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A Mario, Joana, Alberto e Ogeni,

a quem devo quase tudo,

obrigado.

Resumo

O Modelo Padrão (MP) foi a primeira teoria completa (excluída a gravidade),
fundamental da natureza a ser estabelecida. A conquista dessa posição se deve
às previsões revolucionárias, eventualmente confirmadas, surgidas do modelo,
além de sua extraordinária capacidade de acomodar dados de precisão e de
inédita complexidade. Por outro lado, o MP é incapaz de explicar algumas
observações universais – além de apresentar características incômodas. Em
decorrência disso, nova física é necessária, o que, no reino de altas
energias, com poucas exceções, implica em novas partículas. Nesta tese
exploraremos uma tal espécie exótica: o bóson vetorial (ou, simplesmente,
vetor) bileptônico duplamente carregado. Em especial, focamos em seus efeitos
como mediador de Violação de Sabor Leptônico Carregado, o que, por sua vez,
pode representar uma ‘smoking gun’ para sua descoberta ou para a limitação de
seu espaço de parâmetros. Esse tipo de processo ainda não foi ceticamente
explorado, e a mistura leptônica em sua interação com o $U^{\pm\pm}$ não foi
alvo de consideração focada.

Nosso primeiro passo é investigar que limites um processo trimuônico, no LHC,
pode impor no espaço de parâmetros dessa partícula. Os resultados nos
motivaram a buscar por dados de origens alternativas que pudessem, também,
gerar limites úteis. Os sete canais de decaimento leptônico de três corpos se
mostraram ótimos candidatos, e efetuamos uma análise detalhada e independente
de modelo do que eles podem implicar sobre o $U^{\pm\pm}$ e outras duas
espécies exóticas. Atenção especial é dada às interferências relevantes entre
as novas contribuições.

Por fim, seguimos para construir uma análise profunda da evolução dos
acoplamentos do Modelo 3-3-1, principal teoria a conter um vetor bileptônico,
inspecionando de que modo ela é ameaçada por polos de Landau no seu
acoplamento abeliano. Encontramos explicitamente as contribuições de 1-loop
das partículas exóticas aos runnings, obtendo que o regime perturbativo do
modelo é, de fato, consideravelmente maior do que usualmente suposto.

Durate a produção desta tese, o autor contribui para os seguintes trabalhos
originais:

  * •

M.W. Barela and V. Pleitez, Trimuon production at the LHC, Phys. Rev. D
101(2020) 015024.

  * •

M.W. Barela and J. Montaño Domínguez, Constraints on exotic particle masses
from flavor violating charged lepton decays and the role of interference,
Phys.Rev. D 106 (2022) 055013.

  * •

M.W. Barela, On the 3-3-1 Landau pole, Nuclear Physics, Section B, 116475
(2024).

  * •

M.W. Barela and R. Capdevilla, Di-Higgs Signatures in Neutral Naturalness, J.
High Energ. Phys. 2024, 50 (2024).

Este texto é baseado, principalmente, nos três primeiros.

Palavras-chaves: Física de Partículas; Extensões do Modelo Padrão; Vetor
Bileptônico Duplamente Carregado; Modelo 3-3-1; Violação de Sabor Leptônico
Carregado.

Áreas do conhecimento: Física de partículas; Física de altas energias;
Fenomenologia do LHC.

Abstract

The Standard Model stands as the first established, complete (gravity aside),
fundamental theory of nature. This position was merited by the confirmation of
many revolutionary predictions the model implied, besides its ability to fit
high complexity precision data to an extreme degree. In the other hand, the
Standard Model is unable to account for some universal observations – besides
possessing a number of undesirable characteristics. Thus, there is a pressing
need for new physics, which, in the high energy realm, usually imply new
particles. In this thesis, we delve into one such particle: the doubly-charged
vector bilepton. This particle is a singular feature of Beyond the Standard
Model theories and its phenomenology is still incipient. We focus, in
particular, on its power as a mediator of Charged Lepton Flavour Mediation, a
type of process which, in turn, may represent a smoking gun with respect to
its discovery or constraining of its parameters. Such processes have not been
skeptically explored, and the lepton mixing in their interaction with the
$U^{\pm\pm}$ not been properly focused on.

The first step we take is investigating what bounds an LHC trimuon process may
impose on the parameter space of this particle. The results motivate us to
look for alternative experimental sources of limits. The seven 3-body lepton
decay channels delivered a great prospect, and we perform a detailed, model
independent analysis of what they can imply for the $U^{\pm\pm}$ and other two
exotic species. Special attention is paid to occurring interferences between
new contributions.

Finally, we proceed to construct a thorough analysis of the evolution of
couplings in the Minimal 3-3-1 Model, the main theory incorporating a vector
bilepton, examining how it is threatened by Landau poles in its abelian
coupling. We explicitly obtain the 1-loop contributions of exotic particles to
the runnings, finding that the perturbative regime of the model is in fact
considerably larger than conventionally assumed.

During the production of this thesis, the author have contributed to the
following original works:

  * •

M.W. Barela and V. Pleitez, Trimuon production at the LHC, Phys. Rev. D
101(2020) 015024.

  * •

M.W. Barela and J. Montaño Domínguez, Constraints on exotic particle masses
from flavor violating charged lepton decays and the role of interference,
Phys.Rev. D 106 (2022) 055013.

  * •

M.W. Barela, On the 3-3-1 Landau pole, Nuclear Physics, Section B, 116475
(2024).

  * •

M.W. Barela and R. Capdevilla, Di-Higgs Signatures in Neutral Naturalness, J.
High Energ. Phys. 2024, 50 (2024).

This text is mainly based on the first three.

Key-words: Particle Physics; Beyond the Standard Model; Doubly-charged Vector
Bilepton; 3-3-1 Model; Charged Lepton Flavour Violation.

Branches of knowledge: Particle physics; High energy physics; LHC
phenomenology.

Agradecimentos

À minha família, por tudo. Amo vocês.

Ao meu orientador, Vicente Pleitez, pelo conhecimento, gentileza e
compreensão.

Ao meu colaborador, Rodolfo Capdevilla, um co-orientador espontâneo, por tudo
que me ensinou.

Aos meus amigos – Gui, Junior, Polar, Vitor, Gadeia, Lucas e João Paulo –,
que, a maioria desde a infância, integram o conjunto (de baixíssima
cardinalidade) das coisas que importam.

À Emanuele, pelo amor, pelo presente e pelo futuro.

E, principalmente, a Deus, pelas graças não merecidas e, em especial, por não
ter desistido deste filho.

E ao CNPq, pelo suporte financeiro.

Notations and conventions

  * •

Throughout this thesis, repeated Lorentz indices are always summed, as usual:

$a_{\mu}b^{\mu}\coloneqq\sum_{\mu=1}^{4}a_{\mu}b^{\mu}.$

Non-Lorentz indices are sometimes implicitly summed, which should be clear
from the context. The choice between matrix and component notation is also
made at convenience and in an obvious manner.

  * •

We use a mostly minus signature for the metric:

$\eta_{\mu\nu}=\mathrm{diag}\left(+1,-1,-1,-1\right).$

  * •

The gauge covariant derivative is written with a minus sign:

$D_{\mu}=\partial_{\mu}-igA_{\mu}^{a}T_{a}.$

This is consistent with the following definitions for the field strength
tensor,

$F_{\mu\nu}\equiv\frac{i}{g}\left[D_{\mu},D_{\nu}\right]=\partial_{\mu}A_{\nu}^{a}-\partial_{\nu}A_{\mu}^{a}+gf^{abc}A_{\mu}^{b}A_{\nu}^{c},$

and the transformation rules,

$\begin{split}\Psi&\to U\Psi,\\\ A_{\mu}&\to
UA_{\mu}U^{-1}-\frac{i}{g}(\partial_{\mu}U)U^{-1}.\end{split}$

for $\Psi$ in the fundamental and $A_{\mu}$ in the adjoint representation.

  * •

As for the Standard Model, we normalize the weak hypercharge without the
traditional 1/2 factor, i.e., the right-handed charged lepton singlets have
$Y=-1$. Moreover, the vacuum expectation value of the Higgs doublet, sometimes
denoted $v_{W}$ and other times $v_{H}$, is given by

$v_{H}=\frac{|\mu|}{\sqrt{\lambda}}\approx$246\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$,$

without the factor of $1/\sqrt{2}$ that is occasionally implemented.

  * •

Finally (and of little importance), we are idiosyncratic with regard to
relation signs, using the ones below in the corresponding situations:

    * –

$\coloneqq$ is used when a defined property is being explained;

    * –

$\equiv$ is used to define a symbol;

    * –

$\approx$ is used to denote a numerical approximation;

    * –

$\simeq$ symbolizes an analytical approximation;

    * –

$\sim$ denotes a loose similarity, be it depicting a functional dependence or
a numerical proximity or order of magnitude;

    * –

$\propto$ signifies a proportionality.

###### Contents

  1. 1 Introduction
  2. 2 The Standard Model
    1. 2.1 History and development
    2. 2.2 Review
  3. 3 Flaws of the Standard Model
    1. 3.1 Neutrino masses
    2. 3.2 Hierarchy Problem
    3. 3.3 Strong $\mathit{CP}$ problem
    4. 3.4 Dark matter
    5. 3.5 Matter-Antimatter assymetry
  4. 4 The Minimal 3-3-1 Model
    1. 4.1 Particles, particles, particles
    2. 4.2 3-3-1 Model: original motivations
    3. 4.3 The representation content of the minimal version
    4. 4.4 Spontaneous symmetry breaking
    5. 4.5 Vector boson eigenstates and masses
    6. 4.6 Yukawa interactions and fermion masses
    7. 4.7 Anomaly cancellation
    8. 4.8 Closing the m331 numerically at the electroweak scale
  5. 5 LHC phenomenology of flavour violating $U^{\pm\pm}$ processes
    1. 5.1 Review of existing phenomenological literature
    2. 5.2 Process and background
    3. 5.3 Parametrizing and numerically defining the necessary interactions
    4. 5.4 Parameter space and simulation
    5. 5.5 Statistics
    6. 5.6 Results
  6. 6 Model independent constraints on exotic particles from flavour violating lepton decays: Preparations
    1. 6.1 Phenomenological status of the 3-body lepton decays
    2. 6.2 Exotic contributions and model independent interactions
    3. 6.3 On the free parameters
    4. 6.4 Feynman rules and diagrams from Lagrangians with explicit charge conjugation
    5. 6.5 Amplitudes, phase space and method of evaluation
  7. 7 Model independent constraints on exotic particles from flavour violating lepton decays: Results
    1. 7.1 Pure $U$ Scenario
    2. 7.2 $U-s$ Scenario
    3. 7.3 $U-Y$ Scenario
    4. 7.4 $Y-s$ Scenario
    5. 7.5 Analysis and conclusions
  8. 8 The 331 perturbative regime
    1. 8.1 Symmetry Matching Conditions and Symmetric RGE
    2. 8.2 Exotic mass scales and rotations in the m331
    3. 8.3 Approximating the m331 by the SM symmetry
      1. 8.3.1 Matching and strategy
      2. 8.3.2 Results
    4. 8.4 Effective approach
      1. 8.4.1 Matching and strategy
      2. 8.4.2 Renormalization framework
      3. 8.4.3 Parametric structure
      4. 8.4.4 Results
    5. 8.5 Can we set $t_{X}^{2}=\frac{s_{W}^{2}}{1-4s_{W}^{2}}$?
    6. 8.6 Analysis and perspectives
  9. 9 Conclusions
  10. A General exact neutral current couplings
    1. A.1 Minimal Model
      1. A.1.1 General exact neutral current couplings
      2. A.1.2 Neutral current couplings of the known fermions to the SM $Z$ within the closing solution
    2. A.2 Model with right handed neutrinos
      1. A.2.1 General exact neutral current couplings
      2. A.2.2 Neutral current couplings of the known fermions to the SM $Z$ within the closing solution

###### List of Figures

  1. 3.1 Atmospheric neutrino events as a distribution of the zenith angle
  2. 3.2 Quadratically divergent corrections to the Higgs mass.
  3. 4.1 LHC cumulative integrated luminosity.
  4. 4.2 Representation of how unitarity is saved in the production of a $W_{0}W_{0}$ pair by the existence of the $Z$.
  5. (a)
  6. (b)
  7. 4.3 The badly divergent $t$-channel $WV$ production amplitude and the $U$ $s$-channel exchange that cures it.
  8. (a)
  9. (b)
  10. 5.1 SM contributions to $pp\to\ell^{+}\ell^{-}\ell^{\prime+}\ell^{\prime-}$
  11. 5.2 m331 contributions to the trimuon process.
  12. 5.3 Contributions of the m331 exotic quarks to $pp\to\ell^{+}_{1}\ell^{-}_{2}\ell^{+}_{3}\ell^{-}_{4}$.
  13. 5.4 95% exclusion contour on the $M_{U}\times(V_{U})_{e\mu}$ plane given by the trimuon process.
  14. 5.5 Event thresholds on the $M_{U}\times(V_{U})_{e\mu}$ plane.
  15. 6.1 SM contributions to the 3-body charged lepton decays
  16. 6.2 Exotic contributions to the 3-body charged lepton decays.
  17. 6.3 Simple representation of the 3-body decay kinematics.
  18. 7.1 Exclusion contours in the pure $U$ Scenario.
  19. (a) $|V_{Uij}|>10^{-3}$
  20. (b) $|V_{Uij}|>10^{-4}$
  21. (c) $|V_{Uij}|>10^{-5}$
  22. (d) $|V_{Uij}|>10^{-5}$ and $V_{11}=0.85$
  23. 7.2 Heuristic demonstration of the existing interference between the $U$ and $s$ contributions to the 3-body CLFV decays.
  24. 7.3 Exclusion contours on the $M_{U}\times M_{s}$ plane generated by the bounds on the 3-body CLFV decays.
  25. (a) $|O_{sij},V_{Uij}|>10^{-3}$
  26. (b) $|O_{sij},V_{Uij}|>10^{-4}$
  27. (c) $|O_{sij},V_{Uij}|>10^{-5}$
  28. (d) $|O_{sij},V_{Uij}|>10^{-5}$ and $O_{s11}=O_{s22}=1$
  29. 7.4 Density plot of the neutral scalar $s$ contributions to $\mu^{+}\to e^{+}e^{-}e^{+}$ in a model with a $U^{\pm\pm}$.
  30. 7.5 Heuristic demonstration that the interference between the $U$ and $Y$ contributions to the 3-body CLFV decays depend on $m_{e}$.
  31. 7.6 Exclusion contours on the $M_{U}\times M_{Y}$ plane generated by the bounds on the 3-body CLFV decays.
  32. 7.7 Exclusion contours on the $M_{Y}\times M_{s}$ plane for bosons constrained to have similar masses on the optimal point.
  33. (a) $|O_{Yij},O_{sij}|>10^{-2}$
  34. (b) $|O_{Yij},O_{sij}|>10^{-3}$
  35. 8.1 Running of $g_{Y}$ and $g_{X}$ in the SM approximation.
  36. 8.2 Running of $g_{X}$ and $g_{3L}$ in the effective approach.
  37. 8.3 Upper limit of the perturbative range of the m331 as a function of $v_{\chi}$.
  38. 8.4 Complete set of exotic diagrams of the m331 relevant to the RGE analysis of the $\gamma u\bar{u}$ and $Wd\bar{u}$ vertices.

##  1 Introduction

It could be argued that the most impressive success of physics throughout
history was the discovery of the Higgs boson, in 2012, independently confirmed
by two teams, at the Large Hadron Collider (LHC). The reason this feat is so
astounding is that it confirms a prediction made forty years earlier, based
purely on the – relatively new at the time – mechanism of gauge invariant
renormalizability. This observation was the last experimental piece expected
in order to stabilize the Standard Model of Elementary Particles and
Fundamental Interactions (SM) as the accepted established general theory of
particle physics, although understood to be flawed as a complete fundamental
description of reality.

However, the arising low TeV scale has proven to be cryptic, and the period
that followed 2012 is marked, in collider and general experimental particle
physics, by a desert in which neither paradigm changing nor much guiding
evidence of new physics have been observed. Some of the most interesting facts
stemming from data collected is this period are the ratification of the SM
predictions for some Yukawa couplings [1, 2]; and the recently obtained
$W$-boson mass [3], in tension with the SM expectation. With insufficient
data-driven insight to guide new constructions, the theoretical struggle to
find solutions to the various SM issues is made difficult.

This thesis is another attempt at investigating and restricting the Beyond the
Standard Model (BSM) theory space while this research context is not altered
by new discoveries, trying to improve our understanding and intuition
regarding the alternative models, hopefully selecting their best aspects and
constraining their parameter space. The main object of our efforts is the
doubly-charged vector bilepton, perhaps the rarest exotic spin-1 particle to
exist within an ultraviolet complete theory with a non-simple Gauge Group.

The Minimal 3-3-1 Model (m331) is the only theory of this type that features
such a bilepton $U^{\pm\pm}$, and predicts that it couples to charged leptons
through a unitary mixing matrix $V_{U}$. To our knowledge, the effects of this
matrix had not received focused consideration yet. The first tale we shall
tell is motivated by this, and corresponds to a study of the trimuon $pp\to
e3\mu$ LHC process. This reaction is free of irreducible background and allows
for an exploration of lepton mixing through $U^{\pm\pm}$ mediation. Our
analysis is model independent to some extent but evokes the m331 when a
benchmark is necessary, and our results comprise exclusion contours on a bi-
dimensional parameter space formed by the bilepton mass and one of the free
angles that parametrize $V_{U}$. Our findings demonstrate that the trimuon
process provides bounds in agreement with the existing literature, but only in
a small sector of the observed parametric region, in the rest of which
bileptons of very small masses remain possible by this process.

Once Charged Lepton Flavour Violation (CLFV) proved to be a possible smoking
gun for the vector bilepton, we were prompted to look for alternative such
channels to which the $U^{\pm\pm}$ can, in theory, contribute. In this spirit,
our second tale corresponds to a model independent study of the purely
leptonic CLFV 3-body decays. Among the several benefits of this set of
channels as a phenomenological guide is the fact that their branching ratios
are simple well known quantities and have sensitivity expected to increase.
More importantly, the exploration of such processes avoid the worries with
statistical treatment, hadronic and collider physics. We investigate the
parameter space of every exotic degree of freedom that can, in principle,
contribute to these decay channels: a doubly-charged vector bilepton, a
doubly-charged scalar boson, and an exotic neutral scalar. Besides seeking to
uncover the strongest conservative bounds on the contributing particles that
this data is able to originate, we contemplate the following question:

Exclusion contours on BSM parameters are usually derived from a minimal model
that isolates some exotic species. Would the resulting allowed regions be
relevantly different if a second new particle, subdominant within the process
at hand, was included to interfere with the primary contribution?

The last research avenue we shall thread is model specific. As already
defined, the main protagonist of our work, the $U^{\pm\pm}$, is a rare feature
of BSM theories contained most notably in the m331. We thus proceed to deliver
a focused investigation into this theory through a Renormalization Group
analysis of its perturbativity regime. The standing result is that the model,
without additional mechanisms, breaks down at scales of a few TeV because of a
Landau pole in its abelian gauge coupling. We carefully check this claim,
verifying its premises and implications. The position of the Landau pole in
$g_{X}$ is rederived through an explicit, brute force approach, and is shown
to appear at higher scales than originally thought.

This thesis is organized as follows: In Chapter 2 we start with a historic
account of the construction of the Standard Model and general particle
physics, and follow to a necessary review of the SM; Chapter 3 is entirely
devoted to the in-depth description of the most prominent flaws of the SM,
which justify the necessity of BSM physics; In Chapter 4, a complete
exposition of the m331 is undertaken; In Chapter 5, the study of the LHC
trimuon process as a source of bounds for the $U^{\pm\pm}$ is described; In
Chapter 6, we describe the theory behind the 3-body lepton decay phenomenology
that this thesis develops, explaining the definitions, choices, calculational
and computational methods; In Chapter 7 the results of the study are
presented; Chapter 8 corresponds to our last front of investigation, in which
the structure of the Renormalization Group of the gauge couplings of the m331
is investigated; And our conclusions are presented in Chapter 9.

##  2 The Standard Model

The current picture of our understanding of the universe may be, at the most
fundamental level, be subsumed to the paradigms of quantum field theory (QFT)
and general relativity. With the unification of these frameworks within a
single theory seeming, still, a distant dream, we are able to apply QFT in
order to generate predictions regarding phenomena with characteristic energies
much smaller than the Planck scale
$m_{p}\sim$10^{19}\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$. For this,
however, an additional model is required – which is now understood to be
usually composed by: A symmetry structure, including the properties of the
vacuum; A representation of that symmetry, which is commonly a sum of various
irreducible representations and corresponds to the particle content; And an
additional, a priori arbitrary (except for its symmetry), potential
functional. This chapter is devoted to reviewing the characteristics of the
Standard Model of Particle Physics, the ruling theory of Elementary Particles
and Fundamental Interactions, starting with an attempt of recollection of the
enlightening process of its foundation. We make an effort to catalogue the
original references, hoping that this Section could provide a useful
consultation resource. This Chapter then follows to a light-speed review of
the model itself.

### 2.1 History and development

By 1925111Take the dates claimed within this section with a grain of salt, as
distinct sources differ by one or even two years regarding some events., the
reality of the highly non-intuitive nature of physics in the quantum scales
was unavoidable, and the understanding of the nature of quanta itself had
already matured through several phenomenological observations. In order to
produce a retrospective notion, we recall some notable ones, such as the
double-slit experiment of Thomas Young (1801); The advancements in the
spectrometry of hydrogen by Balmer (1885) and Rydberg (1888); The observation
of the Zeeman effect (1896); The explanation of the Larmor Precession (1897);
The progress on black-body radiation, through the suggestion by Max Planck
that the emitted energy was quantized (1900); The photon hypothesis, correctly
proposed by Albert Einstein in an effort to describe the photoelectric
emission spectrum, together with his theory of special relativity, both in
1905; The Stark effect (1913); The Stern-Gerlach experiment (1922), which
resulted in the proposition of the quantization of microscopic angular
momentum; The extension of the wave-particle duality to particles, by Louis de
Broglie (1923); And the Pauli exclusion principle (1924).222These are just a
few unequivocally crucial developments within the history of quantum physics –
there are, of course, dozens of additional key historic events which could be
cited.

Simultaneously, there was a somewhat disjoint ‘basic knowledge of the
structure of matter’. This corresponded to the understanding of a minimal set
of ingredients that would evolve to a theoretical substrate and eventually
become the standard particle theory. The most basic component of this model
was the electron, whose existence was recognized since 1897, when J. J.
Thompson determined the composition of cathode rays. The development of this
matter continued and, in 1911, Ernest Rutherford, Ernest Marsden and Hans
Geiger established that the atom, knew to be neutral, contained a positively
charged core, the nucleus, through their famous experiment which scattered
alpha particles (Helium nuclei) through a thin gold plate. In 1919, the study
of the interaction of the same type of particles with the nitrogen in air led
Rutherford to find that the occurring reaction responsible for the
scintillation was ^14_7N + ^4_2$\alpha$ -> ^17_8O + ^1_1p, thus discovering
the positively charged subatomic particles within nuclei, the protons.

It was in this scientific context that, in 1925, Werner Heisenberg, Max Born
and Pascual Jordan wrote the first specific account of matrix quantum
mechanics, after which the development of the field quickly accelerated. In
1926 Erwin Schrodinger postulated his wave equation, which would become the
central quantitative mathematical postulate of non-relativistic quantum
mechanics. In 1927, Werner Heisenberg formulated the uncertainty principle
and, together with Niels Bohr, proposed the Copenhagen interpretation of
wavefunctions, the standard perspective of quantum physics. Finally, in 1927,
motivated by an attempt to purge the negative probability solutions to the
Klein-Gordon equation, Paul A. M. Dirac proposed a first order differential
equation which was simultaneously compatible with the established quantum
theory and special relativity. The revolutionary quality of Dirac equation
cannot be overstated: not only it formally accounted for the previously ad hoc
introduced spin, as it led to the prediction of antimatter, one of the
greatest achievements in the history of physics.

By that point, the ideas of quantum mechanics (QM) had already began to be
applied to systems with an infinite number of degrees of freedom, as the
quantization of fields (or second quantization) seemed to naturally solve
issues such as causality or multiparticle states. This was a crucial
development and perhaps the effective start of the model building efforts that
would culminate in the SM. After the neutrino was hypothesized by Pauli, in
1930, in order to explain the continuous spectrum of beta decays, in 1933
Enrico Fermi invented his famous weak force theory to explain the same
phenomenon. In 1935, Hideki Yukawa analysed the force binding nuclei together
and, from typical atomic nuclei radius, predicted it to be carried by
intermediate ‘heavy quanta’ mesons (in analogy to the ‘light quanta’ which
carried electromagnetic forces), with a mass of
$\sim$100\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$$. In the 1930s, the menu of
subatomic particles empirically increased, with the observation of the
neutron, by James Chadwick, in 1932, and the observation of the positron
(confirming Dirac theory), in 1932, by Carl Anderson, and of the muon, in
1936, by Carl Anderson and Seth Neddermeyer.

The birth of quantum field theory followed to be marked, throughout most of
the process, by a grave theoretical impediment: the omnipresent infinities
which arose in evaluations of physical quantities. Although even today there
is some reserve against QFT among formalists, the advancements of the late
forties were providential to eventually assuage the apprehension of most
theoreticians with the construction. Effective methods to deal with the
divergences and save perturbative expansions appeared between 1946 and 1948,
when the scientific community was finally vibrant again after World War II. An
approach of perturbative renormalization based purely on the familiar operator
formalism was introduced independently by Sin-Itiro Tomonaga [4] and Julian
Schwinger [5, 6, 7, 8]. In the same year, Richard Feynman stated his path
integral formulation of QM [9, 10], a groundbreaking alternative complete
approach to operational quantum physics, which allowed profound theoretical
matters to be analysed directly, and represented another method of
renormalization. One year later, in 1949, Freeman Dyson proved that the two
distinct approaches, one of Schwinger and Tomonaga and the other of Feynman,
are, in fact, equivalent [11, 12].

Renormalization cemented quantum electrodynamics (QED) as a good theory, the
first block of the SM. Its tremendous success could be visualized through its
achievements on the understanding of the anomalous magnetic moment of the
electron and the Lamb shift in the hydrogen spectrum, which motivated the
search for a QFT of other phenomena, specially beta decay and nuclear forces.
In the year before and in the fourteen years following the finalization of
QED, extra experimental findings favoured the theoretical efforts: in 1947,
had occurred the discovery of the pion (thought to be the meson predicted by
Yukawa) by Lattes, Powell and Occhilini, and of the strange kaon meson, by
Rochester and Butler. Then, in 1956, Cowan and Reines found the electron
neutrino, while the muon one was observed in 1962 by Lederman, Schwartz and
Steinberger. In the side of theory, this period was mainly marked by two
events: in 1953, Kazuhiko Nishijima and Tadao Nakano [13] first quoted an
empirical relation between the quantum numbers of barion number, strangeness
(earlier defined quantity, seemly conserved during collisions but not during
decays) and isospin, independently found in 1956 by Murray Gell-Mann [14]; And
the postulation, in 1957, of neutrino flavour oscillation by Bruno Pontecorvo
[15].

One of the most important steps in the popularization of the modern model
building strategies came in 1962, when Gell-mann [16] and Yuval Ne’eman [17]
independently classified known hadrons according to the Eightfold Way333A
reference to the buddhist doctrine of the Noble Eightfold Path.. The idea
amounted to the grouping of all known mesons and baryons according to
representations of $SU(3)$, with strangeness and electric charge as the
quantum numbers. In total, nature seemed to contain a scalar meson octet, a
scalar meson singlet, a fermionic baryon octet, and a spin-$\frac{3}{2}$
baryon decuplet. The Eightfold Way led to the presentation, in 1964, of the
quark model, made independently by Gell-Mann [18] and George Zweig [19]. The
quark model postulated that the known hadrons were not elementary, but
composed by smaller particles, the quarks. The eightfold way was thus
justified, with the group representations arising therein as composite
representations induced by a fundamental triplet, with the up, down and
strange quarks as components.

Although the quark model helped make sense of the known hadron list (whose
non-‘elementarity’ was not without evidence [20, 21, 22, 23, 24]), the theory
was somewhat discredited by the fact that these quarks had never been observed
in isolation. The idea of symmetry as a fundamental dynamical principle,
however, already occupied a large room in the particle physicist imaginarium.
As QED could have been regarded as a $U(1)$ gauge theory, in the 1950s Chen
Ning Yang and Robert Mills generically proposed a non-abelian gauge group
[25], and the strategy to describe fundamental interactions through symmetric
minimal couplings became standard. Models of this sort could be proposed to
replace, for instance, the Fermi theory of beta decay, which was known to be
non-renormalizable by the formalism of Tomonaga-Schwinger-Feynman. From the
beginning, however, there existed a crucial obstacle to this sort of theory:
masses.

The only massless vector boson known at the time was the photon, and it had
been supposed that any other one would surely have been already observed. The
problem is then stated, because a mass term for a gauge boson put in by hand
would destroy the symmetry and, as was eventually realized [26, 27, 28, 29],
is not renormalizable. The symmetry properties of the vacuum then naturally
became focus of attention, as it was believed that a theory with non-symmetric
vacua (a spontaneously broken symmetry) would present itself in nature as an
approximate symmetry – as was known to be the case of the eightfold way, for
instance, as the masses of quarks within a same multiplet were not exactly
identical. It was, then, disappointing, when Jeffrey Goldstone stated a
theorem [30], in 1961, which claimed that for every dimension of the symmetry
which is not respected by the vacuum, arise a massless scalar boson, none of
which was seen in the universe.

Following the usual scientific route, this setback would ultimately lead to
another one of the greatest achievements in history. In 1964, Peter Higgs [31,
32], and François Englert collaborating with Robert Brout [33], independently
found a mechanism to obtain the benefits of spontaneous symmetry breaking
without its burdens. They showed how, if the broken symmetry is regarded to be
local, the unwanted goldstone bosons may be ‘swallowed’ by the transversal
gauge vector bosons to grant them a mass, turning into their longitudinal
helicity component. Not only did this get rid of the unseen scalars, but also
achieved a massive force carrier, as was wanted in order to model the nuclear
forces by a Yang-Mills theory.

Seeking after underlying symmetry principles for the observed forces was
popular at the mid sixties for yet another, disjoint (or opposite, even),
reason: the hypothesis which suggested the pion as carrier meson of the
nuclear forces had recently been tuned to a successful construction. The
theory with a chiral isotopic symmetry $SU(2)_{L}\times SU(2)_{R}$, explicitly
broken by quark mass terms and spontaneously broken at low energies by non-
perturbative quark condensation, produced the pion as a pseudo-Nambu-Goldstone
boson. This represented a groundbreaking accomplishment and resolved a few
previously unavoidable issues mentioned in this summary. In particular, this
advancement diluted the interest in the Higgs mechanism, as it diminished the
desire to get rid of the Goldstone bosons.

Different constructions of the same idea were tested, still in attempts to
describe the strong force. The most popular tried to fit known mesons as the
gauge bosons of the vector and axial vector symmetries, with the pion
appearing as a leftover goldstone boson. Eventually, however, Abdus Salam and
John Clive Ward employed this program in the context of the electroweak
interactions [34]. In 1964, they proposed a $SU(2)\times U(1)$ group (which
was already mentioned by Sheldon Glashow in 1961 [35]) together with a manual
breaking of symmetry, predicting three massive vector bosons plus the photon.
Independently, in 1967 Steven Weinberg [36] proposed the same group, now
spontaneously broken. The first test the theory faced was the issue of
renormalizability, known since the twenties to be a necessity of an exact
fundamental model. This challenge was overcome in a series of works by t’Hooft
[37], Veltmann and t’Hooft [38], Lee and Zinn-Justin [39, 40, 41, 42] and,
finally, Becchi, Rouet, Stora and Tyutin [43], with the BRST formalism. After
this theoretical matter was resolved, the theory could be confronted with
newly acquired data. In 1973, the neutral electroweak currents were firstly
observed at the Gargamelle bubble chamber [44], at CERN, and, ten years later,
after the availability of the Super Proton Synchroton, the $W$ and $Z$ bosons
were in the UA1 experiment [45, 46, 47]. The standard theory of electroweak
interactions was thus established, and there was a single ingredient missing
confirmation, to which we shall briefly come back.

A detour from model structure must be made in order for its particle
components to be assessed. In 1964, a fourth quark had already been proposed
by James Bjorken and Glashow [48], for the purely aesthetic reason of
parallelism with the leptonic sector, within which four components existed.
Then, in 1970, when the one-family electroweak theory was already well known,
Glashow, John Iliopoulos and Luciano Maiani showed how the existence of a
fourth quark, called charm, would cancel the neutral currents responsible for
the decay of the strange kaon meson $\overline{K}^{0}$ into muons, which,
without this mechanism, should be much more common than observed [49]. Four
years after the charm quark existence had been confidently claimed by the
three collaborators, the charmed $J/\psi$-meson was simultaneously observed by
Burton Richter’s group at SLAC [50] and Samuel Ting’s at BNL [51].

With the electroweak theory consolidated on two families of quarks and
leptons, it remained to describe the force which bounded nuclei, stable cores
of positive and neutral particles, together. Following the trend of the 1960s
and the example of the electroweak theory, a crucial step was taken when –
through the renormalization group methods of Gell-Mann, Low, Callan, Symanzic,
Coleman and Jackiw [52, 53, 54] – Frank Wilczek and David Gross [55], and
David Politzer [56], independently, found out in 1973 that non-abelian gauge
interactions become arbitrarily weak at high energies. If the strong force was
indeed an interaction dictated by a non-abelian gauge theory, asymptotic
freedom not only allowed perturbation theory to be sensible at high energies,
as it predicted that the strength of the corresponding force does not fall
with distance, motivating an explanation for the mysterious refusal of the
quarks to appear isolated. Furthermore, the necessity of a new quantum number
was already known by, for example, the requirement of the Pauli exclusion
principle applied to equal constituents baryons, and was expected to be three
by fittings of some observed meson decay rates. Because of this, it was
natural when, still in 1973, Harald Fritzsch, Heinrich Leutwyler and Gell-Mann
proposed that the new quantum number, color, corresponded to a conserved
$SU(3)$ gauge group [57]. The reason as to why the eight extra massless spin-1
bosons had never been observed was soon understood to be another facet of
confinement: since they interact strongly with particles and, in particular,
themselves, the gluons are trapped inside color singlets [58, 59, 60, 57].

To finalize the proposition of the particle content of the SM, some blocks of
nature had yet to be recognized. The first individual of the third generation
to appear was the tau lepton, discovered between 1974 and 1977, by Martin
Lewis Perl’s group at SLAC [61, 62]. In 1977, Leon Lederman’s team found the
bottom quark, already predicted four years earlier as a mechanism to allow the
experimentally observed $\mathit{CP}$-violation [63] by Kobayashi and Maskawa
[64]. The top quark, much heavier and needed by the electroweak symmetry
structure, was discovered only eighteen years later by the CDF collaboration
[65]. Finally, the tau-neutrino was firstly experimentally inferred in 2000,
by the DONUT experiment at Fermilab [66].

The SM was thus defined as a $SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}$ gauge
theory, with three generations of leptons, each composed of a negatively
charged, colorless, massive Dirac fermion and a massless Weyl neutrino; And
three generations of 3-colored quarks, each containing a $Q=2/3$ and a
$Q=-1/3$ Dirac fermion. The model also features a fundamental scalar doublet
of a complex positively charged and a complex neutral field, which, through
the carefully designed scalar and Yukawa potential, triggers spontaneous
symmetry breaking and grants a mass to the $W$ and $Z$ boson and all the
fermions, keeping the gluons, photons and neutrinos massless. While the
electroweak sector is well understood, the $SU(3)_{c}$ dynamics was known from
the beginning to be highly non-perturbative at low energies, and is to this
day a vibrant research field, even within the SM context.

The glorious last piece of the puzzle would come to its place in 2012, when,
after exhaustive search at the LEP and Fermilab, the spin-0 neutral mass
eigenstate, leftover physical degree of freedom of the SM scalar doublet, was
observed as an excess of events around
$125\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$, by the Atlas and CMS groups, at
the LHC. This was the crowning achievement of the SM and of the theoretical
advancements made during its pursuit, and the last experimental observation to
revolutionize particle physics.

### 2.2 Review

With the familiarity gained through the historical approach above, we are
exceptionally prepared to briefly review the SM [67, 68, 69]. Now, the SM (or,
at least, its basic facts) is a topic extensively understood, which is why we
shall try to subsume it quickly, in an effort not to bore the reader with
unnecessary discussions.

The SM is a model built around the gauge group

$SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}.$ (2.1)

The representation content is, as dictated by the usual programme, built with
a priori knowledge of the electric charge of the degrees of freedom that form
the basis of the representations. The electric charge operator is then given
in terms of $Y$ and of the third component of $SU(2)_{L}$ isospin (the
diagonal generators) as444Notice that many authors chose the alternative
normalization $Y\to 2Y$, in which case $Q=I_{3}+\frac{1}{2}Y$.

$Q=I_{3}+Y.$ (2.2)

Each generation within the leptonic sector is formed by left-handed doublets
of a charged lepton grouped with the corresponding neutrino, plus the right-
handed charged singlet:

$L_{\ell}\equiv\begin{pmatrix}\nu_{\ell}\\\
\ell\end{pmatrix}_{L}\thicksim(\mathbf{2},-1/2),\;\;\;\;\ell_{R}\thicksim(\mathbf{1},-1),\;\;\;\;\ell=e,\mu,\tau.$
(2.3)

The color sector presents three families of left-handed doublets containing an
up- and a down-type quark, plus a right-handed singlet for every flavour.
Every particle is a color triplet, as seen below

$\begin{gathered}Q_{i}\equiv\begin{pmatrix}u_{i}\\\
d_{i}\end{pmatrix}_{L}\thicksim\left(\mathbf{3},\mathbf{2},\frac{1}{6}\right)\\\\[8.53581pt]
u_{iR}\thicksim\left(\mathbf{3},\mathbf{1},\frac{2}{3}\right),\;\;\;\;\;d_{iR}\thicksim\left(\mathbf{3},\mathbf{1},-\frac{1}{3}\right)\\\\[8.53581pt]
i=1,2,3;\;\;\;u_{i}=u,c,t;\;\;\;d_{i}=d,s,b.\end{gathered}$ (2.4)

The scalar sector proposes a single doublet

$\phi\equiv\begin{pmatrix}\phi^{+}\\\
\phi^{0}\end{pmatrix}\thicksim\left(\mathbf{2},+\frac{1}{2}\right),$ (2.5)

whose main purpose of existence is triggering spontaneous symmetry breaking
(SSB). For that end, we also require the classic ‘mexican hat’ scalar
potential

$V_{\phi}=-\mu^{2}(\phi^{\dagger}\phi)+|\lambda|(\phi^{\dagger}\phi)^{2},$
(2.6)

with $\mu>0$. In order to enable the use, without modification, of the
canonical formalism appropriate to derive physical transition rates from
‘unphysical’ perturbative amplitudes, the creation and annihilation operators
must be normalized according to the LSZ reduction formula requirement

$\langle 0|\phi|0\rangle=0.$ (2.7)

This implies that Eq. (2.6) must be expanded around the lowest energy state of
the potential. It is immediate to see that $V_{\phi}$ carries a one-
dimensional manifold of degenerate vacua parametrized by

$\phi_{0}\in\left\\{\begin{pmatrix}\phi^{+}\\\
\phi^{0}\end{pmatrix},\;\;|\phi^{+}|^{2}+|\phi^{0}|^{2}=\frac{\mu^{2}}{2|\lambda|}\right\\}.$
(2.8)

By charge conservation and Lorentz invariance, we favor a point in the pure
neutral direction and rewrite $\phi$ as

$\phi=\frac{1}{\sqrt{2}}\begin{pmatrix}\phi_{1}+i\phi_{2}\\\
v_{H}+h+i\phi_{3}\end{pmatrix},$ (2.9)

where $v_{H}\equiv|\mu|/\sqrt{\lambda}$ is chosen real and a possible source
of $\mathit{CP}$ violation is ignored. It is easy to verify that this vacuum
is only left invariant by transformations generated by the electric charge,
generating the following pattern of SSB

$SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}\to SU(3)_{c}\times
U(1)_{\mathrm{EM}}.$ (2.10)

Another trivial exercise shows that the diagonalization of the gauge sector
can be achieved by

Table 2.1: SM gauge groups and their relevant symbols. | Gauge boson | Coupling constant | Generator
---|---|---|---
$SU(3)_{c}$ | $g^{a}_{\mu}$ | $g_{s}$ | $\frac{1}{2}\lambda^{a}$
$SU(2)_{L}$ | $W^{a}_{\mu}$ | $g$ | $\frac{1}{2}\sigma^{a}$
$U(1)_{Y}$ | $b_{\mu}$ | $g^{\prime}$ | $Y$
$U(1)_{\mathrm{EM}}$ | $A_{\mu}$ | $e$ | $Q$

$\begin{gathered}W_{\mu}^{\pm}=\frac{W_{\mu}^{1}\mp
iW_{\mu}^{2}}{\sqrt{2}},\qquad
Z_{\mu}=\frac{-g^{\prime}b_{\mu}+gW_{\mu}^{3}}{\sqrt{g^{2}+g^{\prime 2}}}\\\
A_{\mu}=\frac{gb_{\mu}+g^{\prime}W_{\mu}^{3}}{\sqrt{g^{2}+g^{\prime
2}}}.\end{gathered}$ (2.11)

where the symbols are defined in Table 2.1. The corresponding masses are given
by

$\begin{split}M_{W}&=\frac{gv_{H}}{2}\\\ M_{Z}&=\frac{\sqrt{g^{2}+g^{\prime
2}}}{g}M_{W}\equiv\frac{M_{W}}{c_{W}}\\\ M_{A}&=0,\end{split}$ (2.12)

where we have introduced the notation $\tan\theta_{W}\equiv
t_{W}=g^{\prime}/g$, which entails the complementary trigonometric definitions
of $s_{W}\equiv\sin\theta_{W}$ and $c_{W}\equiv\cos\theta_{W}$. With this, the
strength of the photon-fermion interaction, immediately identifiable with the
positive fundamental electric charge, may be written as

$e=\frac{gg^{\prime}}{\sqrt{g^{2}+g^{\prime 2}}}=gs_{W}=g^{\prime}c_{W}.$
(2.13)

An additional Lagrangian density is necessary in order for the matter fields
to acquire a mass through the electroweak SSB. This is the Yukawa potential,
and is, by definition, a general renormalizable interaction bilinear on
fermion and linear on scalar fields. Naively, the following expression could
appear sufficient

$\begin{split}\mathcal{L}_{\mathrm{Y}}=-\sum_{\ell}\zeta^{\ell}_{\ell}[\bar{\ell}_{R}(\phi^{\dagger}L_{\ell})+(\bar{L}_{\ell}\phi)\ell_{R}]&{}-\sum_{i}\zeta^{u}_{i}[(\bar{Q}_{i}\tilde{\phi})u_{iR}+\bar{u}_{iR}(\tilde{\phi}^{\dagger}Q_{i})]\\\
&{}-\sum_{d}\zeta^{d}_{i}[(\bar{Q}_{i}\tilde{\phi})d_{iR}+\bar{d}_{iR}(\tilde{\phi}^{\dagger}Q_{i})],\end{split}$
(2.14)

where $\tilde{\phi}=i\sigma_{2}\phi^{*}$ and the $\zeta$ are couplings to be
fitted. If these Yukawa couplings are put to zero, however, the SM, with the
usual diagonal form and standard normalization of the kinetic terms, due to
its particle content and gauge symmetry, can be seen to possess an accidental
global symmetry generated by independent unitary rotations, in flavour space,
of each of its fermion multiplets:

$\begin{split}Q_{i}&\to U^{Q}_{ij}\,Q_{j}\\\ u_{iR}&\to U^{u}_{ij}\,u_{jR}\\\
d_{iR}&\to U^{d}_{ij}\,d_{jR}\\\ L_{i}&\to U^{L}_{ij}\,L_{j}\\\ \ell_{iR}&\to
U^{\ell}_{ij}\,\ell_{jR}.\\\ \end{split}$ (2.15)

A sector of this symmetry is explicitly broken by the Yukawa interactions, and
it is not hard to show by brute force manipulation of the rules that the
resulting physical effect of such a $U(3)^{5}$ transformation on the Yukawa
sector can be equivalently factorized as

$SU(3)^{5}\times U(1)_{Y}\times U(1)_{B}\times U(1)_{L}\times
U(1)_{\mathrm{PQ}}\times U(1)_{\ell_{R}},$ (2.16)

where each $SU(3)$ factor corresponds to a fermion multiplet; $U(1)_{Y}$
coincides with the gauged weak hypercharge group; $U(1)_{B(L)}$ is generated
by the baryon(lepton) number; $U(1)_{\mathrm{PQ}}$ is the Peccei-Quinn
symmetry, which assigns an identical quantum number to $\ell_{R}$ and $d_{iR}$
and the opposite one to $\phi$; and $U(1)_{\ell_{R}}$ rotates only the right-
handed lepton singlet.

The key observation is that the original Lagrangian is insensitive to an
$U(5)$ action, and it turns out (as was already known in the sixties from
strangeness violating decays and postulated by weak universality) that the
correct description of the observable effects of the Yukawa sector does
necessitate the violation of flavour symmetry. To summarize, the ‘symmetry
eigenstates’ do not correspond to the propagating degrees of freedom, and the
Yukawa Lagrangian must be updated to

$\begin{split}\mathcal{L}_{\mathrm{Y}}=-\sum_{\ell}Y^{\ell}_{\ell}[\bar{\ell}_{R}(\phi^{\dagger}L_{\ell})+(\bar{L}_{\ell}\phi)\ell_{R}]&{}-\sum_{i}Y^{u}_{ij}[(\bar{Q}_{i}\tilde{\phi})u_{jR}+\bar{u}_{jR}(\tilde{\phi}^{\dagger}Q_{i})]\\\
&{}-\sum_{d}Y^{d}_{ij}[(\bar{Q}_{i}\tilde{\phi})d_{jR}+\bar{d}_{jR}(\tilde{\phi}^{\dagger}Q_{i})],\end{split}$
(2.17)

where all quark degrees of freedom are now understood to be symmetry
eigenstates. It becomes trivial to visualize, in the unitary gauge, that all
effects of the diagonalization of the quark mass matrices, given by

$\mathcal{M}_{ij}^{u(d)}=Y^{u(d)}_{ij}\frac{v_{H}}{\sqrt{2}},$ (2.18)

can be included by the following redefinitions in Eq. (2.14)

$\begin{split}u_{iL(R)}&\to u_{iL(R)}\\\ d_{iR}&\to d_{iR}\\\
d_{iL}&\to(V_{\mathrm{CKM}})_{ij}\,d_{jL},\end{split}$ (2.19)

where $V_{\mathrm{CKM}}$ is the Cabibo-Cobayashi-Maskawa matrix, written in
terms of the biunitary transformation that diagonalizes the quark mass
matrices as $V_{\mathrm{CKM}}=V_{L}^{u}V_{L}^{d\dagger}$.

Finally, the complete Lagrangian of the theory is then composed as

$\mathcal{L}_{\mathrm{SM}}=\mathcal{L}_{\mathrm{kin}}+\mathcal{L}_{\phi}+\mathcal{L}_{\mathrm{Y}}+\mathcal{L}_{\mathrm{ghosts}}+\mathcal{L}_{\mathrm{gauge-
fixing}},$ (2.20)

where $\mathcal{L}_{\mathrm{ghosts}}$ comprises the ghost Lagrangian, needed,
together with $\mathcal{L}_{\mathrm{gauge-fixing}}$, to quantize the
spontaneously broken non-abelian gauge symmetry of the theory, and where the
kinetic portion is naturally

$\mathcal{L}_{\mathrm{kin}}=\sum_{f}\bar{f}\not{D}f-\sum_{X}\frac{1}{4}X_{\mu\nu}^{a}X^{a\mu\nu}+(D_{\mu}\phi)^{\dagger}D^{\mu}\phi,$
(2.21)

with $f$ running through the matter representation content and $X$ through the
set of gauge multiplets, where
$X_{\mu\nu}^{a}=\partial_{\mu}X^{a}_{\nu}-\partial_{\nu}X^{a}_{\mu}+gf_{abc}X_{\mu}^{b}X_{\nu}^{c}$
denotes the covariant field strength tensor of the spin-1 $X^{a}_{\mu}$ field
transforming in the adjoint representation of some group with structure
constant $f$ (that vanishes for the abelian $U(1)$ gauge field).

The SM has thus been described as an exactly symmetric gauge theory which
undergoes spontaneous symmetry breaking through a potential whose operators
have mass dimensions of two and four. It is known, then, to be in principle
renormalizable. There is a caveat, however: a classical symmetry of the
Lagrangian is not necessarily maintained after quantization. Such anomalies
are rooted, in the functional formalism, in the non-invariance of the complete
path integral measure of the effective action; And, equivalently, can be seen
in the canonical field-perturbative point of view from the non-existence of a
simultaneously gauge and Lorentz invariant regulator for some loop graphs with
chiral vertices [70, 71]. There is nothing inadmissible about an anomalous
global symmetry – in fact, both baryon and lepton number are anomalous within
the SM (this fact will be touched upon next chapter). A gauge symmetry,
however, cannot be anomalous, as its validity is detrimental for the
renormalizability of the theory. Because of its chiral nature, the SM with,
for instance, a single fermion doublet is indeed anomalous. The anomaly of the
3-point function can be show to be proportional to the completely symmetric
gauge theoretic quantity

$\mathfrak{a}_{abc}\equiv\sum_{f}\mathrm{tr}\left[\gamma_{5}\,t_{a}\\{t_{b},t_{c}\\}\right],$
(2.22)

where here $\gamma_{5}$ represents the eigenvalue of $\gamma_{5}$ on a given
fermionic degree of freedom $f$ – i.e., left-handed fields contribute a minus
sign and right-handed ones a plus sign. $t$ is any of the 12 generators of the
SM. Considering its fermion content, let us analyse the cancellation of
anomalies for each combination of generators, which is equivalent to examining
the anomaly of the triangle diagram with every possible arrangement of
incoming gauge bosons.

  1. 1.

$\left[SU(3)_{c}\right]^{3}$:

The anomaly of the triangle with three $SU(3)_{c}$ currents automatically
vanishes because QCD is a vector theory, hence every right-handed contribution
is exactly cancelled by a left-handed one.

  2. 2.

$\left[SU(3)_{c}\right]^{2}\left[U(1)_{Y}\right]$:

Here the anomaly factor reads

$\begin{split}\mathfrak{a}_{Ybc}&=\sum_{f}\frac{1}{4}\mathrm{tr}\left[\gamma_{5}\,Y\\{\lambda_{b},\lambda_{c}\\}\right]=\sum_{f}\frac{1}{4}\gamma_{5}Y\mathrm{tr}\left[\\{\lambda_{b},\lambda_{c}\\}\right]\\\
&=\sum_{q}\frac{1}{2}\gamma_{5}Y\delta_{bc}\\\
&=\frac{1}{2}\delta_{bc}\left[(-1)\times
2\times\left(\frac{1}{6}\right)+1\times 1\times\frac{2}{3}+1\times
1\times\left(-\frac{1}{3}\right)\right]\\\ &=0,\end{split}$ (2.23)

where we have used the ciclicity of the trace and the normalization of the
Dynkin index $\mathrm{tr}\left[t_{a}t_{b}\right]=\frac{1}{2}\delta_{ab}$, with
$t_{a}=\frac{\lambda_{a}}{2}$. In the penultimate line, each term corresponds
to a given contributing multiplet, which are the colored ones alone (notice
that we have rewritten the sum to be over $q$, symbolizing that it should be
restricted to quark fields). For didactic reasons, we have explicitly shown
how the final numeric result is obtained, with the factors in the being
ordered as

$(\mathrm{handedness})\times(\mathrm{multiplicity})\times(Y);$ (2.24)

  3. 3.

$\left[SU(3)_{c}\right]^{2}\left[SU(2)_{L}\right]$:

Since different simple factors of the SM group commute (of course, as the
total group is defined through a direct product), this trace factorizes and we
have

$\begin{split}\mathfrak{a}_{abc}&=\sum_{f}\frac{1}{8}\mathrm{tr}\left[\gamma_{5}\,\sigma_{a}\\{\lambda_{b},\lambda_{c}\\}\right]\\\
&=\frac{1}{8}\sum_{f}\gamma_{5}\,\mathrm{tr}\left[\sigma_{a}\right]\mathrm{tr}\left[\\{\lambda_{b},\lambda_{c}\\}\right]=0,\end{split}$
(2.25)

which vanishes since $\mathrm{tr}\left[\sigma_{a}\right]$ or, more generally,
the $SU(N)$ algebra $\mathfrak{su(n)}$ is the set of hermitian traceless
$n$-dimensional matrices, together with its Lie bracket.

  4. 4.

$\begin{cases}&\left[SU(3)_{c}\right]\left[SU(2)_{L}\right]^{2}\\\
&\left[SU(3)_{c}\right]\left[SU(2)_{L}\right]\left[U(1)_{Y}\right]\\\
&\left[SU(3)_{c}\right]\left[U(1)_{Y}\right]^{2}\\\
&\left[SU(2)_{L}\right]\left[U(1)_{Y}\right]^{2}:\end{cases}$

These configurations vanish by the same argument, i.e., all of them contain
one of the non-abelian groups appearing on exactly one current.

  5. 5.

$\left[SU(2)_{L}\right]^{3}$:

$\begin{split}\mathfrak{a}_{abc}&=\sum_{f}\frac{1}{8}\mathrm{tr}\left[\gamma_{5}\,\sigma_{a}\\{\sigma_{b},\sigma_{c}\\}\right]=\sum_{f}\frac{1}{4}\gamma_{5}\mathrm{tr}\left[\sigma_{a}\delta_{bc}\right]\\\
&=0,\end{split}$ (2.26)

where we have used $\\{\sigma_{a},\sigma_{b}\\}=2\delta_{ab}$.

  6. 6.

$\left[SU(2)_{L}\right]^{2}\left[U(1)_{Y}\right]$:

$\begin{split}\mathfrak{a}_{Ybc}&=\sum_{f}\frac{1}{4}\mathrm{tr}\left[\gamma_{5}\,Y\\{\sigma_{b},\sigma_{c}\\}\right]=\sum_{f}\frac{1}{4}\gamma_{5}Y\mathrm{tr}\left[\\{\sigma_{b},\sigma_{c}\\}\right]\\\
&=\sum_{f}\frac{1}{2}\gamma_{5}Y\delta_{bc}\\\
&=\frac{1}{2}\delta_{bc}\left[1\times(-1)\times
2\times\left(-\frac{1}{2}\right)+3\times(-1)\times
2\times\frac{1}{6}\right]\\\ &=0,\end{split}$ (2.27)

where, now, only the left-handed doublets, which form a non-trivial
representation of $SU(2)_{L}$, contribute. In this and in the next case, the
factors have been ordered as

$(\mathrm{color})\times(\mathrm{handedness})\times(\mathrm{multiplicity})\times(Y).$
(2.28)

  7. 7.

$\left[U(1)_{Y}\right]^{3}$:

The final combination gives

$\begin{split}\mathfrak{a}&=\sum_{f}\gamma_{5}\,Y^{3}\\\ &=1\times 1\times
1\times(-1)^{3}+1\times(-1)\times 2\times\left(-\frac{1}{2}\right)\\\
&\quad+3\times\left[(-1)\times 2\times\left(\frac{1}{6}\right)+1\times
1\times\left(\frac{2}{3}\right)+1\times
1\times\left(-\frac{1}{3}\right)\right]\\\ &=0,\end{split}$ (2.29)

The 3-point functions are linearly divergent, hence the 4-point ones diverge
logarithmically and a shift in the integration variable of the box diagrams
may be carried without repercussions. This means that the integrals may be
regularized and cannot contribute to the anomaly. Even if that was not the
case, the anomaly factor of higher point functions is proportional to that of
the triangle. What any of these two facts imply is that the analysis of the
3-point function is enough to guarantee that a theory is anomaly free. With
this in mind, the complete assessment of the SM anomaly factors just performed
shows that it is, indeed, without gauge anomalies. In particular, it may be
verified that every combination of the gauge currents considered above would
also vanish if a single generation was taken into account. Therefore, the SM
is free of anomalies generation by generation, and is a well-defined,
renormalizable555At least in the weak coupling limit of QCD. theory.

The grandiosity of the complexity and resulting success of the SM is not only
improbable, but hard to satisfactorily narrate. It is a single, closed form
theory that is more complicate (and has more intriguing aspects) than any
other fundamental, supposedly exact and general, theory of nature. Its
accomplishments start by the unimaginable series of predictions recalled last
section, but do not end there. For instance, the theory correctly describes
the plethora of recent LHC data with little deviation [72]. Most
significantly, the SM has been confidently validated by the so-called
electroweak precision tests. One way to explore electroweak precision
observables (EWPOs) in order to stress test the SM takes advantage of the
vector ($g_{V}$) and axial-vector ($g_{A}$)666These parameters are usually
called $v_{f}$ and $a_{f}$, respectively – we chose the alternative naming
which is more appropriate for the case when an exotic $Z^{\prime}$ exists, as
will be the case when these quantities reappear in Section 4.8 and in Appendix
A. couplings of the $Zf\bar{f}$ interactions at the $Z$-pole. These enter the
parametrization of the weak neutral currents, for massless fermions, as

$\mathcal{L}_{Zf\bar{f}}=i\bar{\psi}_{f}\gamma^{\mu}(g^{f}_{V}-g_{A}^{f}\gamma_{5})\psi_{f}Z_{\mu}.$
(2.30)

The analysis take as typical input parameters
$\alpha,\alpha_{s},G_{F},M_{Z},M_{H}$ and $m_{t}$, which are extracted from
data assuming the SM as accurate. Several precision observables are then
obtained to a high accuracy, and are subsequently cast in the form of an
intricate SM prediction in terms of the input and of the vector/axial-vector
parameters. Some instances of such precision observables are the $Z$ decay
width; $e^{+}e^{-}\to f\bar{f}$ cross-sections; The left-right assimetries of
the same processes and etc. The observed $g_{V},g_{A}$ are then compared
against the SM calculation, and the accumulated results attest that the SM
predictions are coherent to a high degree of precision. For great reviews on
this matter, see [73, 74] and references therein.

##  3 Flaws of the Standard Model

Although an outstanding attempt at a fundamental theory of elementary
particles (which can never be sufficiently emphasized), the Standard Model
cannot be the ultimate theoretical expression of nature. The job of describing
the building blocks of matter and their interactions is a difficult one, and
at this point we know of several reasons why the SM must be amended. These
range from theoretical discomforts (such as the arbitrariness in the number of
families) to the unacceptable inability to describe important phenomena.
Notwithstanding, the problems of the SM come with a benefit: they guide us
towards BSM physics, which leads to the next hypotheses, candidates to update
or replace the SM. Accordingly, to understand deeply each of the issues of the
theory is an imperative matter in the search for BSM physics, and that is the
objective of this chapter: to lay down a rudimentary review, as self-contained
as possible, of the major flaws of the SM.

### 3.1 Neutrino masses

The most direct and unassailable argument for the insufficiency of the SM,
even within a pure particle theoretical point of view, is the absence of
neutrino masses. Neutrinos are weakly interacting particles whose existence,
when hypothesized last century, could only be inferred indirectly through
kinematic observations, all of which indicated that its mass should be nearly
zero. This, together with their highly penetrating quality, caused them to be
assumed massless for several decades.

At the same time, it was known, even from simple quantum mechanical arguments
alone, that the phenomenon of neutrino oscillating between its flavours in the
vacuum could only occur if at least one species had a mass. This seems obvious
from the understanding that mass eigenstates are the ones that propagate and
flavour the ones which are produced through interaction, so that propagation
may lead to oscillation only if the two differ. Nevertheless, it is useful to
put forward a simple mathematical argument which is as follows. By definition,
the states which propagate are the ones that diagonalize the unitary evolution
operator $U(t)=e^{-iHt}$, called mass eigenstates. These degrees of freedom do
not need to correspond to symmetry eigenstates, defined as the fields whose
gauge interactions are diagonal in flavour space. If probabilities are to be
conserved through a change of basis, these two sets of eigenvectors must be
related unitarily, that is

$\left|\,\nu_{j},t\,\right\rangle=\sum_{\ell}U_{j\ell}\left|\,\nu_{\ell},t\,\right\rangle,$
(3.1)

where $\ell=e,\mu,\tau$ labels flavour eigenstates and $j=1,2,3$ mass ones,
and

$UU^{\dagger}=U^{\dagger}U=\mathds{1}.$ (3.2)

In the context of the simple extensions of the SM, this matrix is called
Pontecorvo-Maki-Nakagawa-Sakata, and may be found to be
$U_{\mathrm{PMNS}}\equiv(V_{L}^{\ell})^{\dagger}V_{L}^{\nu}$. With this and
the law for time evolution, one may find, at once

$\left|\,\nu_{\ell},t\,\right\rangle=\sum_{\ell^{\prime},j}U^{*}_{\ell
j}e^{-iE_{j}t}U_{\ell^{\prime}j}\left|\,\nu_{\ell^{\prime}},0\,\right\rangle,$
(3.3)

with which the transition amplitude for an $\ell$-flavour neutrino, at time
$t=0$, to oscillate to an $\ell^{\prime}$ one at time $t$ is easily
calculated, and, using
$P\left(\nu_{\ell}(0)\to\nu_{\ell^{\prime}}(0)\right)=\delta_{\ell\ell^{\prime}}$,
leads to a general probability

$P\left(\nu_{\ell}(0)\right.\left.\to\nu_{\ell^{\prime}}(t)\right)=\delta_{\ell\ell^{\prime}}+{}\\\
{}-\sum_{i>j}\left(4\,\operatorname{Re}\left\\{\mathcal{Q}_{\ell\ell^{\prime},ij}\right\\}\sin^{2}{\phi_{ij}}-2\,\operatorname{Im}\left\\{\mathcal{Q}_{\ell\ell^{\prime},ij}\right\\}\sin{2\phi_{ij}}\right),$
(3.4)

where we have defined the quartic $\mathcal{Q}_{\ell\ell^{\prime},ij}\equiv
U_{\ell i}U^{*}_{\ell^{\prime}i}U^{*}_{\ell j}U_{\ell^{\prime}j}$, and the
phases are given by

$\phi_{ij}\equiv\frac{E_{i}-E_{j}}{2}.$ (3.5)

Since $\phi_{ij}=0\Rightarrow
P\left(\nu_{\ell}(0)\to\nu_{\ell^{\prime}}(t)\right)=0$, this finishes a
simple argument to justify the claim at the beginning of this section.

Hard evidence for neutrino oscillation was first published in 1998 [75], after
the Super-Kamiokande experiment observed a large difference between data and
expectation for the ratio of muon to electron atmospheric neutrinos. The
significant muon neutrino deficit is explained by its oscillation during the
course of its travel, and is better fitted by the dominant
$\nu_{\mu}\to\nu_{\tau}$ hypothesis, or even by the muon neutrinos oscillating
preferably to an exotic sterile flavour $\nu_{\mu}\to\nu_{\text{s}}$, but not
by the simplest mode $\nu_{\mu}\to\nu_{e}$. Figure 3.1 shows how oscillation
is necessary for the collected data to be satisfactorily explained.

Figure 3.1: Number of events of atmospheric neutrinos as a distribution of the
zenith angle. The blue histogram corresponds to the predictions of the non-
oscillation hypothesis. The plots show how there is a clear muon neutrino
scarcity and that it cannot be explained by their oscillation into electron
ones. Figure taken from PDG [72] and provided by the Super-Kamiokande
Collaboration.

The paradigm of general neutrino oscillation, unveiled through testing of the
atmospheric neutrino flux, has only been reaffirmed by additional experiment
designs, which investigated analogous phenomena in accelerator [76], reactor
[77] and solar [78] flux, establishing the theoretical expectations of the
normal and inverted hierarchies for neutrino masses. The former postulates
$m_{3}\gg m_{2}\gtrsim m_{1}$, and the latter hypothesizes $m_{2}\gtrsim
m_{1}\gg m_{3}$, with $\nu_{1},\nu_{2},\nu_{3}$ being more
$\nu_{e},\nu_{\mu},\nu_{\tau}$-like, respectively.

It is not difficult to manually solve this issue in the form of a minimal
extension of the SM. In fact, the first possibility does not imply an
increased particle content, as a neutrino mass may be obtained by the
inclusion of the following dimension-5 operator in the theory

$\mathcal{L}_{5}=\frac{c_{5}}{\Lambda}\left(\overline{\tilde{L}}H\right)\left(\tilde{H}^{\dagger}L\right).$
(3.6)

However, since this interaction is non-renormalizable, it represents no
solution at all and does not improve the SM as a fundamental theory.

A better option proposes the inclusion of a right-handed neutrino singlet
(consider a single generation for simplicity)
$\nu_{R}\thicksim(\bm{1},\bm{1},0)$ to the model. This extension allows for
different possibilities, each with its weaknesses:

  1. 1.

Pure Dirac mass:

The singlet allows for a Dirac mass to be generated by the Yukawa term

$\mathcal{L}_{\text{D}}=y_{\nu,\text{D}}\bar{L}\tilde{\phi}\nu_{R}.$ (3.7)

The problem is that neutrino squared mass differences are estimated to be
$\Delta m_{21}^{2}\sim$7.4\times 10^{-5}\text{\,}\mathrm{e}\mathrm{V}^{2}$$
and $\Delta m_{32}^{2}\sim$2.5\times
10^{-3}\text{\,}\mathrm{e}\mathrm{V}^{2}$$ [79], and the absolute bound for
the masses is usually cited, from cosmological analysis, as of the order of
$\sum m_{\nu}<$1\text{\,}\mathrm{e}\mathrm{V}$$. Since $m_{e}^{2}=$2.6\times
10^{5}\text{\,}\mathrm{e}\mathrm{V}^{2}$$, this would imply a hierarchy for
the dimensionless Yukawa parameters within the lepton sector already larger
than the current largest hierarchy in the entire model.

  2. 2.

Pure Majorana mass:

A pure Majorana mass of the form
$m_{\nu,R}\overline{\nu^{c}_{\hphantom{c}R}}\,\nu_{R}$ for the exotic degree
of freedom would not solve the issue, as the right-handed field is a pure
singlet and thus sterile and, therefore, could not grant a mass to the known
neutrinos.

  3. 3.

Seesaw Mechanism:

In the Seesaw Mechanism, with (type-II) or without (type-I) the addition of an
exotic scalar triplet, the Majorana and Dirac mass terms appear together, and
conspire to produce a light and a heavy mass eigenstate. This is because
natural values of $y_{\nu,\text{D}}$ and $m_{\nu,R}$ produce a sufficient
splitting between the two eigenvalues. Although Seesaw models are by far the
most viable, the issue with type-I Seesaw is that a non-explained parametric
hierarchy remains; And with the type-II mechanism is that, with the
introduction of a scalar triplet, unobserved physical goldstones degrees of
freedom enter the theory (which could be avoided if lepton number violation is
included in the potential).

For a nice introductory review of the theoretical and experimental aspects of
neutrino masses see [80] and [81], respectively.

### 3.2 Hierarchy Problem

Another quantum field theoretical trouble of the SM is the Hierarchy Problem
(HP). Although less important than the neutrino masses, it has been
traditionally used as a guide in the pursuit of new physics while new strong
experimental input does not appear. The Higgs mass is radioactively corrected
at 1-loop by the diagrams of Figure 3.2. Parametrizing the divergence in these
contributions through a cutoff regulator, each of them contribute an amplitude
of the form

$i\mathcal{M}\sim\Lambda^{2}.$ (3.8)

Because of its coupling strength, the value of the fermion loop diagram with
$f$ being the top quark is around two orders of magnitude larger than the
others, which is why it is usually considered alone.

The usual line of thought then follows to interpret this result as physically
meaningful by setting the cutoff to a scale of next physics, such as the Grand
Unification $\left($10^{15}\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$\right)$ or
Planck $\left($10^{19}\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$\right)$ scale.
It is then concluded that, to be able to fit the physical mass of the Higgs,
the SM implies a fine tuning of

$\frac{\delta m_{H}^{2}}{m_{H}^{2}}\sim 10^{33}.$ (3.9)

Such a 10 quadrillion fine tuning in the masses violates the principle of
naturalness, which dictates that every dimensionless ratio between parameters
of a same sector must be close to order one.

Figure 3.2: Quadratically divergent corrections to the Higgs mass which are
the perturbative root of the HP. Here $f$ represents any fermion.

This argumentation is a bit misleading in that the cutoff, as presented, has
no physical meaning and is nothing but an artifact used in order to make sense
of intermediate calculations. The precise way to introduce the HP is through
an effective field theory formalism, in which case $\Lambda$ becomes a
physically meaningful Wilsonian cutoff [82]. Another way to perceive the
inaccuracy of the previous rationale comes from trying to replicate it using,
instead of cutoff, dimensional regularization. In that case, the HP does not
manifest and one is tempted to say, as is often done, that it becomes hidden.
This makes no strict sense, and the correct claim is that the HP can only be
addressed in effective constructions. With this in mind, notice that, unlike
in cutoff regularization which presents an explicit avatar of new physics
($\Lambda$), in dimensional regularization the new physics must be put in by
hand, e.g., as new interactions with heavy particles. Indeed, to produce an
example, we calculate the $\overline{\mathit{MS}}$ correction to the Higgs
mass that results from adding a test heavy fermion, with $m_{H}/m_{f}\to 0$,
to interact with the $H$ with Yukawa $\lambda_{f}$. We find, for physical
characteristic energies of the order of the Higgs mass, that the squared mass
of the scalar particle picks up a contribution of

$\delta
m_{H}^{2}(\mu)=\Sigma^{\overline{\mathit{MS}}}=\frac{\lambda_{f}^{2}}{8\pi^{2}}m_{f}^{2}\left[3\log\left(\frac{m_{f}^{2}}{\mu^{2}}\right)-1\right].$
(3.10)

We see that $\delta m_{H}^{2}\sim m_{f}^{2}$, which is how the ‘quadratic
scaling of the renormalized mass with the new physics parameter’ appears
(which one could perhaps argue from dimensional reasoning alone).

Summarizing, we emphasize that the HP is a real issue – the problem, however,
is not ‘the top loop’, but the sensitivity to higher scaled physics.
Furthermore, it is an intrinsic issue of scalar masses, as fermion masses are
protected by the chiral symmetry which is restored if they vanish; and vector
boson masses, on renormalizable theories, are protected by the gauge symmetry
in the same way.

The regular strategy to solve the HP amounts to extending this mechanism to
the scalar masses. In fact, the HP is one of the greatest motivations for
supersymmetry [83], as it is responsible, in the Minimal Supersymmetric
Standard Model (MSSM), for guarding the Higgs mass against quadratic
corrections. Diagrammatically, this occurs through the loop with inverse
statistics (and hence sign) induced by the stops, which exactly cancels the
top loop independently of the masses of the supersymmetric partners, as long
as the relation between couplings is maintained. Unfortunately, the MSSM, even
if eventually proven the right theory, have already become a non-ideal
solution of the HP because the lightest stop mass has been pushed to the TeV
scale by experiment [84, 85], which entails a leftover 1% fine tuning on the
Higgs mass – this is the so-called Little Hierarchy Problem.

Variant attempts at solving the HP comprise models in which the physical Higgs
arises as a pseudo Nambu-Goldstone boson (pNGB), such as Twin [86, 87] or
Composite Higgs [88] models. The intuition behind it is that putting a pNGB
mass to zero suffices to recover the explicitly broken symmetry, so that it
itself is a technically natural parameter. One more possibility is the Folded
Minimal Supersymmetric Standard Model (FMSSM) [89], which has a low energy
representation content similar to that of the MSSM, but in which the colored
superpartners are charged not under QCD, but under a distinct, ‘dark’, QCD′.
In the FMSSM, the ultraviolet supersymmetry still protects the Higgs mass, but
the phenomenological constraints on the stops scale are less severe.

### 3.3 Strong $\mathit{CP}$ problem

If one takes it seriously, the naturalness principle originates a second
source of stress onto the legitimacy of the SM. The requirement that every
dimensionless ratio is of order one implicate the expectation that every term
allowed by the theoretical framework is included in the Lagrangian. Now,
consider the kinetic operator of a gauge boson multiplet in a Yang-Mills
theory, defined to contain every invariant quadratic in the field strength
tensor (since bilinears on the gauge fields can only appear through this
specific combination). In a theory with a semisimple local symmetry group,
this can be shown to be equivalent to [90]

$\mathcal{L}_{\text{kin}}=-\frac{1}{4}F^{a}_{\mu\nu}F^{a\mu\nu}+\frac{\theta}{32\pi^{2}}\epsilon_{\mu\nu\rho\sigma}F^{\mu\nu}F^{\rho\sigma}.$
(3.11)

The first term is the usual gauge boson kinetic term, but the second is
generally omitted. To understand why, we verify

$\begin{split}\frac{1}{8}\epsilon_{\mu\nu\rho\sigma}F^{\mu\nu}F^{\rho\sigma}&=\mathrm{tr}\left[F\wedge
F\right]=\mathrm{tr}\left[(\mathrm{d}A+A\wedge A)\wedge(\mathrm{d}A+A\wedge
A)\right]\\\
&=\mathrm{tr}\left[\mathrm{d}(A\wedge\mathrm{d}A)+\frac{2}{3}\mathrm{d}(A\wedge
A\wedge A)\right]\\\
&=\mathrm{d}\,\mathrm{tr}\left[A\wedge\mathrm{d}A+\frac{2}{3}A\wedge A\wedge
A\right],\end{split}$ (3.12)

where $\mathrm{d}$ is the exterior derivative and we have used the ciclicity
of the trace, the anti-commutativity of the wedge product, and
$\text{d}^{2}=0$. This shows that the so-called $\theta$-term is a total
derivative, and, thus, it is perturbatively inconsequential.

Notwithstanding, the topological defect formalism shows that if euclidean
spacetime is identified with the 3-sphere (which amounts to the
compactification of spacetime by identifying infinity with a point), then for
pure gauge configurations [91, 92, 93]

$g_{\mu}\big{\rvert}_{S^{3}}=iU(x)\partial_{\mu}U^{\dagger}(x)\Rightarrow\int_{S^{3}}d^{4}x\frac{\theta}{32\pi^{2}}\epsilon_{\mu\nu\rho\sigma}F^{\mu\nu}F^{\rho\sigma}=\theta
n,$ (3.13)

where $g_{\mu}$ is the gluon field, $U(x)$ is a local $SU(3)$ gauge
transformation and $n$ is an integer. This implies that a $\theta$-term
generates (or is generated by, if one inverts the reasoning) non-perturbative
solutions of the field equations which interpolates between inequivalent
vacua. Moreover, consider the intuitive abelian electromagnetic situation, in
which case we define the electric and magnetic fields as $E_{i}\equiv F_{0i}$
and $B_{i}\equiv-\frac{1}{2}\epsilon_{ijk}F^{jk}$. In this context, one has

$\epsilon_{\mu\nu\rho\sigma}F^{\mu\nu}F^{\rho\sigma}\propto\mathbf{E}\cdot\mathbf{B}.$
(3.14)

This is the easiest way to arrive at the conclusion that the $\theta$-term
violates $\mathit{P}$ and $\mathit{CP}$ if one recalls from electrodynamics
that $\mathbf{E}$ is a $T$-odd vector while $\mathbf{B}$ is a $T$-even
pseudovector, and that the gluon field is real.

Being non-perturbative, one has no hope to evaluate exact phenomenological
repercussions of the $\theta$-term with the fundamental degrees of freedom,
but the fact that it is the only possible source of $\mathit{CP}$ violation in
the strong sector prompts us to seek evidence of this non-conservation in pure
QCD processes in order to estimate the value of $\theta$. Indeed, this is done
via measurement of the neutron electric dipole moment (EDM), as the existence
of such a permanent dipole of a spin-1/2 particle implicates in $C$ and $T$
violation, and, thus, that also $\mathit{CP}$ is violated. The connection is
made through chiral perturbation theory [94] (for a more modern approach in
lattice QCD, see [95]), which allows a bound on the neutron EDM to be
translated into a bound on $\theta$.

The current result for the EDM bound is [96]

$d_{n}<1.12\times 10^{-26}\,e\hskip 0.80002pt$\mathrm{c}\mathrm{m}$,$ (3.15)

implying a severe fine tuning on the physical parameter of the order [97]

$\theta\lesssim 10^{-10},$ (3.16)

which states the strong $\mathit{CP}$ problem.

Notice that, in principle, similar effects could be present within the
electroweak sector – specifically, a $\theta$-term for the non-abelian $SU(2)$
is also possible. To see why it is unphysical in that context, recall that by
virtue of the anomalous nature of the chiral symmetry, a rotation of the form
$\psi_{f}\to e^{i\alpha\gamma_{5}}\psi_{f}$ on a Dirac fermion $\psi_{f}$
induces an increment on the effective action of the form

$\delta\mathcal{L}_{\text{YM}}=-\frac{\alpha}{16\pi^{2}}\epsilon_{\mu\nu\rho\sigma}F^{\mu\nu}F^{\rho\sigma},$
(3.17)

which translates into a redefinition $\overline{\theta}=\theta-2\alpha$. Such
a rotation, however, is of consequence within the scalar sector, introducing a
complex phase on mass terms of the form $m_{i}\to e^{2i\alpha_{i}}m_{i}$ for a
diagonal matrix. In order to keep masses real, by convenience, one generally
chooses to move all $\mathit{CP}$ violating effects of this type to the
$\theta$-term. Now, in the electroweak theory, the chiral anomaly may be used
to remove the $\theta$-term without repercussion on the mass terms, since any
phase matrix may be absorbed into the right-handed fields, which are weak
isospin singlets and cannot give rise to anomaly corrections (in fact, the
most formal way to prove that an electroweak $\theta$-term is unobservable is
to show that the instantonic configurations posses fermionic zero-modes [98]).
This is not the case of QCD, since it is vector-like.

The traditional solution of the strong $\mathit{CP}$ problem is that of the
Peccei-Quinn (PQ) mechanism [99], which, usually in the context of extended
scalar sectors, propose a spontaneously broken PQ global symmetry, which gives
rise to a (pseudo) Nambu-Goldstone boson, the axion [97], that, through its
effective potential corrections, dynamically sets $\overline{\theta}=0$.

### 3.4 Dark matter

The next call for new physics is interdisciplinary. Several observations
indicate the necessity of an exotic species of matter with specific
characteristics, conventionally called dark matter (DM) [100], for reasons to
become apparent. Among the evidences which originated this understanding,
stand out the rotation curves of galaxies, whose circular velocity radial
pattern generally indicate the existence of unseen spherically symmetric
massive halos, much larger than the primary gaseous disk. Another is the
Cosmic Microwave Background (CMB) anisotropies, which are too small to account
for the currently observed large scale structure of the universe. The CMB
spherical oscillations can be made remarkably compatible with the observations
if we include into the description an additional component of matter which
decoupled from the thermal bath before recombination of electrical subatomic
particles into atoms.

The defining qualities of this specific type of matter make it

  * •

Neutral: In other words, it must be dark. Technically, it is required that DM
was effectively decoupled from the baryon-photon plasma at recombination.

  * •

Stable: Clearly, DM must be incapable to decay in cosmological time scales if
it is to account for its observable gravitational effects since early epochs.

  * •

Cold: In fact, there may be a DM component which is hot, but the totality must
include a large cold component, because, if DM was predominantly hot, the
small scale universal structure could not be explained, as ultra-relativistic
almost inert matter would have free steamed out of dense regions.

  * •

Massive: Another obvious feature, as DM must interact gravitationally. The
constraints on its mass are highly model dependent and vary immensely, but we
quote conservative bounds which state that the DM mass is
${}>$70\text{\,}\mathrm{e}\mathrm{V}$$ [101] if it is fermionic and
${}\gtrsim$10^{-22}\text{\,}\mathrm{e}\mathrm{V}$$ [102] if it is bosonic,
with upper limits going up to several masses of the sun.

The standard cosmological description assumes that at some point in the early
history the reactions between DM and ordinary matter became too rare, which
caused DM to fall out of chemical equilibrium and remain an independent
component of the universal energy density, changing on large scales only due
to the spacetime expansion. This process is called freeze-out. Employing the
Einstein equations together with an ansatz for the spacetime geometry (the
Robertson-Walker metric), another ansatz for the state equation of the gas
components along the evolution of the universe and the assumption that the
variations in the local comoving particle densities, on average, are due only
to collisions (through the Boltzmann equation), it is possible to calculate
the current relic density of dark matter (which makes up $\sim 26$% of the
total energy budget), given a particle physics model, in terms of the freeze-
out temperature and equilibrium density of DM.

One of the most motivated and important classes of DM candidate is composed by
WIMPs, or Weakly Interacting Massive Particles. This is because it has been
show that the DM relic density is naturally accounted for by a DM species
which coannihilate with cross sections of electroweak order and masses within
the approximate range $10-$100\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$. SM
neutrinos are the immediate option for WIMP contender, but are known to have a
mass which is too small and could account for only a small fraction of the
relic density [103]. This settles the SM as unable to propose a DM
description.

Popular BSM solutions include electroweak sector extensions that present more
cosmologically effective WIMPs, such as, for instance, sterile heavy neutrinos
(although this possibility is troubled – see [104] and references therein) or
the lightest neutralino in supersymmetric theories [105]. Another possibility
is given by axionic models [106], already mentioned as a possible solution to
the strong $\mathit{CP}$ problem, with the axion being a viable DM candidate.

### 3.5 Matter-Antimatter assymetry

The last important theoretical shortcoming of the SM that we shall mention is,
again, of cosmological nature. Both experimental (such as the study of the
diffuse $\gamma$-ray background [107]) and trivial everyday interactions with
the universe of today show that the amount of existing antimatter is
negligible. This is in contrast with the natural expectation that the
primordial, hot, dense and in equilibrium universe would produce similar
fractions of charge conjugate types of matter.

The departure from this matter-antimatter ‘democracy’ may be measured by the
ratio of the matter-antimatter asymmetry to the number density of photons

$\eta_{\gamma}\equiv\frac{n_{B}-n_{\bar{B}}}{n_{\gamma}},$ (3.18)

which is expected to be constant since early epochs because both numerator and
denominator only scale with the expanding length parameter $a^{3}$. Another
useful, sometimes preferred, quantity, is the ratio to total entropy
density111Unfortunately, the same letter $\eta$ is usually used to denote both
quantities, even though they differ in value. [108]

$\eta_{s}\equiv\frac{n_{B}-n_{\bar{B}}}{s},$ (3.19)

also stationary. To see this, recall, from basic thermodynamics, $dU=TdS-
PdV+\mu dN$. Now use $U=\rho V$ and the Friedmann equation
$\dot{\rho}+3H(\rho+P)=0$ to obtain $dS=-\frac{\mu}{T}dN$. This implies that,
for non-degenerate matter $\left(\mu/T\ll 1\right)$ or when reactions are
impossible $\left(dN=0\right)$, the entropy is conserved. These two quantities
are related by $\eta_{s}=\eta_{\gamma}\frac{n_{\gamma}}{s}$, where the
proportionality factor may be calculated to be
$\frac{n_{\gamma}}{s}\approx\frac{1}{7.04k_{B}}$ [109].

This input parameter has been consistently calculated both by cosmic microwave
background analysis [110] and by Big Bang nucleosynthesis to be

$\eta_{\gamma}=\frac{n_{B}-n_{\bar{B}}}{n_{\gamma}}\simeq\frac{n_{B}}{n_{\gamma}}\approx
6.12\times 10^{-10},$ (3.20)

which cannot be accounted for by the standard cosmological particle model.

The conditions upon which a matter-antimatter symmetric primordial universe
could give rise to the highly asymmetrical current one had been laid out in a
classical paper by Sakharov [111]. The requirements are

  1. 1.

$C$ and $\mathit{CP}$ violation: This is an obvious requirement, as a
$C$-conserving interaction would compensate any conversion from matter to
anti-matter with the opposite reaction. In the presence of chiral matter, $C$
violation is not enough, as a $C$ violating but $\mathit{CP}$ conserving
interaction could still keep the matter-antimatter balance while only turning
right-handed particles into left-handed ones or vice-versa.

  2. 2.

Baryon number violation: Another trivial necessity, as, if baryon number was
exactly conserved, no individual process that can produce an asymmetry can
exist.

  3. 3.

Departure from thermal equilibrium: If thermal equilibrium is maintained,
$\mathit{CPT}$ would assure that no asymmetry can be generated even if the two
first conditions are met.

The SM can, in principle, accommodate all the Sakharov criteria for
baryogenesis. $C$ and $\mathit{CP}$ violation are well understood to exist
within the electroweak sector. A departure from thermal equilibrium is
possible through the expansion of the electroweak phase transition bubbles
supplied by the symmetry breaking – we shall not discuss this further and
direct the reader to the nice summary (on this and most topics in this
subsection) in [112].

Finally, let us discuss condition 2 at greater detail. Baryon number is a
classical symmetry of the SM and generates a vector rotation of the quark
fields of the form $q_{L(R)}\to e^{i\theta/3}q_{L(R)}$, which leaves the
action invariant even in the presence of mass terms. It turns out that this
symmetry is anomalous, as is lepton number. In fact, we have

$\partial_{\mu}J_{B}^{\mu}=\partial_{\mu}J_{L}^{\mu}=n_{f}\frac{g^{2}}{32\pi^{2}}F_{\mu\nu}^{a}\tilde{F}^{a\mu\nu},$
(3.21)

where $n_{f}$ is the number of families, and $g$ and $F^{a}_{\mu\nu}$ are the
$SU(2)_{L}$ coupling and field strength tensor, respectively222Note that the
$B$ generator is defined with a factor $1/3$ with relation to the $L$ one,
which is compensated by the three quark colors..

However, as has already been pointed out, a pure right-chiral rotation can be
performed to eradicate the theoretical signs and hence, if the theory is
accurate as it stands, the observability of any effect resulting of the
anomaly. Nonetheless, the WKB formalism shows that if such effects were
existent (as can be made to happen by the inclusion of explicit baryon
violating interactions [113]) the rate of instantonic tunneling between vacua
that would mediate a change in $B$ is proportional to [114]

$\Gamma_{\text{inst}}(\Delta B)\propto e^{-S_{\text{E}}^{\text{inst}}}\sim
e^{-16\pi^{2}/g_{2L}^{2}}\approx e^{-164},$ (3.22)

which is to say it is impossible in practice and cannot account for
baryogenesis. If this zero-temperature type process cannot help us, however,
there is another type of topological solution which, at high temperatures,
could possibly provide the desired effect.

Note that the anterior pure Yang-Mills theory has no intrinsic scale, and as
such contains a collective coordinate related to its extension. In fact, this
is what saves the theory from the Derrick’s theorem and allows for topological
solutions – in particular, this means that, although a barrier to be tunnelled
through exists, it can be scaled to be arbitrarily small. Consider now a
spontaneously broken gauge theory comprised of a gauge field and a scalar
multiplet with a potential. Now there is a dimensionful parameter in the
theory (the Higgs VEV), and a fixed energy barrier between classical vacua –
in particular, there is no exact instanton solution. If one can calculate such
potential height, they can understand under which conditions a thermally
induced transition between vacua may occur.

The solutions corresponding to such transitions may be found as follows [92,
115]. Consider every path $\mathcal{C}$ in configuration space which connects
consecutive topologically inequivalent vacua of the theory. Find, in each
path, the configuration (i.e., point) of maximal energy $x^{*}_{\mathcal{C}}$.
The lowest such point $(x^{*}_{\mathcal{C}})_{\mathrm{min}}$, i.e., the point
obeying
$\mathcal{E}((x^{*}_{\mathcal{C}})_{\mathrm{min}})=\mathrm{min}\\{\mathcal{E}(x^{*}_{\mathcal{C}})\\}$,
is, by construction, a strict saddle point of the euclidean action and thus a
solution of the field equations. The saddle point solutions are static,
unstable, finite action configurations which live in the top of the barrier
between classical $n$-vacua, and explicit calculation shows they possess
winding number $N_{\mathrm{CS}}=1/2$. Such a construction may be possible or
not, and has been shown to exist in the standard electroweak model [115].

Note that although $J_{B}^{\mu}$ and $J_{L}^{\mu}$ are individually non-
conserved, it coincidentally occurs that
$\partial_{\mu}\left(J_{B}^{\mu}-J_{L}^{\mu}\right)=0$ and $B-L$ remains
conserved in the quantum theory. Furthermore, from (3.21), using that that the
integral of $(g^{2}/32\pi^{2})F_{\mu\nu}^{a}\tilde{F}^{a\mu\nu}$ is the Chern-
Simons number $N_{\text{CS}}=1/2$, we obtain, for the simplest sphaleron decay
(between consecutive vacua), $\Delta(B+L)=Nn_{f}$, for integer $N$. An example
reaction induced by the sphaleron explosion, with $\Delta(B+L)=6$ is

$\bar{u}+\bar{d}+\bar{c}\,\to\,d+2s+2b+t+\nu_{e}+\nu_{\mu}+\nu_{\tau}.$ (3.23)

Specifically, the sphaleron height has been calculated to be about
$10\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$.

As in the case of dark SM neutrinos, though, the asymmetry generated within
pure SM is not enough. To start, the $\mathit{CP}$ violation provided by the
quark mixing is too weak. Furthermore, to produce the observed asymmetry a
much more violent first phase transition than that provided by the SM (which
might be merely a crossover) is necessary [116, 117, 118].

This topic is the most complex and is less direct to catalogue solutions,
which come in the form of many specific models and mechanisms. These, however,
usually involve Lagrangians with stronger sources of $\mathit{CP}$ violation.
Additionally, the thermal transition may be strengthened by some exotic
particle decay, such as a right-handed neutrino (which could also trigger an
assymetry through leptogenesis). This electroweak first order phase transition
is also boosted in supersymmetric or grand unified extensions.

##  4 The Minimal 3-3-1 Model

One set of new physics we shall present regards the phenomenology of an exotic
sector composed by a neutral scalar $s$, a doubly-charged scalar $Y^{\pm\pm}$
and a doubly-charged vector boson $U^{\pm\pm}$. A theory which accommodates at
least one representative of each of this classes is the Minimal 3-3-1 Model
(m331). Later, we shall directly tackle a problem of this theory: a thorough
analysis of its exact Renormalization Group predictions. Although our results
will prove to be useful in a larger scope – and the phenomenological subject
is, in fact, approached in a model independent manner – it is paramount to
give a detailed exposition of the basics of the model. That is the objective
of this chapter, which begins with a discussion of our motivations and of why
it is interesting to focus on the $U^{\pm\pm}$.

### 4.1 Particles, particles, particles

As expressed in the last Chapter, understanding the problems of the SM is
detrimental in the quest of theorizing the physics out of its grasp. This is,
of course, not a deep philosophical statement (as if the SM was not
specifically understood to be flawed, there would be no need for the
theorizing of supposedly better theories), but it calls attention to the
importance of considering every subtle hint that the failures of the SM can
provide while building or studying new models.

A not so subtle hint that can be gained from the discussions carried in that
chapter is that there probably exists exotic particles in the universe which
are not contemplated by the SM. Almost all usual solutions of its problems
imply their existence: Neutrino masses imply either new right-handed degrees
of freedom or exotic scalars (or both); Solutions to the hierarchy problem
imply particles capable to protect the Higgs mass against its quadratic
sensitivity to high scales, such as sfermions in supersymmetric theories or
scalars in twin Higgs models; The Pecci-Quinn mechanism, which deals with the
Strong $\mathit{CP}$ problem, is usually implemented on enlarged scalar
sectors; Within the particle content of the SM there is no viable and
sufficient candidate for Dark Matter, so that a solution itself amounts to the
introduction of some exotic species; And, finally, models which strengthen
thermal phase transition, thus allowing sufficient baryogenesis, also contain
an enlarged particle spectrum. In fact, the situation is more drastic since
realistic models which feature such mechanisms in order to solve one or,
ideally, many of the issues, generally imply an even larger particle content,
with, many times, extra spin-0, spin-1/2 and spin-1 particles simultaneously.

Figure 4.1: Cumulative integrated luminosity of the Atlas and CMS experiments
along each working year. Taken from [119] and [120], respectively.

The problem is that there is no direct evidence – such as a previously
unobserved excess of events indicating a resonance – of these new elementary
particles. Such experimentation is performed at industrial capacity at the
LHC, currently at Run 3 colliding hadrons with a center-of-mass energy of
$\sqrt{s}=$13.6\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$$. The cumulative
integrated luminosity along the three runs within the Atlas and CMS
experiments (the largest groups operating at the LHC) can be seen in Figure
4.1. The total luminosity delivered by the LHC along all runs sums around
$266\text{\,}\mathrm{f}\mathrm{b}^{-1}$. A better notion of the amount of
collision data produced by this experiment, by far the most energetic
available, can be obtained through the expected average luminosity for run 3:
it implies a data production, at the end of works, of $600$ petabytes. Within
all the information accumulated thus far there is no signs of an exotic
species.

In this difficult scenario, it remains to perform exploratory phenomenology,
taking advantage of the ever increasing bounds on physical quantities
(particularly, the pure and direct cross sections on processes) to impose
restrictions on the parameters of any hypothesized particle, theory or generic
concept. Doing this, we may constrain the viable theory space to a smaller
sector, hopefully guiding both theoretical and experimental efforts towards
(or at least closer to) the right direction. This is exactly the spirit of the
first part of this thesis, in which we do not deal directly with the SM
issues, but, rather, perform skeptical, model-independent particle
phenomenology.

Among the many members of the set of motivated hypothetical particles, one
stands out because of its rarity in BSM models: the doubly-charged vector
bilepton. While most of the other species – specially bosons – are present in
several ultraviolet complete or effective models, the doubly-charged spin-1
boson is known to be contained within the m331 and in a $SU(15)$ grand
unification theory – and that is all. The fact that this species is so unique
in terms of its existence within interesting models means that constraining
its parameter space represents crucial information gain. This entity is the
central object of our phenomenological inquiries.

At this point, because it is the only ‘low energy’ model which contains a
$U^{\pm\pm}$ and because the second part of this text will regard it directly,
an introductory review of the m331 is in order.

### 4.2 3-3-1 Model: original motivations

Although the 3-3-1 group had already been considered as a symmetry for the
electroweak interactions in the 70s [121] (similarly, a $SU(3)$ unified
electroweak interaction had been proposed [122]), it was not until many years
later that the model was fully constructed with the modern representation
content. To understand its original motivation, consider the diagram of Figure
4.2(a), of the first order contribution to the process
$\nu_{e}\bar{\nu}_{e}\to W^{+}W^{-}$ in a phenomenologic 1-family theory of
leptons. The diagram in the left, portraying the $t$-channel electron
exchange, is known to violate unitarity in the production of a longitudinally
polarized $W$ pair, with a cross section behaving at high energies as

$\sigma(\nu\bar{\nu}\to W_{0}^{+}W_{0}^{-})\xrightarrow[s\gg
M_{W}]{}\frac{G_{F}^{2}s}{3\pi},$ (4.1)

where $G_{F}=g^{2}/4\sqrt{2}M_{W}^{2}=v_{W}^{2}/\sqrt{2}$ is the Fermi
constant and $s$ is the center-of-mass energy squared. In the SM, gauge
invariance comes to the rescue, and the diagram in Fig. 4.2(b), of the
$s$-channel exchange of a $Z$-boson, exactly cancel the high energy divergent
$s$-dependence of the cross section above.

(a)

(b)

Figure 4.2: (a): $t$-channel $e$ exchange contribution to the production of a
pair of longitudinally polarized $W$ bosons. This graph is non-unitary if
considered alone. (b): $s$-channel $Z$ boson exchange which saves the
unitarity of the model at high energies.

Now, suppose that an exotic singly charged vector boson $W^{\prime\pm}$,
henceforth called $V^{\pm}$, exists, and that it couples to right-handed SM
currents or, equivalently, violates lepton number. Then, our BSM model
predicts the first order contribution to the process $e^{-}e^{-}\to
W^{-}V^{-}$ that is shown in Fig. 4.3(a). The vertex and phase space structure
of this graph is of the same form of that of 4.2(a), hence the theory inherits
the same violation of unitarity depicted by Eq. (4.1). It also follows that
the high energy perturbativity of the theory may be saved by the same
mechanism of the $s$-channel exchange of a vector boson belonging to the gauge
group of $V^{\pm}$ (since a gauge invariant theory is unitary). The
corresponding graph appears in Fig. 4.3(b). In this case, however, the spin-1
intermediate particle must be doubly-charged.

This mechanism to cure the non-unitarity in right-handed currents of exotic
charged vector bosons motivated Pisano and Pleitez [123] to propose the m331
as an extension of the electroweak theory, independently realized by Frampton
[124], whose interest was in producing a theory with a doubly-charged vector
bilepton, interesting in itself.

(a)

(b)

Figure 4.3: The same mechanism that cured non-unitarity in the $W_{0}W_{0}$
production in the SM now working to cure the bad high energy behaviour in the
production of an exotic $V^{\pm}$. The intermediate $s$-channel particle,
however, now must be doubly-charged.

### 4.3 The representation content of the minimal version

We are interested mainly in the minimal version of the 3-3-1 and begin
describing its representation content. The first novelty relative to the SM is
that all the leptonic degrees of freedom (of a same generation) are grouped
within a same multiplet

$L_{\ell}\equiv\begin{pmatrix}\nu_{\ell}\\\ \ell\\\
\ell^{c}\end{pmatrix}_{L}\thicksim(\mathbf{1},\mathbf{3},0),\;\;\;\;\ell=e,\mu,\tau.$
(4.2)

Note the explicit charge conjugation on the third components: this is the
source of some subtleties in deriving the Feynman rules for the model.

In the quark sector, occurs the introduction of three exotic Dirac fermions:
two $j_{i}$, of electric charge $-4/3$, and the $J$, of electric charge $5/3$.
The complete representation content is given by

$\displaystyle Q_{iL}$ $\displaystyle\equiv\begin{pmatrix}d_{i}\\\ -u_{i}\\\
j_{i}\end{pmatrix}_{L}\thicksim(\mathbf{3},\mathbf{\overline{3}},-1/3),$
$\displaystyle\;\;\;\;\;Q_{3L}\equiv\begin{pmatrix}u_{3}\\\ d_{3}\\\
J\end{pmatrix}_{L}\thicksim(\mathbf{3},\mathbf{3},2/3)$
$\displaystyle\;\;\;\;\;u_{\alpha
R}\thicksim\left(\mathbf{3},\mathbf{1},2/3\right),$ $\displaystyle d_{\alpha
R}\thicksim\left(\mathbf{3},\mathbf{1},-1/3\right)$ (4.3)
$\displaystyle\;\;\;\;\;J_{R}\thicksim\left(\mathbf{3},\mathbf{1},5/3\right),$
$\displaystyle j_{iR}\thicksim\left(\mathbf{3},\mathbf{1},-4/3\right)$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;i=1,2;\;\;\;\,\alpha=$
$\displaystyle 1,2,3.$

Note that the fermion content is not democratic as the left-handed triplets of
the first two quark generations form an anti-fundamental representation,
contrary to the other triplets which are fundamental.

The minimal scalar sector [125, 126] is now understood to be composed of three
triplets

$\begin{split}\eta\equiv\begin{pmatrix}\eta^{0}\\\ \eta_{1}^{-}\\\
\eta_{2}^{+}\end{pmatrix}_{L}\thicksim(\mathbf{1},\mathbf{3},0),\;\;\;\rho\equiv\begin{pmatrix}\rho^{+}\\\
\rho^{0}\\\
\rho^{++}\end{pmatrix}_{L}\thicksim(\mathbf{1},\mathbf{3},1),\;\;\;\chi&\equiv\begin{pmatrix}\chi^{-}\\\
\chi^{--}\\\
\chi^{0}\end{pmatrix}_{L}\thicksim(\mathbf{1},\mathbf{3},-1),\end{split}$
(4.4)

and a sextet

$S\equiv\begin{pmatrix}\sigma_{1}^{0}&\frac{h_{2}^{+}}{\sqrt{2}}&\frac{h_{1}^{-}}{\sqrt{2}}\\\
\frac{h_{2}^{+}}{\sqrt{2}}&H_{1}^{++}&\frac{\sigma_{2}^{0}}{\sqrt{2}}\\\
\frac{h_{1}^{-}}{\sqrt{2}}&\frac{\sigma_{2}^{0}}{\sqrt{2}}&H_{1}^{--}\end{pmatrix}\thicksim(\mathbf{1},\mathbf{\overline{6}},0).$
(4.5)

The electric charge operator, from which the assigned values may be confirmed,
is defined (in units of $e$) as the following combination of the diagonal
generators

Table 4.1: 3-3-1 gauge groups and their relevant symbols. The SM symbols appear as well since the theory is an intermediate stage of the 3-3-1 spontaneous symmetry breaking pattern. Note that gauge bosons of different groups are referred to by the same letter (this is a strict abuse of notation) because there is little opportunity for confusion. Notice, also, that the left-handed couplings are now called $g_{3L}$ and $g_{2L}$ (instead of simply $g$), making explicit reference to the group they relate to – and the same goes for $g_{X}$ and $g_{Y}$ (in place of $g^{\prime}$). | Gauge boson | Coupling constant | Generator
---|---|---|---
$SU(3)_{c}$ | $g^{a}_{\mu}$ | $g_{s}$ | $\frac{1}{2}\lambda_{c}^{a}$
$SU(3)_{L}$ | $W^{a}_{\mu}$ | $g_{3L}$ | $\frac{1}{2}\lambda^{a}$
$U(1)_{X}$ | $b_{\mu}$ | $g_{X}$ | $X$
$SU(2)_{L}$ | $W^{a}_{\mu}$ | $g_{2L}$ | $\frac{1}{2}\sigma^{a}$
$U(1)_{Y}$ | $b_{\mu}$ | $g_{Y}$ | $Y$
$U(1)_{\text{EM}}$ | $A_{\mu}$ | $e$ | $Q$

$Q=T_{3}-\beta T_{8}+X.$ (4.6)

The $\beta$ parameter is defined by the version under consideration and, in
particular, the minimal model is obtained by $\beta=\sqrt{3}$. The generators,
as every other symbol parametrizing the gauge group of the 3-3-1, appear in
Table 4.1.

It is useful to write down the general 3-3-1 covariant derivative

$D_{\mu}=\partial_{\mu}-ig_{s}\frac{\lambda_{c}^{a}}{2}g^{a}_{\mu}-ig_{3L}\frac{\lambda^{a}}{2}W^{a}_{\mu}-ig_{X}XB_{\mu},$
(4.7)

in which appear the vector adjoint multiplets that (besides the unphysical
ghost degrees of freedom) complete the theory:

$\begin{split}g_{\mu}^{a}&\thicksim(\mathbf{8},\mathbf{1},0)\\\
W_{\mu}^{a}&\thicksim(\mathbf{1},\mathbf{8},0)\\\
b_{\mu}&\thicksim(\mathbf{1},\mathbf{1},0).\end{split}$ (4.8)

To finalize this section, we note how the full representation content is
projected onto the SM symmetry in Table 4.2.

Table 4.2: Representation content of the m331 projected onto the SM symmetry. Source multiplet | Projected multiplets
---|---
$L_{\ell}$ | $L^{(2)}_{\ell}\equiv\begin{pmatrix}\nu_{\ell}\\\ \ell\end{pmatrix}_{L}\thicksim(\mathbf{1},\mathbf{2},-1/2)\;\;\;\;\ell_{R}\thicksim\left(\mathbf{1},\mathbf{1},-1\right)$
$Q_{iL}$ |  $Q^{(2)}_{iL}\equiv\begin{pmatrix}d_{i}\\\ u_{i}\end{pmatrix}_{L}\thicksim(\mathbf{3},\mathbf{\overline{2}},1/6)\;\;\;\;u_{iR}\thicksim\left(\mathbf{3},\mathbf{1},2/3\right)\;\;\;\;d_{iR}\thicksim\left(\mathbf{3},\mathbf{1},-1/3\right)$ $j_{iL}\thicksim\left(\mathbf{3},\mathbf{1},-4/3\right)\;\;\;\;j_{iR}\thicksim\left(\mathbf{3},\mathbf{1},-4/3\right)$
$Q_{3L}$ |  $Q^{(2)}_{3L}\equiv\begin{pmatrix}u_{3}\\\ d_{3}\end{pmatrix}_{L}\thicksim(\mathbf{3},\mathbf{2},1/6)\;\;\;\;u_{3R}\thicksim\left(\mathbf{3},\mathbf{1},2/3\right)\;\;\;\;d_{3R}\thicksim\left(\mathbf{3},\mathbf{1},-1/3\right)$ $J_{L}\thicksim\left(\mathbf{3},\mathbf{1},5/3\right)\;\;\;\;J_{R}\thicksim\left(\mathbf{3},\mathbf{1},5/3\right)$
$\eta$ | $\phi_{\eta}\equiv\begin{pmatrix}\eta^{0}\\\ \eta_{1}^{-}\end{pmatrix}\thicksim(\mathbf{1},\mathbf{2},-1/2)\;\;\;\;\eta^{+}_{2}\thicksim(\mathbf{1},\mathbf{1},1)$
$\rho$ | $\phi_{\rho}\equiv\begin{pmatrix}\rho^{+}\\\ \rho^{0}\end{pmatrix}\thicksim(\mathbf{1},\mathbf{2},1/2)\;\;\;\;\rho^{++}\thicksim(\mathbf{1},\mathbf{1},2)$
$\chi$ | $\phi_{\chi}\equiv\begin{pmatrix}\chi^{-}\\\ \chi^{--}\end{pmatrix}\thicksim(\mathbf{1},\mathbf{2},-3/2)\;\;\;\;\chi^{0}\thicksim(\mathbf{1},\mathbf{1},0)$
$S$ |  $\phi_{S}\equiv\begin{pmatrix}h_{1}^{-}\\\ \sigma_{2}^{0}\end{pmatrix}\thicksim(\mathbf{1},\mathbf{\overline{2}},-1/2)\;\;\;\;H^{--}\thicksim(\mathbf{1},\mathbf{1},-2)$ $\Phi_{S}\equiv\begin{pmatrix}\sigma_{1}^{0}&\frac{h_{2}^{+}}{\sqrt{2}}\\\ \frac{h_{2}^{+}}{\sqrt{2}}&H_{1}^{++}\end{pmatrix}\thicksim(\mathbf{1},\mathbf{\overline{3}},1)$

### 4.4 Spontaneous symmetry breaking

The scalar potential is the most general gauge invariant, renormalizable
functional of the scalar degrees of freedom. The true maximal functional that
fulfil these requisites, however, is not the most frequently postulated
because it prevents the possibility of conferring a consistent lepton number
to each scalar. The popular potential which does allow for generalized lepton
number conservation is

$\begin{split}V_{1}(\eta,\rho,\chi,S)&=\mu_{1}^{2}\eta^{\dagger}\eta+\mu_{2}^{2}\rho^{\dagger}\rho+\mu_{3}^{2}\chi^{\dagger}\chi+\lambda_{1}(\eta^{\dagger}\eta)^{2}+\lambda_{2}(\rho^{\dagger}\rho)^{2}\\\
&+\lambda_{3}(\chi^{\dagger}\chi)^{2}+\lambda_{4}(\eta^{\dagger}\eta)(\rho^{\dagger}\rho)+\lambda_{5}(\eta^{\dagger}\eta)(\chi^{\dagger}\chi)+\lambda_{6}(\rho^{\dagger}\rho)(\chi^{\dagger}\chi)\\\
&+\lambda_{7}(\eta^{\dagger}\rho)(\rho^{\dagger}\eta)+\lambda_{8}(\eta^{\dagger}\chi)(\chi^{\dagger}\eta)+\lambda_{9}(\chi^{\dagger}\rho)(\rho^{\dagger}\chi)\\\
&+f_{1}(\eta\rho\chi+\text{H.C.})\\\
&+\mu_{4}^{2}\text{Tr}(S^{\dagger}S)+\lambda_{10}[\text{Tr}(S^{\dagger}S)]^{2}+\lambda_{11}\text{Tr}[(S^{\dagger}S)^{2}]\\\
&+[\lambda_{12}(\eta^{\dagger}\eta)+\lambda_{13}(\rho^{\dagger}\rho)+\lambda_{13}(\chi^{\dagger}\chi)]\text{Tr}(S^{\dagger}S)\\\
&+f_{2}(\rho\chi S+\text{H.C.})\\\
&+\lambda_{15}\eta^{\dagger}SS^{\dagger}\eta+\lambda_{16}\rho^{\dagger}SS^{\dagger}\rho+\lambda_{17}\chi^{\dagger}SS^{\dagger}\chi\\\
&+(\lambda_{19}\rho^{\dagger}S\rho\eta+\lambda_{20}\chi^{\dagger}S\chi\eta+\lambda_{21}\eta^{2}SS+\text{H.C.}),\end{split}$
(4.9)

where each term schematically represent the invariant contraction – for
instance, the term $\eta\rho\chi$ is expanded out as
$\epsilon^{ijk}\eta_{i}\rho_{j}\chi_{k}$, which is invariant since
$\epsilon^{ijk}\eta_{i}\rho_{j}\sim(\mathbf{3}\otimes\mathbf{3})_{A}=\bar{\mathbf{3}}$,
where the $A$ subscript here indicates that the irreducible anti-symmetric
part is taken. The last two lines of the potential above are usually omitted,
although there is no good first principle reason to do so (additional discrete
symmetries that could be enforced to forbid that portion of the potential
would accidentally forbid other terms and harmfully enhance the symmetry of
the potential, generating further unwanted Goldstone bosons). The elimination
of some parameters through minimization of the potential above and the
obtaining and diagonalization of the mass matrices are straightforward and,
because they do not pertain to our study and, in particular, are not necessary
for the understanding of the symmetry breaking pattern, are not performed
here. For a complete treatment of these matters we refer the reader to [127].

To get a first sense of what to expect of the SSB, let us recall the
projection of the m331 scalars capable of acquiring a VEV onto the SM:

$\begin{gathered}\phi_{\eta}\equiv\begin{pmatrix}\eta^{0}\\\
\eta_{1}^{-}\end{pmatrix}_{L}\thicksim(\mathbf{1},\mathbf{2},-1/2),\;\;\;\phi_{\rho}\equiv\begin{pmatrix}\rho^{+}\\\
\rho^{0}\end{pmatrix}_{L}\thicksim(\mathbf{1},\mathbf{2},1/2),\;\;\;\phi_{S}\equiv\begin{pmatrix}h_{1}^{-}\\\
\sigma_{2}^{0}\end{pmatrix}_{L}\thicksim(\mathbf{1},\mathbf{\overline{2}},-1/2)\\\
\Phi_{S}\equiv\begin{pmatrix}\sigma_{1}^{0}&\frac{h_{2}^{+}}{\sqrt{2}}\\\
\frac{h_{2}^{+}}{\sqrt{2}}&H_{1}^{++}\end{pmatrix}\thicksim(\mathbf{1},\mathbf{\overline{3}},1),\;\;\;\;\chi^{0}\thicksim(\mathbf{1},\mathbf{1},0).\end{gathered}$
(4.10)

Looking above, it may be realized that the condensation of
$\phi_{\eta},\phi_{\rho},\phi_{S}$ and $\Phi_{S}$ all trigger the pattern
$SU(2)_{L}\times U(1)_{Y}\to U(1)_{\mathrm{EM}}$, i.e., the SM SSB. In the
other hand, $\chi^{0}$ is absolutely neutral from the SM point of view and
cannot contribute to the SSB. This hints that the $\chi$-triplet alone is
responsible for the m331 descent into the SM.

With this finding in mind, consider the theory at high energies, in which case
the active symmetry is the full $SU(3)_{c}\times SU(3)_{L}\times U(1)_{X}$.
The VEV to be examined is

$\langle\chi\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}0\\\ 0\\\
v_{\chi}\end{pmatrix}\thicksim(\mathbf{1},\mathbf{3},-1).$ (4.11)

and the complete set of generators of the electroweak symmetry read

$\displaystyle T_{1}$ $\displaystyle=\frac{1}{2}\begin{pmatrix}0&1&0\\\
1&0&0\\\ 0&0&0\end{pmatrix},\qquad T_{2}$
$\displaystyle=\frac{1}{2}\begin{pmatrix}0&-i&0\\\ i&0&0\\\
0&0&0\end{pmatrix},\qquad T_{3}$
$\displaystyle=\frac{1}{2}\begin{pmatrix}1&0&0\\\ 0&-1&0\\\
0&0&0\end{pmatrix}$ $\displaystyle T_{4}$
$\displaystyle=\frac{1}{2}\begin{pmatrix}0&0&1\\\ 0&0&0\\\
1&0&0\end{pmatrix},\qquad T_{5}$
$\displaystyle=\frac{1}{2}\begin{pmatrix}0&0&-i\\\ 0&0&0\\\
i&0&0\end{pmatrix}$ $\displaystyle T_{6}$
$\displaystyle=\frac{1}{2}\begin{pmatrix}0&0&0\\\ 0&0&1\\\
0&1&0\end{pmatrix},\qquad T_{7}$
$\displaystyle=\frac{1}{2}\begin{pmatrix}0&0&0\\\ 0&0&-i\\\
0&i&0\end{pmatrix},\qquad T_{8}$
$\displaystyle=\frac{1}{2\sqrt{3}}\begin{pmatrix}1&0&0\\\ 0&1&0\\\
0&0&-2\end{pmatrix}$ $\displaystyle X=\begin{pmatrix}1&0&0\\\ 0&1&0\\\
0&0&1\end{pmatrix}.$

It is imediate to see that

$\begin{split}T_{1}\langle\chi\rangle&=0\\\ T_{2}\langle\chi\rangle&=0\\\
T_{3}\langle\chi\rangle&=0\\\
\left(-\sqrt{3}T_{8}+X\right)\langle\chi\rangle&=0.\\\ \end{split}$ (4.12)

The four above are the only generators which annihilate the vaccum, and the
symmetry generated by the five remaining ones is spontaneously broken.

Now, the conserved $\\{T_{1},T_{2},T_{3}\\}$ manifestly generate a $SU(2)$
restricted to the uppermost two components of the triplets. Furthermore, from
the definitions of the charge operators within the m331 and the SM as seen in
Eqs. (2.1) and (4.6), respectively, we find that the embedding of the weak
hypercharge into the m331 is given by

$Y=-\sqrt{3}T_{8}+X,$ (4.13)

which corresponds precisely to the fourth conserved generator!

We have thus found that the SSB of the m331 proceeds through the acquisition
of a VEV by $\chi$, triggering $SU(3)_{c}\times SU(3)_{L}\times U(1)_{X}\to
SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}$. And, through the VEVs of every
other neutral scalar, the usual SM breaking is guaranteed.

### 4.5 Vector boson eigenstates and masses

The representation of the adjoint charge operator may be readily obtained as

$\mathfrak{ad}_{Q}=\begin{pmatrix}0&-i&&&&&&&\\\ i&0&0&&&&&&\\\ &0&0&0&&&&&\\\
&&0&0&i&&&&\\\ &&&-i&0&0&&&\\\ &&&&0&0&2i&&\\\ &&&&&-2i&0&0&\\\ &&&&&&0&0&0\\\
&&&&&&&0&0\end{pmatrix}$ (4.14)

where the basis have been ordered as $\\{T_{1},T_{2},\cdots,T_{8},X\\}$.
$\mathfrak{ad}_{Q}$ is easily diagonalized, with eigenvectors

$\begin{split}T_{\pm}&=\frac{T_{1}\pm iT_{2}}{\sqrt{2}}\\\
T^{\prime}_{\pm}&=\frac{T_{4}\mp iT_{5}}{\sqrt{2}}\\\
T_{\pm\pm}&=\frac{T_{6}\mp iT_{7}}{\sqrt{2}},\end{split}$ (4.15)

plus a degenerate three-dimensional eigenspace of neutral generators. Now, the
gauge boson eigenstates associated to the generators above can be found by
decomposing the corresponding sector of the algebra valued gauge multiplet
$W^{i}_{\mu}T_{i}$ in terms of the generators with definite charge:

$\begin{split}\smash[b]{\sum_{T_{i}\in
B_{\mathfrak{g}_{c}}}T_{i}W^{i}_{\mu}}&\equiv
T_{+}W^{+}_{\mu}+T_{-}W^{-}_{\mu}\\\
&\quad+T^{\prime}_{+}V^{+}_{\mu}+T^{\prime}_{-}V^{-}_{\mu}\\\
&\quad+T_{++}U^{++}_{\mu}+T_{--}U^{--}_{\mu}\,,\end{split}$ (4.16)

where $B_{\mathfrak{g}_{c}}=\\{T_{1},T_{2},T_{4},T_{5},T_{6},T_{7}\\}$ is a
basis of the charged sector of the algebra. The result for the charged gauge
boson mass eigenstates in terms of the symmetry eigenstates, finally, reads

$\begin{split}W_{\mu}^{\pm}&=\frac{W_{\mu}^{1}\mp iW_{\mu}^{2}}{\sqrt{2}}\\\
V_{\mu}^{\pm}&=\frac{W_{\mu}^{4}\pm iW_{\mu}^{5}}{\sqrt{2}}\\\
U_{\mu}^{\pm\pm}&=\frac{W_{\mu}^{6}\pm iW_{\mu}^{7}}{\sqrt{2}}.\\\
\end{split}$ (4.17)

Denoting the VEVs by $\langle\eta^{0}\rangle=\frac{v_{\eta}}{\sqrt{2}}$,
$\langle\rho^{0}\rangle=\frac{v_{\rho}}{\sqrt{2}}$,
$\langle\chi^{0}\rangle=\frac{v_{\chi}}{\sqrt{2}}$,
$\langle\sigma_{2}^{0}\rangle=v_{s2}$ and
$\langle\sigma_{1}^{0}\rangle=\frac{v_{s}}{\sqrt{2}}$, their masses may be
directly calculated from the scalar kinetic terms, to give

$\begin{split}M_{W}^{2}&=\frac{1}{4}g^{2}v_{W}^{2}\\\
M_{V}^{2}&=\frac{1}{4}g^{2}(v_{\eta}^{2}+2v_{s}^{2}+v_{\chi}^{2})\\\
M_{U}^{2}&=\frac{1}{4}g^{2}(v_{\rho}^{2}+2v_{s}^{2}+v_{\chi}^{2}).\end{split}$
(4.18)

The diagonalization of the neutral sector is more involved, and there is no
way to escape the mass matrix, given by

$M^{2}=\frac{g^{2}v_{\chi}^{2}}{4}\begin{pmatrix}\bar{v}_{W}^{2}&\frac{1}{\sqrt{3}}(\bar{v}_{W}^{2}-2\bar{v}_{\rho}^{2})&-2t\bar{v}_{\rho}^{2}\\\
\frac{1}{\sqrt{3}}(\bar{v}_{W}^{2}-2\bar{v}_{\rho}^{2})&\frac{1}{3}(\bar{v}_{W}^{2}+4)&\frac{2}{\sqrt{3}}t(\bar{v}_{\rho}^{2}+2)\\\
-2t\bar{v}_{\rho}^{2}&\frac{2}{\sqrt{3}}t(\bar{v}_{\rho}^{2}+2)&4t^{2}(\bar{v}_{\rho}^{2}+1)\end{pmatrix},$
(4.19)

where the overbar indicates the ratio by $v_{\chi}$ as
$\bar{v}_{\alpha}\equiv\frac{v_{\alpha}}{v_{\chi}}$. To enhance clarity, in
this chapter we avoid abbreviations expressing the relation between mass and
symmetry eigenstates, which we calculate to be

$\begin{split}Z_{1\mu}&=\frac{Z_{13}W_{3\mu}+Z_{18}W_{8\mu}+Z_{1B}B_{\mu}}{N_{1}}\\\
Z_{2\mu}&=\frac{Z_{23}W_{3\mu}+Z_{28}W_{8\mu}+Z_{2B}B_{\mu}}{N_{2}}\\\
A_{\mu}&=\frac{A_{3}W_{3\mu}+A_{8}W_{8\mu}+A_{B}B_{\mu}}{N_{A}},\\\
\end{split}$ (4.20)

where $Z_{1\mu}$ components are given by

$\begin{split}Z_{13}&=\frac{-\bar{v}_{W}^{4}+\left[(3t^{2}+2)\bar{v}_{\rho}^{2}+3t^{2}+1+R\right]\bar{v}_{W}^{2}-2(3t^{2}+1)\bar{v}_{\rho}^{4}}{t\left[(3t^{2}+1)\bar{v}_{\rho}^{4}+\left(3t^{2}+1-R\right)\bar{v}_{\rho}^{2}-\bar{v}_{W}^{2}\right]}\\\
Z_{18}&=\frac{-\bar{v}_{W}^{4}+\left[(3t^{2}+2)\bar{v}_{\rho}^{2}+3t^{2}-1+R\right]\bar{v}_{W}^{2}-2\bar{v}_{\rho}^{2}\left(R-3t^{2}-1\right)}{\sqrt{3}t\left[(3t^{2}+1)\bar{v}_{\rho}^{4}+\left(3t^{2}+1-R\right)\bar{v}_{\rho}^{2}-\bar{v}_{W}^{2}\right]}\\\
Z_{1B}&=2;\end{split}$ (4.21)

and the $Z_{2\mu}$ ones read

$\begin{split}Z_{23}&=\frac{-\bar{v}_{W}^{4}+\left[(3t^{2}+2)\bar{v}_{\rho}^{2}+3t^{2}+1-R\right]\bar{v}_{W}^{2}-2(3t^{2}+1)\bar{v}_{\rho}^{4}}{t\left[(3t^{2}+1)\bar{v}_{\rho}^{4}+\left(3t^{2}+1+R\right)\bar{v}_{\rho}^{2}-\bar{v}_{W}^{2}\right]}\\\
Z_{28}&=\frac{-\bar{v}_{W}^{4}-\left[-(3t^{2}+2)\bar{v}_{\rho}^{2}-3t^{2}+1+R\right]\bar{v}_{W}^{2}+2\bar{v}_{\rho}^{2}\left(R+(3t^{2}+1)\right)}{\sqrt{3}t\left[(3t^{2}+1)\bar{v}_{\rho}^{4}+\left(3t^{2}+1+R\right)\bar{v}_{\rho}^{2}-\bar{v}_{W}^{2}\right]}\\\
Z_{2B}&=2;\end{split}$ (4.22)

for $A_{\mu}$, we have

$\begin{split}A_{3}&=t\\\ A_{8}&=-\sqrt{3}t\\\ A_{B}&=1.\end{split}$ (4.23)

The normalization factors have been naturally defined as

$N_{X}=\sqrt{c_{X3}^{2}+c_{X8}^{2}+c_{XB}^{2}},$ (4.24)

for $X_{\mu}=(c_{X3}W_{3\mu}+c_{X8}W_{8\mu}+c_{XB}B_{\mu})/N_{X}$. The $R$
factor, appearing in most expressions, is given by

$R\equiv\left\\{\left[\bar{v}_{W}^{2}-(3+6t^{2})\bar{v}_{\rho}^{2}-1-6t^{2}\right]\bar{v}_{W}^{2}\right.+\\\
\left.{}+(1+3t^{2})\left[3(1+t^{2})\bar{v}_{\rho}^{4}+6t^{2}\bar{v}_{\rho}^{2}+1+3t^{2}\right]\right\\}^{1/2},$
(4.25)

with which the masses may be written

$\begin{split}M_{A}^{2}&=0\\\
M_{Z_{1}}^{2}&=\frac{g_{3L}^{2}}{6}v_{\chi}^{2}\left(\bar{v}_{W}^{2}+3t^{2}\bar{v}_{\rho}^{2}+1+3t^{2}-R\right)\\\
M_{Z_{2}}^{2}&=\frac{g_{3L}^{2}}{6}v_{\chi}^{2}\left(\bar{v}_{W}^{2}+3t^{2}\bar{v}_{\rho}^{2}+1+3t^{2}+R\right).\end{split}$
(4.26)

### 4.6 Yukawa interactions and fermion masses

The Yukawa Lagrangian is composed by the most general renormalizable, gauge
invariant functional bilinear on fermion fields and linear on scalars. Its
primordial objective is that of generating fermion masses through SSB in a
renormalizable fashion. Let us begin treating the leptonic Yukawa sector,
which is instructive because it sheds light over the scalar sector of the
m331.

Note first that a mass term could be projected out of the product
$\mathbf{3}\otimes\mathbf{3}=\bar{\mathbf{3}}_{A}\oplus\mathbf{6}_{S}$, which
already represents great motivation for the existence of triplets and sextets
within the scalar sector of the model. Furthermore, notice that leptons are
$X=0$ representations of $U(1)_{X}$, which means they can only form an
invariant with $\eta$ and $S$. Since $\bar{\mathbf{3}}$ is the anti-symmetric
part of $\mathbf{3}\otimes\mathbf{3}$, the triplet term is given by

$\begin{split}G_{ab}^{\eta}\overline{(L_{ai})^{c}}L_{bj}\epsilon^{ijk}\eta_{k}+\mathrm{H.C.}&=G^{\eta}_{ab}\begin{pmatrix}\overline{\nu_{aR}^{c}}&\overline{\ell_{aR}^{c}}&\overline{\ell_{aR}}\end{pmatrix}\begin{pmatrix}0&0&0\\\
0&0&1\\\ 0&-1&0\end{pmatrix}\begin{pmatrix}\nu_{bL}\\\ \ell_{bL}\\\
\ell_{bL}^{c}\end{pmatrix}\frac{v_{\eta}}{\sqrt{2}}\\\
&=G_{ab}^{\eta}\frac{v_{\eta}}{\sqrt{2}}\left[\overline{\ell^{c}_{aR}}\ell^{c}_{bL}-\overline{\ell_{aR}}\ell_{bL}\right]+\mathrm{H.C.}.\end{split}$
(4.27)

where we have projected out the vacuum component of $\eta$. In the expression
above, $G^{\eta}$ is one a priori arbitrary Yukawa matrix which, from the
symmetry structure of the starting expression, may be taken symmetric. Notice
also that

$\begin{split}\overline{\ell_{aR}^{c}}\ell^{c}_{bL}&=\overline{\ell_{aL}^{\hphantom{aL}c}}\ell^{\hphantom{bR}c}_{bR}=\left(\overline{\ell_{aL}^{\hphantom{aL}c}}\ell^{\hphantom{bR}c}_{bR}\right)^{T}=-\left(\ell^{\hphantom{bR}c}_{bR}\right)^{T}\gamma_{0}^{T}\left(\ell_{aL}^{\hphantom{aL}c}\right)^{*}\\\
&=-\left(\ell^{\hphantom{bR}c}_{bR}\right)^{T}\gamma_{0}^{T}C^{*}\left(\overline{\ell_{aL}}^{T}\right)^{*}=-\left(\ell^{\hphantom{bR}c}_{bR}\right)^{T}\gamma_{0}^{T}C^{*}\gamma_{0}^{\dagger}\ell_{aL}\\\
&=-\left(\ell^{\hphantom{bR}c}_{bR}\right)^{T}CC^{-1}\gamma_{0}^{T}C\gamma_{0}^{\dagger}\ell_{aL}=\left(\ell^{\hphantom{bR}c}_{bR}\right)^{T}C\ell_{aL}\\\
&=\overline{\ell_{bR}}\ell_{aL},\end{split}$ (4.28)

where we have used $(\ell_{L})^{c}=(\ell^{c})_{R}$, and
$\ell^{c}=C\bar{\ell}^{T}$. The charge conjugation matrix obeys
$C^{-1}\gamma_{\mu}C=-\gamma_{\mu}^{T}$ and
$C=C^{*}=-C^{T}=-C^{\dagger}=-C^{-1}$. Using the equation above, together with
the anti-symmetry of $G^{\eta}$, we find

$G_{ab}^{\eta}\overline{(L_{ai})^{c}}L_{bj}\epsilon^{ijk}\eta_{k}+\mathrm{H.C.}=-\frac{2v_{\eta}}{\sqrt{2}}G_{ab}^{\eta}\left(\overline{\ell_{aR}}\ell_{bL}+\mathrm{H.C.}\right),$
(4.29)

stating clearly that it gives a mass term.

Moving on to the term generated by the sextet, we have

$\begin{split}G_{ab}^{S}\overline{(L_{ai})^{c}}L_{bj}S^{ij}&=G^{S}_{ab}\begin{pmatrix}\overline{\nu_{aR}^{c}}&\overline{\ell_{aR}^{c}}&\overline{\ell_{aR}}\end{pmatrix}\begin{pmatrix}\frac{v_{s}}{\sqrt{2}}&0&0\\\
0&0&\frac{v_{s_{2}}}{\sqrt{2}}\\\
0&\frac{v_{s_{2}}}{\sqrt{2}}&0\end{pmatrix}\begin{pmatrix}\overline{\nu_{bL}}\\\
\overline{\ell_{bL}}\\\ \overline{\ell_{bL}^{c}}\end{pmatrix}\\\
&=G_{ab}^{S}\frac{v_{s_{2}}}{\sqrt{2}}[\overline{(\ell^{c}_{a})_{R}}(\ell^{c}_{b})_{L}+\overline{\ell_{aR}}\ell_{bL}]+G_{ab}^{S}\frac{v_{s}}{\sqrt{2}}\overline{(\nu^{c}_{a})_{R}}\nu_{bL},\end{split}$
(4.30)

which, with the same manipulations of the last case, morphs into

$\begin{split}G_{ab}^{S}\overline{(L_{ai})^{c}}L_{bj}S^{ij}+\mathrm{H.C.}&=\frac{2v_{s_{2}}}{\sqrt{2}}G_{ab}^{S}\left(\overline{\ell_{aR}}\ell_{bL}+\mathrm{H.C.}\right)\\\
&\quad{}+G_{ab}^{S}\frac{v_{s}}{\sqrt{2}}\overline{(\nu^{c}_{a})_{R}}\nu_{bL}+\mathrm{H.C.},\end{split}$
(4.31)

so that the sextet generates a contribution to the leptonic Dirac masses and,
if $v_{s}\neq 0$, a Majorana neutrino mass. The leptonic Yukawa Lagrangian as
may then be written as

$\mathcal{L}_{\ell}^{\text{Y}}=\frac{1}{2}G_{ab}^{\eta}\overline{(L_{ai})^{c}}L_{bj}\epsilon^{ijk}\eta_{k}+\frac{1}{2}G_{ab}^{S}\overline{(L_{ai})^{c}}L_{bj}S^{ij}+\text{H.C.}$
(4.32)

from which, and from the equations above, arises the lepton mass matrix

$M_{\ell}=G_{ab}^{\eta}\frac{v_{\eta}}{\sqrt{2}}+G_{ab}^{S}\frac{v_{s2}}{2}.$
(4.33)

This matrix is, in principle, general, and may be diagonalized by a biunitary
transformation like

$\mathrm{diag}(m_{e},m_{\mu},m_{\tau})=V_{R}^{\ell\dagger}M_{\ell}V_{L}^{\ell},$
(4.34)

where $V_{L(R)}^{\ell}$ relates symmetry and mass eigenstates as

$\ell^{\prime}_{L(R)}=V^{\ell}_{L(R)}\ell_{L(R)}.$ (4.35)

This is a good point to seek insight into the scalar build of the model.
Consider a theory in which the sextet has been omitted. In that case,
$M_{\ell}$ is anti-symmetric and hence has a spectrum of the form
$\\{0,-m,m\\}$, where the minus sign of the second eigenvalue may be removed
by a chiral rotation. It is thus impossible to fit the three nonzero and non-
equal lepton masses in a theory without this multiplet.

Eq. (4.32) also gives rise to the following Majorana mass matrix for the
neutrinos

$M_{\nu}=G_{ab}^{S}\frac{v_{s}}{\sqrt{2}},$ (4.36)

by which another model building fact may be perceived. Suppose that the $X=0$
triplet had not been introduced, or that $v_{\eta}=0$ had been set. Then the
lepton and neutrino mass matrices would be proportional $M_{\ell}\propto
M_{\nu}$, and would thus be diagonalized by the same biunitary transformation.
This, in turn, implies that the PMNS matrix obeys
$V_{\text{PMNS}}=V_{L}^{\ell\dagger}V_{L}^{\nu}=\mathbb{1}$. Since this is
phenomenologically unacceptable, the $\eta$-triplet is indispensable.

To finalize the lepton sector discussion, let us consider the addition of a
right-handed neutrino to the model. One may then add the extra piece to the
Lagrangian

$\mathcal{L}_{\nu_{R}}^{\text{Y}}=\frac{1}{2}G_{ab}^{\nu}\overline{L_{ai}}\nu_{bR}\eta_{i}+\frac{1}{2}m^{\nu_{R}}_{ab}\overline{(\nu_{aR})^{c}}\nu_{bR}+\text{H.c.},$
(4.37)

giving, in total, the following neutrino mass matrix

$(M_{\nu})=\begin{pmatrix}G^{S}_{3\times 3}\frac{v_{s}}{\sqrt{2}}&G_{3\times
3}^{\nu}\frac{v_{\eta}}{\sqrt{2}}\\\ G_{3\times
3}^{\nu}\frac{v_{\eta}}{\sqrt{2}}&m_{3\times 3}^{\nu_{R}}\end{pmatrix},$
(4.38)

in the $\\{\nu_{L},\nu_{R}^{\hphantom{R}c}\\}$ basis.

Once the lepton masses are fitted, it becomes clear that an extra triplet is
necessary to arrange the quark masses. With this understood, we simply mention
the results for the colored particles. The Yukawa Lagrangian is given by

$\mathcal{L}_{q}^{\text{Y}}=\overline{Q_{mL}^{\prime}}[G_{m\alpha}^{u}U^{\prime}_{\alpha
R}\rho^{*}+G_{m\alpha}^{d}D^{\prime}_{\alpha
R}\eta^{*}]+\overline{Q_{3L}^{\prime}}[F_{3\alpha}^{u}U^{\prime}_{\alpha
R}\rho+F_{3\alpha}^{d}D^{\prime}_{\alpha R}\eta],$ (4.39)

where $G^{u(d)}$ are $2\times 3$ and $F^{u(d)}$ $1\times 3$ matrices. The
resulting mass matrix for the known up-type quarks in the basis
$\\{-u,-c,t\\}$ is

$M_{u}=\begin{pmatrix}rG^{u}_{11}&rG^{u}_{12}&rG^{u}_{13}\\\
rG^{u}_{21}&rG^{u}_{22}&rG^{u}_{23}\\\ F^{u}_{31}&F^{u}_{32}&F^{u}_{33}\\\
\end{pmatrix}v_{\eta};$ (4.40)

and the one for the down-type quarks in the $\\{d,s,b\\}$ basis reads

$M_{d}=\begin{pmatrix}r^{-1}G^{d}_{11}&r^{-1}G^{d}_{12}&r^{-1}G^{d}_{13}\\\
r^{-1}G^{d}_{21}&r^{-1}G^{d}_{22}&r^{-1}G^{d}_{23}\\\
F^{d}_{31}&F^{d}_{32}&F^{d}_{33}\\\ \end{pmatrix}|v_{\rho}|;$ (4.41)

where we have defined $r\equiv v_{\rho}/v_{\eta}$.

The Yukawa lagrangian for the exotic quarks is easiest to construct

$\mathcal{L}_{j_{m},J}^{\text{Y}}=K^{j}_{ab}\overline{Q_{aL}^{\prime}}\chi^{*}j_{bR}+y^{J}\overline{Q_{3L}^{\prime}}\chi
J_{R},$ (4.42)

giving the following $2\times 2$ mass matrix for the $j_{i}$

$M^{j}_{ab}=K^{j}_{ab}\frac{v_{\chi}}{\sqrt{2}}$ (4.43)

and, for the $J$ mass,

$M^{J}=y^{J}\frac{v_{\chi}}{\sqrt{2}}.$ (4.44)

### 4.7 Anomaly cancellation

Finally, it is instructive to verify that the gauge invariance of the m331 is
unharmed by anomalies. Repeating the process of the end of Section 2.2, we
consider every triangle at a time

  1. 1.

$\left[SU(3)_{c}\right]^{3}$:

This combination trivially vanishes by arguments already exhaustively
explored.

  2. 2.

$\left[SU(3)_{c}\right]^{2}\left[U(1)_{X}\right]$:

The corresponding anomaly factor reads

$\begin{split}\mathfrak{a}_{Xbc}&=\sum_{f}\frac{1}{4}\gamma_{5}X\mathrm{tr}\left[\\{\lambda_{b},\lambda_{c}\\}\right]=\sum_{q}\frac{1}{2}\gamma_{5}X\delta_{bc}\\\
&=\frac{1}{2}\delta_{bc}\left[{}-2\times
3\times\left(-\frac{1}{3}\right)-1\times 3\times\frac{2}{3}+3\times
1\times\frac{2}{3}\right.\\\ &\hskip 39.83368pt\left.{}+3\times
1\times\left(-\frac{1}{3}\right)+1\times 1\times\frac{5}{3}+2\times
1\times\left(-\frac{4}{3}\right)\right]=0,\end{split}$ (4.45)

where handedness is this time included in the presign and the other factors
have been ordered as

$(\mathrm{\\#\hskip 2.5ptof\hskip
2.5ptmultiplets})\times(\mathrm{multiplicity})\times(X);$ (4.46)

The sum is only over quark fields. In particular, it may be noticed that the
anomaly vanishes for any generation in isolation, even though they are treated
differently.

  3. 3.

$\begin{cases}&\left[SU(3)_{c}\right]^{2}\left[SU(3)_{L}\right]\\\
&\left[SU(3)_{c}\right]\left[SU(3)_{L}\right]^{2}\\\
&\left[SU(3)_{c}\right]\left[SU(3)_{L}\right]\left[U(1)_{X}\right]\\\
&\left[SU(3)_{c}\right]\left[U(1)_{X}\right]^{2}\\\
&\left[SU(3)_{L}\right]\left[U(1)_{X}\right]^{2}:\end{cases}$

The combinatorics above all vanish because they are of the form

$\mathfrak{a}\propto\mathrm{tr}\left[\sigma\right]\hskip
3.99994pt\mathrm{or}\hskip 4.19998pt\mathrm{tr}\left[\lambda\right].$ (4.47)

  4. 4.

$\left[SU(3)_{L}\right]^{2}\left[U(1)_{X}\right]$:

This is similar to case 2, but now the sum runs over left-handed triplets

$\begin{split}\mathfrak{a}_{Xbc}&=\sum_{f}\frac{1}{4}\gamma_{5}X\mathrm{tr}\left[\\{\lambda_{b},\lambda_{c}\\}\right]=\sum_{L}\frac{1}{2}\gamma_{5}X\delta_{bc}\\\
&=\frac{1}{2}\delta_{bc}\left[{}-2\times\left(-\frac{1}{3}\right)-1\times\frac{2}{3}{}\right]=0,\end{split}$
(4.48)

  5. 5.

$\left[U(1)_{X}\right]^{3}$:

Straightforwardly:

$\begin{split}\mathfrak{a}&=\sum_{f}\gamma_{5}\,X^{3}\\\ &={}-2\times
3\times\left(-\frac{1}{3}\right)^{3}-1\times
3\times\left(\frac{2}{3}\right)^{3}+3\times
1\times\left(\frac{2}{3}\right)^{3}\\\ &\quad+3\times
1\times\left(-\frac{1}{3}\right)^{3}+2\times
1\times\left(-\frac{4}{3}\right)^{3}+1\times
1\times\left(\frac{5}{3}\right)^{3}=0.\end{split}$ (4.49)

  6. 6.

$\left[SU(3)_{L}\right]^{3}$:

Lastly, to analyse this case, consider

$\begin{split}\mathrm{tr}\left[\gamma^{5}\bar{t}^{a}\left\\{\bar{t}^{b},\bar{t}^{c}\right\\}\right]&=-\mathrm{tr}\left[\gamma^{5}\left(t^{a}\right)^{T}\left\\{\left(t^{b}\right)^{T},\left(t^{c}\right)^{T}\right\\}\right]\\\
&=-\mathrm{tr}\left[\gamma^{5}\left(\left\\{t^{c},t^{b}\right\\}t^{a}\right)^{T}\right]=-\mathrm{tr}\left[\gamma^{5}\left\\{t^{c},t^{b}\right\\}t^{a}\right]\\\
&=-\mathrm{tr}\left[\gamma^{5}t^{a}\left\\{t^{b},t^{c}\right\\}\right].\end{split}$
(4.50)

Above, $\bar{t}$ are the generators of the anti-fundamental, given by
$\bar{t}^{a}=-\left(t^{a}\right)^{T}$, and we have used the invariance of the
trace by transposition. What this shows is that components of an anti-
fundamental contribute the opposite of those of a fundamental representation.