[description=number of isosinglet $q$-type vector-like quarks
($q=u,d$)]nq$n_{q}$ [description=quark mixing matrix, of dimension
$(3+n_{u})\times(3+n_{d})$]V$V$ [description=$3\times 3$ CKM matrix (upper-
left block of $V$, not necessarily unitary)]VCKM$V_{\textrm{CKM}}$
[description=exact matrix parameterizing deviations from unitarity]X$X$
[description=approximate matrix parameterizing deviations from
unitarity]Theta$\Theta$ [description=matrices controlling FCNC
($q=u,d$)]Fq$F^{q}$ [description=$q$-type quark mass matrix
($q=u,d$)]Mq$\mathcal{M}_{q}$ [description=first 3 rows of
$\mathcal{V}^{q}_{\chi}$ ($q=u,d$; $\chi=L,R$)]A$A^{q}_{\chi}$
[description=last $n_{q}$ rows of $\mathcal{V}^{q}_{\chi}$ ($q=u,d$;
$\chi=L,R$)]B$B^{q}_{\chi}$ [description=$(3+n_{q})$-dimensional unitary
diagonalization matrix ($q=u,d$; $\chi=L,R$)]mV$\mathcal{V}^{q}_{\chi}$

UNIVERSIDADE DE LISBOA

INSTITUTO SUPERIOR TÉCNICO

Non-Unitary $3\times 3$ Mixing in Majorana Neutrinos and Vector-like Quark
Models
Pedro Manuel Ferreira Pereira

Supervisor: Doctor Maria Margarida Nesbitt Rebelo da Silva
---
Co-Supervisor: Doctor Gustavo da Fonseca Castelo Branco

Thesis approved in public session to obtain the PhD Degree in

Physics

Jury final classification: Pass with Distinction

2024

UNIVERSIDADE DE LISBOA

INSTITUTO SUPERIOR TÉCNICO
Non-Unitary $3\times 3$ Mixing in Majorana Neutrinos and Vector-like Quark
Models
Pedro Manuel Ferreira Pereira
Supervisor: Doctor Maria Margarida Nesbitt Rebelo da Silva Co-Supervisor:
Doctor Gustavo da Fonseca Castelo Branco

Thesis approved in public session to obtain the PhD Degree in
Physics
Jury final classification: Pass with Distinction

Jury
Chairperson: Doctor Mário João Martins Pimenta, Instituto Superior Técnico,
Universidade de Lisboa Member of the Committee: Doctor Francisco Jose Botella
Olcina, Institut de Física Corpuscular, Universitat de València, Espanha
Doctor Mário João Martins Pimenta, Instituto Superior Técnico, Universidade de
Lisboa Doctor António Joaquim Onofre Abreu Ribeiro Gonçalves, Escola de
Ciências, Universidade do Minho Doctor Maria Margarida Nesbitt Rebelo da
Silva, Instituto Superior Técnico, Universidade de Lisboa Doctor Ricardo Jorge
Gonzalez Felipe, Instituto Superior de Engenharia de Lisboa - Instituto
Politécnico de Lisboa Doctor Ivo de Medeiros Varzielas, Instituto Superior
Técnico, Universidade de Lisboa Funding Institution

FCT: Fundação para a Ciência e a Tecnologia, Fellowship SFRH/BD/145399/2019

2024

"Às vezes, quando ergo a cabeça estonteada dos livros em que escrevo as contas
alheias e a ausência de vida própria, sinto uma náusea física, que pode ser de
me curvar, mas que transcende os números e a desilusão. A vida desgosta-me
como um remédio inútil. E é então que eu sinto com visões claras como seria
fácil o afastamento deste tédio se eu tivesse a simples força de o querer
deveras afastar.

Vivemos pela acção, isto é, pela vontade. Aos que não sabemos querer — sejamos
génios ou mendigos — irmana-nos a impotência. De que me serve citar-me génio
se resulto ajudante de guarda-livros? Quando Cesário Verde fez dizer ao médico
que era, não o Sr. Verde empregado no comércio, mas o poeta Cesário Verde,
usou de um daqueles verbalismos do orgulho inútil que suam o cheiro da
vaidade. O que ele foi sempre, coitado, foi o Sr. Verde empregado no comércio.
O poeta nasceu depois de ele morrer, porque foi depois de ele morrer que
nasceu a apreciação do poeta.

Agir, eis a inteligência verdadeira. Serei o que quiser. Mas tenho que querer
o que for. O êxito está em ter êxito, e não em ter condições de êxito.
Condições de palácio tem qualquer terra larga, mas onde estará o palácio se o
não fizerem ali?"111Livro do Desassossego por Bernardo Soares.Vol.I. Fernando
Pessoa. (Recolha e transcrição dos textos de Maria Aliete Galhoz e Teresa
Sobral Cunha. Prefácio e Organização de Jacinto do Prado Coelho.) Lisboa:
Ática, 1982. - 85.

A minha mãe,

Que sempre me fez acreditar que seria o que quisesse.

Que nunca leu Piaget,

Nem nenhum desses supostos entendidos em psicologia do desenvolvimento,

E que sempre soube mais que eles todos juntos.

### Acknowledgments

Gostaria de agradecer a todos os meus amigos e familiares que, de alguma
forma, contribuíram para o sucesso deste doutoramento e para a conclusão desta
tese. Sem o seu apoio emocional nada teria sido possível.

Deixo também um agradecimento muito especial aos meus orientadores por todos
estes anos de orientação e aconselhamento. Estendo esse agradecimento a todas
as pessoas que tive o prazer de conhecer graças à Física e a todos com que
tive o prazer de colaborar. Obrigado.

Este trabalho teve o apoio financeiro da Fundação para a Ciência e Tecnologia
(FCT) através da bolsa de Doutoramento SFRH/BD/145399/2019.

### Resumo

Matrizes de mistura $3\times 3$ não unitárias estão presentes em muitas
extensões do Modelo Padrão. Uma das extensões mais simples é a adição de
$n_{R}$ neutrinos de direita sem impor a Conservação do Número Leptónico
($n_{R}\nu$SM). Na região do espaço de parâmetros onde a conhecida aproximação
de seesaw é válida, os desvios de unitariedade da matriz de mistura leptónica
$3\times 3$ são desprezáveis, existindo três neutrinos leves e dois
($n_{R}=2$) ou três ($n_{R}=3$) neutrinos pesados com massas próximas da
Escala GUT. Apesar desta diferença de escala de massa, os acoplamentos de
Yukawa são de ordem 1, fornecendo uma explicação natural para a pequenez das
massas dos neutrinos leves - daí o nome seesaw.

Outras regiões do espaço de parâmetros, com neutrinos pesados mais leves e
maiores desvios da unitariedade da matriz de mistura leptónica $3\times 3$,
são ainda permitidas pela experiência. De facto, sempre que as massas dos
neutrinos pesados são muitas ordens de magnitude abaixo da escala GUT, há
implicações fenomenológicas muito interessantes a baixas energias. Para
analisar tais regiões do espaço de parâmetros com precisão, é preciso uma
parametrização exata, pois as aproximações válidas no cenário seesaw comum
falham. Esta tese consistirá num estudo e classificação de regiões do espaço
de parâmetros do 3$\nu$SM à luz de uma parametrização exata recém-
desenvolvida.

Além disso, também incluirá um estudo de modelos com vector-like quarks, outra
extensão simples do SM, que contém uma matriz de mistura de quarks $3\times 3$
não-unitária. O estudo das regiões mais interessantes do espaço de parâmetros
dos modelos com vector-like quarks também beneficia do uso da parametrização
exata mencionada anteriormente. Essas regiões são ricas em fenomenologia e
podem explicar certas questões em aberto como o problema da unitariedade
CKM/anomalia do ângulo de Cabibbo.

Esta tese também inclui um capítulo onde se discute as implicações
experimentais destes modelos.

Palavras-chave: Modelo Padrão, Neutrinos, Vector-like Quarks, Matriz de
Mistura, Desvios de Unitariedade.

### Abstract

Non-unitary $3\times 3$ mixing matrices are present in many extensions of the
Standard Model. One of the most simple extensions is the addition of $n_{R}$
right-handed neutrinos without imposing Lepton Number Conservation
($n_{R}\nu$SM). In the region of the parameter space where the usual seesaw
approximation is valid, deviations from $3\times 3$ unitarity of the leptonic
mixing matrix are negligible, while one has three light neutrinos and two
($n_{R}=2$) or three ($n_{R}=3$) heavy neutrinos with masses close to the GUT
scale. This gap between the mass scales occurs while having order 1 Yukawa
couplings, providing a natural explanation for the smallness of light neutrino
masses - hence the name seesaw.

Other regions of the parameter space, with lighter heavy neutrinos - with an
eV or keV mass, for instance - and bigger deviations from unitarity of the
leptonic mixing matrix, are still allowed by experiment. In fact, whenever
heavy neutrino masses are many orders of magnitude below the GUT scale there
are interesting phenomenological implications at low energies. To analyse such
regions with precision, one needs an exact parameterisation, since
approximations that were valid in the usual seesaw case now fail. This thesis
will consist of a study and classification of regions of the 3$\nu$SM
parameter space in the light of a newly-developed exact parameterisation.

Furthermore, it will also include a study of models with vector-like quarks,
another simple extension of the SM, that contains a $3\times 3$ non-unitary
quark mixing matrix. Studying the most interesting regions of the parameter
space of vector-like quarks models also benefits from the use of the
aforementioned exact parameterisation. These regions are phenomenologically
rich and may explain certain open questions like the CKM unitarity
problem/Cabibbo angle anomaly.

A chapter discussing the experimental implications of these models is also
included.

Keywords: Standard Model, Neutrinos, Vector-like Quarks, Mixing Matrix,
Deviations from Unitarity.

### List of Publications

This thesis is based on the following publications, listed in chronological
order:

  * •

N. R. Agostinho, G. C. Branco, P. M. F. Pereira, M. N. Rebelo and J. I. Silva-
Marcos, Can one have significant deviations from leptonic $3\times 3$
unitarity in the framework of type I seesaw mechanism?, Eur. Phys. J. C78
(2018) no.11, 895 doi:10.1140/epjc/s10052-018-6347-2 [arXiv:1711.06229 [hep-
ph]] 222Strictly speaking, this paper constitutes the research conducted for
my Master’s degree.

  * •

G. C. Branco, J. T. Penedo, P. M. F. Pereira, M. N. Rebelo and J. I. Silva-
Marcos, Type-I Seesaw with eV-Scale Neutrinos, JHEP 07 (2020), 164
doi:10.1007/JHEP07(2020)164 [arXiv:1912.05875 [hep-ph]].

  * •

G. C. Branco, J. T. Penedo, P. M. F. Pereira, M. N. Rebelo and J. I. Silva-
Marcos, Addressing the CKM unitarity problem with a vector-like up quark, JHEP
07 (2021), 099 doi:10.1007/JHEP07(2021)099 [arXiv:2103.13409 [hep-ph]].

The research performed by the author during his PhD studies has also led to
the publication of the Review

  * •

J. M. Alves, G. C. Branco, A. L. Cherchiglia, J. T. Penedo, P. M. F. Pereira,
C. C. Nishi, M. N. Rebelo and J. I. Silva-Marcos, Vector-like Singlet Quarks:
a Roadmap, Phys. Rept. 1057 (2024) 1-69 doi:10.1016/j.physrep.2023.12.004
[arXiv: 2304.10561 [hep-ph]]

as well as to the following contribution to conference proceedings:

  * •

P. M. F. Pereira, Non-Unitary Mixing Matrices in Neutrino and Vector-like
Quark Models, in Proceedings of the Corfu Summer Institute “School and
Workshops on Elementary Particle Physics and Gravity” (CORFU2021) Corfu,
Greece, 29 August-8 September 2021, [arXiv: 2205.05101].

###### Contents

  1. Acknowledgments
  2. Resumo
  3. Abstract
  4. List of Publications
  5. 1 Introduction and Motivation
    1. 1.1 Thesis Outline
    2. 1.2 Brief Summary of the Standard Model of Particle Physics
  6. 2 The Standard Model + $n_{R}$ right-handed neutrinos ($n_{R}\nu$SM)
    1. 2.1 The Quasi-Decoupled Parameter Space: $M\sim$ GeV/TeV
      1. 2.1.1 Status of Neutrino Physics and DU for $M>M_{W}$
      2. 2.1.2 Benchmarks and Perturbativity Remarks
    2. 2.2 The Non-Decoupled Parameter Space: $M\sim$ eV/keV
      1. 2.2.1 Deviations from Unitarity and Loop Corrections
      2. 2.2.2 Benchmarks
        1. 2.2.2.1 Case I: $\lambda_{\nu}=(1,1,0)$
        2. 2.2.2.2 Case II: $\lambda_{\nu}=(1,-1,-1)$, $M_{1}\ll M_{2}\sim M_{3}$
    3. 2.3 General Remarks
      1. 2.3.1 On CP Violating Phases
      2. 2.3.2 On the Parameter Space
  7. 3 The Standard Model + $n_{u}$ up-type vector-like quarks + $n_{d}$ down type vector-like quarks ($n_{u}$SM$n_{d}$)
    1. 3.1 The Most General Case ($n_{u}$SM$n_{d}$)
      1. 3.1.1 Weak Bases and Physical Parameters in the presence of VLQs
        1. 3.1.1.1 Useful weak bases
        2. 3.1.1.2 Parameter counting
      2. 3.1.2 Deviations from Unitarity, FCNC and Perturbativity
      3. 3.1.3 CP Violation in the Presence of VLQs
    2. 3.2 Models with $n_{d}=0$ or $n_{u}=0$
      1. 3.2.1 Model with $n_{u}=1$, $n_{d}=0$
        1. 3.2.1.1 New Physics and Global Fit
      2. 3.2.2 Model $n_{d}=1$, $n_{u}=0$
  8. 4 New Physics and Experimental Signals
    1. 4.1 Neutrinos
      1. 4.1.1 Neutrinoless double beta decay
        1. 4.1.1.1 $M\gtrsim$ MeV
        2. 4.1.1.2 $M\lesssim$ MeV $\sim$ eV, keV
      2. 4.1.2 Prompt Decay Searches at LHC: ATLAS, CMS and LHCb
        1. 4.1.2.1 $M\sim$ GeV/TeV
        2. 4.1.2.2 $M\lesssim$ GeV
      3. 4.1.3 LFV, LFNU and Electroweak Precision Observables
        1. 4.1.3.1 Muon Decay: $G_{F}$
        2. 4.1.3.2 Invisible $Z$ width: $N_{\nu}$
        3. 4.1.3.3 $W$ Decay to leptons: $W\rightarrow l_{\alpha}\nu_{\alpha}$
        4. 4.1.3.4 LFNU: Lepton Flavour Universality Tests
        5. 4.1.3.5 Unitarity of the CKM Matrix
        6. 4.1.3.6 LFV: Rare Charged Lepton Decays, $\mu\rightarrow e\gamma$
      4. 4.1.4 Signals in Neutrino Oscillation Experiments
        1. 4.1.4.1 Effects in LBL Experiments
        2. 4.1.4.2 Effects in SBL Experiments
      5. 4.1.5 Beta Decay
        1. 4.1.5.1 $M\sim$ eV
        2. 4.1.5.2 $M\sim$ keV
        3. 4.1.5.3 $M>$ keV
      6. 4.1.6 CP Violation Effects at low-energy
        1. 4.1.6.1 $M\sim$ eV, keV
      7. 4.1.7 Displaced Vertex Signatures
        1. 4.1.7.1 $M\in[1,50]$ GeV
      8. 4.1.8 Signals at Future Experiments: FCC, DUNE, HyperK
    2. 4.2 Vector-like Quarks
      1. 4.2.1 Direct Searches
        1. 4.2.1.1 Searches for Pair-Produced VLQs that Decay Predominantly to $t,b$ Quarks
        2. 4.2.1.2 Searches for Pair-Produced VLQs that Decay Predominantly to $u,d,s$ Quarks
        3. 4.2.1.3 Searches for Singly-Produced VLQs that Decay Predominantly to $u,d,s$ Quarks
        4. 4.2.1.4 Searches for Singly-Produced VLQs that Decay Predominantly to $t,b$ Quarks
      2. 4.2.2 Effects on Observables
        1. 4.2.2.1 Rare Top Decays $t\to q\leavevmode\nobreak\ Z/h$
        2. 4.2.2.2 $D^{0}$-$\overline{D}^{0}$ Mixing
        3. 4.2.2.3 $B_{d,s}^{0}$-$\overline{B}_{d,s}^{0}$ mixing
        4. 4.2.2.4 $K^{0}$-$\overline{K}^{0}$ mixing
        5. 4.2.2.5 Scalar Mediated Box Diagrams: Effects of Very Heavy VLQs
        6. 4.2.2.6 CP Violation: The $K^{0}$ system, $\epsilon_{K}$ and $\epsilon^{\prime}/\epsilon$
        7. 4.2.2.7 $B_{q}$ Decays
        8. 4.2.2.8 $K$ Decays
        9. 4.2.2.9 $Z\to bb$: $R_{b}$
        10. 4.2.2.10 Oblique Parameters
      3. 4.2.3 Future search prospects
  9. 5 Conclusions

###### List of Tables

  1. 1.1 Quantum Numbers and Representations of every SM Particle
  2. 1.2 The two particle/anti-particle states of a massless SM without Yukawa couplings. $P_{L}$ and $P_{R}$ are chirality projection operators - $P_{L}l_{L}=l_{L}$, $P_{L}l_{R}=0$, and so on.
  3. 2.1 Neutrino oscillation parameter summary from 2018. For $\Delta m^{2}_{31}$, $\sin^{2}\theta_{23}$ , $\sin^{2}\theta_{13}$, and $\delta$ the upper (lower) row corresponds to normal (inverted) neutrino mass hierarchy.
  4. 2.2 Neutrino oscillation parameter summary from 2020. For $\Delta m^{2}_{31}$, $\sin^{2}\theta_{23}$ , $\sin^{2}\theta_{13}$, and $\delta$ the upper (lower) row corresponds to normal (inverted) neutrino mass hierarchy.
  5. 2.3 Examples of deviations from unitarity, with light neutrino masses of $m_{1}=0.005050$ eV, $m_{2}=0.01005$ eV, and $m_{3}=0.05075$ eV for the case of NO, are expressed by the Hermitian matrix $\eta$. Three different hierarchies for the heavy Majorana neutrino masses $M_{i}$ are considered. For each hierarchy, two examples are given where the choice of the matrix $X$ is varied. The third column shows the Dirac-type neutrino mass matrix expressed in GeV units, the fifth column displays the Majorana-type neutrino mass matrix expressed in units of the top quark mass, and the fourth column displays the trace of $mm^{\dagger}$ in units of the squared top quark mass.
  6. 2.4 Examples of deviations from unitarity for the case of IO are provided for light neutrino masses of $m_{1}=0.05064\leavevmode\nobreak\ eV$ , $m_{2}=0.05138\leavevmode\nobreak\ eV$ , $m_{3}=0.00864\leavevmode\nobreak\ eV$. These deviations are quantified by the Hermitian matrix $\eta$. Three different hierarchies are considered for the heavy Majorana neutrino masses $M_{i}$, and for each hierarchy, two examples are given by varying the choice of the matrix $X$. The third column shows the Dirac-type neutrino mass matrix in GeV units, and the fifth column displays the Majorana-type neutrino mass matrix in units of the top quark mass. The fourth column presents the trace of the product $mm^{\dagger}$ in units of the squared top quark mass.
  7. 2.5  The counting of physical parameters in a type-I seesaw scenario with three sterile neutrinos involves 15 variables. These include 9 lepton masses (3 for charged leptons and 6 for neutrinos) and 6 mixing angles. There are also 6 physical phases, taking into account that rephasing the charged leptons can eliminate 3 phases from the matrix $V$. It’s worth mentioning that in the basis under consideration, $m_{l}$ is real and diagonal.
  8. 6a Numerical benchmark for case Ia. The ordering of light neutrinos is NO. From the input matrices $m$ and $M$, and taking into account one-loop corrections, the other quantities here listed follow. It should be noted that $O_{c}$ of Eq. (2.20) is only defined at tree level. Values for the mixing angles and CPV phases of the $3\nu$-framework in the standard parametrisation are extracted by identifying the unitary matrix $V$ with a unitary $3\times 3$ PMNS mixing matrix.
  9. 6b The same as Table 6a for case Ib.
  10. 6c The same as Table 6a for case II. For this benchmark, $M_{3}-M_{2}\simeq 7.6$ eV $\ll M_{2,3}$.
  11. 3.1  The only VLQ multiplets that couple to the SM quarks through Yukawa interactions ($Q=I_{3}+Y$).
  12. 3.2 Physical parameter count in the presence of isosinglet VLQs ($n_{f}=3$).
  13. 4.1 Constraints from neutral meson observables on products of mixing matrix elements ($\Theta=\arg V_{Ts}^{*}V_{Td}$) for two benchmark masses of the new heavy top quark.
  14. 4.2 Constraints from indirect CP Violation in the kaon system on products of mixing matrix elements ($\Theta=\arg\lambda^{T}_{ds}=\arg(V_{Ts}^{*}V_{Td})$) for two benchmark masses of the new heavy top quark.

###### List of Figures

  1. 1.1 List of SM particles. Until 2012, all of them but the Higgs ($H$) were experimentally discovered. Image is not property of this thesis’ author. Licensed under Creative Commons Attribution 3.0 Unported license, one is free to to copy, distribute, transmit and adapt the work. Numerical values taken from Particle Data Group Booklet 2016.
  2. 2.1 Maximum deviations from unitarity as a function of $M_{1}$, generated under the condition that Tr$(mm^{\dagger})\leq m_{t}^{2}$ and $|\eta_{12}|\leq 10^{-4}$.
  3. 2.2 Active-sterile mixing measure $\sin^{2}2\vartheta_{\mu e}^{(4)}$ versus the lightest neutrino mass $m_{\text{min}}$ from a scan of the case-Ia parameter space, with NO ($m_{\text{min}}=m_{1}$). The heavy spectrum at tree level has $M_{1}=1$ eV and $M_{2}=1$ keV, while three values of the heaviest mass are considered, $M_{3}=10^{13}\,(10^{14})\,\\{5\times 10^{14}\\}$ GeV, corresponding to the black (dark blue) {light blue} points in the scatter plot. The horizontal green band shows the $99.7\%$ CL interval, while the vertical red exclusion band is obtained by combining the most stringent bound on the sum of light neutrino masses from cosmology, $\sum_{i}m_{i}<0.12$ eV ($95\%$ CL), with the $3\sigma$ ranges of mass-squared differences. The dark green contour delimits the region inside which loop-stable points have been found (see text), while the benchmark of Table 6a is marked in yellow.
  4. 2.3 The average of $\sin^{2}2\vartheta_{\mu e}^{(4)}$ and $\sin^{2}2\vartheta_{\mu e}^{(5)}$ versus the lightest neutrino mass $m_{\text{min}}$, from a scan of the case-Ib parameter space, with NO ($m_{\text{min}}=m_{1}$). The heavy spectrum at tree level has $M_{1}=3.00$ eV and $M_{2}=3.16$ eV, while three values of the heaviest mass are considered, $M_{3}=10^{13}\,(10^{14})\,[5\times 10^{14}]$ GeV, corresponding to the black (dark blue) [light blue] points in the scatter plot. The vertical red band corresponds to the cosmological constraint, as in Figure 2.2. The dark green contour delimits the region inside which loop-stable points have been found, while the benchmark of Table 6b is marked in yellow.
  5. 2.4 Active-sterile mixing measure $\sin^{2}2\vartheta_{\mu e}^{(4)}$ versus the lightest neutrino mass $m_{\text{min}}$ from a scan of the case-II parameter space, with NO ($m_{\text{min}}=m_{1}$). The heavy spectrum at tree level has $M_{1}=1$ eV, while three values of the heaviest quasi-degenerate masses are considered, $M_{2}\simeq M_{3}=8\,v\,(14\,v)\,\\{20\,v\\}$, corresponding to the black (dark blue) {light blue} points in the scatter plot. Here, $v\simeq 174$ GeV is the Higgs VEV. The horizontal green band shows the $99.7\%$ CL interval, and the vertical red band corresponds to the cosmological constraint, as in Figure 2.2. The dark green contour delimits the region inside which loop-stable points have been found, while the benchmark of Table 6c is marked in yellow.
  6. 2.5 Projection of the parameter space of the $n_{R}\nu$SM in the deviations from unitarity - heavy neutrino masses space. The pink and orange lines are just placeholders to depict the log scale used and to show how fast $m_{1}$ approaches zero for lighter tones of blue.
  7. 3.1  Masses $m_{T}$ of the new heavy quark and first-row deviations from unitarity ($\sqrt{\delta}\simeq\theta_{14}$) compatible with the bounds on $|V_{ij}|$, $\gamma$, meson mixing, $|\epsilon_{K}|$ and with the requirement of perturbativity. The latter constraint imposes an upper limit on $m_{T}$, while lower bounds are set by ATLAS and CMS (95% CL).
  8. 3.2 Branching ratios for the rare top decays $t\rightarrow uZ$ as a function of first-row deviations from unitarity $\delta$ ($\sqrt{\delta}\simeq\theta_{14}$). These may considerably exceed the SM expectations Br$(t\to uZ)_{\text{SM}}\sim 10^{-16}$ (not shown). Dashed contours refer to different values for $\theta_{34}$, in the small angle approximation.
  9. 3.3 Branching ratios for the rare top decays $t\rightarrow cZ$ as a function of first-row deviations from unitarity $\delta$ ($\sqrt{\delta}\simeq\theta_{14}$). These may considerably exceed the SM expectations Br$(t\to cZ)_{\text{SM}}\sim 10^{-14}$ (not shown). Dashed contours refer to different values for the product $\theta_{24}\,\theta_{34}$, in the small angle approximation.
  10. 4.1 $m_{\beta\beta}$ as given in eq. (4.6) as a function of the lightest neutrino mass. The regions below the horizontal lines are allowed at 90% C.L. with $\textrm{Xe}^{136}$ from KamLAND-Zen, the side panel shows the corresponding limits for $\textrm{Xe}^{136}$, $\textrm{Ge}^{76}$ , and $\textrm{Te}^{130}$.
  11. 4.2 The CMS experiment has established exclusion regions at a 95% confidence level (CL) for heavy neutrinos in the $|V_{eN}|^{2}$ vs. $m_{N}$ (left) and $|V_{\mu N}|^{2}$ vs. $m_{N}$ (right) planes. In these plots, the dashed black curve represents the expected upper limit, accompanied by dark green and light yellow bands denoting one and two standard-deviation uncertainties, respectively. The solid black curve represents the observed upper limit, while the dotted black curve represents the observed limit assuming prompt decays of heavy neutrinos. Additionally, the plots include the best upper limits at a 95% CL obtained from other collider searches conducted by L3, DELPHI, ATLAS (8 TeV), and CMS (8 TeV).
  12. 4.3 The CMS experiment has established exclusion regions at a 95% confidence level (CL) for heavy neutrinos in the $|V_{eN}|^{2}$ vs. $m_{N}$ (upper) and $|V_{\mu N}|^{2}$ vs. $m_{N}$ (lower) planes. In these plots, the dashed black curve represents the expected upper limit, accompanied by dark green and light yellow bands denoting one and two standard-deviation uncertainties, respectively. The solid black curve represents the observed upper limit and the dashed cyan line shows constraints from EWPD. Also shown are the upper limits from other direct searches: DELPHI, L3, ATLAS, and the upper limits from the CMS $\sqrt{s}=8$ TeV 2012 data and the trilepton analysis based on the same 2016 data set as used in this analysis.
  13. 4.4 Observed and expected 95% CL upper limits on the heavy Majorana neutrino mixing element $|V_{\mu N}|^{2}$ function of $m_{N}$ in the Phenomenological Type-I Seesaw model. The one and two standard deviation bands of the expected limit are indicated in green and yellow, respectively.
  14. 4.5 Upper limit on $|V_{\mu 4}|^{2}$ at 95% confidence level from the LHCb experiment. The dashed line shows the limit from. The solid line shows the limit that would be extracted using the decay width formulae in this paper. For comparison, the lower dotted line shows the recently revised limit from Belle. All three limit curves are constructed with the assumption $V_{e4}=V_{\tau 4}=0$.
  15. 4.6 Observed upper limit on the mixing parameter $|V_{\mu N}|^{2}$ between a heavy neutrino and a muon neutrino in the mass range $5-50$ GeV for same-sign and opposite-sign muons in the final states with and without lifetime correction.
  16. 4.7 Distribution of muon neutrino events plotted as a function of reconstructed energy in the MINOS far detector at a distance of $735$ km. The solid grey line shows the expected spectrum under no oscillation hypothesis. The depletion of events at $\gtrsim 1.5$ GeV and the shape of the depleted spectrum is consistent with dominant $\nu_{\mu}\rightarrow\nu_{\tau}$ oscillations with maximal mixing.
  17. 4.8 The results of the MiniBooNE-only and combined fits with MicroBooNE’s CCQE sample (top) and MicroBooNE’s Inclusive sample (bottom). The likelihood is obtained by profiling over all parameters $\Delta m^{2}$ and $\sin^{2}(2\theta_{\mu e})\equiv\sin^{2}2\vartheta^{(4)}_{\mu e}$. The two best-fit points are shown as appropriately colored stars, and the contours are obtained by comparing the profile-likelihood-ratio test-statistic to the asymptotic distribution provided by Wilks’ theorem, and assuming a difference of two degrees of freedom.
  18. 4.9 The combined limits at 99% C.L.(CLs) on $\sin^{2}(2\theta_{\mu e})\equiv\sin^{2}2\vartheta{\alpha\beta}$ using the CLs method (red). The exclusion contours are compared to the LSND and MiniBooNE allowed regions. Also shown are the allowed regions from global fits by Gariazzo et al. [105] and Dentler et al. [78]. Note that $\sin^{2}(2\theta_{\mu e})\equiv\sin^{2}2\vartheta^{(4)}_{\mu e}$. Image taken from ref. [273].
  19. 4.10 MicroBooNE constraints on sterile neutrino parameter space at 3$\sigma$ C.L. (blue, Inclusive and orange, CCQE). For reference, the MiniBooNE 1-, 2-, and 3-$\sigma$ preferred regions is shown in shades of grey, the future sensitivity of the three SBN detectors (pink), and existing constraints from KARMEN (green) and OPERA (gold).
  20. 4.11 Illustration of the shape distortion induced by a non-zero $m_{\beta}$ on the tritium $\beta$-decay spectrum.
  21. 4.12 Illustration of the almost-sterile neutrino signature in a differential tritium $\beta$-decay spectrum. For the sake of clarity, a almost-sterile neutrino with a mass of $10$ keV and an unrealistic large mixing angle is considered.
  22. 4.13 eV-scale sterile neutrino exclusion contours at 95% C.L. for the first and second campaign of KATRIN. Note that $\sin^{2}(\theta_{ee})$ in the figure refers to $|V_{14}|^{2}$ in the text. The combined results of the two campaigns and the projected final sensitivity are shown. These exclusions are compared to the most recent results from reactor, gallium, beta decay and double beta decay experiments.
  23. 4.14 95 % C.L. exclusion limit obtained based on the first tritium data set of KATRIN with respect to the best fit (BF) (blue). The exclusion limit obtained by comparing the $\chi^{2}$ values to the null hypothesis (NH) (orange) is in reasonable agreement with the sensitivity (dashed black). The current laboratory limits were improved (colored shaded areas) on the active-to-sterile mixing amplitude in a mass range of $0.1$ keV $<m_{4}<1.0$ keV by up to an order of magnitude. The main result is given by the exclusion limit with respect to the best fit (blue line).
  24. 4.15 The heavy neutrino life time as a function of its mass and coupling calculated with MadWidth.
  25. 4.16 Observed 95% confidence-level exclusion in $|V_{\mu 4}|^{2}$ versus the heavy neutrino mass for the prompt signature (the region above the black line is excluded) and the displaced signature (the region enclosed by the red line is excluded). The solid lines show limits assuming lepton-number violation (LNV) for 50% of the decays and the long-dashed line shows the limit in the case of lepton-number conservation (LNC). The dotted lines show expected limits and the bands indicate the ranges of expected limits obtained within $1\sigma$ and $2\sigma$ of the median limit, reflecting uncertainties in signal and background yields.
  26. 4.17 The 95% CL limits on $|V_{4e}|^{2}$ (left) and $|V_{4\mu}|^{2}$ (right) as functions of $m_{N}$ for a Majorana heavy neutrino. The area inside the solid (dashed) black curve indicates the observed (expected) exclusion region. Older results from the DELPHI and the CMS Collaborations are shown for reference.
  27. 4.18 The observed and expected 95% CL limits on $|V_{N\alpha}|^{2}$ vs. $m_{N}$ in the Majorana-limit case, with green and yellow bands showing the one and two standard deviation spreads for the expected limits. (b,c) The observed limits in the 2QDH scenario with inverted (IH) and normal (NH) mass hierarchy, and in 1SFH scenarios. 1SFH refers to the model where the heavy neutrino only couples with one flavour of active neutrinos. 2QDH refers to the more realistic case of two quasi-degenerate neutrinos with masses in the range between $1$ and $20$ GeV that couple to all active neutrino flavours.
  28. 4.19 Allowed branching ratios for singlet VLQs $T$ (left) and $B$ (right) assuming they only couple with the third family quarks. VLQs in non-trivial $SU(2)_{L}$ multiplets are also shown. Red dots correspond to $M_{T,B}=2\leavevmode\nobreak\ \mathrm{TeV}$ and are representative of the asymptotic high-mass value. The crosses indicate $M_{T}=640\leavevmode\nobreak\ \mathrm{GeV}$ or $M_{B}=590\leavevmode\nobreak\ \mathrm{GeV}$. The blue lines refer to intermediate masses.
  29. 4.20 The $95\%$ CL expected (left) and observed (right) lower mass limits on pair-produced $T$ (upper) and $B$ (lower) quark masses, from the combined fit to all channels, as functions of their branching ratios to H and W bosons. Mass contours are shown with lines of various styles.
  30. 4.21 Expected (left) and observed (right) mass limits for $T\overline{T}$ (upper row) and $B\overline{B}$ (lower row) production. The mass limit is calculated using the NN giving the most stringent expected limit at each signal mass and branching ratio point. The white lines indicate mass exclusion contours. The black markers indicate the branching ratios for the SU(2) singlet and doublet scenarios for masses above 800 GeV, where they are approximately independent of the VLQ mass.
  31. 4.22 The (left) upper and (right) lower bounds on the range of heavy quark masses observed to be excluded at $95\%$ CL, as a function of the branching ratio of the heavy quark to $Wq$ versus $hq$, with the branching ratio to $Zq$ fixed by the requirement $\textrm{BR}(Q\rightarrow Zq)=1-\textrm{BR}(Q\rightarrow Wq)-\textrm{BR}(Q\rightarrow hq)$. The region above the diagonal is forbidden by unitarity.
  32. 4.23  The median expected (left) and observed (right) combined lower mass limits represented in a triangular form, where each point of the triangle corresponds to a given set of branching fractions for a VLQ decaying into a boson and a first-generation quark. The limit contours are determined assuming that $\tilde{k}_{W}$ and $\tilde{k}_{Z}$ are so small that the single-production modes can be neglected, and therefore that the heavy quarks can only be produced in pairs via strong interaction. The white area in the triangle with expected limits indicates mass limits below 400 GeV.
  33. 4.24  The median expected (left) and observed (right) combined lower mass limits represented in a triangular form, where each point of the triangle corresponds to a given set of branching fractions for the decay of a VLQ into a boson and a first-generation quark. The limit contours are determined assuming that $\tilde{k}_{W}=1.0$, which means that the signal is dominated by electroweak single production.
  34. 4.25  Observed and expected $95\%$ CL upper limit on the product of cross section and branching fraction for $tW\rightarrow B$ production mode in the singlet LH scenario, with the $B$ decaying to $bZ$ with "SU(2) singlet BRs". The $68\%$ and $95\%$ expected bands are shown. The branching fraction $\textrm{BR}(Q\rightarrow Zq)$ is $0.25$.
  35. 4.26  Observed $95\%$ CL exclusion limits on the cross section times branching ratio of single $T$-quark production as a function of the universal coupling constant $\kappa$, defined in eq. (4.67) for singlet $T$ VLQs, and the $T$-quark mass in the "SU(2) singlet scenario". The red hashed area around the observed limit corresponds to the theoretical uncertainty of the NLO cross-section prediction. All values of $\kappa$ above the black contour line are excluded at each mass point. The purple contour lines denote exclusion limits of equal cross section times branching ratio in units of fb.
  36. 4.27 NP contribution to $D^{0}$-$\overline{D}^{0}$ mixing via $Z$-mediated FCNC, in the presence of one up-type VLQ.
  37. 4.28  Leading contributions to $B_{d,s}^{0}$-$\overline{B}_{d,s}^{0}$ mixing in the presence of one up-type VLQ, including the effect of the new heavy quark, $u_{i,j}=u,c,t,T$.
  38. 4.29 Leading contributions to $K^{0}$-$\overline{K}^{0}$ mixing in the presence of one up-type VLQ, including the effect of the new heavy quark, $u_{i,j}=u,c,t,T$.
  39. 4.30 Scalar-box contributions to neutral meson mixing, where $\psi^{q}$ ($q=u,d$), with $\psi^{u}=(u,c,t,U_{1},\ldots,U_{n_{u}})^{T}$ and $\psi^{d}=(d,s,b,D_{1},\ldots,D_{n_{d}})^{T}$.

## Nomenclature

  * Abbreviations
  * $n_{R}$$\nu$SM

SM with the addition of $n_{R}$ right-handed neutrinos

  * $n_{u}$SM$n_{d}$

SM with the addition of $n_{u}$ up-type VLQs and $n_{d}$ down-type VLQs

  * BAU

Baryon Asymmetry of the Universe

  * BR

Branching Ratio

  * CKM

Cabibbo-Kobayashi-Maskawa

  * CPV

CP Violation

  * DU

Deviations from Unitarity

  * FCNC

Flavour-Changing Neutral Coupling

  * LBL

Long Baseline

  * LFNU

Lepton Flavour Non-Universality

  * LFV

Lepton Flavour Violation

  * LHC

Large Hadron Collider

  * NP

New Physics

  * PMNS

Pontecorvo-Maki-Nakagawa-Sakata

  * QFT

Quantum Field Theory

  * SBL

Short Baseline

  * SM

Standard Model

  * SSB

Spontaneous Symmetry Breaking

  * VEV

Vacuum Expectation Value

  * VLQ

Vector-like Quark

  * WB

Weak Basis

  * WBT

Weak Basis Transformation

  * Subscripts
  * $\alpha,\beta,\gamma,\delta$

Flavour indices.

  * $\mu,\nu,\sigma,\rho$

Lorentz indices.

  * $a,b,c$

Gauge group Generators indices.

  * $i,j,k,l$

Mass state indices.

  * $u,v,w$

Color indices.

  * Superscripts
  * $\dagger$

Conjugate Transpose.

  * *

Conjugate.

  * T

Transpose.

##  Chapter 1 Introduction and Motivation

The very precise measurements of the quark mixing matrix - the CKM matrix [1,
2] and its (apparent) consistency with unitarity [3] was one of the main
motivations for the prediction of the top quark, which was discovered in 1995
[4, 5]. Since then, the unitarity of the mixing matrices of the Standard Model
(SM) is almost taken for granted.

In the leptonic sector, with massless neutrinos and only the three known
charged leptons - $e$, $\mu$ and $\tau$, the leptonic mixing matrix was forced
to be the identity matrix, as it could be rotated away via weak-basis
transformations (WBT).

With the discovery of neutrino oscillations [6, 7], which implies that at
least two have non-zero mass [8, 9, 10, 11], there was finally room for a non-
trivial leptonic mixing matrix \- the PMNS matrix [12]. The study of this
matrix was supported by the work of B. Pontecorvo and his peers [8, 9, 10, 11,
13], which during decades questioned the hypothesis of all neutrinos being
massless. These neutrinos are Dirac particles [14, 15] if right-handed
neutrinos are added to the SM and lepton number Conservation is imposed,
yielding a PMNS that is a unitary $3\times 3$ matrix.

However, if one does not impose lepton number conservation - the approach with
the least number of assumptions - the Majorana mass term is allowed and
neutrinos are Majorana particles. In this case, the $3\times 3$ PMNS is non-
unitary in general, as there will be extra neutrino states.

Furthermore, current oscillation experiments, both short and long baseline,
measure the entries of the PMNS matrix with very low precision, while many
results one finds in the literature are obtained assuming the unitarity of
PMNS. Hence, there is no experimental indication of a unitary PMNS and a non-
unitary PMNS is still allowed [16, 17].

Thus, considering the Majorana neutrinos hypothesis - which given the number
of papers in the literature that use it, should be what the HEP community
believes is more likely to be true - a leptonic $3\times 3$ non-unitary mixing
matrix has to be incorporated in the SM, which spoils the presumed feature
that all the mixing matrices in the SM are unitary matrices.

This would be enough to motivate this thesis but recently a new anomaly was
discovered in the quark sector - known as the CKM unitarity problem or the
Cabibbo angle anomaly [18, 19, 20, 21, 22, 23].

This anomaly stems from new measurements of $|V_{us}|$ and $|V_{ud}|$ that
indicate that the unitarity of the first row of the CKM matrix may be violated
\- $|V_{ud}|^{2}+|V_{us}|^{2}+|V_{ub}|^{2}<1$ \- at two or three standard
deviations. This deficit results from new theory calculations of the SM
radiative corrections to $\beta$-decay processes [24, 25]. The fact that a
unitary CKM matrix has been so successful until this anomaly made an
appearance shows that deviations from $3\times 3$ unitarity, if present,
should be small. One must keep in mind that, until now, a large number of
experimental data both on quark mixing and CP violation is consistent with a
unitary CKM matrix.

It is then curious that an extension of the SM almost as simple, and in many
ways similar, to adding right-handed neutrinos can provide an answer to this
anomaly. The addition of particles known as vector-like quarks has the notable
feature of leading to a naturally suppressed violation of $3\times 3$
unitarity as well as to naturally suppressed flavour changing neutral
couplings (FCNC) at tree level.

Since this thesis deals with unitarity, it is relevant to stress the
differences between Unitarity in the QFT sense and what is meant by unitarity
of the mixing matrices of the SM. Unitarity in the QFT sense usually refers to
quantities being bounded from above - see ref. [26] where the absence of the
Higgs results in some cross sections growing with energy ad infinitum - due to
the unitarity of the $S$ matrix, a matrix connecting sets of asymptotically
free particle states (the in-states and the out-states) in the Hilbert space
of physical states. Unitarity is essentially the condition that the time
evolution of a quantum state according to a given evolution equation is
mathematically represented by a unitary operator (i.e. there is conservation
of probability) - in QFT, this operator is the $S$ matrix. Unitarity is
usually taken as an axiom or basic postulate of quantum mechanics and there is
no experimental evidence of its violation [26, 27, 28, 29]. The thing to keep
in mind is that, in this thesis, what is being discussed is the violation of
the unitarity of what the HEP community, at the time of writing of this
thesis, defines as the mixing matrices of the SM in the leptonic and quark
sectors, which are $3\times 3$ matrices. This does not violate the unitarity
of the $S$ matrix of the SM + massive neutrinos or SM + vector-like quarks.
The complete mixing matrices must be unitary and probability is conserved when
one takes into account the new extra states and the new decays the standard
particles may have. It just might be a possibility that what until now was
considered to be the full mixing matrix is actually a $3\times 3$ non-unitary
part of a bigger matrix that is unitary.

###  1.1 Thesis Outline

Although more significant in the quark sector [17, 22], the experimental hints
of deviations from unitarity of the mixing matrices on both fermionic sectors
of the SM show that studying models with $3\times 3$ non-unitary mixing
matrices is a relevant topic.

In the published work that this thesis is based on, an exact parameterisation
of these non-unitary mixing matrices was developed, such that one could study
more exotic regions of the parameter space of these models without recurring
to approximations, which usually fail out of these standard areas.

This work is organized as follows. In the rest of this section a brief review
of the SM will be made, in order to fix notation and introduce concepts. In
chapter 2, the Standard Model + $n_{R}$ right-handed neutrinos ($n_{R}\nu$SM)
will be covered, using the parameterisation first presented in ref. [30].
Results from refs. [30, 31] are discussed, including some unpublished work
regarding CP Violating phases and the classification of the whole parameter
space of $n_{R}\nu$SM.

In chapter 3, the Standard Model + $n_{u}$ up-type vector-like quarks +
$n_{d}$ down-type vector-like quarks ($n_{u}$SM$n_{d}$) will be covered, using
notations and definitions from refs. [32, 33]. Some emphasis will be placed on
models with only up vector-like quarks $n_{d}=0$, ($n_{u}$SM$0$), with a
discussion of results from ref. [34]. Models with only down vector-like quarks
$n_{u}=0$, ($0$SM$n_{d}$) will also be examined, using results from ref. [33].

Chapter 4 will include the most important results of these models and the
experimental signals they can leave on current and future experiments, as
discussed in refs [30, 31, 33, 34]. To conclude, a final chapter with the main
conclusions of the published works and some future prospects is included.

###  1.2 Brief Summary of the Standard Model of Particle Physics

This section consists of a brief review of the SM and its main features and is
directed to someone who is not familiar with the topic. For a more complete
treatment, with emphasis on the technical subjects, refer to refs. [27, 28,
29, 35, 36, 37, 38].

The SM is a Quantum Field Theory (QFT) that introduces fundamental
interactions (strong, weak and electromagnetic) such that the Lagrangian is
locally invariant under Gauge Symmetries - where every field is in a
representation of the Gauge Symmetry Group. Thus, the SM is a Gauge Theory. In
QFT, particles are seen as excitations (quanta) of a field that permeates the
universe. Going from a Classical Field Theory to a QFT does not involve a
modification of the Lagrangian or of the field equations, but rather a
reinterpretation of the field variables.

The SM particles currently known can be divided into Bosons and Fermions. In
the Bosons category there is a scalar (spin $0$) boson - $H$ \- the Higgs
particle, responsible for the mass generation mechanism, and 4 kinds of gauge
bosons with spin $1$: $W^{\pm}$, $Z$ and $\gamma$ (the photon) - responsible
for electro-weak interactions - and $G$ (the gluons), responsible for strong
interactions. Fermions can be divided into Quarks and Leptons and there are
three families of each. Each family of Quarks has an up (electric charge
$+2/3$) and down (electric charge $-1/3$) type quark with 3 colours each,
while for Leptons each family has a charged lepton (electric charge $-1$) and
the correspondent neutrino (electric charge $0$). Although a fermion is any
particle with a half-integer spin - $1/2,3/2,5/2,...$ \- fundamental fermions
with spins different from $1/2$ were never discovered in nature. Hence, all SM
fermions have spin $1/2$.

Figure 1.1: List of SM particles. Until 2012, all of them but the Higgs ($H$)
were experimentally discovered. Image is not property of this thesis’ author.
Licensed under Creative Commons Attribution 3.0 Unported license, one is free
to to copy, distribute, transmit and adapt the work. Numerical values taken
from Particle Data Group Booklet 2016.

Fig. 1.1, illustrates all the SM particles. It is clear that the only
characteristic that distinguishes particles with the same quantum numbers and
from different families is their mass. Each of these fundamental fermions can
be thought as a unique particle species and is usually referred to as a
flavour. Hence, flavour physics is the study of how these different particle
species mix with each other and flavour changing interactions are interactions
where one species turns into another - one down quark transforms into a charm
quark, for instance.

In gauge theories, masses cannot be introduced in the free theory because they
would break the required local gauge invariance. Hence, masses are introduced
via Spontaneous Symmetry Breaking (SSB) of the SM Group

$G_{SM}=SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}\leavevmode\nobreak\ .$ (1.1)

The massless Lagrangian for this theory is locally invariant under
transformations of this group.

One can define the construction of the SM into two parts: Before SSB, where
every field is massless and every weak interaction is diagonal - this is named
the interaction or flavour basis -, and after SSB, where some fields become
massive and mixing occurs - named mass or physical basis.

The quantum numbers of the particle spectrum of the SM in the interaction
basis 111Note that in the interaction basis $G^{+}$ and $G^{0}$ compose the
Higgs doublet and $B$, $W^{1}$, $W^{2}$ and $W^{3}$ are linear combinations of
the physical bosons $W^{\pm}$, $Z$ and $\gamma$. are given in [Table 1.1].

Table 1.1: Quantum Numbers and Representations of every SM Particle

Field | $l^{\alpha}_{L}$ | $l^{\alpha}_{R}$ | $\nu^{\alpha}_{L}$ | $u^{\alpha}_{L}$ | $u^{\beta}_{R}$ | $d^{\beta}_{L}$ | $d^{\beta}_{R}$ | $G^{+}$ | $G^{0}$ | $G$ | $B$ | $W^{1,2}$ | $W^{3}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$T_{3}$ | -1/2 | 0 | 1/2 | 1/2 | 0 | -1/2 | 0 | 1/2 | -1/2 | 0 | 0 | $\pm 1$ | 0
Y | -1/2 | -1 | -1/2 | 1/6 | 2/3 | 1/6 | -1/3 | 1/2 | 1/2 | 0 | 0 | 0 | 0
Q | -1 | -1 | 0 | 2/3 | 2/3 | -1/3 | -1/3 | 1 | 0 | 0 | 0 | $\pm 1$ | 0
$SU(3)_{c}$ Rep. | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 8 | 1 | 1 | 1
Lorentz-Poincare Rep. | (1/2,0) | (0,1/2) | (1/2,0) | (1/2,0) | (0,1/2) | (1/2,0) | (0,1/2) | (0,0) | (0,0) | (1/2,1/2) | (1/2,1/2) | (1/2,1/2) | (1/2,1/2)

Where $\alpha=1,2,3$ is the leptonic generation index and $\beta=1,2,3$ is the
quark generation index- one can define that to the top quark ($t_{R,L}$)
corresponds $u^{3}_{R,L}$ and to the muon ($\mu_{R,L}$) corresponds
$l^{2}_{R,L}$, and so on. $Y$ is defined by the relation $Q=Y+T_{3}$, which
will be explained later.

The quantum numbers of the particle spectrum of the SM are chosen such that
they are in the correct representation of the gauge group and that the
conserved quantum number (after SSB) - electric charge $Q$ \- has the correct
value for each one. $Y$ is named Weak Hypercharge and is the Quantum Number
corresponding to $U(1)_{Y}$, $T_{3}$ is named Weak Isospin and is the quantum
number corresponding to $SU(2)_{L}$. The Quantum Number corresponding to
$SU(3)_{c}$ is colour. Quarks can have three colours (red, green and blue) and
anti-quarks can have three anti-colours (anti-red, anti-green and anti-blue).
They are in the triplet representation of $SU(3)_{c}$. Gluons can have the 8
independent combinations of these 6 (3+3) colours and anti-colours. However,
every object observed in nature at low energy is colourless (a singlet of
$SU(3)_{c}$, like an electron or a proton), which means quarks and gluons
aren’t asymptotic states and don’t have a spectral representation, only
hadrons (combinations of these) do. This happens due to a special property of
QCD: Asymptotic freedom [39, 40].

The chirality of a particle also plays a major role, albeit being a
particularly abstract concept. The chirality of a particle is positive
(negative) if the particle transforms in a right-handed (left-handed)
representation of the Poincaré group. A related and easier to understand
concept is helicity. Mathematically, helicity is the sign of the projection of
the spin vector onto the momentum vector: “left” is negative, “right” is
positive. Although similar, Chirality and Helicity only match if a particle is
massless [28, 29]. Understandably, helicity is an easier concept to understand
in classical terms - and actually is a constant of motion, since it commutes
with the Hamiltonian, and thus, in the absence of external forces, is time-
invariant. Nonetheless, one should use chirality when working in particle
physics, since it is Lorentz invariant. Often times in the literature (and in
this thesis) one refers to left-handed or right-handed particles, it should be
understood as negative or positive chirality particles. Thus, if one considers
the chirality state of a particle, positive and negative chirality states
could be considered different particles, since they have different
interactions (the SM is a chiral QFT, more on that later). In a massless SM
without Yukawa couplings, this would be the case, as one wouldn’t have other
choice but to identify particles based on their interactions. One wouldn’t
have 3 families of Quarks and Leptons, but just one, with the double amount of
fields - 1 Dirac field can be decomposed into two Weyl fields (chiral massless
fields) [14]:

| $P_{L}$ | $P_{R}$
---|---|---
$l$ | $l_{L}$ | $l_{R}$
$\overline{l}$ | $\overline{l}_{R}$ | $\overline{l}_{L}$

Table 1.2: The two particle/anti-particle states of a massless SM without
Yukawa couplings. $P_{L}$ and $P_{R}$ are chirality projection operators -
$P_{L}l_{L}=l_{L}$, $P_{L}l_{R}=0$, and so on.

This happens because the SM is a chiral theory - it treats differently
particles with different chirality. For instance, considering again the
massless SM without Yukawa couplings, this time with the addition of positive
chirality neutrinos, $\nu_{R}$. In this model, the $\nu_{R}$ would be a
sterile particle, as it doesn’t interact with anything, while the $\nu_{L}$
would interact via weak interaction. In the case of the SM, which is a theory
with massive particles with only Dirac fermions, a particle is considered a
state with definite mass which is a superposition of the negative and positive
chirality states - a Dirac mass term in the SM Lagrangian can be interpreted
as a interaction term between the positive chirality state and the negative
chirality state. Thus, SSB bounds the two chirality states into what is
defined as a particle state.

To obtain a Lagrangian that is locally invariant under transformations of the
group defined in eq. (1.1), one must introduce gauge fields and correspond to
each one of them a generator of the group. The SM group has $12$ generators -
$8$ $SU(3)$ bosons ($G^{a}$), $3$ $SU(2)$ bosons ($W^{b}$) and $1$ $U(1)$
boson ($B$).

Gauge bosons are in the adjoint representation of their corresponding gauge
group (octet for $SU(3)_{c}$ and triplet for $SU(2)_{L}$) and in the vector
representation of the Lorentz-Poincare group, quarks and leptons are in the
fundamental or anti-fundamental representation of the Lorentz-Poincare Group,
while the negative chirality ones are in in the fundamental representation of
$SU(2)_{L}$ (doublet):

$\Psi^{\alpha}_{L}=\begin{pmatrix}\nu^{\alpha}_{L}\\\
l^{\alpha}_{L}\end{pmatrix}\leavevmode\nobreak\ ,\leavevmode\nobreak\
\Upsilon^{\beta}_{L}=\begin{pmatrix}u^{\beta}_{L}\\\
d^{\beta}_{L}\end{pmatrix}\leavevmode\nobreak\ .$ (1.2)

Note that $\alpha$ and $\beta$ are flavour indices. The positive chirality
ones are in the singlet representation of $SU(2)_{L}$:

$l^{\alpha}_{R}\leavevmode\nobreak\ ,\leavevmode\nobreak\
u^{\beta}_{R}\leavevmode\nobreak\ ,\leavevmode\nobreak\
d^{\beta}_{R}\leavevmode\nobreak\ .$ (1.3)

Regarding $SU(3)_{c}$, Leptons are in the singlet representation - don’t take
part in coloured interactions - while quarks are in the triplet
representation:

$Q^{\gamma}=\begin{pmatrix}q^{\gamma}_{r}\\\ q^{\gamma}_{g}\\\
q^{\gamma}_{b}\end{pmatrix}\leavevmode\nobreak\ ,$ (1.4)

where $\gamma=1,2,..,6$ is the quark flavour index - implicitly defining the
up quark ($u$) as the triplet $Q^{1}$ with entries $q^{1}_{i}$, the down quark
($d$) as the triplet $Q^{2}$ with entries $q^{2}_{i}$, the charm quark ($c$)
as the triplet $Q^{3}$ with entries $q^{3}_{i}$, and so on. Note that
$i=r,g,b$ is a colour index and that the order is irrelevant.

The gauge fields are necessarily bosons because to get gauge invariance it is
necessary to add fields that transform like the derivative - creating what is
known as the Covariant Derivative

$D_{\mu}^{q}=\partial_{\mu}+ig_{s}\sum_{a=1}^{8}G_{\mu}^{a}\frac{\lambda^{a}}{2}\leavevmode\nobreak\
,\leavevmode\nobreak\
D_{\mu}^{L}=\partial_{\mu}+ig\sum_{b=1}^{3}W_{\mu}^{b}\frac{\sigma^{b}}{2}+ig^{\prime}YB_{\mu}\leavevmode\nobreak\
,\leavevmode\nobreak\
D_{\mu}^{R}=\partial_{\mu}+ig^{\prime}YB_{\mu}\leavevmode\nobreak\ .$ (1.5)

where the $g_{s}$ are the couplings of each
interaction,$\frac{\lambda^{a}}{2}$ are the 8 generators of SU(3) (Gell-Mann
matrices) and $T^{b}=\frac{\sigma^{b}}{2}$ are the 3 generators of SU(2)
(Pauli matrices). The used sign notation is consistent with [41], taking all
$\eta_{i}=1$. Since the derivative transforms like a vector under the Lorentz-
Poincare group, gauge fields must have the same behaviour, and thus, be vector
fields, with integer non-zero spin (vector bosons). This covariant derivative
generates the interactions between the gauge bosons and the fermionic fields:

$\begin{split}\mathcal{L}_{\textrm{Fermion}}=&\sum_{\gamma}\overline{Q^{\gamma}}i\gamma^{\mu}D_{\mu}^{q}Q^{\gamma}+\sum_{\alpha}\left(\overline{\Psi^{\alpha}_{L}}i\gamma^{\mu}D_{\mu}^{L}\Psi^{\alpha}_{L}+\overline{l^{\alpha}_{R}}i\gamma^{\mu}D_{\mu}^{R}l^{\alpha}_{R}\right)\\\
&+\sum_{\beta}\left(\overline{\Upsilon^{\beta}_{L}}i\gamma^{\mu}D_{\mu}^{L}\Upsilon^{\beta}_{L}+\overline{u^{\beta}_{R}}i\gamma^{\mu}D_{\mu}^{R}u^{\beta}_{R}+\overline{d^{\beta}_{R}}i\gamma^{\mu}D_{\mu}^{R}d^{\beta}_{R}\right)\leavevmode\nobreak\
,\end{split}$ (1.6)

where the indices $\alpha$, $\beta$ and $\gamma$ have the same meaning as in
Table 1.1 and in eq. (1.4).

The gauge boson self-interactions come from their kinetic terms, which are of
the form:

$\mathcal{L}_{\textrm{kin}}=-\frac{1}{4}G^{a\mu\nu}G_{a\mu\nu}-\frac{1}{4}W^{b\mu\nu}W_{b\mu\nu}-\frac{1}{4}B^{\mu\nu}B_{\mu\nu}\leavevmode\nobreak\
,$ (1.7)

where

$G^{a}_{\mu\nu}=\partial_{\mu}G^{a}_{\nu}-\partial_{\nu}G^{a}_{\mu}-g_{s}f^{acd}G_{c\mu}G_{d\nu}\leavevmode\nobreak\
,W^{b}_{\mu\nu}=\partial_{\mu}W^{b}_{\nu}-\partial_{\nu}W^{b}_{\mu}-gf^{bcd}W_{c\mu}W_{d\nu}\leavevmode\nobreak\
,B_{\mu\nu}=\partial_{\mu}B_{\nu}-\partial_{\nu}B_{\mu}$ (1.8)

and $a=1,...,8$ is summed implicitly and runs over the number of gauge bosons
of $SU(3)$ and $b=1,2,3$ is summed implicitly and runs over the number of
gauge bosons of $SU(2)$. $f^{abc}$ are the structure constants for the gauge
group - for $SU(2$) $f^{abc}=\epsilon^{abc}$. For abelian gauge groups - like
$U(1)$ \- these are zero. These terms generate self-interactions (in the gluon
case) but also interactions between different gauge bosons. After electroweak
symmetry breaking the physical states $Z$, $\gamma$, $W^{+}$, $W^{-}$ are
revealed to be linear combinations of the gauge fields $B$, $W^{1}$, $W^{2}$,
$W^{3}$. It is easy to see from eqs. (1.7) (1.8) that this generates triple
and quartic interactions between $Z$, $\gamma$, $W^{+}$ and $W^{-}$.

When electroweak symmetry breaking happens the mass of fundamental particles
is generated, and here the Higgs is the protagonist. The existence of the
Higgs is one of the simplest ways of generating mass for fundamental particles
without explicitly breaking the gauge symmetry while keeping the unitarity of
the SM [42, 26]. This ’spontaneous’ symmetry breaking means that the vacuum of
the theory at a certain point in time - spontaneously - stops having the same
symmetry as the Lagrangian. The Higgs mechanism spontaneously breaks $G_{SM}$
into $SU(3)_{C}\times U(1)_{Q}$, in order for this to happen, a scalar doublet
of $SU(2)_{L}$ \- $\phi$ \- is added to the theory:

$\phi=\begin{pmatrix}G^{+}\\\ G^{0}\end{pmatrix}\leavevmode\nobreak\
,\leavevmode\nobreak\
V(\phi)=-\mu^{2}(\phi^{\dagger}\phi)+\lambda(\phi^{\dagger}\phi)^{2}\leavevmode\nobreak\
,\leavevmode\nobreak\
\mathcal{L}_{\textrm{Higgs}}=(D_{\mu}^{L}\phi)^{\dagger}\cdot(D^{\mu
L}\phi)-V(\phi)\leavevmode\nobreak\ ,$ (1.9)

where $V(\phi)$ is the most general renormalizable potential that can be added
to the Lagrangian. Considering $<\phi>$ the value of $\phi$ for which
$V(\phi)$ is minimal, if $\mu^{2}>0$ and $\lambda>0$, the potential has an
absolute minimum for $<\phi>\neq\begin{pmatrix}0\\\ 0\end{pmatrix}$. From the
minimization equation one gets
$<\phi^{\dagger}\phi>=\frac{\mu^{2}}{2\lambda}$. All other SM fields with spin
different from zero are compelled to have a zero vacuum expectation value, by
Lorentz invariance. This vev can be parameterised in the following way:

$<\phi>=\frac{1}{\sqrt{2}}\begin{pmatrix}0\\\ v\end{pmatrix},$ (1.10)

which gives

$<\phi^{\dagger}\phi>=\frac{\mu^{2}}{2\lambda}=\frac{v^{2}}{2}\implies
v^{2}=\frac{\mu^{2}}{\lambda}\leavevmode\nobreak\ .$ (1.11)

To get a proper spectrum it’s useful to write the $\phi$ field as a
perturbation around its vev, taking into account all its degrees of freedom:

$\phi=<\phi>+\begin{pmatrix}G^{+}\\\
\frac{H+iG^{0}}{\sqrt{2}}\end{pmatrix}\leavevmode\nobreak\ .$ (1.12)

One can parameterise three degrees of freedom in the form of a global
$SU(2)_{L}$ transformation:

$\phi=e^{i\frac{\sigma^{b}}{2}\omega_{b}}\begin{pmatrix}0\\\
\frac{v+H}{\sqrt{2}}\end{pmatrix}\leavevmode\nobreak\ ,$ (1.13)

and then use the freedom to apply a global $SU(2)_{L}$ transformation to
absorb them. This is known as going to the unitary gauge:

$\phi\xrightarrow{}e^{-i\frac{\sigma^{b}}{2}\omega_{b}}\phi=\begin{pmatrix}0\\\
\frac{v+H}{\sqrt{2}}\end{pmatrix}=<\phi>+\begin{pmatrix}0\\\
\frac{H}{\sqrt{2}}\end{pmatrix}\leavevmode\nobreak\ .$ (1.14)

In this gauge, the $H$ field parameterises the deviations from the value of
$\phi$ that minimizes the potential. It will correspond to the Higgs field.
Note that the vacuum $<\phi>$ is the kind of vacuum we need for SSB because
it’s not invariant under $SU(2)_{L}$ transformations anymore:

$e^{i\frac{\sigma^{b}}{2}\theta_{b}}\cdot<\phi>\neq<\phi>\implies\frac{\sigma^{b}}{2}\cdot<\phi>\neq\begin{pmatrix}0\\\
0\end{pmatrix}\leavevmode\nobreak\ .$ (1.15)

However, it’s invariant under the combination that represents electric charge,
given by $Q=T_{3}+Y$ [43], the Gell-Mann-Nishijima relation, where $Y$ is a
diagonal matrix in flavour space:

$Q<\phi>=\left(\frac{\sigma^{3}}{2}+Y\cdot\mathds{1}_{2\times
2}\right)\cdot<\phi>=\begin{pmatrix}1/2+Y&0\\\
0&-1/2+Y\end{pmatrix}\cdot\frac{1}{\sqrt{2}}\begin{pmatrix}0\\\
v\end{pmatrix}=\begin{pmatrix}0\\\ 0\end{pmatrix}=\bf{0}$ (1.16)

which is zero if $Y\phi=\frac{1}{2}\phi$. This proves that in the SM with only
one Higgs doublet it is impossible to break electric charge conservation.

Thus, this vacuum spontaneously breaks part of the electroweak gauge symmetry,
after which only one neutral Higgs scalar, $H$, remains in the physical
particle spectrum. However note that the number of scalars in the theory is
not constrained and there can be more than one Higgs-like particle if one
introduces more than one Higgs-like doublet. This itself is another topic of
theoretical and experimental research. The Covariant derivative acting on
$\phi$ (kinetic term on $\mathcal{L}_{Higgs}$) generates electroweak mixing
among the bosons. After a basis rotation, the gauge bosons mass matrices - a
collection of all terms that are order two in the gauge boson fields - become
diagonal, revealing the masses of the gauge bosons. This identifies the
physical states and gives a relation for their couplings

$Z_{\mu}=\cos{\theta_{\textrm{W}}}W^{3}_{\mu}-\sin{\theta_{\textrm{W}}}B_{\mu}\leavevmode\nobreak\
,\leavevmode\nobreak\
A_{\mu}=\cos{\theta_{\textrm{W}}}B_{\mu}+\sin{\theta_{\textrm{W}}}W^{3}_{\mu}\leavevmode\nobreak\
,\leavevmode\nobreak\ W^{\pm}_{\mu}=\frac{1}{\sqrt{2}}(W^{1}_{\mu}\mp
iW^{2}_{\mu})\leavevmode\nobreak\ ,$ (1.17)
$sin^{2}(\theta_{\textrm{W}})=\frac{g^{\prime 2}}{g^{2}+g^{\prime
2}}\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\
m_{W}=m_{Z}\cos{\theta_{\textrm{W}}}\leavevmode\nobreak\ ,\leavevmode\nobreak\
e=g\sin{\theta_{\textrm{W}}}=g^{\prime}\cos{\theta_{\textrm{W}}}\leavevmode\nobreak\
,$ (1.18) $m_{W}^{2}=\frac{1}{4}v^{2}g^{2}\leavevmode\nobreak\
,\leavevmode\nobreak\ m_{Z}^{2}=\frac{1}{4}v^{2}(g^{2}+g^{\prime
2})\leavevmode\nobreak\ ,\leavevmode\nobreak\
m_{H}^{2}=2\mu^{2}\leavevmode\nobreak\ .$ (1.19)

Note that $A_{\mu}$ is the traditional name for the $\gamma$ (the photon)
field. The initial vacuum was identically zero and was left invariant under
the 4 generators of $SU(2)_{L}\times U(1)_{Y}$. Now, it is left invariant
under one combination of two of these generators. From the Goldstone Theorem
one should have $3\times(4-1)$ Nambu-Goldstone bosons (massless scalar
bosons), but there are $3$ massive gauge bosons instead. In a pictoric
language it is said the gauge bosons "eat" the Nambu-Goldstone bosons. If the
broken symmetry was global (and not gauge/local) one would have 3 massless
scalars [44, 45] \- $G^{\pm}$ (2, real and imaginary part) and $G^{0}$ (1).

After SSB one can go the physical basis and write the covariant derivative in
terms of the physical bosons

$D_{\mu}=\partial_{\mu}+i\sum_{a=1}^{8}G_{\mu}^{a}\frac{\lambda^{a}}{2}+i\frac{g}{\sqrt{2}}(W_{\mu}^{+}T_{+}+W_{\mu}^{-}T_{-})+i\frac{g}{\cos{\theta_{\textrm{W}}}}(T_{3}-Q\sin^{2}{\theta_{\textrm{W}}})Z_{\mu}+ieQA_{\mu}\leavevmode\nobreak\
,$ (1.20)

where $T_{\pm}=T_{1}\pm iT_{2}=\frac{\sigma_{1}}{2}\pm
i\frac{\sigma_{2}}{2}=\sigma_{\pm}$.

It’s useful to rewrite eq. (1.5)222$D^{q}_{\mu}$ stays the same after
electroweak symmetry breaking. $SU(3)_{c}$ is not broken.:

$D_{\mu}^{L}=\partial_{\mu}+i\frac{g}{\sqrt{2}}(W_{\mu}^{+}T_{+}+W_{\mu}^{-}T_{-})+i\frac{g}{\cos{\theta_{\textrm{W}}}}(T_{3}-Q\sin^{2}{\theta_{\textrm{W}}})Z_{\mu}+ieQA_{\mu}\leavevmode\nobreak\
,$ (1.21)
$D_{\mu}^{R}=\partial_{\mu}+i(Q-T_{3})\frac{g}{\cos{\theta_{\textrm{W}}}}(\sin{\theta_{\textrm{W}}}\cos{\theta_{\textrm{W}}}A_{\mu}-\sin^{2}{\theta_{\textrm{W}}}Z_{\mu})\leavevmode\nobreak\
.$ (1.22)

To generate Dirac fermion masses one needs to create an $SU(2)_{L}\times
U(1)_{Y}$ invariant term, using $\phi$ and the fermion fields, that after SSB
generates their mass terms. Using Table 1.1 it is easy to see that a term like

$Y^{l}_{\delta\sigma}\begin{pmatrix}\overline{\nu^{\delta}_{L}}&\overline{l^{\delta}_{L}}\end{pmatrix}\phi
l^{\sigma}_{R}=Y^{l}_{\delta\sigma}\begin{pmatrix}\overline{\nu^{\delta}_{L}}&\overline{l^{\delta}_{L}}\end{pmatrix}\begin{pmatrix}G^{+}\\\
v+H+iG^{0}\end{pmatrix}l^{\sigma}_{R}\xrightarrow[\text{\leavevmode\nobreak\
}]{\text{SSB}}Y^{l}_{\delta\sigma}\frac{v}{\sqrt{2}}\overline{l^{\delta}_{L}}l^{\sigma}_{R}\leavevmode\nobreak\
,$ (1.23)

is invariant because it has $Y=1/2+1/2-1=0$, hence being a $SU(2)_{L}\times
U(1)_{Y}$ singlet. After SSB this generates a mass term proportional to $v$.
The proportionality constant is
$Y^{l}_{\delta\sigma}=\frac{\sqrt{2}m^{l}_{\delta\sigma}}{v}$, where
$m^{l}_{\delta\sigma}$ is the $(\delta,\sigma)$ entry of the Dirac mass matrix
for the charged leptons. $Y^{l}_{\delta\sigma}$ are known as the Yukawa
couplings, and they parameterise the couplings between the Higgs field and
fermions before SSB. Their origin and possible connection to other couplings
is an object of study [46, 47]. This method also works for other fermions.
However, for up quarks (the ones with $T_{3}=\frac{1}{2}$) one needs to define
the adjoint doublet $\tilde{\phi}=\begin{pmatrix}v+H-iG^{0*}\\\
-G^{-}\end{pmatrix}$ which has $Y=-\frac{1}{2}$. After SSB, one has
$<\tilde{\phi}>=\frac{1}{\sqrt{2}}\begin{pmatrix}v\\\ 0\end{pmatrix}$.

Thus, after SSB one gets a Dirac mass matrix for up quarks, down quarks and
for charged leptons:

$\mathcal{L}_{\textrm{Yukawa}}=-\overline{\Psi^{\delta}_{L}}Y^{l}_{\delta\sigma}\phi
l^{\sigma}_{R}-\overline{\Upsilon^{\delta}_{L}}Y^{d}_{\delta\sigma}\phi
d^{\sigma}_{R}-\overline{\Upsilon^{\delta}_{L}}Y^{u}_{\delta\sigma}\tilde{\phi}u^{\sigma}_{R}+h.c.\leavevmode\nobreak\
.$ (1.24)

Going from the interaction basis to the mass basis requires the
diagonalisation of these mass matrices, with the new rotated states becoming
the physical states. This generates mixing - mass states being linear
combinations of interaction states - which makes the interaction Lagrangian in
eq. (1.6) no longer diagonal in the fields.

In the SM, this only happens in the quark sector, as neutrinos are massless
and it is possible to remove the mixing in the leptonic sector.

Let us rewrite eq. (1.24) after SSB, including only the mass terms

$\mathcal{L}_{\textrm{FMass}}=-\overline{l_{L}^{0}}M^{l}l_{R}^{0}-\overline{d_{L}^{0}}M^{d}d_{R}^{0}-\overline{u_{L}^{0}}M^{u}u_{R}^{0}+h.c.\leavevmode\nobreak\
,$ (1.25)

where $M=\frac{v}{\sqrt{2}}Y$. Thus, in the SM, a general Dirac mass matrix is
a $3\times 3$ matrix. Since in the SM there are no $\nu_{R}$, there are only
mass matrices for up and down quarks as well as for charged leptons. After one
diagonalises these mass matrices one finds new states - mass states that
correspond to the physical particles.

The diagonalisation of a mass matrix cannot be performed with just one
diagonalising matrix. One needs a unitary bi-diagonalisation (or as
mathematicians call it - Singular Value Decomposition), because one needs to
diagonalise a matrix while rotating two different fields - one with positive
chirality and another with negative chirality - cf. eq. (1.25). Thus, the
matrices acting on them are, in general, different. Then, one has:

${U^{l}_{L}}^{\dagger}M^{l}U^{l}_{R}=d_{l}\leavevmode\nobreak\
,\leavevmode\nobreak\
{U^{u}_{L}}^{\dagger}M^{u}U^{u}_{R}=d_{u}\leavevmode\nobreak\
,\leavevmode\nobreak\
{U^{d}_{L}}^{\dagger}M^{d}U^{d}_{R}=d_{d}\leavevmode\nobreak\ ,$ (1.26)

where $d_{u}$, $d_{d}$ and $d_{l}$ are diagonal $3\times 3$ matrices with
positive real entries, which contain the masses of the quarks and charged
leptons, respectively, in the diagonal. From eq. (1.25) one notes that to go
to this mass basis, the fields must transform as

$l^{\delta 0}_{R,L}=({U^{l}_{R,L}})^{\delta k}l^{k}_{R,L}\leavevmode\nobreak\
,\leavevmode\nobreak\ u^{\delta 0}_{R,L}=({U^{u}_{R,L}})^{\delta
k}u^{k}_{R,L}\leavevmode\nobreak\ ,\leavevmode\nobreak\ d^{\delta
0}_{R,L}=({U^{d}_{R,L}})^{\delta k}d^{k}_{R,L}\leavevmode\nobreak\ ,$ (1.27)

where the left hand side corresponds to interaction states ($\delta$=$e$,
$\mu$, $\tau$, for the charged leptons, for instance) and the right hand side
corresponds to mass states ($k=1,2,3$).

It is now possible to write the Lagrangian in eq. (1.6) in the physical basis,
highlighting the terms that contain interactions with the physical electroweak
gauge bosons. For charged interactions with leptons,

$\mathcal{L}^{l}_{W}=-\frac{g}{\sqrt{2}}\left[W^{+}_{\mu}\overline{\nu^{j}_{L}}\delta_{ij}\gamma^{\mu}l^{i}_{L}+W^{-}_{\mu}\overline{l^{i}_{L}}\delta_{ij}\gamma^{\mu}\nu^{j}_{L}\right]\leavevmode\nobreak\
,$ (1.28)

where the matrix $U^{l}_{L}$ was rotated away by an unphysical redefinition of
the $\nu_{L}$ fields. And for leptonic interactions with the photon and $Z$
boson,

$\mathcal{L}^{l}_{A,Z}=-\frac{g}{2\cos{\theta_{\textrm{W}}}}\left[Z_{\mu}(\overline{\nu^{i}_{L}}\delta_{ij}\gamma^{\mu}\nu^{j}_{L}-\overline{l^{i}_{L}}\delta_{ij}\gamma^{\mu}l^{j}_{L})\right]-\left[(\frac{g\sin^{2}{\theta_{\textrm{W}}}}{\cos{\theta_{\textrm{W}}}}Z_{\mu}-eA_{\mu})(\overline{l^{i}_{L}}\delta_{ij}\gamma^{\mu}l^{j}_{L}+\overline{l^{i}_{R}}\delta_{ij}\gamma^{\mu}l^{j}_{R})\right]\leavevmode\nobreak\
,$ (1.29)

where $e$ is the modulus of the electron charge. For charged interactions of
quarks, one has

$\mathcal{L}^{q}_{W}=-\frac{g}{\sqrt{2}}\left[W^{+}_{\mu}\overline{u^{j}_{L}}V_{\text{CKM}}\gamma^{\mu}d^{i}_{L}+W^{-}_{\mu}\overline{d^{i}_{L}}V_{\text{CKM}}^{\dagger}\gamma^{\mu}u^{j}_{L}\right]\leavevmode\nobreak\
,$ (1.30)

where the matrix $V_{\text{CKM}}=U^{u\dagger}_{L}U^{d}_{L}$ is the unitary
$3\times 3$ quark mixing matrix. The existence of this matrix implies that the
otherwise diagonal charged interactions become non-diagonal, giving origin to
what is known as mixing. In the quark sector, the mixing matrix is known as
$V_{\textrm{CKM}}$. As for the neutral interactions among quarks and the $Z$
and $\gamma$

$\begin{split}\mathcal{L}^{q}_{A,Z}=&-\frac{g}{2\cos{\theta_{\textrm{W}}}}\left[Z_{\mu}(\overline{u^{i}_{L}}\delta_{ij}\gamma^{\mu}u^{j}_{L}-\overline{d^{i}_{L}}\delta_{ij}\gamma^{\mu}d^{j}_{L})\right]\\\
&-\left[(-\frac{g\sin^{2}{\theta_{\textrm{W}}}}{\cos{\theta_{\textrm{W}}}}Z_{\mu}+eA_{\mu})\left(\frac{2}{3}\right)(\overline{u^{i}_{L}}\delta_{ij}\gamma^{\mu}u^{j}_{L}+\overline{u^{i}_{R}}\delta_{ij}\gamma^{\mu}u^{j}_{R})\right]\\\
&-\left[(\frac{g\sin^{2}{\theta_{\textrm{W}}}}{\cos{\theta_{\textrm{W}}}}Z_{\mu}-eA_{\mu})\left(\frac{1}{3}\right)(\overline{d^{i}_{L}}\delta_{ij}\gamma^{\mu}d^{j}_{L}+\overline{d^{i}_{R}}\delta_{ij}\gamma^{\mu}d^{j}_{R})\right]\leavevmode\nobreak\
,\end{split}$ (1.31)

which shows that neutral currents remain diagonal, which means that there are
no flavour Changing Neutral Currents (FCNC) at tree level in the SM - in any
sector. Beyond tree level they are highly suppressed - cf.
$K^{0}_{L}\rightarrow\mu^{+}\mu^{-}$ \- this suppression stems from what is
known as the GIM mechanism [48].

The quark mixing matrix $V_{\textrm{CKM}}\equiv V$ is complex but some of its
phases have no physical meaning, due to the fact that one has the freedom to
rephase mass eigenstates, $u^{\prime}_{i}=e^{i\phi_{i}}u_{i}$ and
$d^{\prime}_{j}=e^{i\phi_{j}}d_{j}$ , transforming the entries of the mixing
matrix:

$V^{\prime}_{ij}=e^{i(\phi_{j}-\phi_{i})}V_{ij}\leavevmode\nobreak\ .$ (1.32)

Thus, it’s useful to look for rephasing invariants such as the the moduli of
each entry, $|V_{ij}|$, or the quartets:

$Q_{ijkl}=V_{ij}V_{kl}V^{*}_{il}V^{*}_{kj}\leavevmode\nobreak\ ,$ (1.33)

because it can be proved that invariants of higher order can always be written
as combinations of quartets and the moduli.

The fact that V is complex, in general, means that CP Violation can exist.
Performing a CP transformation is performing a Charge transformation -
transforming a particle in its anti-particle - followed by a Parity
transformation - flipping the sign of the spatial coordinate, which means
changing the chirality of a field, since axial vectors get an extra sign under
parity transformations. One can check section 13.2 of [35] to see how SM
fields transform under CP transformations. Performing a CP transformation to
the SM Lagrangian after SSB, one obtains the condition for CP invariance of
the SM:

$V^{*}_{ij}=e^{i(\xi_{W}+\xi_{j}-\xi_{i})}V_{ij}\leavevmode\nobreak\ ,$ (1.34)

where $\xi_{\alpha}$ are spurious CP phases that arise from the
transformation. One can make eq. (1.34) always true for a single entry of $V$,
however, if one considers all of the entries, one is forced to conclude that
all quartets must be real and, thus, all other rephasing invariants. Hence, CP
violation stems from the non-removable phases of $V_{\textrm{CKM}}$. Thus, it
is of uttermost importance to determine how many physical CP violating phases
might exist in $V$. Due to the rephasing invariance, for $N_{f}$ generations
($V$ is an $N_{f}\times N_{f}$ unitary matrix parametrized by $N_{f}^{2}$
parameters) one can remove $2N_{f}-1$ phases, making the total number of
parameters $(N_{f}-1)^{2}$. $N_{f}(N_{f}-1)/2$ of these parameters will be
angles, while $(N_{f}-1)(N_{f}-2)/2$ will be phases. If one takes $N_{f}=3$,
one obtains that there is only one phase. This is a CP violating phase,
Kobayashi and Maskawa [49] arrived to the above conclusion, proving that only
for $N_{f}\geq 3$ one has CP Violating phases in the quark sector. For
$N_{f}=3$ the imaginary part of all quartets are equal, up to a sign

$J=|Im[Q]|\leavevmode\nobreak\ .$ (1.35)

From the unitarity constraints on the entries of V one can define what is
known as unitarity triangles - cf. section 13.5 and 13.6 of [35]. From these
one gets a remarkable geometrical interpretation to $J$ \- it is twice the
area of any of the six possible unitarity triangles.

For an extensive treatment of CP Violation refer to ref. [35]. The standard
parametrization [3] of the CKM mixing matrix is

$\displaystyle
V=\left(\begin{array}[]{ccc}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta}\\\
-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta}&\quad
c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta}&s_{23}c_{13}\\\
s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta}&-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta}&c_{23}c_{13}\end{array}\right)\leavevmode\nobreak\
,$ (1.39)

where $c_{ij}=\cos\theta_{ij}$, $s_{ij}=\sin\theta_{ij}$, and $\delta$ is a
Dirac-type CP violating phase. In this parameterisation,

$J=c_{12}c_{23}c^{2}_{13}s_{12}s_{23}s_{13}\sin{\delta}\leavevmode\nobreak\ .$
(1.40)

However, note that $\delta$ is not a rephasing invariant and is only
meaningful under this parametrization. It is possible to parametrize the
mixing matrix using a non optimal number of parameters with every single one
of them being a rephasing invariant [3]. The current best-fit values for the
quark mixing matrix are:

$V_{\textrm{CKM}}=\left(\begin{array}[]{ccc}0.97373\pm 0.00031&0.2243\pm
0.0008&0.00382\pm 0.00020\\\ 0.221\pm 0.004&0.975\pm 0.006&0.0408\pm 0.0014\\\
0.0086\pm 0.0002&0.0415\pm 0.0009&1.014\pm
0.029\end{array}\right)\leavevmode\nobreak\ .$ (1.41)

The unitarity of $V_{\textrm{CKM}}$ is heavily constrained [3], contrarily to
the leptonic sector where sizable deviations from unitarity of the mixing
matrix are not ruled out.

This difference stems from the fact that there are many hadron decay processes
which enable the direct measurement of individual $V_{\textrm{CKM}}$ entries.
However, in the leptonic sector this is not possible. There is not enough
precision to detect neutrino mass states (their mass scale is too small) in
leptonic weak decays, so in each process one can only know the produced
interaction state, which is a linear combination of the mass states, with
certainty. Currently, one of the most reliable ways to get information
regarding the mixing matrix are oscillation experiments and, even in those,
one only has access to the first row and the last column of the $3\times 3$
mixing matrix. Furthermore, what is measured in those cases are combinations
of the entries of the mixing matrix, and not individual entries like in the
quark case [16].

In chapter 2, in the discussion of the $n_{R}\nu$SM - the SM with right-handed
neutrinos-, the emergence of a non-trivial mixing matrix in the leptonic
sector will be studied and the steps are very similar to what has been done
here for the quark sector. The emergence of a $(3+n_{u})\times(3+n_{d})$
unitary mixing matrix in the $n_{u}$SM$n_{d}$ \- the SM with the addition of
$n_{u}$ up vector-like quarks and $n_{d}$ down vector-like quarks - will also
be discussed in chapter 3. There are only two pieces left to have the SM
Lagrangian completely defined:

$\mathcal{L}_{\textrm{SM}}=\mathcal{L}_{\textrm{Fermion}}+\mathcal{L}_{\textrm{Kin}}+\mathcal{L}_{\textrm{Higgs}}+\mathcal{L}_{\textrm{Yukawa}}+\mathcal{L}_{\textrm{GF}}+\mathcal{L}_{\textrm{{Ghost}}}\leavevmode\nobreak\
.$ (1.42)

The result in eq. (1.14) is gauge dependent. A gauge independent formulation
of the SM should use eq. (1.12). However, with this definition,
$\mathcal{L}_{Higgs}$ will generate mixed quadratic terms in fields, with the
three Goldstone bosons $G^{0}$ and $G^{\pm}$, that complicate the definition
of the gauge boson propagators. Using the gauge independent eq. (1.12) in
$\mathcal{L}_{Fermion}$ also introduces interactions between fermions and the
unphysical Goldstone bosons. These should be taken into account when
performing calculations in a general gauge. To cancel the mixed quadratic
terms that arise from $\mathcal{L}_{Higgs}$, one should include a new term in
the SM Lagrangian:

$\mathcal{L}_{\textrm{GF}}=-\frac{1}{2\xi_{G}}F_{G}^{a}F_{Ga}-\frac{1}{2\xi_{A}}F_{A}^{2}-\frac{1}{2\xi_{Z}}F_{Z}^{2}-\frac{1}{\xi_{W}}F_{-}F_{+},$
(1.43)

where $F_{G}^{a}=\partial^{\mu}G_{\mu}^{a}$, $F_{A}=\partial^{\mu}A_{\mu}$ ,
$F_{Z}=\partial^{\mu}Z_{\mu}+\xi_{Z}m_{Z}G^{0}$,
$F_{\pm}=\partial^{\mu}W_{\mu}^{\pm}\pm i\xi_{W}m_{W}G^{\pm}$.
$\mathcal{L}_{GF}$ are, actually, the gauge breaking terms in
$\mathcal{L}_{SM}$.

The last piece is the Ghost Lagrangian. Faddeev-Popov Ghosts are unphysical
particles that violate the Spin-Statistics Theorem. In theories like the SM
they are bosonic (spin 0) with anti-comutation relations (fermionic). Every
gauge boson correspond to a non-Abelian Gauge Group will have a Ghost - 8 for
$SU(3)$ , 4 for $SU(2)\times U(1)$. These ghost fields are necessary to
achieve a linear gauge fixing condition like in $\mathcal{L}_{GF}$, generating
gauge field propagators with transverse and longitudinal component, thus,
invertible. If the SM gauge group was abelian there would be no need for
Faddeev-Popov Ghosts [27].

$\mathcal{L}_{\textrm{Ghost}}=\sum_{i=1}^{4}\left[\overline{c}_{+}\frac{\partial(\delta
F_{+})}{\partial\alpha_{i}}+\overline{c}_{-}\frac{\partial(\delta
F_{-})}{\partial\alpha_{i}}+\overline{c}_{Z}\frac{\partial(\delta
F_{Z})}{\partial\alpha_{i}}+\overline{c}_{A}\frac{\partial(\delta
F_{A})}{\partial\alpha_{i}}\right]c_{i}+\sum_{a,b=1}^{8}\overline{\omega}^{a}\frac{\partial(\delta
F^{a}_{G})}{\partial\beta_{b}}\omega^{b},$ (1.44)

where $\alpha=1,2,3,4$ and $\beta=1,...,8$ are parameters of the correspondent
gauge transformations. For a more technical treatment of Faddeev-Popov Ghosts
refer to Appendix A of [41], sections 16.2 and 21.1 of [28] and chapter 6 of
[27]. Since ghosts don’t couple to matter fields, their contribution to one-
loop corrections of physical processes involving fermions in external lines is
zero, hence explaining ghosts further goes beyond the scope of this work.

Adding the final two terms in eq. (1.42), one can check that the Unitary gauge
corresponds to $\xi_{Z,W}\xrightarrow{}\infty$ and $\xi_{A,G}\xrightarrow{}1$
and that the Goldstone bosons correspondent to massive gauge bosons - $G^{0}$
and $G^{\pm}$ \- acquire gauge dependent masses. The unitary gauge - used in
eq. (1.14) - only contains physical particles because having infinite masses
means a decoupling from the theory. The formulation of the SM in a general
gauge will be needed in later parts of this work.

To conclude, one final note regarding the SM. The SM has 19 free parameters: 9
fermion masses, 3 quark mixing angles, 1 CP violating phase in the quark
sector, 1 Higgs mass, 1 Higgs vaccum expectation value ($v$), 1 QCD Vacuum
angle ($\Theta_{\text{QCD}}$), 1 weak mixing angle333Instead of considering
$\theta_{\textrm{W}}$ one may consider the $U(1)_{Y}$ gauge coupling
($g^{\prime}$) or the fine-structure constant ($\alpha$), as they are all
proportional to each other. ($\theta_{\textrm{W}}$), 1 $SU(2)_{L}$ gauge
coupling ($g$) and 1 $SU(3)_{c}$ gauge coupling ($g$). These are the
parameters that need to be fitted to experimental data.

Hence, the SM does not predict fermion masses nor gives an explanation to the
number of generations of these. The only known thing regarding the generation
number is that the number of generations of leptons and quarks must be equal
in order to cancel gauge anomalies that appear at one-loop corrections - cf.
Chapter 19 section 19.4 and chapter 20 section "Anomaly Cancellation" of [28].

Furthermore, the SM does not include gravity nor particles that can be dark
matter candidates. Also, it does not explain dark energy.

In what regards CP Violation, it was not seen in strong sector (QCD) while
there is nothing that inhibits it. In addition, the CP Violation detected in
the electroweak sector is not large enough to explain the matter-antimatter
asymmetry observed in the universe.

There are also some theoretical-inspired shortcomings that haunt the SM. One
is the hierarchy problem, related to the fine-tuning that needs to happen in
higher order calculations to achieve a Higgs mass near the electroweak scale
($v$) and the fact that the gauge couplings do not unify at high-energy,
unlike what happens in some GUT models.

All these drawbacks lead to the belief that the SM is not the final theory of
everything (TOE) but just a low-energy effective theory of it. The SM can be
summarized in its Feynman Rules, which can be found in ref. [41]. In this
work, these were used setting all $\eta_{i}=1$.

##  Chapter 2 The Standard Model + $n_{R}$ right-handed neutrinos
($n_{R}\nu$SM)

This chapter includes the study of a model where $n_{R}$ neutrinos are added
to the SM without imposing Lepton Number Conservation, allowing for Majorana
mass terms. This extension of the SM will be treated exactly using the
parameterisation first devised in [30], in the more succinct form of [32].

In a WB where the charged leptons are diagonal and their masses are in the
diagonal of the matrix $d_{l}$, the neutrino Dirac mass matrix is labeled $m$,
and is a $3\times n_{R}$ matrix and the symmetric Majorana mass matrix, $M$,
is an $n_{R}\times n_{R}$ matrix, the leptonic mass Lagrangian is given by

$\displaystyle\mathcal{L}_{m}$
$\displaystyle=\,-\left[\frac{1}{2}n_{L}^{T}C^{*}\mathcal{M}^{*}n_{L}+\overline{l_{L}}d_{l}l_{R}\right]+\text{h.c.}\,,$
(2.1)

where $n_{L}^{T}=(\nu_{L}^{T}\,\,\,\overline{\nu_{R}}\,C^{T})$,
$n_{L}=(\nu_{L}\,,\,\,C\,\overline{\nu_{R}}^{T})$ is a column vector and $C$
is the charge conjugation matrix. The $(3+n_{R})\times(3+n_{R})$ neutrino mass
matrix is

$\displaystyle\mathcal{M}\,=\,\left(\begin{array}[]{c;{2pt/2pt}c}\,\textbf{0}&\,m\\\\[4.2679pt]
\hdashline[2pt/2pt]{\,\,\,m^{T}\,\,\,}\\!&{\,\,\,M\,\,\,}\\!\\\\[2.84526pt]
\end{array}\right)\,.$ (2.4)

And its diagonalisation equation, using the Takagi decomposition, a special
case of singular value decomposition for symmetric matrices, is

$\mathcal{V}^{T}\mathcal{M}^{*}\mathcal{V}=\mathcal{D}\leavevmode\nobreak\ ,$
(2.5)

where $\mathcal{D}$ is a diagonal matrix and has the masses of the 3 light
neutrinos and the $n_{R}$ heavy neutrinos in the diagonal and $\mathcal{V}$ is
a unitary $(3+n_{R})\times(3+n_{R})$ matrix that can be written as

$\mathcal{V}\,=\,\left(\begin{array}[]{c}{}\hfil\\\\[-11.38109pt] \qquad
A\\\\[5.69054pt] \hdashline[2pt/2pt]{}\\\\[-11.38109pt] \qquad
B\\\\[5.69054pt] \end{array}\right)\,,$ (2.6)

which combined with eq. (2.4) enables the rewriting of the diagonalisation
equations in eq. (2.5) as

$\begin{split}&\textbf{0}=A\,\mathcal{D}\,A^{T}\,,\quad\\\
&m=A\,\mathcal{D}\,B^{T}\,,\quad\\\ &M=B\,\mathcal{D}\,B^{T}\,.\end{split}$
(2.7)

These are exact equations for the mass matrices that will be very useful when
$A$ and $B$ are properly defined. The leptonic charged current Lagrangian in
the physical basis takes the form of

$\mathcal{L}_{W}^{\nu}=-\frac{g}{\sqrt{2}}\overline{l}_{L}\,V\,\gamma^{\mu}\begin{pmatrix}n_{L}\\\\[5.69054pt]
N_{L}\end{pmatrix}W_{\mu}\,+\,\text{h.c.}\leavevmode\nobreak\ ,$ (2.8)

where $n$ ($N$) are light (heavy) neutrino mass states and

$V=A\leavevmode\nobreak\ ,$ (2.9)

plays the role of the physical mixing matrix. Note that $V$ is a
$3\times(3+n_{R})$ matrix. The neutral interaction - involving the $Z$ and the
Higgs boson - Lagrangians can be written as

$\begin{split}\mathcal{L}_{Z}^{\nu}&=-\frac{g}{2\cos{\theta_{W}}}Z_{\mu}[\left(\begin{array}[]{cc}\overline{n_{L}}&\overline{N_{L}}\end{array}\right)\leavevmode\nobreak\
F\leavevmode\nobreak\ \gamma^{\mu}\left(\begin{array}[]{cc}n_{L}\\\
N_{L}\end{array}\right)]+\,\text{h.c.}\leavevmode\nobreak\ ,\\\ \end{split}$
(2.10)

and

$\mathcal{L}_{H}^{\nu}\,=\,-\frac{h}{v}\bigg{[}\begin{pmatrix}\overline{n}_{L}&\overline{N}_{L}\end{pmatrix}\,F\leavevmode\nobreak\
\mathcal{D}\,\begin{pmatrix}n_{L}^{c}\\\\[5.69054pt]
N_{L}^{c}\end{pmatrix}\,\bigg{]}\,+\,\text{h.c.}\,,$ (2.11)

where

$F=A^{\dagger}A\leavevmode\nobreak\ .$ (2.12)

Note that this matrix $F$ is only the identity matrix if $A$ is unitary on the
left, which will be proved to be only possible when the heavy neutrinos are
completely decoupled from the light ones, being completely sterile. Actually,
$A$ is only unitary on the right. Hence, $F$ controls the Flavour Changing
Neutral Couplings (FCNCs) of this model. Given the unitarity constraints, it
is possible to write the matrix $\mathcal{V}$ defined in eq. (2.4) as

$\displaystyle\mathcal{V}\,=\,\left(\begin{array}[]{c;{2pt/2pt}c}\,\,\,K&\,\,\,K\,X^\textdagger\\\\[4.2679pt]
\hdashline[2pt/2pt]{-\overline{\\!K}\,\,{X}\,\,}\\!&{\quad\,\,\overline{\\!K}\,\quad\,\,}\\!\\\\[2.84526pt]
\end{array}\right)\leavevmode\nobreak\ ,$ (2.15)

for a non-singular $3\times 3$ general complex matrix $K$, a non-singular
$n_{R}\times n_{R}$ general complex matrix $\overline{\\!K}\,$ and a
$n_{R}\times 3$ matrix, $X$, that will be defined next. This parameterisation
is completely general and exact, under the aforementioned conditions, and it
was first introduced in ref. [30]. From eq. (2.6) one can identify

$A=(K\leavevmode\nobreak\ \leavevmode\nobreak\
KX^{\dagger})\leavevmode\nobreak\ ,\leavevmode\nobreak\
B=(-\overline{\\!K}\,X\leavevmode\nobreak\ \leavevmode\nobreak\
\overline{\\!K}\,)\leavevmode\nobreak\ ,$ (2.16)

where the unitarity of $\mathcal{V}$ yields

$\displaystyle\mathcal{V}\mathcal{V}^{\dagger}=\begin{pmatrix}\,A\,\\\\[5.69054pt]
B\end{pmatrix}\begin{pmatrix}\,{A}^{\dagger}&{B}^{\dagger}\,\end{pmatrix}\,$
$\displaystyle=\,\begin{pmatrix}\,A\,{A}^{\dagger}&A\,{B}^{\dagger}\,\\\\[5.69054pt]
\,B\,{A}^{\dagger}&B\,{B}^{\dagger}\,\end{pmatrix}\,=\,\begin{pmatrix}\,\mathds{1}_{3}\,&\,0\,\\\\[5.69054pt]
\,0\,&\,\mathds{1}_{n_{R}}\,\end{pmatrix}\,,$ (2.17)
$\displaystyle\mathcal{V}^{\dagger}\mathcal{V}=\begin{pmatrix}\,{A}^{\dagger}&{B}^{\dagger}\,\end{pmatrix}\begin{pmatrix}\,A\,\\\\[5.69054pt]
B\end{pmatrix}\,$
$\displaystyle=\,\,{A}^{\dagger}\,A+{B}^{\dagger}\,B\,\,=\,\,\mathds{1}_{3+n_{R}}\,.$
(2.18)

Note that by eq. (2.18), $A$ is only unitary on the left if
$\overline{\\!K}\,$ and $X$ are exactly zero. This corresponds to the case of
having heavy fully sterile neutrinos, decoupled from the rest of the theory.
Using eq. (2.16) and the unitarity relations in eqs. (2.17) and (2.18), eq.
(2.7) becomes

$\displaystyle\textbf{0}=d+X^{\dagger}DX^{*}\leavevmode\nobreak\ ,$ (2.19)
$\displaystyle
m\,=\,K\,X^{\dagger}D\left(\overline{\\!K}\,^{-1}\right)^{*}\leavevmode\nobreak\
,$ $\displaystyle
M\,=\,\overline{\\!K}\,\,(D+X\,d\,X^{T})\,\overline{\\!K}\,^{T}\,.$

The last two equations are exact equations for the $3\times n_{R}$ Dirac mass
matrix, $m$, and the $n_{R}\times n_{R}$ Majorana mass matrix, $M$. The
equation involving the null matrix has the solution

$\displaystyle X\,=\pm\,i\,\sqrt{D^{-1}}\,O_{c}\,\sqrt{d}\,,$ (2.20)

where $O_{c}$ is an orthogonal complex matrix and $d$ ($D$) is a diagonal
matrix with the masses of the light (heavy) neutrinos in the diagonal. This
unique solution defines the matrix $X$. From the unitarity relations of
$\mathcal{V}$ one can also obtain the following definitions

$\begin{split}&K=U_{K}(\mathds{1}_{3}+X^{\dagger}X)^{-1/2}\leavevmode\nobreak\
,\\\
&\overline{\\!K}\,=U_{\overline{\\!K}\,}(\mathds{1}_{n_{R}}+XX^{\dagger})^{-1/2}\leavevmode\nobreak\
,\\\ &K_{PMNS}=K\leavevmode\nobreak\ ,\end{split}$ (2.21)

where $U_{K}$ and $U_{\overline{\\!K}\,}$ are unitary matrices, since the
unitarity relations only define $K$ and $\overline{\\!K}\,$ up to a unitary
matrix on the left. $K$ will play the role of the PMNS mixing matrix and is
only unitary when $X\xrightarrow[]{}\textbf{0}$.

Furthermore, using this parameterisation is also possible to rewrite the
matrix that controls FCNCs, defined in eq. (2.12), as

$F=\begin{pmatrix}(\mathds{1}_{3}+X^{\dagger}X)^{-1}&(\mathds{1}_{3}+X^{\dagger}X)^{-1}X^{\dagger}\\\
X(\mathds{1}_{3}+X^{\dagger}X)^{-1}&X(\mathds{1}_{3}+X^{\dagger}X)^{-1}X^{\dagger}\\\
\end{pmatrix}\leavevmode\nobreak\ .$ (2.22)

The complete form of the $(3+n_{R})\times(3+n_{R})$ unitary matrix
$\mathcal{V}$ is written as

$\mathcal{V}=\begin{pmatrix}U_{K}(\mathds{1}_{3}+X^{\dagger}X)^{-1/2}&U_{K}(\mathds{1}_{3}+X^{\dagger}X)^{-1/2}X^{\dagger}\\\
-U_{\overline{\\!K}\,}(\mathds{1}_{n_{R}}+XX^{\dagger})^{-1/2}X&U_{\overline{\\!K}\,}(\mathds{1}_{n_{R}}+XX^{\dagger})^{-1/2}\\\
\end{pmatrix}\leavevmode\nobreak\ .$ (2.23)

With all the relevant matrices defined in a general and exact way, it is now
possible to proceed to the study of the model in interesting regions of the
parameter space. Note that parameterisations with a similar structure, in the
leptonic sector, existed in the literature prior to this work [50, 51], but
are either approximations or a special case of this one.

In the rest of this chapter and remaining of this thesis, in the context of
neutrinos and whenever there is no risk of confusion, $M$ should refer to the
scale of the masses of the heavy neutrinos, while $m$ refers to the scale of
the masses of the light neutrinos. Where there is risk of confusion, it will
be specifically stated if one is referring to the mass matrix or to the mass
of a specific species of neutrino.

###  2.1 The Quasi-Decoupled Parameter Space: $M\sim$ GeV/TeV

In ref. [30] one tried to argue that a region of the parameter space where all
the heavy neutrinos have a GeV-scale mass and the deviations from unitarity
(DU) of the $3\times 3$ leptonic mixing matrix match the most stringent
experimental bounds at the time [52], was allowed and still provided a
reasonable explanation to the smallness of light neutrinos masses, given that
the Yukawa couplings were $\mathcal{O}(1)$.

In the literature [3] the PMNS matrix is usually parameterised as

$\displaystyle
U_{\textrm{PMNS}}=\left(\begin{array}[]{ccc}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta}\\\
-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta}&\quad
c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta}&s_{23}c_{13}\\\
s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta}&-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta}&c_{23}c_{13}\end{array}\right)\cdot
P\leavevmode\nobreak\ ,$ (2.27)

with $P$ given by

$P=\mathrm{diag}\ (1,e^{i\alpha_{21}},e^{i\alpha_{31}})\leavevmode\nobreak\ ,$
(2.28)

where $c_{ij}=\cos\theta_{ij}$, $s_{ij}=\sin\theta_{ij}$ and $\delta$ is a
Dirac-type CP violating phase, while $\alpha_{21}$, $\alpha_{31}$ denote
Majorana phases. The PMNS matrix has this form in the framework of a low-
energy effective theory, where the deviations from unitarity of the $3\times
3$ mixing matrix are negligible and the heavy neutrinos are so heavy that it
is possible to integrate them out. In this effective theory, only mass terms
involving $\nu_{L}$ exist, preventing the rephasing of the neutrino field,
leading to the presence of two additional Majorana phases.

####  2.1.1 Status of Neutrino Physics and DU for $M>M_{W}$

When it is necessary to go beyond this level of approximation, the consensus
in the literature [53, 54, 55, 56, 52, 57] dictates that one should write the
$3\times 3$ leptonic mixing matrix as the product of a hermitian matrix by a
unitary matrix:

$K=(\mathds{1}_{3}-\eta)U_{K}\leavevmode\nobreak\ ,$ (2.29)

where $U_{K}$ is a unitary matrix and $\eta$ is a hermitian matrix with small
entries, that are proportional to the deviations from unitarity. Note that
under the previous definition of the $3\times 3$ leptonic mixing matrix in eq.
(2.21), the matrices $U_{K}$ and $\eta$ can be written as

$\eta=\mathds{1}_{3}-U_{K}(\mathds{1}_{3}+X^{\dagger}X)^{-1/2}U_{K}^{\dagger}\approx\frac{1}{2}U_{K}\left(X^{\dagger}X\right)U_{K}^{\dagger}\leavevmode\nobreak\
.$ (2.30)

In ref. [30], the matrix $U_{K}$ is then fixed making use of the, at the time
of publication, most updated best-fit values obtained from a global analysis
based on the assumption of unitarity, present in Table 2.1. These were the
values used in the published results.

Table 2.1: Neutrino oscillation parameter summary from [58]. For $\Delta m^{2}_{31}$, $\sin^{2}\theta_{23}$ , $\sin^{2}\theta_{13}$, and $\delta$ the upper (lower) row corresponds to normal (inverted) neutrino mass hierarchy. Parameter | Best fit |  $1\sigma$ range
---|---|---
$\Delta m^{2}_{21}$ $[10^{-5}eV^{2}]$ | 7.56 | 7.37 – 7.75
$|\Delta m^{2}_{31}|$ $[10^{-3}eV^{2}]$ (NO) | 2.55 | 2.41 – 2.59
$|\Delta m^{2}_{31}|$ $[10^{-3}eV^{2}]$ (IO) | 2.49 | 2.45 – 2.53
$\sin^{2}\theta_{12}$ | 0.321 | 0.305 – 0.339
$\sin^{2}\theta_{23}$ (NO) | 0.430 | 0.412 – 0.450
$\sin^{2}\theta_{23}$ (IO) | 0.596 | 0.576 – 0.614
$\sin^{2}\theta_{13}$ (NO) | 0.02155 | 0.02080 –0.02245
$\sin^{2}\theta_{13}$ (IO) | 0.02140 | 0.02055 – 0.02222
$\delta$ (NO) | 1.40 $\pi$ | 1.20 –1.71 $\pi$
$\delta$ (IO) | 1.44 $\pi$ | 1.70 –1.21 $\pi$

For comparison, at the time of writing of this thesis, the most updated best-
fit values obtained from a global analysis based on the assumption of
unitarity are present in Table 2.2. These values were not used in any of the
published results.

Table 2.2: Neutrino oscillation parameter summary from [59]. For $\Delta m^{2}_{31}$, $\sin^{2}\theta_{23}$ , $\sin^{2}\theta_{13}$, and $\delta$ the upper (lower) row corresponds to normal (inverted) neutrino mass hierarchy. Parameter | Best fit |  $1\sigma$ range
---|---|---
$\Delta m^{2}_{21}$ $[10^{-5}eV^{2}]$ | 7.50 | 7.30 – 7.72
$|\Delta m^{2}_{31}|$ $[10^{-3}eV^{2}]$ (NO) | 2.55 | 2.52 – 2.57
$|\Delta m^{2}_{31}|$ $[10^{-3}eV^{2}]$ (IO) | 2.45 | 2.42 – 2.47
$\sin^{2}\theta_{12}$ | 0.318 | 0.302 – 0.340
$\sin^{2}\theta_{23}$ (NO) | 0.574 | 0.560 – 0.588
$\sin^{2}\theta_{23}$ (IO) | 0.578 | 0.568 – 0.561
$\sin^{2}\theta_{13}$ (NO) | 0.02200 | 0.02138 –0.02269
$\sin^{2}\theta_{13}$ (IO) | 0.02225 | 0.02155 – 0.02289
$\delta$ (NO) | 1.08 $\pi$ | 0.96 –1.21 $\pi$
$\delta$ (IO) | 1.58 $\pi$ | 1.42 –1.73 $\pi$

According to [52], it is advantageous, from a phenomenological perspective, to
represent $K$ with a unitary matrix on the right, as it is not possible to
determine which physical light neutrino is produced in experiments. Therefore,
one must sum over all neutrino indices. As a result, most observables are
dependent on $KK^{\dagger}$, which depends on the following combination

$\left(KK^{\dagger}\right)_{\alpha\beta}=\delta_{\alpha\beta}-2\eta_{\alpha\beta}+\mathcal{O}(\eta^{2}_{\alpha\beta})\leavevmode\nobreak\
.$ (2.31)

A fit of 28 observables, including the W boson mass and the effective mixing
weak angle $\theta_{W}$, was used to derive global constraints on the matrix
$\eta$ in ref. [52]. The fit considered several ratios of $Z$ fermionic
decays, the invisible width of the $Z$, ratios of weak decays that constrain
EW universality, weak decays that constrain CKM unitarity, and some radiative
lepton flavour violating (LFV) processes.

$\displaystyle\left|2\eta_{\alpha\beta}\right|\leq\left(\begin{array}[]{ccc}2.5\times
10^{-3}&2.4\times 10^{-5}&2.7\times 10^{-3}\\\ 2.4\times 10^{-5}&4.0\times
10^{-4}&1.2\times 10^{-3}\\\ 2.7\times 10^{-3}&1.2\times 10^{-3}&5.6\times
10^{-3}\end{array}\right)\leavevmode\nobreak\ .$ (2.35)

It is crucial to emphasize that this result only holds when all heavy
neutrinos possess masses above the electroweak (EW) scale [57]. In other
words, if $M$ represents the mass of the heavy neutrinos, it is required that
all of them satisfy $M>M_{W}$, where $M_{W}$ denotes the mass of the $W$
boson. This is the case in ref. [30].

####  2.1.2 Benchmarks and Perturbativity Remarks

Before examining the benchmarks that demonstrate the feasibility of having GeV
heavy neutrinos with substantial deviations from the unitarity of the $3\times
3$ leptonic mixing matrix while still having $\mathcal{O}(1)$ Yukawa
couplings, it is crucial to understand the theoretical basis behind this
possibility. By combining equations (2.19), (2.20), and (2.21), one can
reformulate the $3\times 3$ Dirac mass matrix for neutrinos as

$\displaystyle m\,=\
V\,\sqrt{\left(\mathds{1}+X^{\dagger}X\right)^{-1}}\,X^{\dagger}D\,\sqrt{\mathds{1}+X^{*}X^{T}}U_{\overline{\\!K}\,}^{T}\,,$
(2.36)

which, using eq. (2.20) becomes

$\displaystyle
m\,=\,-i\,U_{K}\,\sqrt{\left(\mathds{1}_{3}+X^{\dagger}X\right)^{-1}}\,\sqrt{d}\,O_{c}^{\dagger}\sqrt{D}\,\sqrt{\mathds{1}_{n_{R}}+X^{*}X^{T}}\leavevmode\nobreak\
U_{\overline{\\!K}\,}^{T}\,.$ (2.37)

Taking $U_{\overline{\\!K}\,}=\mathds{1}_{n_{R}}$, since this matrix is
unphysical at low energy, the equation for $m$ in first order approximation in
$X$ (or $Oc$) takes the form

$\displaystyle m\,\approx\,-i\,U_{K}\,\,\sqrt{d}\,O_{c}^{\dagger}\sqrt{D}.$
(2.38)

It is evident that for $\mathcal{O}(1)$ Yukawa couplings and fixed light
neutrino masses, there is a seesaw relationship between the magnitude of
deviations from unitarity, the singular values of $O_{c}$, and the scale of
heavy neutrino masses in $D$. This relationship becomes clearer when one
defines the singular value decomposition of $X$

$d_{X}=W^{\dagger}XU\leavevmode\nobreak\ ,$ (2.39)

and calculates the quantity

$Tr\left[mm^{\dagger}\right]=Tr\left[\left(\sqrt{\left(\mathds{1}_{3},+d_{X}^{2}\right)}\
\right)^{-1}d_{X}\ W^{\dagger}\ D\ W^{\ast}\
\left(\mathds{1}_{3}\,+d_{X}^{2}\right)\ W^{T}\ D\ W\ d_{X}\
\left(\sqrt{\left({1\>\\!\\!\\!\mathrm{I}}\,+d_{X}^{2}\right)}\
\right)^{-1}\right]\leavevmode\nobreak\ ,$ (2.40)

which to a very good approximation yields

$Tr\left[mm^{\dagger}\right]=Tr\left[d_{X}\ W^{\dagger}D^{2}W\
d_{X}\right]\leavevmode\nobreak\ .$ (2.41)

This can be rewritten as

$\begin{array}[]{l}Tr\left[mm^{\dagger}\right]=d_{X_{1}}^{2}\left(M_{1}^{2}\left|W_{11}\right|^{2}+M_{2}^{2}\left|W_{21}\right|^{2}+M_{3}^{2}\left|W_{31}\right|^{2}\right)+\\\
d_{X_{2}}^{2}\left(M_{1}^{2}\left|W_{12}\right|^{2}+M_{2}^{2}\left|W_{22}\right|^{2}+M_{3}^{2}\left|W_{32}\right|^{2}\right)+\\\
d_{X_{3}}^{2}\left(M_{1}^{2}\left|W_{13}\right|^{2}+M_{2}^{2}\left|W_{23}\right|^{2}+M_{3}^{2}\left|W_{33}\right|^{2}\right)\leavevmode\nobreak\
.\end{array}$ (2.42)

It can be demonstrated that, through the use of the properties of orthogonal
complex matrices, only the largest of the $d_{X_{i}}$, represented by
$d_{X_{3}}$, can have a substantial value (for example, $d_{X_{3}}\approx
10^{-2}$) while the other two are negligible. As a result, in good
approximation,

$Tr\left[mm^{\dagger}\right]=d_{X_{3}}^{2}\left(M_{1}^{2}\left|W_{13}\right|^{2}+M_{2}^{2}\left|W_{23}\right|^{2}+M_{3}^{2}\left|W_{33}\right|^{2}\right)\leavevmode\nobreak\
,$ (2.43)

and using the unitarity of $W$

$Tr\left[mm^{\dagger}\right]=d_{X_{3}}^{2}M_{1}^{2}\left(1+\left(\frac{M_{2}^{2}}{M_{1}^{2}}-1\right)\left|W_{23}\right|^{2}+\left(\frac{M_{3}^{2}}{M_{1}^{2}}-1\right)\left|W_{33}\right|^{2}\right)\leavevmode\nobreak\
,$ (2.44)

which, with the choice $M_{3}\geq M_{2}\geq M_{1}$, leads to

$d_{X_{3}}^{2}M_{1}^{2}\leq
Tr\left[mm^{\dagger}\right]=\sum_{i,j}\left|m_{ij}\right|^{2}\sim\left(\frac{v}{\sqrt{2}}\right)^{2}\approx
m_{t}^{2}\leavevmode\nobreak\ ,$ (2.45)

for $\mathcal{O}(1)$ Yukawa couplings. This seesaw between the deviations from
unitarity and the scale of the heavy neutrino mass is easy to see if one plots
$\frac{1}{2}d_{X_{3}}^{2}$ against $M_{1}$, as in fig. 2.1.

Figure 2.1: Maximum deviations from unitarity as a function of $M_{1}$,
generated under the condition that Tr$(mm^{\dagger})\leq m_{t}^{2}$ and
$|\eta_{12}|\leq 10^{-4}$.

It should now be clear that by decreasing the mass scale of the heavy
neutrinos while keeping the Yukawa couplings $\mathcal{O}(1)$, the deviations
from $3\times 3$ unitarity can be increased.

The benchmarks for NO (IO) of the light neutrino masses, where the 3 heavy
neutrinos have masses in the range of $500-1500$ GeV and the $\eta$ matrix is
consistent with the most stringent upper bounds from [52] as given in equation
(2.35) for $\textrm{Tr}(mm^{\dagger})\sim(0.03-0.5)\leavevmode\nobreak\
m_{t}^{2}$, can be found in Table 2.3 (2.4).

Table 2.3: Examples of deviations from unitarity, with light neutrino masses
of $m_{1}=0.005050$ eV, $m_{2}=0.01005$ eV, and $m_{3}=0.05075$ eV for the
case of NO, are expressed by the Hermitian matrix $\eta$. Three different
hierarchies for the heavy Majorana neutrino masses $M_{i}$ are considered. For
each hierarchy, two examples are given where the choice of the matrix $X$ is
varied. The third column shows the Dirac-type neutrino mass matrix expressed
in GeV units, the fifth column displays the Majorana-type neutrino mass matrix
expressed in units of the top quark mass, and the fourth column displays the
trace of $mm^{\dagger}$ in units of the squared top quark mass.

Heavy Neutrino Masses | $|\eta|$ | $|m|(GeV)$ | $\textrm{Tr}(mm^{\dagger})(m_{t}^{2})$ | $|M|(m_{t})$
---|---|---|---|---
$\begin{pmatrix}M_{1}=3\ m_{t}\\\ M_{2}=(3+1\times 10^{-7})\ m_{t}\\\ \ M_{3}=9\ m_{t}\ \end{pmatrix}$ | $\begin{pmatrix}1.12\times 10^{-3}&8.92\times 10^{-6}&1.36\times 10^{-3}\\\ .&7.11\times 10^{-8}&1.08\times 10^{-5}\\\ .&.&1.65\times 10^{-3}\end{pmatrix}$ | $\begin{pmatrix}24.6&6.88\times 10^{-11}&2.71\times 10^{-5}\\\ 0.196&3.43\times 10^{-10}&1.69\times 10^{-4}\\\ 29.9&3.03\times 10^{-10}&1.49\times 10^{-4}\end{pmatrix}$ | 0.0497 | $\begin{pmatrix}9.99\times 10^{-8}&2.99&7.68\times 10^{-6}\\\ 2.99&1.99\times 10^{-11}&7.66\times 10^{-6}\\\ 7.68\times 10^{-6}&7.66\times 10^{-6}&8.99\end{pmatrix}$
$\begin{pmatrix}M_{1}=3\ m_{t}\\\ M_{2}=(3+1\times 10^{-7})\ m_{t}\\\ \ M_{3}=9\ m_{t}\ \end{pmatrix}$ | $\begin{pmatrix}5.43\times 10^{-7}&9.57\times 10^{-6}&3.18\times 10^{-5}\\\ .&1.69\times 10^{-4}&5.61\times 10^{-4}\\\ .&.&1.86\times 10^{-3}\end{pmatrix}$ | $\begin{pmatrix}0.542&1.21\times 10^{-10}&4.92\times 10^{-5}\\\ 9.57&3.69\times 10^{-10}&1.85\times 10^{-4}\\\ 31.78&3.41\times 10^{-10}&1.32\times 10^{-4}\end{pmatrix}$ | 0.0366 | $\begin{pmatrix}1.00\times 10^{-7}&2.99&8.72\times 10^{-6}\\\ 2.99&2.40\times 10^{-11}&8.70\times 10^{-6}\\\ 8.72\times 10^{-6}&8.70\times 10^{-6}&8.99\end{pmatrix}$
$\begin{pmatrix}M_{1}=3\ m_{t}\\\ M_{2}=(3+1\times 10^{-7})\ m_{t}\\\ \ M_{3}=100\ m_{t}\ \end{pmatrix}$ | $\begin{pmatrix}1.12\times 10^{-3}&8.92\times 10^{-6}&1.36\times 10^{-3}\\\ .&7.11\times 10^{-8}&1.08\times 10^{-5}\\\ .&.&1.65\times 10^{-3}\end{pmatrix}$ | $\begin{pmatrix}24.6&6.88\times 10^{-11}&5.40\times 10^{-5}\\\ 0.196&3.43\times 10^{-10}&5.61\times 10^{-4}\\\ 29.9&3.03\times 10^{-10}&6.11\times 10^{-4}\end{pmatrix}$ | 0.0497 | $\begin{pmatrix}1.00\times 10^{-7}&2.99&3.73\times 10^{-5}\\\ 2.99&1.99\times 10^{-11}&3.72\times 10^{-5}\\\ 3.73\times 10^{-5}&3.72\times 10^{-5}&99.9\end{pmatrix}$
$\begin{pmatrix}M_{1}=3\ m_{t}\\\ M_{2}=(3+1\times 10^{-7})\ m_{t}\\\ \ M_{3}=100\ m_{t}\ \end{pmatrix}$ | $\begin{pmatrix}5.43\times 10^{-7}&9.57\times 10^{-6}&3.18\times 10^{-5}\\\ .&1.69\times 10^{-4}&5.61\times 10^{-4}\\\ .&.&1.86\times 10^{-3}\end{pmatrix}$ | $\begin{pmatrix}0.542&1.21\times 10^{-10}&1.64\times 10^{-4}\\\ 9.57&3.69\times 10^{-10}&6.02\times 10^{-4}\\\ 31.8&3.41\times 10^{-10}&5.72\times 10^{-4}\end{pmatrix}$ | 0.0366 | $\begin{pmatrix}1.00\times 10^{-7}&2.99&4.23\times 10^{-5}\\\ 2.99&2.40\times 10^{-11}&4.22\times 10^{-5}\\\ 4.23\times 10^{-5}&4.22\times 10^{-5}&99.9\end{pmatrix}$
$\begin{pmatrix}M_{1}=3\ m_{t}\\\ M_{2}=9\ m_{t}\\\ \ M_{3}=(9+1\times 10^{-7})\ m_{t}\ \end{pmatrix}$ | $\begin{pmatrix}1.12\times 10^{-3}&8.92\times 10^{-6}&1.36\times 10^{-3}\\\ .&7.11\times 10^{-8}&1.08\times 10^{-5}\\\ .&.&1.65\times 10^{-3}\end{pmatrix}$ | $\begin{pmatrix}73.8&6.89\times 10^{-11}&2.73\times 10^{-5}\\\ 0.589&3.43\times 10^{-10}&9.70\times 10^{-5}\\\ 89.6&3.03\times 10^{-10}&1.33\times 10^{-4}\end{pmatrix}$ | 0.447 | $\begin{pmatrix}1.00\times 10^{-7}&8.97&2.01\times 10^{-5}\\\ 8.97&1.85\times 10^{-11}&8.85\times 10^{-6}\\\ 2.01\times 10^{-5}&8.85\times 10^{-6}&2.99\end{pmatrix}$
$\begin{pmatrix}M_{1}=3\ m_{t}\\\ M_{2}=9\ m_{t}\\\ \ M_{3}=(9+1\times 10^{-7})\ m_{t}\ \end{pmatrix}$ | $\begin{pmatrix}5.23\times 10^{-7}&9.22\times 10^{-6}&3.06\times 10^{-5}\\\ .&1.62\times 10^{-4}&5.41\times 10^{-4}\\\ .&.&1.80\times 10^{-3}\end{pmatrix}$ | $\begin{pmatrix}1.60&1.23\times 10^{-10}&2.85\times 10^{-5}\\\ 28.2&3.75\times 10^{-10}&1.02\times 10^{-4}\\\ 93.6&3.47\times 10^{-10}&1.29\times 10^{-4}\end{pmatrix}$ | 0.317 | $\begin{pmatrix}1.00\times 10^{-7}&8.98&2.40\times 10^{-5}\\\ 8.98&2.49\times 10^{-11}&1.04\times 10^{-5}\\\ 2.40\times 10^{-5}&1.04\times 10^{-5}&2.99\end{pmatrix}$

Table 2.4: Examples of deviations from unitarity for the case of IO are
provided for light neutrino masses of $m_{1}=0.05064\leavevmode\nobreak\ eV$ ,
$m_{2}=0.05138\leavevmode\nobreak\ eV$ , $m_{3}=0.00864\leavevmode\nobreak\
eV$. These deviations are quantified by the Hermitian matrix $\eta$. Three
different hierarchies are considered for the heavy Majorana neutrino masses
$M_{i}$, and for each hierarchy, two examples are given by varying the choice
of the matrix $X$. The third column shows the Dirac-type neutrino mass matrix
in GeV units, and the fifth column displays the Majorana-type neutrino mass
matrix in units of the top quark mass. The fourth column presents the trace of
the product $mm^{\dagger}$ in units of the squared top quark mass.

Heavy Neutrino Masses | $|\eta|$ | $|m|(GeV)$ | $\textrm{Tr}(mm^{\dagger})(m_{t}^{2})$ | $|M|(m_{t})$
---|---|---|---|---
$\begin{pmatrix}M_{1}=3\ m_{t}\\\ M_{2}=(3+1\times 10^{-7})\ m_{t}\\\ \ M_{3}=9\ m_{t}\ \end{pmatrix}$ | $\begin{pmatrix}1.18\times 10^{-3}&1.00\times 10^{-5}&1.20\times 10^{-3}\\\ .&8.50\times 10^{-8}&1.02\times 10^{-5}\\\ .&.&1.22\times 10^{-3}\end{pmatrix}$ | $\begin{pmatrix}25.2&5.41\times 10^{-10}&1.67\times 10^{-4}\\\ 0.215&2.51\times 10^{-10}&1.59\times 10^{-4}\\\ 25.8&3.22\times 10^{-10}&1.64\times 10^{-4}\end{pmatrix}$ | 0.0431 | $\begin{pmatrix}1.00\times 10^{-7}&2.99&1.24\times 10^{-7}\\\ 2.99&5.01\times 10^{-11}&1.23\times 10^{-7}\\\ 1.24\times 10^{-7}&1.23\times 10^{-7}&8.99\end{pmatrix}$
$\begin{pmatrix}M_{1}=3\ m_{t}\\\ M_{2}=(3+1\times 10^{-7})\ m_{t}\\\ \ M_{3}=9\ m_{t}\ \end{pmatrix}$ | $\begin{pmatrix}1.33\times 10^{-7}&5.36\times 10^{-6}&1.09\times 10^{-5}\\\ .&2.16\times 10^{-4}&4.38\times 10^{-4}\\\ .&.&8.85\times 10^{-4}\end{pmatrix}$ | $\begin{pmatrix}0.269&9.83\times 10^{-10}&1.63\times 10^{-4}\\\ 10.8&3.19\times 10^{-10}&1.17\times 10^{-4}\\\ 21.91&2.47\times 10^{-10}&9.96\times 10^{-6}\end{pmatrix}$ | 0.0198 | $\begin{pmatrix}1.00\times 10^{-7}&2.99&7.49\times 10^{-6}\\\ 2.99&4.04\times 10^{-11}&7.48\times 10^{-6}\\\ 7.49\times 10^{-6}&7.48\times 10^{-6}&8.99\end{pmatrix}$
$\begin{pmatrix}M_{1}=3\ m_{t}\\\ M_{2}=(3+1\times 10^{-7})\ m_{t}\\\ \ M_{3}=100\ m_{t}\ \end{pmatrix}$ | $\begin{pmatrix}1.16\times 10^{-3}&9.90\times 10^{-6}&1.19\times 10^{-3}\\\ .&8.43\times 10^{-8}&1.01\times 10^{-5}\\\ .&.&1.21\times 10^{-3}\end{pmatrix}$ | $\begin{pmatrix}25.1&5.43\times 10^{-10}&5.56\times 10^{-4}\\\ 0.214&2.52\times 10^{-10}&5.31\times 10^{-4}\\\ 25.6&3.24\times 10^{-10}&5.47\times 10^{-4}\end{pmatrix}$ | 0.0428 | $\begin{pmatrix}1.00\times 10^{-7}&2.99&3.20\times 10^{-7}\\\ 2.99&5.05\times 10^{-11}&3.19\times 10^{-7}\\\ 3.20\times 10^{-7}&3.19\times 10^{-7}&100\end{pmatrix}$
$\begin{pmatrix}M_{1}=3\ m_{t}\\\ M_{2}=(3+1\times 10^{-7})\ m_{t}\\\ \ M_{3}=100\ m_{t}\ \end{pmatrix}$ | $\begin{pmatrix}1.33\times 10^{-7}&5.36\times 10^{-6}&1.09\times 10^{-5}\\\ .&2.16\times 10^{-4}&4.38\times 10^{-4}\\\ .&.&8.85\times 10^{-4}\end{pmatrix}$ | $\begin{pmatrix}0.269&9.83\times 10^{-10}&5.44\times 10^{-4}\\\ 10.8&3.19\times 10^{-10}&4.31\times 10^{-4}\\\ 21.9&2.47\times 10^{-10}&1.13\times 10^{-4}\end{pmatrix}$ | 0.0198 | $\begin{pmatrix}1.00\times 10^{-7}&3.00&3.63\times 10^{-5}\\\ 3.00&4.04\times 10^{-11}&3.63\times 10^{-5}\\\ 3.63\times 10^{-5}&3.63\times 10^{-5}&99.9\end{pmatrix}$
$\begin{pmatrix}M_{1}=3\ m_{t}\\\ M_{2}=9\ m_{t}\\\ \ M_{3}=(9+1\times 10^{-7})\ m_{t}\ \end{pmatrix}$ | $\begin{pmatrix}1.11\times 10^{-3}&9.46\times 10^{-6}&1.14\times 10^{-3}\\\ .&8.05\times 10^{-8}&9.65\times 10^{-6}\\\ .&.&1.16\times 10^{-3}\end{pmatrix}$ | $\begin{pmatrix}73.7&5.56\times 10^{-10}&9.63\times 10^{-5}\\\ 0.626&2.58\times 10^{-10}&9.20\times 10^{-5}\\\ 75.2&3.31\times 10^{-10}&9.48\times 10^{-5}\end{pmatrix}$ | 0.367 | $\begin{pmatrix}1.00\times 10^{-7}&8.98&1.68\times 10^{-7}\\\ 8.98&5.29\times 10^{-11}&6.42\times 10^{-8}\\\ 1.68\times 10^{-7}&6.42\times 10^{-8}&2.99\end{pmatrix}$
$\begin{pmatrix}M_{1}=3\ m_{t}\\\ M_{2}=9\ m_{t}\\\ \ M_{3}=(9+1\times 10^{-7})\ m_{t}\ \end{pmatrix}$ | $\begin{pmatrix}1.32\times 10^{-7}&5.34\times 10^{-6}&1.08\times 10^{-5}\\\ .&2.15\times 10^{-4}&4.35\times 10^{-4}\\\ .&.&8.80\times 10^{-4}\end{pmatrix}$ | $\begin{pmatrix}0.804&9.85\times 10^{-10}&9.54\times 10^{-5}\\\ 32.4&3.20\times 10^{-10}&1.27\times 10^{-4}\\\ 65.5&2.48\times 10^{-10}&1.25\times 10^{-4}\end{pmatrix}$ | 0.177 | $\begin{pmatrix}1.00\times 10^{-7}&8.99&2.43\times 10^{-5}\\\ 8.99&4.06\times 10^{-11}&2.08\times 10^{-5}\\\ 2.43\times 10^{-5}&2.08\times 10^{-5}&2.99\end{pmatrix}$

Note that at least two of the heavy neutrinos must be quasi-degenerate. This
is due to the light neutrino mass one-loop corrections. In order for these not
to be too large, a lepton-number-like $U(1)$ symmetry must be present and
softly broken. This will be discussed in more detail in the next section.

###  2.2 The Non-Decoupled Parameter Space: $M\sim$ eV/keV

In this section, a study of a region of the parameter space where some heavy
neutrinos have eV and keV mass is done, based in ref. [31]. This region is
interesting because of the anomalies found in Short-baseline experiments, that
may be hinting at the existence of an eV almost sterile neutrino [60, 61, 62,
63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73]. One exciting feature of this
region of the parameter space is also that the deviations from $3\times 3$
unitarity can be much larger than the region studied in the previous section,
avoiding the bounds from ref. [52], as discussed in [57], which enables the
detection of its effects in current and future neutrino oscillation
experiments.

In the presence of deviations from unitarity, the probabilities of neutrino
oscillations are altered, as documented in refs. [54, 57]. If $n$ heavy
neutrinos can be produced in the beam used in oscillation experiments, the
calculation of oscillation probabilities involves a $3\times(3+n)$ matrix,
$\Theta$, which is a submatrix of $\mathcal{V}$ defined in equation (2.23),

$\Theta\,=\,\left(\begin{array}[]{cc}K&(KX^{\dagger})_{3\times
n}\end{array}\right)\,.$ (2.46)

$\Theta$ is the matrix defined by the first $3+n$ columns of $A$, defined in
eq. (2.16). The probability of transition between flavour (anti-)neutrinos
[-.7pt]$\nu$ ( – ) α and [-.7pt]$\nu$ ( – ) β, or of survival for a given
flavour ($\alpha=\beta$), with $\alpha,\beta=e,\mu,\tau$, can be shown to take
the form

$\displaystyle
P_{\stackon[-.7pt]{$\nu$}{\scalebox{0.3}{(}\raisebox{-2.0pt}{\bf--}\scalebox{0.3}{)}}_{\alpha}\rightarrow\stackon[-.7pt]{$\nu$}{\scalebox{0.3}{(}\raisebox{-2.0pt}{\bf--}\scalebox{0.3}{)}}_{\beta}}(L,E)\,=\,\frac{1}{(\Theta\Theta^{\dagger})_{\alpha\alpha}(\Theta\Theta^{\dagger})_{\beta\beta}}\Bigg{[}\left|(\Theta\Theta^{\dagger})_{\alpha\beta}\right|^{2}$
$\displaystyle-4\sum_{i>j}^{3+n}\,\operatorname{Re}\left(\Theta_{\alpha
i}^{*}\,\Theta_{\beta i}\,\Theta_{\alpha j}\,\Theta_{\beta
j}^{*}\right)\sin^{2}\Delta_{ij}$ (2.47) $\displaystyle\pm
2\sum_{i>j}^{3+n}\,\operatorname{Im}\left(\Theta_{\alpha i}^{*}\,\Theta_{\beta
i}\,\Theta_{\alpha j}\,\Theta_{\beta j}^{*}\right)\sin
2\Delta_{ij}\Bigg{]}\,,$

where the plus or minus sign in the second line refers to neutrinos or anti-
neutrinos, respectively. Here, $L$ denotes the source-detector distance, $E$
is the (anti-)neutrino energy, and one has defined $\begin{aligned}
\Delta_{ij}\,\equiv\,\frac{\Delta
m^{2}_{ik}\,L}{4E}\,\simeq\,1.27\,\frac{\Delta
m_{ij}^{2}[\text{eV}^{2}]\,L[\text{km}]}{E[\text{GeV}]}\,,\end{aligned}$ with
mass-squared differences $\Delta m_{ij}^{2}\equiv m_{i}^{2}-m_{j}^{2}$, as
usual.

The equation (2.47) is a formula for the probability of neutrino oscillation,
which takes into account deviations from a unitary mixing matrix. When $n=3$,
the equation reduces to the usual formula for unitary mixing matrices. The
normalization term in the equation will be close to unity in all cases, and
the term proportional to $|(\Theta\Theta^{\dagger})_{\alpha\beta}|^{2}$ is
referred to as the "zero-distance" term, which is a result of deviations from
$3\times 3$ unitarity. This term will also be negligible. The equation can be
simplified for specific cases, such as short-baseline (SBL) and long-baseline
(LBL) experimental setups. However, matter effects must be considered in LBL
experiments.

The SBL anomalies is the name given to the consistent measurements, in the
last few decades, of anomalous excesses/deficits of
$\nu_{e}$/$\overline{\nu}_{e}$ at $L/E\sim 1\leavevmode\nobreak\
\textrm{m/MeV}$ from $\nu_{\mu}/$ $\overline{\nu}_{\mu}$ and $\nu_{e}/$
$\overline{\nu}_{e}$ sources. It first started with the LSND experiment [74,
75], which observed an excess of $\overline{\nu}_{e}$ at the detector, from a
$\overline{\nu}_{\mu}$ source. Given the $L/E$ settings of the detector, this
result pointed towards a large $\Delta m^{2}$, if interpreted as a two
neutrino oscillation $\overline{\nu}_{\mu}\xrightarrow{}\overline{\nu}_{e}$.
Some years after the LSND results came the Reactor anomaly [76, 77] and the
Radioactive Source anomalies [78]. These experiments operated in nuclear
reactors, NEOS and DANSS, or used radioactive $\nu_{e}$ sources, GALLEX and
SAGE, and measured a deficit of $\nu_{e}$ or $\overline{\nu}_{e}$ in the
detector. Recently, the MiniBOONE experiment, which has about the same $L/E$
as LSND, but has different energy, beam and detector systematics and different
event signatures and backgrounds, obtained results consistent with the ones
from LSND [79, 80]. Combining all other experimental constraints together with
the results from these experiments seems to indicate a preference for $(3+1)$
scenario, where one has 3 light neutrinos and 1 $eV$ mass neutrino [72].
Although this is the case, it is important to note that a closer examination
on the data set reveals tension between the appearence and disappearence
datasets and the neutrino and anti-neutrino datasets. [81].

####  2.2.1 Deviations from Unitarity and Loop Corrections

These anomalies may be explained if the spectra of heavy neutrinos includes at
least one eV mass neutrino and the deviations from $3\times 3$ unitarity (DU)
are sizeable. These DU need to be larger than in the previous studied region
of the parameter space. Thus, they must violate the experimental bounds from
ref. [52], cf. eq. (2.35). This can be done without much effort because the
bounds from ref. [52] are only valid when all heavy neutrinos have a mass
larger than the W boson mass, $M_{W}$. Hence, constraints on the entries of
$\eta$ depend on the mass scale of the new neutrinos.

Therefore, to limit the parameter space of spectra with light heavy neutrinos,
different sets of experimental constraints must be considered based on the
spectrum. For light heavy neutrinos with eV-scale masses, the strongest
constraints come from oscillation experiments [57]. At the time of writing of
ref. [31], the most stringent bounds came from BUGEY-3 [82], MINOS [83], NOMAD
[84, 85] and Super-Kamiokande [86]. In those cases, observables constrain
directly the entries of $KX^{\dagger}$, and not just the product
$KX^{\dagger}XX^{\dagger}$. To take constraints from these experiments, the
relevant exclusion curves in the $\sin^{2}2\vartheta_{\alpha\beta}$ – $\Delta
m^{2}$ planes are considered and translated into constraints on the elements
of the mixing matrix block $KX^{\dagger}$. For keV or GeV-TeV light heavy
neutrinos, $\beta$-decay experiments (see e.g. [87] and references within) and
from LHC searches for heavy Majorana neutrinos [88, 89, 90, 91, 92, 93, 94]
should be taken into account. Another crucial experimental input, also taken
into account in this analysis, is the limit on the $\mu\rightarrow e\gamma$
branching ratio obtained by the MEG Collaboration, $BR(\mu\rightarrow
e\gamma)<4.2\times 10^{-13}$ ($90\%$ CL) [95], as it is one of the strictest
bounds on lepton flavour violation. For the MeV-GeV intermediate mass range,
constraints can be found in [96].

For more details regarding new physics, effects on physical observables, the
best experimental constraints and the most stringent bounds on a given spectra
of heavy almost-sterile neutrinos see section 4.1.

Deviations from $3\times 3$ unitarity can cause problems when calculations are
done at one loop. At one loop, the $\mathbf{0}_{3\times 3}$ matrix in the
upper-left corner of $\mathcal{M}$, defined in eq. (2.4), is no longer a null
matrix and is corrected. These corrections can be very significant and
increase the light neutrino masses at tree level by orders of magnitude. The
one-loop corrected matrix is given by

$\delta M_{L}\,=\,\delta M_{L}^{Z}+\delta M_{L}^{H}\leavevmode\nobreak\ ,$
(2.48)

where $\delta M_{L}^{Z}$ and $\delta M_{L}^{H}$ represent contributions
depending on the $Z$ and Higgs boson masses, $m_{Z}$ and $m_{H}$,
respectively. Explicitly, one has (see also Appendix A of Ref. [97]):

$\displaystyle\delta M_{L}^{Z}$
$\displaystyle\,=\,\frac{3}{32\pi^{2}\,v^{2}}\,\left(\begin{array}[]{cc}K&KX^{\dagger}\end{array}\right)\,\frac{\mathcal{D}^{3}}{\mathcal{D}^{2}/{m_{Z}^{2}}-\mathds{1}}\,\log\left(\frac{\mathcal{D}^{2}}{m_{Z}^{2}}\right)\,\left(\begin{array}[]{c}K^{T}\\\
(KX^{\dagger})^{T}\end{array}\right)\,,$ (2.49) $\displaystyle\delta
M_{L}^{H}$
$\displaystyle\,=\,\frac{1}{32\pi^{2}\,v^{2}}\,\left(\begin{array}[]{cc}K&KX^{\dagger}\end{array}\right)\,\frac{\mathcal{D}^{3}}{\mathcal{D}^{2}/{m_{H}^{2}}-\mathds{1}}\,\log\left(\frac{\mathcal{D}^{2}}{m_{H}^{2}}\right)\,\left(\begin{array}[]{c}K^{T}\\\
(KX^{\dagger})^{T}\end{array}\right)\,,$

in a generic weak basis, with $v\simeq 174$ GeV being the Higgs VEV and with
$\mathcal{D}$, $K$ and $KX^{\dagger}$ given in Eqs. (2.5) and (2.15). This
result can be cast in a simple form:

$\displaystyle\delta
M_{L}\,=\,K\,f(d)\,K^{T}+KX^{\dagger}\,f(D)\,(KX^{\dagger})^{T}\,,$ (2.50)

where naturally $f$ is applied element-wise to diagonal matrices, with

$\displaystyle
f(m)\,\equiv\,\frac{m^{3}}{(4\pi\,v)^{2}}\left(\frac{3\log(m/m_{Z})}{m^{2}/m_{Z}^{2}-1}+\frac{\log(m/m_{H})}{m^{2}/m_{H}^{2}-1}\right)\,.$
(2.51)

The following section describes the steps taken to obtain numerical results
that are accurate to one-loop corrections. The tree-level form of
$\mathcal{M}$ in Eq.(2.4) can either be derived from a specific model or
constructed using the matrices $d$, $D$, $V$, and $O_{c}$, as shown in Table
2.5. These matrices, along with $m_{l}$, encompass the entire physical
parameter space.

| $\,\,\,\,m_{l}\,\,\,\,$ | $\,\,\,\,d\,\,\,\,$ | $\,\,\,\,D\,\,\,\,$ | $\,\,\,\,O_{c}\,\,\,\,$ | $\,\,\,\,V\,\,\,\,$ | Total
---|---|---|---|---|---|---
Moduli | 3 | 3 | 3 | 3 | 3 | 15 = 9 + 6
Phases | $-3$ | 0 | 0 | 3 | 6 | 6
Table 2.5:  The counting of physical parameters in a type-I seesaw scenario
with three sterile neutrinos involves 15 variables. These include 9 lepton
masses (3 for charged leptons and 6 for neutrinos) and 6 mixing angles. There
are also 6 physical phases, taking into account that rephasing the charged
leptons can eliminate 3 phases from the matrix $V$. It’s worth mentioning that
in the basis under consideration, $m_{l}$ is real and diagonal.

It is then straightforward to obtain $K$ and $X$. One can then compute $\delta
M_{L}$ and proceed to diagonalise

$\displaystyle\mathcal{M}^{\text{1-loop}}\simeq\left(\begin{array}[]{cc}\delta
M_{L}&m\\\ m^{T}&M\end{array}\right)\,,$ (2.54)

obtaining the loop-corrected $K$, $X$, $d$ and $D$. It is reasonable to
neglect one-loop corrections to $m$ and $M$ [98, 99].

The mass loop-corrections can be controlled by incorporating a lepton-number-
like symmetry and softly breaking it. Light sterile neutrinos can emerge
naturally in a seesaw framework with an approximately conserved lepton number
[100, 101, 102]. An exact $U(1)_{L}$ symmetry imposes specific textures on the
mass matrices $m$ and $M$, which can be slightly altered when the symmetry is
only approximate, allowing for non-zero Majorana neutrino masses and non-
trivial mixing.

####  2.2.2 Benchmarks

The goal of this subsection is to examine the parameter space where at least
one mostly-sterile neutrino is light, with a mass of $\mathcal{O}$(eV), to
establish a connection to the Short-Baseline (SBL) anomalies. In order to
achieve Yukawa couplings that are of order one, one must have substantial
deviations from $3\times 3$ unitarity, as explained in the end of section 2.1.
It is interesting to note that these large deviations from unitarity are
necessary to connect to the SBL anomalies. All of this is only possible in the
presence of a softly broken lepton-number-like $U(1)$ symmetry. The choice of
lepton charges should then be such that, in the limit of exact conservation:

  1. 1.

$M$ has zero determinant.

  2. 2.

Not all entries of $m$ are small. One needs a couple of $\mathcal{O}(1)$
Yukawa couplings.

The available choices for $U(1)_{L}$ charges are restricted by these
conditions. The study in the manner of reference [15] is carried out in a
specific ‘symmetry’ weak basis where the lepton charge vectors $\lambda_{\nu}$
and $\lambda_{L}$ are assigned to the three right-handed neutrino singlets and
the three lepton doublets respectively. As $\overline{\\!K}\,$ is not included
in the Lagrangian and thus not a physical matrix, in many cases it is
convenient to set $U_{\overline{\\!K}\,}$ equal to the identity matrix.
Nevertheless, when textures are considered, it is typical to have
$U_{\overline{\\!K}\,}\neq\mathds{1}$ as specified in equation (2.21). In the
exact conservation limit, there are only four non-trivial choices of
$U(1)_{L}$ charges for permutations that result in an $M$ matrix with a zero
determinant: $\lambda_{\nu}=(1,1,0)$, $\lambda_{\nu}=(1,-1,-1)$,
$\lambda_{\nu}=(1,1,1)$ and $\lambda_{\nu}=(0,0,1)$. Out of these four
options, the choice of $\lambda_{\nu}=(1,1,1)$ is not feasible as it leads to
an $M$ matrix equal to zero, and $\lambda_{\nu}=(0,0,1)$ is disregarded since
the requirement for controlled loop corrections within the framework
effectively reduces it to the case with $\lambda_{\nu}=(1,-1,-1)$. The
remaining two options $\lambda_{\nu}=(1,1,0)$ and $\lambda_{\nu}=(1,-1,-1)$
are used in what follows. The choice of $\lambda_{L}$ is determined based on
the conditions that the seesaw mechanism is effective for all light neutrinos
and that all left-handed neutrinos are able to couple with the right-handed
ones, as stated in [15]. The numerical benchmarks in the study described in
section 2.1 and based on ref. [30] were obtained with the $U(1)_{L}$ charge
assignment $\lambda_{\nu}=(1,-1,0)$ and $\lambda_{L}=(1,1,1)$. However, this
leads to $\det,M\neq 0$ in the symmetric limit. Given these considerations,
the following study is divided into three possibilities: Ia , Ib and II.

For each, an exploration of the parameter space of qualitatively similar
seesaw structures will be done, providing an illustrative and representative
numerical benchmark in the end. By approaching it in this manner, the
possibility of a significant active-sterile mixing only within a finely tuned
region of parameter space is ruled out. As anticipated in the beginning of
this section, section 2.2, approximate forms of the transition probabilities
of muon to electron (anti-)neutrinos,
$P_{\stackon[-.7pt]{$\nu$}{\scalebox{0.3}{(}\raisebox{-2.0pt}{\bf--}\scalebox{0.3}{)}}_{\mu}\rightarrow\stackon[-.7pt]{$\nu$}{\scalebox{0.3}{(}\raisebox{-2.0pt}{\bf--}\scalebox{0.3}{)}}_{e}}$,
obtained from Eq. (2.47) while having in mind SBL and LBL setups, for each of
the three scenarios will be provided. Considering the most recent global fits
available at the time [103, 104], which strongly disfavor a light neutrino
mass spectrum with inverted ordering (IO) compared to one with normal ordering
(NO) beyond a $3\sigma$ significance level, the numerical examples are
confined to the case of normal ordering (NO).

Before proceeding, it is important to note that the three scenarios of
interest bear some resemblance to the commonly considered $3+1+1$ (case Ia ),
$3+2$ (case Ib ), and $3+1$ (case II) schemes, as discussed in ref. [72].
While the connection to the latter schemes is not exact, especially in the
case of scenario Ib which does not have a typical $3+2$ spectrum, it could
still be beneficial to consider quantities defined within those schemes in the
analysis, as outlined in [105].

$\displaystyle\sin^{2}2\vartheta^{(k)}_{\mu e}\,\equiv\,4\,\big{|}\Theta_{\mu
k}\big{|}^{2}\,\big{|}\Theta_{ek}\big{|}^{2}\,,$ (2.55)

with $k=4$ in the$3+1$ case, while $k=4,5$ for the other two cases. According
to the global fit to SBL data of Ref. [105], explaining the observed anomalies
requires $\Delta m^{2}_{41}\in[0.87,\,2.04]$ eV2 and
$\sin^{2}2\vartheta^{(4)}_{\mu e}\in[6.5\times 10^{-4},\,2.6\times 10^{-3}]$
($99.7\%$ CL) in the$3+1$ scheme. This result may also be of relevance in
the$3+1+1$ scheme. These intervals are taken as guidelines in the numerical
explorations, since they encompassed the consensus of global-fits at the time
of writing of ref. [31]. Thus, the sterile neutrino parameter space is
constrained through the conservative bounds $\sum_{i}|R_{\alpha i}|^{2}<0.1$
($\alpha=e,\mu,\tau$), and via the constraints of [84, 85, 87] on mixing
matrix elements corresponding to large mass-squared differences $\Delta
m^{2}\sim 10\text{ eV}^{2}-1\text{ keV}^{2}$, as anticipated in subsection
2.2.1.

#####  2.2.2.1 Case I: $\lambda_{\nu}=(1,1,0)$

For this case, the only sensible choice for the doublet charges is
$\lambda_{L}=(0,0,0)$. The mass matrices in the symmetric limit read:

$\displaystyle m\,=\,\left(\begin{array}[]{ccc}0&0&a\\\ 0&0&b\\\
0&0&c\end{array}\right)\,,\quad M\,=\,\left(\begin{array}[]{ccc}0&0&0\\\
0&0&0\\\ 0&0&M_{3}\end{array}\right)\,.$ (2.62)

Breaking the symmetry generates the light neutrino masses, two (mainly)
sterile states with masses $M_{1}$ and $M_{2}$ that can be significantly
smaller than $M_{3}$, and a heavy sterile with a mass close to $M_{3}$.

As expected, some Yukawa couplings remain at $\mathcal{O}(1)$, which can be
explained by the dependence of the Dirac mass matrix $m$ on the sterile masses
contained in $D$, as seen in Eq. (2.36).

This scenario is further divided into two subcases: a scenario with a
hierarchy of $M_{2}\gg M_{1}$ (case Ia), which may occur in a stepwise
symmetry breaking scenario, or a scenario with a single new light-sterile
scale, with $M_{1}\sim M_{2}$ (case Ib).

Case Ia: $M_{1}\ll M_{2}\ll M_{3}$

|  Case Ia numerical benchmark
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