0 0 a1 a1 a1 t ~a ~a 0 = U3(t)~a = 0 ω 0 a = ωa = a0 , (B.15) −→    2  2   2 0 0 ω2 a ω2a a0    3  3  3         00 1 2 2 a1 a1 + 2a2 + 2a3 a1 s 00 3 1 − 1 − 00 ~a ~a = U (s)~a =  2 1 2  a2 =  2a1 a2 + 2a3  = a2  . (B.16) −→ 3 − 3 − 00 2 2 1 a3 2a1 + 2a2 a3 a  −     −   3          We now need to see how the tensor product ~a ~b transforms: ⊗ t 0 ~a ~b ~a ~b = U3⊗3(t) ~a ~b , (B.17) ⊗ −→ ⊗ ⊗ ~a ~b (a1b1, a1b2, a1b3, a2b1, a2b2, a2b3, a3b1, a3b2, a3b3), (B.18) ⊗ ≡ 0 0 0 0 0 0 0 0 0 0 ~a ~b (a1b1 , a1b2 , a1b3 , a2b1 , a2b2 , a2b3 , a3b1 , a3b2 , a3b3 ). (B.19) ⊗ ≡

94 3⊗3 To determine the shape of U (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 0 1 1 ~ 3⊗3 ~ 3 3 ~ ~a b = U (t) ~a b = U (t)~a U (t) b =  ωa2   ωb2  ⊗ ⊗ ⊗ 2 ⊗ 2 (B.20) ω a3 ω b3     2 2 2     = (a1b1, ω a1b2, ω a1b3, ω a2b1, ω a2b2, a2b3, ω a3b1, a3b2, ω a3b3).

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):

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 − − − −  U3 2 U3 2 U3  2 4 4 1 2 2 2 4 4  − s s s − − − −  3⊗3 1 3 3 3 1   U (s) = 2 Us Us 2 Us  =  4 2 4 2 1 2 4 2 4  . (B.21) 3 − 9  − − − −  2 U3 2 U3 U3  4 4 2 2 2 1 4 4 2  s s − s   − − − −       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     − − − −  3⊗3 3⊗3 We now need to ﬁnd the vectors vi that make the change of basis which leaves U (t) and U (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 v as written in the current basis (no tilde): i e e

| | | 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, v = S−1v , correspond (by deﬁnition of a change of e i e i basis) to vectors ei with zeros in every entry except for a one in the j-th position. We construct the vi e 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:

e 1  3×3  1 2 2  1 −    3⊗3   2 1 2   U (s) =  3 −  . (B.23)  2 2 1    −      1 2 2  e    1 −    2 1 2   3 −   2 2 1    −     One may ﬁrst determine vi with i = 1, 2, 3 by noticing that these vectors must be eigenvectors of both 3⊗3 3⊗3 0 0 0 2 00 00 00 U (t) and U (s) with eigenvalues λ1 = 1, λ2 = ω, λ3 = ω and λ1 = λ2 = λ3 = 1, respectively. 3⊗3 Let us determine v1. One sees that the eigenvectors of U (t) with eigenvalue 1 are clearly linear 3⊗3 combinations of e1, e6 and e8. At the same time, v1 must be an eigenvector of U (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 , (B.24) −

p4 = 1, 0, 1, 1, 0, 1, 0, 0, 0 , − −

p5 = 1, 1, 0, 1, 1, 0, 0, 0, 0. − − 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:

1 v1 = 1, 0, 0, 0, 0, 1, 0, 1, 0 . (B.25) √3 3⊗3 The vector v2 will also be a linear combination of the pi, and simultaneously, by looking at U (t), 3⊗3 a combination of e2, e4 and e9 – the eigenvectors of U (t) with eigenvalue ω. Now, p2 through p5 are excluded, leading to the conclusion that v2 p1, and so: ∝ 1 v2 = 0, 1, 0, 1, 0, 0, 0, 0, 1 . (B.26) √3 3⊗3 Finally, by looking at U (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:

1 v3 = 0, 0, 1, 0, 1, 0, 1, 0, 0 . (B.27) √3 1 To ﬁnd out the remaining vi, we restrict ourselves to the space spanned by the orthogonal 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), − (B.28) 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). − 3⊗3 3⊗3 Consider now the eﬀect of U (t) on v4. One sees that v4 and v7 are eigenvectors of U (t) with eigenvalue 1. Thus, v4 and v7 will be linear combinations of e1, e6 and e8 orthogonal to v1. Looking at e e e e e 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 , − − (B.29) v7 =x ¯ q1 +y ¯ q2 = y¯ x,¯ 0, 0, 0, 0, x¯ +y, ¯ 0, 2¯y, 0. − − 1 The qi vectors are not only orthogonal between themselves but also span the space orthogonal to the one of v1, v2 and v3. Normalization of these vectors is avoided as it is cumbersome to work with.

96 The values of x, y, x¯ andy ¯ 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: ⊗ ∗ a1b1 ———– v1 ———– a1b1 ∗ a1b2 ———– v ———– a1b2 ^ †    2    ~a ~b = S ~a ~b g. = . . . (B.30) ⊗ ⊗ ⇔ . . .  .   .   .   g      a b  ———– v∗ ———– a b   3 3  9   3 3       Looking at (B.30) and the previous expressionsg (B.29) for v4 and v7, one sees that:

∗ ∗ ∗ a2b1 = (y x) a1b1 + (x + y) a2b3 + ( 2y) a3b2, − − (B.31) ∗ ∗ ∗ a3b1 = (¯y x¯) a1b1 + (¯x +y ¯) a2b3 + ( 2¯y) a3b2. g − −

In the tilde basis, (a2b1, ag2b2, 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 a b + a b in the g g g g g g 2 1 3 1 indexes i, j of the a b . One then obtains constraints on x, y, x¯ andy ¯ by assuming that v leads to a i j g 4 g symmetric a2b1 and v7 to an antisymmetric a3b1. Once one has obtained v and v , the remaining vectors v will follow since, in the tilde basis, the g 4 7 g i 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 S a b , a3b1 A a b . (B.32) ≡ ij i j ≡ ij i j i,j i,j X X g g One starts by expanding the sum of a2b1 and a3b1 in terms of the aibj, as implied by (B.31):

σ a b S a b + g A a bg= a2b1 + a3b1 i,j ij i j ≡ i,j ij i j i,j ij i j ∗ ∗ ∗ P =P (y x) a1b1 +P (x + y) a2b3 + (g2y) ag3b2 − − (B.33) ∗ ∗ ∗ + (¯y x¯) a1b1 + (¯x +y ¯) a2b3 + ( 2¯y) a3b2 − − ∗ ∗ ∗ = (y +y ¯ x x¯) a1b1 + (x +x ¯ + y +y ¯) a2b3 + ( 2y 2¯y) a3b2. − − − −

To constrain v4 and v7, (anti)symmetrization conditions are imposed:

∗ ∗ ∗ S11 = σ11 A11 = 0 (y x) = (y +y ¯ x x¯) (¯y x¯) = 0 ∧ − − − ∧ − ∗  σ23 + σ32  ∗ ∗ (x +x ¯ y y¯)  S23 = S32 =  (x + y) = ( 2y) = − − . (B.34)  2 ⇒  − 2  σ23 σ32  (x +x ¯ + 3y + 3¯y)∗ A23 = A32 = − (¯x +y ¯)∗ = (2¯y)∗ = 2  −  2     Solving the previous system yieldsx ¯=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 andx ¯ = 1/ 2√2 : | | | | − p 1 v4 = 2, 0, 0, 0, 0, 1, 0, 1, 0 , (B.37) √6 − − 1 v7 = 0, 0, 0, 0, 0, 1, 0, 1, 0 . (B.38) √2 − 3⊗3 Regarding the remaining vi vectors, one sees that v5 and v8 are eigenvectors of U (t) with eigenvalue 3⊗3 2 ω and that v6 and v9 are eigenvectors of U (t) with eigenvalue ω . Thus, v5 and v8 will be linear e e e combinations of e2, e4, and e9, orthogonal to v2, whereas v6 and v9 will be linear combinations of e3, e5, e e e 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:

3⊗3 1 2 2 3⊗3 e−1 1 2 2 U (s) v4 = v4 + v5 + v6 S U (s) S S v4 = S v4 + S v5 + S v6 −3 3 3 ⇒ −3 3 3 (B.40) 3⊗3 1 2 2 e Ue (s) v4 = v4 + v5 + v6 . e e e e ⇒ −3e 3 e 3 e e

Then, by (B.29):

3⊗3 3⊗3 3 x U (s) q1 + 3 y U (s) q2 = v4 + 2v5 + 2v6 − (B.41) 00 00 3 x q + 3 y q = x q1 y q2 + 2 γ q3 + 2 δ q4 + 2 ε q5 + 2 η q6. ⇒ 1 2 − −

3⊗3 00 00 From the explicit form of U (s) in Eq. (B.21) one reads oﬀ q1 and q2 , which can be decomposed 2 as a linear combination of qi vectors as follows :

00 1 00 1 q = q1 + q3 + q4 + q5 + q6 , q = q2 + 3q3 + q4 3q5 + q6 . (B.42) 1 −3 2 −3 −

Inserting this result in (B.41), and using the fact that the qi are linearly independent, gives:

1 3 x 3y = 2γ γ = 2 x 2 y − − −1 −1  x y = 2δ  δ = 2 x 2 y − 1 − 3 . (B.43)  x + 3y = 2ε ⇒  ε = x + y  −  − 2 2 x y = 2η η = 1 x 1 y − − − 2 − 2     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 andx ¯, it follows that:

γ = 0 γ¯ = 1 x¯ 3 y¯ = 2¯x = √1 − 2 − 2 − − 2 2 √1 ¯ 1 1  δ = 3 x =  δ = x¯ y¯ = 0 − 6 2 − 2 . (B.44)  3 √1  1 3 √1  ε = x = 2 ∧  ε¯ = 2 x¯ + 2 y¯ =x ¯ =  − 6  − 2 2 η = 1 x = 1 √1 η¯ = 1 x¯ 1 y¯ = x¯ = √1 − 3 2 6 − 2 − 2 − − 2 2     2   Notice 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:

1 v5 = 0, 1, 0, 1, 0, 0, 0, 0, 2 , (B.45) √6 − − 1 v6 = 0, 0, 1, 0, 2, 0, 1, 0, 0 , (B.46) √6 − − 1 v8 = 0, 1, 0, 1, 0, 0, 0, 0, 0 , (B.47) √2 − 1 v9 = 0, 0, 1, 0, 0, 0, 1, 0, 0 . (B.48) √2 − 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:

√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    −1 † 1  − −  S = S =  0 1 0 1 0 0 0 0 2  . (B.49) √6  − −   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     −  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:

1 √ (a1b1 + a2b3 + a3b2) a1b1 3 1  √ (a1b2 + a2b1 + a3b3)  a1b2 3 g √1 (a b + a b + a b ) a1b3  3 1 3 2 2 3 1   g   √1  a2b1  6 (2a1b1 a2b3 a3b2)  ^ †    1 − −  ~ ~  g   √ ( a b a b + 2a b ) ~a b = S ~a b a2b2 =  6 1 2 2 1 3 3  . (B.50) ⊗ ⊗ ⇒  g   1 − −  a2b3  √ ( a1b3 + 2a2b2 a3b1)    6  g − 1 − a3b1  √     2 (a2b3 a3b2)   g   1 −  a3b2  √ (a1b2 a2b1)   g   2 −  a3b3  1     √ ( a1b3 + a3b1)   g   2 −    g This, for example, means that when one applies any group transformation, a1b1 remains invariant, which means √1 (a b + a b + a b ) transforms trivially (common factors like 1/√3 can be ig- 3 1 1 2 3 3 2 g nored). Thus, considering (B.14), one ﬁnally extracts the Clebsch-Gordan decomposition for the 3 3 = ⊗ 0 00 1 1 1 3s 3a tensor product representation from (B.50): ⊕ ⊕ ⊕ ⊕

a1b1 + a2b3 + a3b2 1, 2a1b1 a2b3 a3b2 a2b3 a3b2 ∼ 0 − − − a1b2 + a2b1 + a3b3 1 ,  a1b2 a2b1 + 2a3b3 3s,  a1b2 a2b1  3a. (B.51) ∼ 00 − − ∼ − ∼ a1b3 + a2b2 + a3b1 1 , a1b3 + 2a2b2 a3b1 a1b3 + a3b1 ∼ − −  −     

99 100