Neutrino Masses and the Hierarchy Problem

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Master Thesis

Neutrino Masses and the Hierarchy

Problem

Filip Allard

Theoretical Elementary Particle Physics, Department of Physics, School of Engineering Sciences

KTH Royal Institute of Technology, SE-106 91 Stockholm, Sweden Stockholm, Sweden 2017

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Akademisk avhandling f¨or avl¨aggande av masterexamen i teknisk fysik.

Scientific thesis for the degree of Master of Engineering in the subject area of Theoretical physics.

TRITA-FYS-2017:42 ISSN 0280-316X

ISRN KTH/FYS/--17:42--SE c

Filip Allard, June 2017

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Abstract

With the detection of the Higgs boson at CERN in 2012 the final piece of the Standard Model (SM) of particle physics is in place. The mass was measured at 125 GeV which is interesting in the light of the discovery of neutrino oscillations, which imply massive neutrinos and is an important clue for physics beyond the SM. In this thesis extensions of the SM, able to explain the smallness of the neu-trino masses, are studied through the concept of counting the number of physical parameters in the lepton sector which quantify the complexity of the models. Fur-thermore, general formulas for the one loop Higgs bare mass correction are derived in the Feynman–’t Hooft gauge for a hard cutoff and using the MS renormalization scheme. The possible usage of the formulas is exemplified by the derivation of an upper bound for the type I seesaw heavy neutrino mass with one generation which shows the connection between the two discoveries, a concept often referred to as the hierarchy problem of the SM.

Key words: Higgs physics, beyond the Standard Model, neutrino physics, hierar-chy problem, renormalization.

Sammanfattning

Med upptäckten av higgsbosonen vid CERN 2012 är den sista delen av partikel-fysikens standardmodell (SM) p˚a plats. Massan mättes till 125 GeV vilket är in-tressant i ljuset av upptäckten av neutrinooscillationer, som visar att neutriner har massa och utgör en viktig ledtr˚ad till fysik bortom SM. I det här arbetet studeras modeller bortom SM som kan förklara att neutrinomassorna är s˚a sm˚a genom kon-ceptet att räkna antalet fysikaliska parametrar i leptonsektorn, vilket kvantifierar modellernas komplexitet. Vidare härleds allmänna uttryck för korrektionen av den icke-renormerade massan i Feynman–’t Hoofts guage med en övre energigräns samt i MS-renormering. Det potentiella användningsorm˚adet för formlerna exemplifieras genom en härledning av en övre gräns för den tunga neutrinomassan i typ I-seesaw med en generation vilket visar kopplingen mellan de tv˚a upptäckterna, ett koncept som ofta benämns som hierarkiproblemet i SM.

Nyckelord: Higgsfysik, fysik bortom standardmodellen, neutrinofysik, hieraripro-blemet, renormering.

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Preface

This thesis is the result of full time work from February to August 2016 and addi-tional ten months of part time work between September 2016 and June 2017, for the degree of Master of Science in Engineering, in the Theoretical Particle Physics group at the Department of Physics, KTH, Sweden. The work concerns the hier-archy problem in extensions of the SM.

Overview of the Thesis

The structure of the thesis is as follows. In Ch. 1 a brief introduction to the history of modern physics is given together with the background of the hierarchy problem in the Standard Model (SM) of Particle Physics. Chapter 2 introduces the SM, some of its shortcomings and continues with extensions of the SM focusing on the smallness of the neutrino masses. The hierarchy problem is discussed in Ch. 3 and formulas for the Higgs bare mass correction are derived. In Ch. 4 the possible use of the formulas is exemplified through a numerical analysis of the type I seesaw model. Finally the work is summarized in Ch. 5.

Notation and Conventions

The summation convention introduced by Einstein is used throughout the thesis if not otherwise stated, thus, we sum over repeated indices. Natural units, in which c = ~ = 1, are used for clarity and convenience. The sign convention for the Minkowski metric tensor is given by

(gµν) = diag (1, −1, −1, −1) . (1)

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Acknowledgements

I would like to thank Professor Tommy Ohlsson for giving me the opportunity to do this work for my diploma thesis in the Theoretical Particle Physics group. I am the most grateful to my two supervisors, Dr. Mattias Blennow who always have had time for my questions and have enlightened me by posing fundamental questions, and Dr. Juan Herrero-Garcia for constantly encouraging me and endlessly discussing and teaching the field of particle physics to me. I would also like to thank all the thesis workers, PhD students and postdocs for interesting conversations and inspiration during my time. Finally, my deepest thanks goes to my family who always have encouraged and supported me in any possible way and to my dearly beloved finac´ee Sofia Skagertun for all your love and support at all times and for tirelessly listening to me trying to explain the world. You are my everything.

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Introduction

“I was born not knowing and have had only a little time to change that here and there.”

– Richard P. Feynman There is nothing more natural than the curiosity of a child. Questions like how and why drives the thirst for new knowledge and the understanding of the world we live in – so also in science. The questions regarding the constituents of Nature, from the length and time scales of the smallest building blocks, elementary particles, to the size and age of the Universe, have given birth to the branch of physics in science. Already at the dawn of physics it was clear that the tool of choice for the journey ahead was mathematics, or to use the words of Galileo Galilei describing Nature, “a book written in the language of mathematics”. However, the beautiful theories and their predictions must also withstand tests in terms of experiments. The importance of a theory being falsifiable cannot be exaggerated enough. In the early days the person who had created a theory was also the one performing the experiments, which then would be verified by others. As time went on the experiments needed in order to test new theories became too involved which lead to two different branches in physics, theoretical and experimental.

One of the most profound contributions in the history of physics is the Principia Mathematica [1] published in 1687 and written by Isaac Newton. With Newton’s three laws of motion the movements of an object, including both earth bound and celestial bodies, could be explained. Nature seemed like a clockwork which during the following 200 years was studied with only a few exceptions left to resolve.

A persistent idea during this period of time, tracing back to philosophical thoughts in ancient cultures, is the notion of matter consisting of indivisible build-ing blocks termed atoms. In 1897 J.J. Thomson discovered the electron, which pre-sumably was smaller than the atom, and when Marie Sklodovska (Madame Curie) and her husband Pierre Curie the following year found that small pieces of their

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newly discovered chemical element, radium, where hurled out the question marks regarding the clockwork of Nature started to pile up.

That the field of physics was in fact facing a revolution became evident with the new century when Max Planck published his work on the energy distribution law of the radiation from a black body in the year of 1901. Introducing the concept of granularity also for energy, thus, postulating a smallest amount for energy, the energy spectrum found experimentally fit perfectly with his theory. Hence, what previously had been thought of as a continuous stream of energy, e.g., the sunlight pouring through a window, could no longer be divided into ever smaller pieces.

In the beginning of the 20th century one of the most brilliant and daring physi-cists in the history of mankind made his profound contributions. In 1905 Albert Einstein took the most essential part from Planck one step further and suggested that light, with all its well established wave properties, was really particles [2] giv-ing rise to the wave- particle duality. Returngiv-ing to the continuous light stream through a window it was literally a hailstorm of quanta or photons, which are the elementary particles of light.

During that same year Einstein also introduced special relativity [3, 4] after solving the question on why James Clerk Maxwell’s equations for electricity and magnetism did not agree with experiments when applied to moving objects. This discovery fundamentally changed our view of space and time and was generalized in 1915 to include gravity, a theory we now by the name of general relativity. Ein-stein’s ability to think completely different in already well explored fields is certainly demonstrated here in which massive objects affects the geometry of the four dimen-sional spacetime, introduced in special relativity. The theory of Newton which had been unquestioned for more than two centuries had been replaced. However, it is important to emphasize that even though Einstein’s theory of relativity is closer to what is seen in Nature, when considering problems in the classical limit, far from the speed of light, Newton’s theory is in excellent agreement with experiment and can be deduced from Einstein’s theory.

Another limit in which Newton’s laws of motions seemed not to agree with experiment was at very small length scales.1 In 1925 Niels Bohr, famous for his atomic model solving several fundamental questions in the field of physics, wel-comed a young fellow physicist by the name of Werner Heisenberg to Copenhagen. During this year Heisenberg developed a mathematical model of quantum mechan-ics and two years later he had formulated Bohr’s quantum model of the atom and introduced the world to his uncertainty principle. At the same time Erwin Schr¨odinger presented a different formulation of quantum mechanics and Wolfgang Pauli published his exclusion principle. In 1930 Pauli also solved the mystery of missing energy in the process of β-decay by proposing a new particle - the neu-trino. The two branches which are the cornerstones of modern physics had thus been discovered - special relativity and quantum mechanics.

1_{As we will see high velocity and small length scale both correspond to high energy, thus, one}

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3 1930 was also the year that Paul Dirac performed his seminal work combining the two theories presenting his famous equation bearing his name. At this time the revolution was in full swing and the interest for the quantum theory of particles, Quantum Electrodynamics (QED), was immense. However, the theorists where struggling with infinite masses and coupling constants which we will consider in detail in this thesis. And it was not until years later that Richard P. Feynman, Shin-Itiro Tomonaga and Julian Schwinger had solved the fundamental questions of QED also found in other Quantum Field Theories (QFT). Feynman formulated QED both in terms of path-integrals and his well known Feynman diagrams which is a graphical representation of mathematical terms in a perturbation series. Another major contribution was made in the 1960s when Murray Gell-Mann and George Zweig independently proposed that hadrons where not elementary particles but constituted by quarks, a theory called Quantum Chromodynamics (QCD). During this decade Sheldon Glashow, Abdus Salam, and Steven Weinberg also presented the electroweak theory and three independent groups, Robert Brout and Fran¸cois Englert, Peter Higgs and Gerald Guralnik, C. R. Hagen, and Tom Kibble presented the Higgs mechanism which completed the fundamental pieces of the theory of the SM. The model was finally completed in the mid 1970s and has given us remarkable predictions. It also fits astonishingly well to experiments performed over the years, e.g., using the Tevatron at Fermilab and the Large Hadron Collider (LHC) at CERN.

Some essential pieces in Nature are though still left unanswered. The SM in-cludes electromagnetic-, strong- and weak interactions, hence, one of the four known forces is missing – gravity. The thought of a theory of everything (ToE) is tanta-lizing and considering how Maxwell unified electricity and magnetism into electro-magnetism followed by Glashow, Salam, and Weinberg who unified QED and weak interactions theorists speculate in one grand unified theory (GUT) which could have been present immediately after the Big Bang. Another aspect is that neither dark matter nor dark energy which together constitute more than 95% of the energy in the Universe is included. Furthermore, no explanation for the anti-symmetry between matter and anti-matter is given. As a final example of shortcomings of the SM we consider the neutrino oscillations found experimentally by the Super-Kamiokande Collaboration in 1998 and by the SNO Collaboration in 2001 demon-strating that neutrinos are in fact massive particles. Different extensions of the SM are able to explain the smallness of the neutrino masses, however, as a side effect it is often hard to preserve the smallness of the Higgs boson mass found at CERN in 2012. The relation between the smallness of the neutrino masses and the Higgs boson mass is the inclusion of heavy particles in the extensions of the SM, e.g., the different seesaw models. Unlike for fermions and gauge bosons, there is no symmetry preserving the scale of the Higgs boson mass. This particular aspect of the SM and extensions of it is called the hierarchy problem.

The hierarchy problem has driven much research and lead the particle physics community into the field of super symmetric (SUSY) models. The contributions to the Higgs boson mass come with different signs for bosons and fermions, thus,

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in SUSY models the smallness of the Higgs boson mass is preserved by adding SUSY partners to all the scalars and fermions in the SM. Hence, for every scalar there is a corresponding fermion and vice versa. Their contributions to the Higgs boson mass would then approximately cancel if the difference in mass between the corresponding partners is not too large. The introduced symmetry between bosons and fermions in SUSY models would be broken at some scale resulting in the SUSY partners being slightly more massive than their SM partners. However, if this difference in mass is too large, and thus the scale of the breaking is too high, including the SUSY partners does not solve the hierarchy problem since the contributions would no longer cancel. At the time of writing no SUSY particles has been found and several of the proposed SUSY models are ruled out by experiments. We will not consider any SUSY models in this thesis.

Another mystery in the field of physics is the cosmological constant which is related to the energy of the vacuum and is ridiculously small, less than 10−120[5]. The arguments outlined for the Higgs boson mass and the hierarchy problem also applies for this measurable quantity of nature. The magnitude of the mystery has lead some in science to believe that the smallness is simply a coincidence, often referred to as the anthropic principle.

Returning to the smallness of the neutrinos masses there is actually no need to introduce additional particles to explain their masses, however, this would require coupling constants between them and the Higgs boson to be unnaturally small. The concept of a model being unnatural is often measured by fine tuning. If a small change in one of the parameters of a model leads to a large difference in the measurable quantities the model is viewed as fine tuned and not as probable.

The aim of this thesis is to study different extensions in the light of the hierarchy problem and fine tuning. The models treated in this thesis are the Dirac model, type I and II seesaw, the left right symmetric seesaw model and effective field theory obtained by adding the Weinberg operator to the SM.

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Chapter 2

The Standard Model and

Beyond

The SM is a quantum field theory (QFT) describing physics at high energies, or equivalently, at small length scales. The success of the SM is breathtaking consid-ering the many predictions, including the existence of several particles such as the top quark, the W and Z bosons and the Higgs boson [6–10] and can be described using only 19 parameters. Considering the large variety of possible measurements with high precision, the SM is an over-constrained system that has proven to be ex-tremely viable. Among the precision tests we find the measurements of the electron magnetic dipole moment and the muon lifetime [11, 12].

In this chapter we give an introduction to the SM, some aspects where it falls short and some of the possible extensions of it that include possible solutions to these shortcomings. We also introduce the concepts of renormalization and effective field theory, which will be of high importance when studying the hierarchy problem in the following chapter. However, in order to introduce the the different QFTs the concept of QFT itself must first be introduced.

2.1 Quantum Field Theory

QFT combines two of the most prominent discoveries of the 20th century, special relativity and quantum mechanics. It is formulated in terms of a Langrangian density (often referred to as just Lagrangian), L [φi(x), ∂µφi(x)], which is a function

of fields and the first order derivatives of the fields in the theory. The fields, in turn, depend on the space-time position x. As in classical analytical mechanics, the dynamics of the theory are governed by the action, which is defined as

S [φ] = ˆ

d4xL. (2.1)

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Through the principle of least action a generalization of the Euler-Lagrange equa-tions ∂L ∂φi − ∂µ ∂L ∂(∂µφi) = 0, (2.2)

yield the classical equations of motion for the fields. In classical mechanics the principle of least action reveals the unique evolution of the fields chosen by nature whereas in a QFT contributions from all possible configurations of the fields de-scribed by the action are needed. This remarkable property of QFT, among others, takes some time to sink in, or to quote Richard P. Feynman on how people often react when facing aspects of QFT for the first time in QED ”What I would like to do now is show you how this model of the world, which is so utterly different from anything you’ve ever seen before (that perhaps you hope never to see it again), can explain all the simple properties of light that you know” [13]. The fields in the Lagrangian take on values at every spacetime coordinate. However, it is not the values themselves that are of interest but rather quantities such as correlation func-tions and S-matrix elements, from which observable quantities like cross-secfunc-tions and decay rates of particles can be calculated.

Throughout history the relation between physics and symmetry has become more and more intimate and in QFT the central role of symmetry is evident. The theory is effectively defined through the Lagrangian, which is constructed based on symmetries. The symmetries coupled to the spacetime coordinates restrict the fields in the Lagrangian. In a relativistic QFT this implies that the fields must transform under some representation of the Lorentz group.1 _{The representations}

considered in this thesis can be specified through the spin of the fields, scalar fields has spin 0, spinor fields has spin 1/2 and vector fields has spin 1.

Another type of symmetries of high importance is internal symmetries that restrict the interactions among the different fields in the Lagrangian, hence, the dynamical degrees of freedom (DoF). Among these symmetries we find the gauge symmetries which have played a central role in constructing the SM as well as extensions of it. The word gauge stems from electrodynamics where the electric field is defined as the divergence of the potential. This implies that adding a constant to the potential does not change the electric field or its dynamics, hence, the theory is invariant under that transformation. This is referred to as a gauge invariance or that the theory is invariant under a gauge symmetry.

The concept of conserved quantities, such as energy and momentum, as well as the concept of symmetry are often introduced in the early school years. However, the close relation between the two, that for every continuous symmetry of the action there is a corresponding conserved quantity, derived by Emmy Noether in ref. [14], is not as well known. The connection between invariance of the action under time and space translations and the conservation of energy and momentum,

1_{The Lorentz group includes rotations and boosts, which distinguishes the different types of}

fields from each other. However, the fields also transform under translations together with which the Poincar´e group is formed.

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2.2. The Standard Model 7 respectively, is truly beautiful. In the context of gauge invariance there are also corresponding conserved quantities, the simplest example being electromagnetic (EM) charge which corresponds to a U (1) gauge group.

The theories considered in this thesis require the Lagrangian to be local, meaning that the Lagrangian only depend on the fields and their derivatives at a single spacetime point. This is essential, e.g., when constructing the interactions of the gauge bosons and for the speed of light to be finite.

2.2 The Standard Model

The central role of symmetries and group theory in QFT is hopefully evident by now. This makes the group of gauge symmetries, or gauge group for short, together with the collection of fields and how they transform under it especially suitable for describing the SM. How the particles acquire mass is, as we will see, the core of the hierarchy problem and we will also introduce the concept of particle mixing in this section.

2.2.1 The SM Gauge Group

The gauge group of the SM is neatly written as

SU (3)C × SU (2)L × U (1)Y, (2.3)

where SU (3)C and SU (2)L × U (1)Y correspond to the strong and electroweak

sector, respectively. Starting with the simplest of the gauge groups, U (1)Y, it

corresponds to the conserved quantity hypercharge and the invariance of the action under a phase shift, or a rotation about the origin in the complex plane. The Lie algebra of the group is one-dimensional, which is related to the number of generators of the group. Furthermore, each generator corresponds to a gauge boson or, in other words, a force carrier. The generator of U (1)Y can simply be written as T = Y₂,2

where Y is the hypercharge of the particle.

The composition of U (1)Yand SU (2)Lforms the electroweak gauge group which

is described by the Glashow-Weinberg-Salam theory [15–17]. The subscript L stands for left-handed and refers to chirality. The fermions in the SM can be categorized into left- and right-handed particles w.r.t. chirality, corresponding to two different representations of the Lorentz group. Considering a Dirac fermion field ψ the left- and right-handed states, also known as Weyl fermions, are obtained by using the projection operators

PL,R=

1 ∓ γ5

2 , (2.4)

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where γ5 is the fifth gamma matrix,3 according to

ψ = (PL+ PR) ψ = ψL+ ψR. (2.5)

Only the left-handed fermions transform non-trivially under the three dimensional group SU (2)L, or equivalently, have a non-zero weak isospin charge, which is the

conserved quantity of SU (2)L. The generators of SU (2)L can be written Ti= iτ

i

2 ,

where i ∈ {1, 2, 3} and τi _{are the Pauli matrices.}

The group SU (3)C is the gauge group of QCD. The subscript C here stands

for the conserved quantum number color which is divided into red, green and blue. The gauge group is eight dimensional with generators given by ta= iλ₂a, where λa are the Gell-Mann matrices.

2.2.2 The SM Lagrangian and Particle Content

The terms in the Lagrangian can be ordered into terms of different types; kinetic terms, which are quadratic in a single field and include derivatives, mass terms, which are quadratic in a single field without any derivatives, and, interaction terms, which involve more than two fields and possibly a derivative. The different types of terms in the SM Lagrangian can be divided into five different parts according to LSM= Lgauge+ Lfermion+ Lscalar+ LYukawa+ Lghost. (2.6)

Structuring the Lagrangian in this way helps the understanding of the origin of the different terms. Let us go through the different terms and how the particle content of the SM is ordered into them.

Gauge Bosons

The term Lgauge contains the kinetic terms and the self-interactions among the

gauge bosons which can be written Lgauge= − 1 4G a µνGa,µν− 1 4W i µνWi,µν− 1 4BµνB µν_, _(2.7)

with the indices a ∈ {1, 2, ..., 8} and i ∈ {1, 2, 3} and Ga_µν = ∂µGaν− ∂νGaµ+ g3fabcGbµG c ν, (2.8) W_µνi = ∂µWνi− ∂νWµi+ g ijk_Wj µW k ν, (2.9) Bµν = ∂µBν− ∂νBµ, (2.10)

where g3 (g) and fabc (ijk) are the coupling constant and structure constant of

SU (3)C(SU (2)L), respectively. The lack of self-interactions for the B gauge bosons

is due to U (1) being an Abelian group, while SU (2) and SU (3) are non-Abelian.

3_{In the Dirac basis γ}

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2.2. The Standard Model 9 Fermions

In Lfermion we find kinetic terms for the fermions on the form

Lψ= i ¯ψ /Dψ, (2.11)

where ¯ψ = ψ†γ0, is the Dirac adjoint spinor and the Feynman-slash notation, /

a ≡ γµaµ for any Lorentz vector aµ, is used for the covariant derivative Dµ. For

all fermion fields in the SM Lagrangian the covariant derivative can be written Dµ = ∂µ− ig0BµY − igWµiτ

i_{− ig} 3Gaµt

a_, _(2.12)

where g0 is the coupling constant of U (1)Y. If the field ψ transforms trivially, i.e.,

is a singlet, under either one of the gauge groups the generator of that group is zero when acting on ψ. Thus, only if ψ has a non-zero quantum number w.r.t. a specific gauge groups will it interact with the gauge bosons of that group, and the interaction terms then originate from the kinetic terms of the fermion.

Up until now we have manage to avoid the question on how these generators of different dimensionality (the Pauli matrices are 2 × 2 and the Gell-Mann matrices are 3 × 3) act on the fermion fields which, except for the 4 components related to the spinor properties and the γ matrices under the Lorentz group, seem to only have one component under the gauge groups of the SM. Starting with SU (2)L the

left-handed projections of the quarks, up (u), down (d), charm (c), strange (s), top (t) and bottom (b) are placed into doublets while the right-handed counterparts are singlets according to u d L ,c s L ,t b L , uR, bR, cR, sR, tR, bR. (2.13)

In Sec. 2.2.3 we will see how the conserved quantum number EM charge, that we know of from our daily life, is formed, however, at this point we simply state this characterizing quantity of the particles. All the up type quarks, u, c and t, carries EM charge of Q = +2/3 while the down type quarks, d, s and b, have Q = −1/3, regardless of chirality. Finally, the complete set of the quarks are in the fundamental representation of SU (3)Cwhich means that each quark field in Eq. (2.13) is actually

a triplet corresponding to the three colors, red, green and blue.

The left-handed leptons are likewise placed in doublets pairing the electron (e) with the electron neutrino (νe), the muon (µ) with the muon neutrino (νµ) and the

tau (τ ) with the tau neutrino (ντ), according to

νe e L ,νµ µ L ,ντ τ L , eR, µR, τR, (2.14)

where we note that there are no right-handed neutrinos in the SM. Only the lower components of the lepton left-handed doublets and their right-handed counterparts carries EM charge Q = −1, while the neutrinos have Q = 0 and, thus, does not

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interact with photons which is the reason for them being so elusive. The leptons are singlets under SU (3)C.

We also introduce the short hand notation qL,i, uR,i, dR,i and `L,i, eR,i for the

quarks and leptons, respectively, where the index i ∈ {1, 2, 3} refers to the genera-tion.

Scalars

In the SM there is one scalar SU (2)L doublet

Φ =φ

+

φ0

, (2.15)

often referred to as the Higgs field with hypercharge Y = 1. The + and 0 notation on the two complex field components refers to the EM charge. The scalar part of the Lagrangian of the SM is given by

Lscalar= |DµΦ| 2

− V (|Φ|) , (2.16)

where the covariant derivative is defined in Eq. (2.12) and the Higgs potential is given by4

V (|Φ|) = −m2|Φ|2+ λ |Φ|4. (2.17)

The Higgs field is a singlet under SU (3)C, and hence, does not interact with the

gluons. In general, just as for the fermions, the covariant derivative gives rise to kinetic terms and interaction terms between the scalars and the gauge bosons while the potential includes interaction terms among the scalars in the theory. This is the most general potential that can be written with the scalar particle content of the SM.

Yukawa Interactions

The Yukawa part of the Lagrangian includes, in general, interaction terms between the scalar fields and the fermion fields. In the case of the SM we find that

LYukawa= −yuijqL,iΦ u˜ R,j− yijdqL,iΦ dR,j− yije`L,iΦ eR,j+ h.c., (2.18)

where yij _{are general complex Yukawa matrices including the coupling constants,}

˜

Φ = iτ2Φ∗ is the charge conjugated Higgs field, τ2 is the second Pauli matrix,

i, j ∈ {1, 2, 3} are generation indices and h.c. is short for Hermitian conjugate. In Sec. 2.2.3 we will understand the importance of these Yukawa terms.

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2.2. The Standard Model 11 Ghosts

As mentioned in Sec. 2.1 the name gauge in the local symmetries stems from classi-cal electrodynamics and refers to the fact that the EM field is not uniquely defined, hence, we are free to add terms to the Lagrangian that do not affect the dynamics of the field. Just as in classical electrodynamics this freedom is not physical and in order to calculate physical parameters using the theory we must fix the gauge with some constraint, e.g., ∂µAµ= 0 which defines the Lorenz gauge. This applies

also to QFT and the gauge bosons described in Sec. 2.2.2. The constraint is added to the Lagrangian and acts as a Lagrangian multiplier. However, in general, this introduces a new auxiliary field called Faddeev-Popov ghost for each gauge boson in order to preserve unitarity. We will encounter other types of ghost particles which in a more general formulation can be described as states with negative norm, and thus, are non-physical.

Instead of directly choosing a specific gauge, the decision may be postponed by introducing gauge fixing terms dependent on the gauge parameter, ξ, with which the family of gauges called the Rξgauges are defined. The gauge fixing terms added

to the Lagrangian are

Lghost= − 1 2ξ ∂ µ_Va µ 2 + ca _−∂µ_Dac µ c c_, _(2.19) where Va

µ is the corresponding vector boson, ca and ccare the anti-ghost and ghost

fields and

D_µac= −∂µδac− gfabcVµb, (2.20)

is the covariant derivative, with g being the corresponding coupling constant. The ghosts are complex scalar fields but anti-commute, thus, they violate the spin-statistics relation, which further strengthens their non-physicality.

As we will see in the following section, some of the gauge bosons in the SM will acquire mass through the spontaneously breaking of the gauge symmetries discussed in Sec. 2.2.1. The Fadeev-Popov quantization procedure then becomes a bit more involved.

2.2.3 Masses and Mass Bases in the SM

The notion of promoting a global symmetry of the Lagrangin to a local gauge symmetry resulting in guage boson fields corresponding to the generators of the group may seem like magic at first sight. In the Lagrangian all the possible terms obeying the symmetries should be included. This however gives rise to a problem in theories with massive particles, since mass terms often break the so successful local gauge symmetries used when creating the theories. Mass terms can be added by hand, thereby breaking the symmetry, but there is an alternative solution found in the context of particle physics by three independent groups consisting of Robert Brout and Francois Englert, Peter Higgs and Gerald Guralnik, C. R. Hagen, and Tom Kibble and is often referred to as the Higgs mechanism [18–20]. According to

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the Higgs mechanism the gauge symmetry is spontaneously broken as a consequence of the shape of the potential, which in Eq. (2.17) is shaped like a hat. Since |Φ| = 0 is not a minimum, as the Universe expands and the energy decreases, the state of the Universe falls into one of the infinite numbers of minima.

In the SM, the real part of the neutral component of the Higgs doublet that acquires a vev, v, according to

Φ = _φ+ 1 √ 2 φ 0 r+ iφ0i = _φ+ 1 √ 2 h + v + iφ 0 i , (2.21)

where, h is the field corresponding to the Higgs boson found at CERN in 2012 [21, 22]. When this occurs the gauge group of the SM is broken according to

SU (3)C × SU (2)L × U (1)Y→ SU (3)C × U (1)Q, (2.22)

where U (1)Qis the EM symmetry corresponding to EM charge. Let us now go into

detail on how the masses in the SM are generated and how the different mass bases arise.

Gauge Boson Masses and Mass Basis

In the SM there are three massive vector bosons, W_µ± and Z0

µ, where ± and 0

again refers to EM charge. The mass terms are generated from the interaction terms involving the Higgs doublet and the gauge bosons which originates from the covariant derivative term in Eq. (2.16). Inserting the vev of the Higgs doublet we find Lscalar⊃ 1 2 v2 4 W 1 µ Wµ2 Wµ3 Bµ     g2 ₀ ₀ ₀ 0 g2 0 0 0 0 g2 −gg0 0 0 −gg0 _g02         W1 µ Wµ2 W3 µ Bµ     , (2.23)

which are the mass terms of the gauge bosons. As we can see in Eq. (2.23) the sub-matrix for the states Wµ3and Bµ is not diagonal and these states are therefor

not mass eigenstates. Diagonalizing the mass matrix, we find the three massive states W±= √1 2 W 1 µ∓ iW 2 µ and Z_µ0 =_p 1 g2_{+ g}02 gW 3 µ − g0Bµ , (2.24)

with masses mW = gv₂ and mZ =

p

g2_{+ g}02 v

2, respectively. These massive gauge

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2.2. The Standard Model 13 The last gauge boson in the electroweak sector, which remains massless, is the photon, which written in terms of Wi

µ and Bµ is given by Aµ= 1 p g2_{+ g}02 gW 3 µ+ g0Bµ , (2.25)

and corresponds to the generator Q = τ3_{+ Y , which yields the conserved quantum}

number EM charge that has the coupling constant

e = gg

0

p

g2_{+ g}02. (2.26)

With Q = −1 for the electron we see that e is the electron charge. Since the Higgs doublet is a singlet under SU (3)C, the covariant derivative of the Higgs doublet

does not involve the gluons, and thus, they also remain massless. Fermion Masses and Mass Bases

Considering the terms in the Yukawa sector and focusing on the terms including the vev, we see from Eq. (2.18) that they take the form

LYukawa⊃ v √ 2 −yij

uuL,iuR,j− yij_ddL,idR,j− yeijeL,ieR,j

+ h.c., (2.27) which we identify as Dirac mass terms with mass matrices

mij_u = vy ij u √ 2, m ij d = vyij_d √ 2, and m ij e = vyij e √ 2. (2.28)

Since the Yukawa matrices are general complex matrices the same applies to the mass matrices. However, they can be diagonalized by a bi-unitary transformation

mij_f = U†f,L_ik m˜kl_fU_ljf,R, (2.29)

where f ∈ {u, d, e}, ˜mf is real and diagonal and Uf,Land Uf,Rare unitary matrices.

In order for the Lagrangian to be invariant under the transformation we must also rotate the fermion fields according to

f_L,i0 = U_ijf,LfL,j and fR,i0 = U f,R

ij fR,j, (2.30)

with which the mass terms are given by

LYukawa⊃ − ˜mijuu0L,iu0R,j− ˜mij_dd0L,id0R,j− ˜mijee0L,ie0R,j+ h.c., (2.31)

where u0, d0 and e0 are fermion fields with definite masses. Hence, this new basis introduced in Eq. (2.30) is called the mass basis.

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Scalar Masses and Mass Basis

Since the real part of the neutral component in the Higgs doublet acquires a vev the scalar particles in the world as we know it are obtained by expanding the scalar fields close to this minimum just as we have done in Eq. (2.21). By applying the minimization condition ∂V (|Φ|) ∂ |Φ| |Φ0| = 0, (2.32)

we can express the vev in terms of the parameters of the potential as

|Φ0| =

r m2

2λ ≡ v ≈ 246 GeV. (2.33)

By applying the minimization condition we also find that it is only the Higgs field h that acquires a mass. The other DoF in the Higgs doublet become so called Goldstone bosons, which are eaten by the massive vector bosons. This can be understood by massless gauge bosons having two DoF while massive analogues have three. Since the number of DoF are set by the fields included in the theory, in order for there to be massive gauge bosons after the spontaneous symmetry breaking, some other DoF must contribute. Denoting the eaten Goldstone bosons by ϕ, the three present in the SM are

ϕ±= φ± and ϕ0= φ0_i, (2.34)

which correspond to the longitudinal components of the Wµ± and Z0, respectively.

Ghost and Goldstone Boson Masses and Mass Basis

Starting with the Goldstone bosons, the first term in Eq. (2.19) will be corrected by a gauge fixing term according to

Lghost⊃ − 1 2ξ ∂ µ_Va µ − ξgTijaφ0,jϕi 2 , (2.35)

where φ0,j is the vev of the system of scalar fields, which in the SM corresponds to

hφ0

ri = v and Tadenotes the generator corresponding to the vector boson Vµa. The

cross term is tailored to cancel terms appearing from the scalar sector while the second added term is a mass term for the Goldstone bosons which depend on the gauge parameter ξ. The physical scalar states remaining after the breaking, which in the SM just correspond to h, are in fact orthogonal to the space Ta

ijφ0,j spanned

by the eaten Goldstone bosons.

As mentioned in Sec. 2.2.2 the Faddeev-Popov procedure is a bit more involved in spontaneously broken theories and the subject is well described in ref. [23, 24]. In

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2.2. The Standard Model 15 this case the second term in Eq. (2.19) will also be corrected due to the spontaneous breaking and can be written as

Lghost⊃ ca− ∂µDacµ − ξg 2 _Ta

ijφ0,j Tikb (φ0+ χ)_k cc, (2.36)

where χ refers to the expanded mass basis of the scalars after the spontaneous breaking including both physical and eaten Goldstone bosons. Thus we see that also the ghost fields will have mass terms dependent on a gauge parameter ξ. In addition the ghosts will interact with the physical scalars, which in the SM means the Higgs boson.

Some common choices of the gauge parameter come with different simplifica-tions, ξ = 0 is called the Lorentz gauge in which e.g. the Goldstone bosons and ghost fields are massless and the couplings involving these non-physical fields are zero since they are proportional to ξ. The limit ξ → ∞ is called the unitary gauge in which the masses of the non-physical fields go to infinity and they stop propa-gating. With ξ = 1 we find ourselves in the Feynman–’t Hooft gauge in which the masses of the non-physical fields are the same as their corresponding gauge bosons. This gauge is convenient for higher order computations and usually implemented in computer programs that calculate properties in different QFT models. It will therefor be the gauge chosen for the calculations performed in this thesis.

2.2.4 Mixing in the SM

When rotating the fermion fields into the mass bases the interaction terms with the gauge bosons in Lfermion will in general also be affected. Consider the quark-W

interaction term Lfermion⊃ g √ 2W + µuL,iγµdL,i+ h.c. = √g 2W + µ U u,L_(Ud,L₎† iju 0 L,iγµd0L,j+ h.c., (2.37)

where the matrix UCKM

ij = Uu,L(UdL)†

ij is the Cabibbo–Kobayashi–Maskawa

(CKM) matrix [25, 26], also known as the quark mixing matrix. Since the CKM matrix is constructed from 2 unitary matrices and unitary matrices form a group under matrix multiplication, the CKM matrix is also unitary. A general n × n unitary matrix has n2_{parameters, n(n − 1)/2 of these are mixing angles and n(n +}

1)/2 are complex phases. Of the complex phases 2n−1 can be removed by rephasing the left handed quark fields according to

u0_L,i→ eiζi_u0

L,i , d0L,j → e iωj_u0

L,j, (2.38)

and the number of mixing angles and physical complex phases we are left with are then given by n(n − 1)/2 and (n − 1)(n − 2)/2, respectively. In order not to affect the mass terms in the Lagrangian when we perform the same rephasing for the right-handed fields. In the case of 3 generations this implies 3 mixing angles and

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6 complex phases to start with, after the rephasing of the fields we are left with 3 mixing angles and 1 complex phase. Each rephasing of the 6 left-handed fields removes 1 complex phase from the CKM matrix, however, if we were to remove all 6 the conservation of baryon number would be violated, which is an accidental symmetry of the SM, hence, we are left with one complex phase.

The common parametrisation of the CKM matrix after having rephased the fields accoding to Eq. (2.38) can then be written

UCKM =   c12c13 s12c13 s13e−iδ q −s12c23− c12s23s13eiδ q c12c23− s12s23s13eiδ q s23c13 s12s23− c12c23s13eiδ q −c12s23− s12c23s13eiδ q c23c13  , (2.39) where cij = cos θ q ij and sij = sin θ q ij and θ q

ij are the three quark mixing angles and

δq _{is the complex quark phase.}

In the case of the leptons, we have only rotated the charged leptons in order to obtain states with definite masses. However, since in the SM there are no neutrino masses we are free to apply the same rotation for the neutrinos as for the left-handed charged leptons

ν_L,i0 = U_ije,LνL,j. (2.40)

Considering the corresponding interaction term as in Eq. (2.37) for the leptons we obtain Lfermion⊃ g √ 2W + µνL,iγµeL,i+ h.c. = √g 2W + µ U e,L_(Ue,L₎† ijν 0 L,iγµe0L,j+ h.c. = √g 2W + µν0L,iγµe0L,i+ h.c., (2.41)

hence, in the SM there is no lepton mixing matrix, thus the flavor basis and the mass basis in the lepton sector is one and the same. The close connection between the mixing in the lepton sector and the massless neutrinos in the SM is of highest interest when considering the extensions of the SM that we will discuss in the following section.

2.3 Extensions of the Standard Model

Having praised the SM and its success it is time to consider its shortcomings, the most obvious being the lack of gravity. Even though the subject of a theory including all four forces found in nature is alluring it will not be treated in this thesis. We instead turn to the highly current subject of neutrino masses. The 2015 Nobel Prize in physics was awarded to Takaaki Kajita and Arthur B. McDonald “for the discovery of neutrino oscillations, which shows that neutrinos have mass” [27]. As we saw in the previous section, the lack of mixing in the lepton sector is due to

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2.3. Extensions of the Standard Model 17 the neutrinos being massless in the SM. The finding that there in fact exists mixing, or neutrino oscillations, thus shows that the at least two of the three neutrinos are massive even though the masses are orders of magnitude smaller than those of the other particles in the SM.

The extensions treated in the following sections are the Dirac model, a general Majorana mass term, the type I and II seesaw models and the left-right symmetric seesaw model. In a larger perspective, these extensions can in turn be extended to so called grand unified theories (GUTs) at a very high energy scale in which the different interactions in the SM and the possible additional gauge groups unite to one. In this thesis we will not consider any GUT in particular, but refer to the supposed high energy scale at which they are thought to appear. However, before going into the details of the extensions we take a step back and consider fermion masses in general and how mass terms can be formed.

2.3.1 Dirac and Majorana Masses

In Sec. 2.2.3 we saw how the massive fermions in the SM acquired Dirac masses from the Yukawa matrices due to the spontaneous breaking of the electro-weak gauge group. In general, the left- and right-handed chiral components of the same Dirac field are used to form a Dirac mass terms according to

LDirac = −mD ψRψL+ ψLψR , (2.42)

where we have written the Hermitian conjugate explicitly. At this point this might seem like the only possibility since

ψL,(R)ψL,(R)= ψPR,(L)PL,(R)ψ = 0. (2.43)

However, through the charge conjugation operator

C : ψ → ψC = CψT, (2.44)

we may form a right-handed field from a left-handed one and vice versa according to (ψL) C = CψL T = C ψPR T = CPRψ T = PRCψ T = PRψC, (2.45) and thus (ψL) C = ψC_R and (ψR) C = ψC_L. (2.46)

In order to simplify the notation we introduce ψC

L ≡ (ψL) C and ψC R ≡ (ψR) C . We can now form a different type of mass term, named after Ettore Majorana, according to LMajorana= − mL 2 ψC LψL+ ψLψLC −mR 2 ψC RψR+ ψRψCR . (2.47)

However, note that since, in general, mL is not equal to mR, we conclude that ψL

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which explains the factors of 1/2 in Eq. (2.47). Furthermore, in order for these mass terms not to break any gauge symmetries the fields must be total singlets, or in other words not carry any charge, under the gauge symmetries. Thus, neglecting a possible phase factor, a field of this type obeys the Majorana condition

ψ = ψC, (2.48)

and is called a Majorana field.

We also make the observation that in the case of mL,R being mass matrices,

e.g., in flavor space, using that each individual term in the Lagrangian is a scalar and thus invariant under transposition we find that

ψC PmPψP= ψC PmPψP T = −ψTPC−1mPψP T = ψC P(mP) T ψP, (2.49)

where P ∈ {L, R} and we have used ψC _{= −ψ}T_C−1 _{in the second step and the}

anti-commutativity of Weyl fermions in the third step. Hence, mL,R are in general

complex symmetric mass matrices. As we will see in the following sections when introducing the different extensions of the SM, both Dirac- and Majorana mass terms may be present at the same time.

2.3.2 Dirac

The most straight forward way for the neutrinos to acquire mass is by introducing right-handed neutrino fields νR, which are singlets under the entire SM gauge group.

Apart from the kinetic terms, Yukawa terms would then be added according to LYukawa⊃ −yijν`L,iΦν˜ R,j+ h.c.. (2.50)

The masses of the neutrinos are then generated in complete analogy to those of the up-type quarks in the SM yielding a mass matrix

mij_D =vy

ij ν

√

2, (2.51)

where D denotes Dirac.

A lepton mixing matrix UPMNS, also known as the Pontecurvo-Maki-Nakagawa-Sakata (PMNS) matrix, corresponding to the quark mixing matrix UCKM would then appear. In the case of three generations, this can be parametrized in the same way as the CKM matrix in Eq. (2.39), but with cij = cos θ`ij and sij = sin θ`ij and

θ`_ij being the three lepton mixing angles and δ`the complex lepton phase.

In this scheme the neutrinos are of Dirac type, like the other fermions in the SM, and to use the same mechanism for generating their masses may seem like the most natural path. However, the fact that the neutrino masses experimentally are orders of magnitude smaller than the masses of the other fermions in the SM implies that the Yukawa couplings for the neutrinos must be extremely small in order to

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2.3. Extensions of the Standard Model 19 account for this fact. There is no known explanation for this large difference. Even though the differences in masses of the other fermions in the SM range over several orders of magnitude, from the electron mass of 0.511 MeV to the top quark mass of 174 GeV, the masses of the particles within the same generation differ less, which would not be the case if the neutrinos were included in this scheme. Considering that the upper bound from Planck for the sum of the neutrino masses is 0.23 eV [28] the difference would increase considerably. Furthermore, since the right-handed neutrinos are singlets, there is no symmetry forbidding a Majorana mass term, and, as argued earlier all possible terms allowed by the gauge group should be added. Thus, after all, it is not as natural as it first may seem.

2.3.3 Majorana

With the knowledge about Majorana masses from Sec. 2.3.1 we may notice that the introduction of right-handed neutrinos is not necessary since neutrino mass terms can be formed according to

LMajorana= −

mij_L 2 ν

C

L,iνL,j+ h.c., (2.52)

where mij_L is a Majorana mass matrix. Just as for the Dirac fermions in the SM we may transform to the mass basis of the Majorana neutrinos by using a unitary transformation

ν_L,i0 = U_ijν,LνL,j, (2.53)

however, since the charge conjugate transforms as

ψC→ C(U ψ)T = U∗ψC (2.54)

the diagonalized Majorana mass matrix is given by mL= Uν,L

T ˜

mLUν,L, (2.55)

where ˜mL is real and diagonal.

The mixing in the lepton sector with Majorana neutrinos is slightly different from the one found with Dirac mass terms. Considering the lepton-W interaction once again we find that the mixing matrix is still given by U = Uν,L Ue,L†. However, when rephasing the lepton fields as in Eq. (2.38) the Majorana mass term in Eq. (2.52) is not invariant. Hence, trying to remove a phase in U , it would appear in the Majorana mass matrix which then no longer would be real and diagonal. Thus, we cannot use the same parametrization as in UP M N S _since

the complex phases that we are unable to remove thus are physical. In the case of three generations the parametrization of the mixing matrix with Majorana masses is often written as

U = UPMNSK, (2.56)

where K = diag 1, eiα1_{, e}iα2 includes the additional complex phases that we are

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As in the case of adding Dirac mass terms by hand, the Majorana mass term in Eq. (2.52) breaks the electro-weak gauge group, SU (2)L × U (1)Y, of the SM. Thus,

solely extending the SM with Eq. (2.52) is not considered as a full theory, however, it is useful to consider since the masses of the neutrinos originates uniquely from Majorana mass terms and proves that neutrinos can be massive without adding right-handed neutrinos. Furthermore, in the following sections we will see examples on how Majorana mass terms can be added without breaking the SM gauge group by hand.

2.3.4 Type I Seesaw

The name seesaw alludes to a mechanism of accounting for the smallness of the neutrino masses by including other heavy particles in the theory. In this first seesaw model we return to the possibility of adding right-handed neutrinos, just as in the Dirac model, however now, in addition to the Yukawa term in Eq. (2.50), we also add the Majorana mass term

LMajorana= −

mij_R 2 ν

C

R,iνR,j+ h.c., (2.57)

excluded in the Dirac model. The components of mRare expected to be very large,

above the electroweak symmetry breaking scale where some GUT with larger gauge group is broken.

Remembering that the charge conjugate of a right-handed field is left-handed and vice versa according to Eq. (2.46), it is possible to write both the Dirac mass term and the Majorana mass term as one united Majorana mass term according to LM,D= − 1 2 νC L νR ₀ _m_D mT D mR νL νC R + h.c. = −1 2N C L mM,D NL+ h.c., (2.58)

where we have suppressed the generational indices. Thus, NL have 2n components

and mM,D is a 2n × 2n matrix.

When transforming to the mass basis of the neutrinos, thus diagonalizing mM,D,

we note that, since the right-handed Majorana masses in Eq. (2.57) are much larger than the Dirac masses, the united mass matrix mM,Dcan be approximately

block-diagonalized by a unitary matrix U = 1 m∗_Dm∗−1_R −m−1_R mT D 1 ≡ 1 α −α† ₁ , (2.59)

which gives at leading order ˜ mM,D= −mDm−1R mTD 0 0 mR ≈ UT_m M,DU, (2.60)

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2.3. Extensions of the Standard Model 21 where ˜mM,D is block diagonal. With the transformation of the fields according to

N_L,i0 = Uij†

NL,j (2.61)

we see that we are left with only Majorana mass terms for the physical neutrino fields, which therefor are Majorana particles and superpositions of the left-handed SM neutrino fields and the heavy right-handed neutrino fields according to

N_L0 ≈ νL,i− α ij_νC R,j αij† νL,j+ νR,iC ! . (2.62)

However, considering that α is small due to its inverse dependence on the large mR

the right-handed neutrino fields constitute only a small part of the light neutrino fields. Furthermore, by inspection of ˜mM,Dthe masses of these fields are suppressed

by the inverse dependence on mRwhich gives a natural explanation for the smallness

of the SM neutrino masses. As a final step in the transformation to the mass basis the block diagonal mass matrix in Eq. (2.60) is made diagonal by the unitary transformation U0 =U P M N S_K ₀ 0 UR . (2.63)

Another interesting aspect of the type I seesaw is that it can be used to explain the baryogenesis, or in other words the asymmetry of matter and antimatter seen in the Universe, through leptogenesis for mRlarger than O(108− 109GeV)[29, 30].

2.3.5 Type II Seesaw

In the type II seesaw model the scalar spectrum is extended by a heavy SU (2)L

triplet−→∆Lwith hypercharge 2, instead of introducing right-handed neutrinos. The

triplet is often written in the 2 × 2 traceless matrix representation as ∆L= δ+ L/ √ 2 δ++_L δ0 L −δ + L/ √ 2 , (2.64)

where {0, +, ++} refer to the EM charges of the components that relate to the vector components as δ_L++=√1 2(∆ 1 L− i∆ 2 L), δ + L = ∆ 3 L, δ 0 L= 1 √ 2(∆ 1 L+ i∆ 2 L). (2.65)

When including this triplet, there will be additional terms in the scalar sector of the Lagrangian Lscalar= |Dµ∆L| 2 + |DµΦ| 2 − V (Φ, ∆L) , (2.66)

where the covariant derivative of the triplet is given by Dµ∆L= ∂µ∆L− ig0BµY − igWµiθ

i_{, ∆} L

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and the terms V (Φ, ∆L) ⊃ m2∆Tr h ∆†_L∆L i +hµ∆ΦC†∆†LΦ + h.c. i + λ1Φ†ΦTr h ∆†_L∆L i + λ2 Trh∆†_L∆L i2 + λ3Tr ∆†_L∆L 2 + λ4Φ†∆L∆†LΦ, (2.68)

where m∆ is the mass of the triplet, are added to the SM potential. In addition,

the Yukawa part of the Lagrangian is adjusted by the inclusion of the terms LYukawa⊃ y∆ijg`L,i∆L`L,j+ h.c., (2.69)

where we have introduced g`L,i≡ iτ2`CL,i for the lepton doublets.

When the Higgs doublet acquires its vev, a term linear in δ0_L is obtained in the potential V (Φ, ∆L) ⊃ − 1 2√2µ∆v 2_δ0 L,r+ 1 2m 2 ∆ δ 0 L,r 2 (2.70) where δ0

L,r is the real part of δ 0

L. Hence, the minimum of the potential is not at

∆L= 0 and a vev for δL,r0 is obtained

h∆Li = 0 0 vt/ √ 2 0 (2.71)

where vt≡ µ∆v2/ 2m2∆, using the terms of the effective potential in Eq. (2.70).

Inserting this vev into the Yukawa term in Eq. (2.69) we find

LYukawa⊃ y_∆ijvt √ 2 ν C L,iνL,j+ h.c.. (2.72)

Hence, a Majorana mass matrix which for a large triplet mass m2

∆will be suppressed

and result in small neutrino masses.

Another important aspect of the type II seesaw is the additional physical scalars introduced by ∆L. Before the breaking, there are 10 degrees of freedom, 6 from the

triplet and 4 from the Higgs doublet. After the breaking the gauge bosons acquire mass by eating massless Goldstone bosons, just as in the SM. However ϕ± and ϕ0

will in general not correspond directly to the components in the Higgs doublet, but a combination including components from the triplet

ϕ0= cos βφ0i + sin βδ0L,i and ϕ±= cos β0φ±+ sin β0δL±, (2.73)

where the angles β and β0 depend on the vevs v and vt. The other 7 scalar DoF

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2.3. Extensions of the Standard Model 23 correspond to combinations of the high energy components with h0 _being

approx-imately the SM Higgs boson and A0 _{is distinguished by it being odd under the}

discrete composite CP symmetry of charge conjugation together with parity. Finally, we note that the accidental global symmetry B − L of the SM, where B and L refer to the baryon and lepton numbers, respectively, is violated by the Yukawa interaction in Eq. (2.69) unless the triplet is assigned a lepton number of −2. However, the violation is then transferred to the term µ∆ΦC†∆†LΦ in the

potential [31]. The only possibility in order to conserve B − L is to remove this term in the potential, which is not possible due to the mass of A0 _{which is given}

by [32] m2_A 0 = µ∆ v2+ 4v2t √ 2vt , (2.74)

thus, if µ∆ is set to 0 then A0 would be massless and travel at the speed of light.

The predictions of the theory would then be in conflict with experimental data since, e.g., interaction terms with the gauge bosons like ∂µh0 A0Zµ would lead

to missing energy from Z decays. In addition, considering the neutrino masses in Eq. (2.72) and the relation between µ∆ and vt we see that the neutrinos would be

massless with µ∆= 0. Thus, we conclude that B − L is not preserved in the type

II seesaw.

2.3.6 Left-Right Symmetric Seesaw

In the left-right symmetric seesaw (LRSS) model, the SM is enlarged with right-handed neutrinos together with a spectrum of scalar fields. The right-handed fermions are placed in doublets just as the left-handed ones which transform under an additional gauge group SU (2)R with gauge bosons WR,µi . This sets the

left-and right-hleft-anded particles on an equal footing left-and at a high energy scale there is a symmetry between the two chiral counterparts, as the name indicates. The smallest gauge group having this left-right (LR) symmetry is [31]

SU (3)C × SU (2)L × SU (2)R × U (1)B-L, (2.75)

where B and L in U (1)B-L denotes the baryon and lepton numbers, respectively,

and the covariant derivative with the total gauge group in Eq. (2.75) can be written Dµ = ∂µ− igLWLµi − igRWRµi − igB−L

B − L

2 Bµ. (2.76)

The fermion fields are thus placed in doublets according to qL= u d L : (2, 1, 1/3) , qR= u d R : (1, 2, 1/3) `L = ν e L : (2, 1, −1) , `R= ν e R : (1, 2, −1) , (2.77)

where the notation used for the representations under the gauge groups are SU (2)L,

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since the fields belong to the same representations as in the SM, and thus only the quarks transform non-trivially.

The Higgs sector of this high energy theory must also be LR symmetric. Since the Higgs field of the SM transforms under SU (2)L, the scalar sector must also be

adjusted to this additional symmetry in order for the complete theory to be LR symmetric. This is done by introducing a bidoublet instead of the Higgs doublet according to Φ = φ 0 1 φ + 1 φ−₂ φ0 2 : (2, 2, 0), (2.78)

together with two scalar triplets − → ∆L : (3, 1, 2) and − → ∆R : (1, 3, 2), (2.79)

which are often written in the traceless matrix representation, as in the type II seesaw. The LR symmetry of the model demands that the theory is invariant under the discrete exchange

Φ ←→ Φ†, −→∆L ←→

− →

∆R, ΨL ←→ ΨR, (2.80)

where ΨL,R = {qL,R, `L,R} are the left- and right handed fermion doublets.

The Yukawa interaction terms in the LRSS Lagrangian can be written as LYukawa= qL f Φ + g eΦqR+ `L f Φ + g eΦ`R + e`Lh∆L`L+ e`Rk∆R`R+ h.c., (2.81)

where eΦ = τ2Φ∗τ2. Applying the discrete transformation in Eq. (2.80) we find that

the matrices f and g are Hermitian and h = k, which by transposition is found to be complex symmetric.5

In the scalar part of the Lagrangian, kinetic and interaction terms for the scalar multiplets in Eq. (2.78) and Eq. (2.79) are found together with a considerably more complicated potential which, due to its size, is written in App. A.1.

The symmetry breaking of the LRSS model is done in stages. Due to the vacuum being neutral under EM charge Q, which in the LRSS is given by

Q = I3L+ I3R+

B − L

2 , (2.82)

the four neutral fields δ0

L,R, φ0 and φ00 can acquire vevs. By rephasing the lepton

doublets, two out of the four vevs may be taken to be real. As is common in the literature, see e.g. ref. [33], we choose the vevs as

h∆Ri = vR √ 2 0 0 1 0 , h∆Li = vLeiαL √ 2 0 0 1 0 and hΦi = √1 2 κ1 0 0 κ2eiαΦ , (2.83)

5_{Note that the Yukawa coupling matrix h should not be confused with the Higgs boson sharing}

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2.3. Extensions of the Standard Model 25 where the physical complex phases are written out explicitly.

The breaking of the symmetries start at a high energy scale at which the LR symmetry is broken by ∆R aquiring its vev. From the kinetic term of the

right-handed triplet, we find masses for the right-right-handed gauge bosons according to Lscalar⊃ htr h (Dµ∆R)†(Dµ∆R) i i = v 2 R 4 h gR2WR+µWR−µ+ 2 gR2 + g2B−L Zµ0 2i , (2.84) and the right-handed triplet term in the Yukawa sector gives rise to the masses of the right-handed neutrinos as

LY ukawa⊃ hf`Rk∆R`Ri + h.c. = − kvR √ 2ν C RνR+ h.c., (2.85)

i.e., a Majorana mass term. Since the breaking takes place at a high energy scale, the vev will be large, which results in heavy masses for these particles. The vev spontaneously breaks the gauge group to

SU (2)R× U (1)B−L→ U (1)Y, (2.86)

where Y is the hypercharge of the SM.

Subsequently, the remaining symmetry SU (2)L× U (1)Y is broken due to the

vevs of Φ which result in Dirac masses for the quarks and leptons and mass terms for the left-handed gauge bosons W_L+, W_L− and ZL. In addition there will be mass

terms for the right-handed gauge bosons as well, however, since this takes place at a lower energy scale these contributions will be smaller than the terms generated by the vev of the right-handed triplet. Finally, the vev of the left-handed triplet is at an even lower energy scale, contributing to the masses of the left-handed gauge bosons and neutrinos.

The same representation as used in the type I seesaw, see Eq. (2.58), with vectors NL for the neutrinos may be used which results in a mass matrix

mM,D= mL mD mT D mR , (2.87) with mL,R = √

2kvL,R. Since in the type II seesaw only the mL component was

present, we see that the LRSS incorporates a combination of the two seesaw types encountered in the previous sections. Using the same transformation as in the type I seesaw mM,D can be block diagonalized to

˜ mM,D=mL − mDm−1_R mTD 0 0 mR , (2.88)

where we see that the requirement of small masses for the left-handed neutrinos actually can be obtained through a cancellation between the two terms mL and

mDm−1R m T D.

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2.4 Effective Field Theories

When calculating physical quantities in a QFT such as scattering amplitudes, or S-matrix elements, this is done by using the LSZ6reduction formula which considers time-ordered n-point correlation functions, also known as Green’s functions. An important feature of the LSZ reduction formula is that the fields used in the time-ordered correlation functions need not be elementary particles as long as we restrict ourselves to the energy scale at which the theory is valid [24]. As an example the amplitudes for non-relativistic scattering of a proton may be calculated with great precision using a field treating the proton as pointlike since the inner structure will not be probed. Likewise, excluding heavy fields from the theory will, to a high precision, not affect the scattering amplitudes as long as we are far below the energy scale at which these fields start propagating. Taking advantage of these features, a theory excluding higher energy phenomena may be used and is generally termed an effective field theory (EFT).

In Sec. 2.6 the concept of renormalization, which removes divergences in a QFT, will be introduced. Historically only renormalizable theories where thought to be of interest. This excluded operators of dimension d ≥ 5 from the Lagrangian. However, according to the discussion above, it is perfectly fine to treat the SM as an EFT and include higher energy phenomena in such operators. These operators are then suppressed by the energy scale at which the new physics becomes significant according to

LEFT= LSM+ δLd=5+ δLd=6+ ..., (2.89)

where the additional terms are often referred to as a tower of effective operators. One way of obtaining an EFT is through the concept of integrating out the heavy DoF from a specific theory which is well explained in ref. [34]. As an example the dimensional 5 operator δLd=5= −1 2κij `C L,iΦe∗ e Φ†`L,j + h.c., (2.90) where κ = yT

νm−1R yν is found in type I seesaw and a similar operator is found also

in type II seesaw [35].

Considering that several different high-energy theories give rise to similar effec-tive operators, one can use a bottom up approach instead of the top town approach described above, in which general higher dimensional operators are included in the EFT without deriving them from a specific extension. One of the most common operators to include is the general version of Eq. (2.90), which often is referred to as the Weinberg operator [36] and is inversely proportional to some energy scale Λ, which can be compared to the right-handed neutrino mass mR. Inserting the Higgs

vev, a Majorana mass term is obtained for the left-handed neutrinos.

6_{The formula is named after the three German physicists Harry Lehmann, Kurt Symanzik and}

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2.5. Counting Parameters 27 However, even the non-renormalizable operators are not required in order to treat a theory as effective. A momentum limit, or cutoff, may be introduced for the divergences which may be seen as incorporating the higher energy phenomena rather than considering the theory to be final [37]. Thus, the cutoff at which the high-energy phenomena become important is interesting by itself. The concept of cutoffs has been introduced also in renormalization schemes and will be treated in more detail in Sec. 2.6.

2.5 Counting Parameters

As seen in the previous sections, there are several models that explain the neutrino masses experimentally found with varying complexity. According to Occam’s razor, the model with the fewest assumptions, normally the model with the least number of parameters, is considered to be the most favorable [38]. Thus, a more complex model should be chosen only if it fits the data significantly better than a simpler model. In order to distinguish the complexity, we introduce the concept of counting the number of physical parameters needed in each extension using symmetry as our tool.

The number of physical parameters needed to describe a model can be difficult to extract, especially considering models that involve spontaneous symmetry breaking. In general the high-energy theory contains parameters that can be removed by rephasing and mixing the fields of the theory, leaving only the physical parameters. Since the models considered in this thesis are constructed in order to describe the smallness of the neutrino masses, we will focus on the lepton Yukawa sector. The standard method when calculating the number of physical parameters is to, after the spontaneous symmetry breaking, diagonalize the mass matrices of all the fields and subsequently reabsorb as many phases as possible by rephasing the fields. This method was used when obtaining the common parametrization of the CKM matrix in Sec. 2.2.4. However, it can be confusing, especially when considering different models, and can easily lead to errors.

In ref. [39] a method based on symmetry in which the number of physical pa-rameters is calculated before spontaneous symmetry breaking is presented. In the symmetry method the Yukawa terms together with terms added in the extension will typically break a chiral symmetry present in the covariant derivative terms of the Lagrangian. The counting of the number of physical parameters NP in this

sector can then be performed using the formula

NP= NO− (NG− NH), (2.91)

where NO is the sum of the number of parameters contained in the terms breaking

the symmetry, NG is the number of parameters contained in the matrices of the

chiral symmetry group G and NH is the number of parameters contained in the

matrices of the subgroup H that remains unbroken by the mass matrices. In order to demonstrate the symmetry method, we will begin by applying it to the Dirac

Neutrino Masses and the Hierarchy Problem

Master Thesis