University of Aveiro Physics Department
2020
Jo˜
ao Pedro Pino
Gon¸
calves
Explorando f´ısica nova num modelo com trˆ
es
dubletos de Higgs e fermi˜
oes do tipo vetorial
Exploring new physics scenarios in a
University of Aveiro Physics Department
2020
Jo˜
ao Pedro Pino
Gon¸
calves
Explorando f´ısica nova num modelo com trˆ
es
dubletos de Higgs e fermi˜
oes do tipo vetorial
Exploring new physics scenarios in a
three-Higgs-doublet model with vector-like fermions
Disserta¸
c˜
ao apresentada `
a Universidade de Aveiro para cumprimento dos
requisitos necess´
arios `
a obten¸
c˜
ao do grau de Mestre em Engenharia F´ısica,
realizada sob a orienta¸
c˜
ao cient´ıfica do Doutor Ant´
onio Morais,
Investi-gador Doutorado do Departamento de F´ısica da Universidade de Aveiro e
sob a coorienta¸
c˜
ao do Doutor Roman Pasechnik, Professor Associado do
Departamento de Astronomia e F´ısica Te´
orica da Universidade de Lund.
O trabalho nesta tese foi suportado pelo projeto From Higgs Phenomenology
to the Unification of Fundamental Interactions com referˆ
encia
PTDC/FIS-PAR/31000/2017.
o j´uri / the jury
presidente / president Margarida Fac˜ao
Assistant Professor at the University of Aveiro
vogais / examiners committee Ant´onio Onofre
Associate Professor at the University of Minho (examiner)
Ant´onio Pestana Morais
agradecimentos / acknowledgements
I would first like to thank everyone involved in the work of this thesis. First, my supervisor Ant´onio Morais, for his invaluable support and assistance in this thesis and his constant availability for assisting me. I would also like to thank Roman, especially in the final phase, for the enlightening discussions we had over Skype. I would like to leave a special thanks to Felipe Freitas, post-doc in Aveiro, for immense help in the numerical part of this thesis, whose help and support is most appreciated.
For my friends, I would like to first thank Pedro Rodrigues, a fellow master’s student in particle physics, for the help provided in initial phase as I acclimated to this new environment. I would also like to mention my closest friends Teresa, Diogo, Sara, Raquel and Pedro. While I might not be the most talkative person, your friendship and assistance has been more helpful than you can imagine.
And to finalise, my family, for their support during my path in University, from when I suddenly decided to switch degrees from computer science to physics, to when I decided to pursue theoretical physics, despite the fact I am in an engineering degree.
Resumo O sucesso do Modelo Padr˜ao (MP) n˜ao deve ser substimado, no entanto, a existˆencia tanto de quest˜oes te´oricas como de observa¸c˜oes experiementais sem res-posta no MP favorecem o estudo de teorias com Nova F´ısica. Nesta tese um novo modelo contendo trˆes dubletos de Higgs e fermi˜oes do tipo vetorial ´e apresentado. Este ´e baseado em princ´ıpios de grande unifica¸c˜ao supersim´etrica e, em particular, inspirado pela unifica¸c˜ao das intera¸c˜oes fundamentais sob o grupo de simetriaE8,
que contem o grupo de gauge do MP,G = SU(3)C× SU(2)L× U(1)Y. O limite supersim´etrico de alta energia ´e apresentado de modo a contextualizar e justificar as escolhas de parˆametros e escalas de massa envolvidas nos estudos num´ericos. O limite de baixas energia deste modelo ´e constru´ıdo de forma a ser invariante sob transforma¸c˜oes do grupo de gauge G. Al´em de novos escalares e trˆes gera¸c˜oes de fermi˜oes do tipo vetorial, o modelo tamb´em oferece um vasto setor de neutrinos com doze novos estados, sendo o mais leve est´eril. Nesta tese, e recorrendo a m´etodos baseados em aprendizagem profunda, apresentamos a primeira an´alise fe-nomenol´ogica deste modelo focando-nos no estudo de novos lept˜oes carregados (do tipo vetorial) e suas poss´ıveis assinaturas no grande colisor de prot˜oes do CERN, o LHC. A nossa an´alise num´erica considera eventos de sinal para topologias de fus˜ao de bos˜oes vetoriais e topologias associadas `a produ¸c˜ao de pares de lept˜oes do tipo vetorial, ambas envolvendo um estado final contendo um par de lept˜oes carregados bem como dois neutrinos estereis que atuam como energia em falta. Para processos de fundo do MP consideramos eventos de cromodinˆamica quˆantica (CDQ) (t¯t) e eventos de fus˜ao (W+W−). As simula¸c˜oes de Monte Carlo s˜ao
realizadas para colis˜oes prot˜ao-prot˜ao como no LHC, a uma energia de centro de massa√s = 14 TeV. Todas as observ´aveis calculadas s˜ao fornecidas para an´alise de aprendizagem profunda, onde uma rede neural com 5 camadas (1 entrada + 3 escondidas + 1 sa´ıda) ´e constru´ıda. A rede neural separa com sucesso eventos de fundo de eventos de sinal com uma precis˜ao de quase 100% e uma ´arvore de decis˜ao aprimorada (ADA) ´e implementada de modo a proporcionar uma melhor identifica¸c˜ao dos diferentes tipos de fundo e de sinal. Tendo em conta as duas topologias analisadas, a significˆancia m´axima combinada para a observa¸c˜ao de no-vos lept˜oes do tipo vetorial no LHC ´e de13.62σ se a sua massa for de 200 GeV, decaindo a mesma para valores inferiores a 2σ se a massa destes estiver acima de 1 TeV. Os resultados obtidos oferecem a possibilidade de testar o modelo em futuras runs do LHC e, em particular, constranger a escala de energia `a qual novas simetrias de gauge emergentes do cen´ario de Unifica¸c˜ao se poder˜ao manifestar.
Abstract The success of the Standard model (SM) can not be understated, however, the existence of both theoretical questions as well as experimental observations without an answer in the SM favours the study of New Physics theories. In this thesis, a new model featuring three Higgs doublets and vector-like fermions is presented. It is based upon a supersymmetric Grand Unified Theory where all fundamental interactions described by the SM gauge groupG = SU(3)C× SU(2)L× U(1)Y as well as the observed flavour structure in nature are ultimately unified under theE8
symmetry group. The supersymmetric high-energy limit of this theory is provided in order to motivate the various parameter choices and mass scales involved in the numerical studies. The low-energy effective limit of this model is constructed to be invariant under the transformations of the gauge group G. Besides new scalars and three generations of vector-like fermions, the model also provides a rich neutrino sector, with twelve new states, where the lightest one is sterile. In this thesis, and using deep learning techniques, we perform the first phenomenological analysis of this model focusing on the study of new charged vector-like leptons (VLLs) and their possible signatures at CERN’s Large Hadron Collider (LHC). In our numerical analysis, we consider signal events for vector-boson fusion and VLL pair production topologies, both involving a final state containing a pair of charged leptons and two sterile neutrinos that act as missing energy. For the SM background processes we consider the quantum chromodynamics (QCD) (t¯t) and the W+W−fusion events.
Monte Carlo simulations are performed for LHC proton-proton collisions at a total centre of mass energy √s = 14 TeV. All calculated observables are provided as data sets for deep learning analysis, where a neural network with 5 layers (1 input + 3 hidden + 1 output) is constructed. The neural network successfully separates background from signal events with approximately 100% accuracy and where a boosted decision tree (BDT) is implemented in order to offer a better identification of the different types of background and signal. Taking into account the effect of both analysed topologies, the maximal combined significance for the observation of new VLLs at the LHC is of13.62σ if their mass is 200 GeV, becoming smaller than 2σ if the lightest VLL mass is beyond 1 TeV. The results obtained show that our model can be tested at future LHC runs and, in particular, constraining the energy scale for which new gauge symmetries emergent from the unification picture can be manifest.
Lista de Figuras
4.1 Schematic representation of the overall flow of the numerical analysis. We start off with the implementation of the model, at the Lagrangian density level, with SARAH, followed by computation of matrix elements via MadGraph. Afterwards, we proceed to hadronization phase, via the Monte Carlo software, PYTHIA8. Finally, at the end, we use deep learning analysis for significance studies. . . 21 4.2 Leading-order Feynman diagram for the ZA topologies. q and ¯q correspond to
quarks originating from the initial protons. V`` represents VLLs and νBSM
deno-tes the lighdeno-test BSM neutrino. There are two purely SM leptonic final channels identified with ` and ν`. . . 22
4.3 Leading-order Feynman diagrams for the VBF topologies. The same nomenclature as seen in Fig. 4.2 applies here. νBSM0 correspond to any BSM neutrino. . . 23
4.4 t¯t background topologies. All particles shown here are purely SM ones. The two final leptonic channels are chosen to be equivalent to signal events final leptonic states. . . 24 4.5 W+W−background topologies. All particles shown here are purely standard model
ones. The two final leptonic channels are chosen to be equivalent to signal events final leptonic states. . . 24 4.6 Schematic representation of two learning rate scenarios. In the left, a constant rate
∆R is considered, which leads to the machine being stuck in a loop between to points. In the right, a varying ∆R is considered, leading to the global minima. . . 26 4.7 A simple example of BDT. Here, we give a vector input, and through magnitude
comparisons we get to the various outputs. In our case, the Input file will be the kinematic features in conjunction with the machine learning predictions. . . 28 4.8 Plots of loss and accuracy as a function of the epoch. On top, we have results for the
ZA topologies, while in the bottom we present the corresponding results for VBF topologies. In orange, validation data set is used while in blue the training data set is plotted. For all scenarios, the overall accuracy (loss) decreases (increases) with epochs. For both scenarios, the machine’s accuracy (loss) reaches almost 100% (0%). 31 4.9 Performance results for the neural network model. ROC plots on the left hand side
and predicted confidence scores for data on the right hand side, where 0 represents background and 1 represents signal. Top two plots are representative of ZA events, while the bottom two are related to VBF events. In red we have signal events, in blue t¯t background and in green W+W− events. AUC stands for area under the
curve. . . 32 4.10 Asimov significance for each signal topology. The two top ones are representative
of ZA events, while the two ones in the bottom are representative of VBF events. . 33 4.11 Significance as a function of VLL (on the left) and neutrino mass (on the right)
for a luminosity ofL = 3000 fb−1. Green curves are representative of VBF events,
4.12 Performance results for the BDT algorithm. ROC plots on the left hand side and BDT predicted confidence scores for data on the right hand side. Top two plots are representative of ZA events, while the bottom two are related to VBF events. In red we have signal events, in blue t¯t background and in green W+W− background
events. AUC stands for area under the curve. . . 35 4.13 Asimov significance for each signal topology. On the left we have results for ZA
events, while on the right we present for VBF events. . . 35 4.14 Representative plot of the relative importance given by BDT when analysing the
events, based on the different features. In the y axis we have all the features (kinematic variables plus deep learning predictions) and in the x axis we have the relative importance. In green, we have VBF signal events and in red, ZA signal events. . . 36 4.15 Total cross section (in fentobar) as a function of the mass of e4(left) and e5 (right)
for a luminosity ofL = 3000 fb−1. Green curves are representative of VBF events,
while red curves correspond to ZA topologies. The y axis is showed in logarithmic scale. . . 36 4.16 In the left, we show a contour plot of the e4 mass against the e5 mass. The colour
scale represents the ZA event signal e4 significance for a luminosity of L = 3000
fb−1. On the right, we plot the significance of e5 as a function of the me5, for the
same luminosity ofL = 3000 fb−1. In red, we have ZA events, while green points are representative of VBF topologies. . . 37 4.17 Asimov significance as a function of the beam luminosity in fb−1. We consider
me4= 200 GeV and me5 = 3.2 TeV, with 1% systematics. The left plot represents
VBF signals, while while on the right we show our results for ZA topologies. The x axis is showed in logarithmic scale. . . 37 4.18 All observables for ZA topologies (in red), with t¯t (in blue) and W+W− (in green)
backgrounds where it is considered 25 bins for all histograms. In the y axis we have normalised events, in arbitrary units. Reading from top to bottom and left to right, we have distributions for e4and ¯e4mass, cos(θν¯ee−), cos θνµµ+
, cos(θW+ν4),
cos(θW−ν4), cos(θW+W−), pseudo-rapidity for e−, µ+, W+, W−, e4 and ¯e4,
trans-verse mass for W+ and W−, transverse momentum for e−, µ+, W+, W−, e
4 and
¯
e4, missing energy M ET and ∆Re−ν¯
e, ∆Rµ+νµ, ∆RW+ν4, ∆RW−ν4. . . 38
4.19 All observables for VBF topologies (in red), with W+W− (in green) background
where it is considered 25 bins for all histograms. In the y axis we have normalised events, in arbitrary units. Reading from top to bottom and left to right, we have distributions for e4 and ¯e4 mass, cos(θν¯ee−), cos θνµµ+
, cos(θW+ν4), cos(θW−ν 4),
cos(θW+W−), pseudo-rapidity for e−, µ+, W+, W−, e4 and ¯e4, transverse mass
for W+ and W−, transverse momentum for e−, µ+, W+, W−, e4 and ¯e4, missing
energy M ET and ∆Re−¯ν
e, ∆Rµ+νµ, ∆RW+ν4, ∆RW−ν4. . . 39
4.20 Boosted observables for VBF topologies (in red) and ZA topologies (in red), with W+W− (in green) background where it is considered 25 bins for all histograms.
VBF topologies correspond to plots without t¯t background, while ZA events have both. In the y axis we have normalised events, in arbitrary units. Reading from top to bottom and left to right, for both topologies, we have distributions for cos(θν¯ee−),
cos θνµµ+
, cos(θW+ν4), cos(θW−ν4), pseudo-rapidity for e−, µ+, e4 and ¯e4 and
transverse momentum for e−, µ+, e
Conte´
udo
1 Introduction 1
2 The Standard Model 2
2.1 Gauge symmetries and quantum numbers . . . 2
2.2 Quantum chromodynamics . . . 3
2.3 Electroweak Sector . . . 4
2.4 Yukawa Sector . . . 6
3 3HDM-SHUT model: Theoretical studies 10 3.1 A Motivation . . . 10
3.2 High-Energy limit . . . 11
3.3 Low-Energy effective limit . . . 14
3.3.1 Physically viable benchmark scenarios for masses . . . 17
3.3.2 Physically viable benchmark scenarios for couplings . . . 18
4 3HDM-SHUT model: Numerical Studies 20 4.1 Analysis and Numerical procedure . . . 20
4.2 Results . . . 29
5 Conclusions and Outlook 41
A Notation and unit conventions 43
B 3HDM-SHUT model Feynman rules 44
Cap´ıtulo 1
Introduction
The ultimate goal of any scientific endeavour is to uncover the mysteries of the universe and the world around us and, so far, the best model that we devised to describe all the matter that surrounds us is modestly called the Standard Model (SM) of particle physics. A misleading name for a model whose total understanding requires the user to be comfortable enough in various areas of advanced calculus/algebra and quantum mechanics. The SM is a quantum field theory (QFT) whose predictions and results have matched the stringiest of tests and predictions [1–7]. However, there are clear indications that something is missing, from the fact that neutrinos have mass, as confirmed by the neutrino oscillation phenomena [8] and that it does not take into account the existence of dark matter [9]. Besides such experimental evidences, there are also theoretical motivations, as, e.g. the origin of the family replication found in nature, the fermion masses and mixing hierarchies and the origin of the SM gauge structure, where a consensual understanding is still lacking.
So, we can safely establish that there is indeed a missing piece of the puzzle, which leaves us with the obvious question: what is missing and how to fix it? Well, so far, the most exotic theories have been put into the forefront, ranging from models where extra spacetime dimensions exist [10] to models with new symmetries between bosons and fermions [11]. While somewhat separate, these theories have an underlying idea in common. The SM is an effective description of a more fundamental theory and only valid up to a certain energy scale beyond which New Physics is needed. Therefore, the problems of the SM all result from our lack of understanding of what such theory really is and at which energy scale it becomes manifest.
Theories like string theory and supersymmetry (SUSY) might cause heart attacks on how mathematically complex they are, but they do provide a solid theoretical framework from which one can build upon. However, like all scientific theories, if they do not provide an accurate description of reality, then they become useless.
With this being said, in this thesis we present a novel supersymmetric model named 3HDM-SHUT1 based upon Grand Unification principles where all matter and fundamental interactions
including flavour are unified in a single framework inspired by the E8 symmetry. The model at
unification scale has already been properly analysed in [12–15]. Therefore, while a short description of the main results will be given, the main focus of this thesis will be to construct and study the low-energy limit of such a framework which offers interesting phenomenological implications for future runs at the LHC. In fact, this model provides well motivated new particles, including new vector-like leptons (VLLs) [16–19] as well as new scalars [20–23]. The numerical analysis will be performed using standard Monte Carlo simulators employed by the particle physics community, where the final step of our analysis consists in applying deep learning techniques for significance studies.
This thesis is organized as follows. First, in chapter 2, we introduce the Standard Model, briefly discussing the main problems that it cannot address. Afterwards, in chapter 3, we introduce the new 3HDM-SHUT model. Here, we briefly review the basic properties of the model both at the unification scale as well as its low energy limit, motivating the parameter choices used in the numerical analysis. The latter represents the main focus of this work which is performed in chapter 4 where a detailed description of the methods employed in our analysis and the results obtained is given. This thesis is finalised in chapter 5, where we take our conclusions and discuss future work and research directions.
Cap´ıtulo 2
The Standard Model
The SM of particle physics is perhaps the most successful theory describing subatomic pheno-mena. However, despite its great success, it still suffers from some inconsistencies that need to be addressed. In this section, we present the SM as well as comment on its shortcomings in order to motivate the need for beyond the SM (BSM) theories.
2.1
Gauge symmetries and quantum numbers
The SM is a gauge QFT, that is, is a quantum theory invariant under local field transformations. In particular, the gauge group under which the SM is invariant corresponds to the direct product of three distinct symmetries, GSM= U(1)Y× SU(2)L× SU(3)C, where Y represents the hypercharge,
L left chirality and C colour [24–26]. In other words, that the underlying theory is invariant under simultaneous transformations of those groups that read
U(1) : ψ(x)→ ψ0(x) = exp iY 2α(x) ψ(x), SU(2) : ψ(x)→ ψ0(x) = exp iωa(x) 2 σ a ψ(x), SU(3) : ψ(x)→ ψ0(x) = exp iβa(x) 2 λ a ψ(x), (2.1.1)
where α(x), ωa(x) and βa(x) are a set of local, i.e. that depend on the spacetime position x,
constant parameters, Y is the hypercharge, σa, a = 1, 2, 3 are the Pauli matrices and λa, a =
1, . . . , 8 are the Gell-Mann matrices.
The gauge quantum numbers for the SM are:
Field SU(3)C SU(2)L U(1)Y # of generations
Aa 1 2 0 1
B 1 1 0 1
G 8 1 0 1
φ 1 2 1 1
Tabela 2.1: Bosonic sector of the SM.
Field SU(3)C SU(2)L U(1)Y # of generations
Q 3 2 1/3 3
uR 3 1 4/3 3
dR 3 1 −2/3 3
L 1 2 -1 3
eR 1 1 -2 3
The SU(2)L doublets Qiand Li are defined as follows, Qi= uL dL i , Li= νL eL i . (2.1.2)
We can divide the model into three distinct sectors: Quantum chromodynamics (QCD), that describes the strong interaction, Electroweak (EW) sector, that describes the weak and electro-magnetic interactions, and the Yukawa sector, responsible for interactions between matter and the Higgs. Each sector is presented in the subsequent sections.
2.2
Quantum chromodynamics
The QCD sector of the standard model describes the strong force and, in particular, interactions among quarks. Formaly, QCD is a non-abelian SU(3) gauge theory [27] and can be described by the following Lagrangian:
LQCD= i ¯Q ij / DkiQkj+ i(¯uR)ijD/ k i(uR)kj+ i ¯dR ij / Dki(dR)kj− 1 4G a µνGµνa , (2.2.1)
where i, k represent colour indices in the fundamental representation of SU(3)C, j is a flavour index, a represents SU(3)Cindices in the adjoint representation and /D = γµD
µ. Summation over
repeated indices is implied. The field strength tensor Ga
µν and the covariant derivative Dµ are
defined as1 Gaµν = ∂µAaν− ∂νAµa− gsfabcAbµAcν, Dµ = ∂µ+ igs λa 2 A a µ, (2.2.2) where gs is the strong gauge coupling, fabc are the SU(3)C structure constants such that the λ
matrices obey a Lie algebra
[λa, λb] = 2ifabcλc. (2.2.3)
By looking at Tabs. 2.1 and 2.2, one immediately notices that all leptons (L and eR) are singlets
of SU(3), so they don’t interact via the strong force. Only quarks which transform under the fundamental representation of SU(3)C, are allowed to couple to gluons, thus strongly interacting. Quark-quark-gluon vertices can be obtained by expanding the kinetic terms in (2.2.1) from where we obtain uβj, dβj ¯ dαi, ¯uαi Aa µ = igs(γµ)αβ (λa) ij 2 . (2.2.4)
The non-abelian nature of the theory implies an extra term in the definition of the field strength tensor of the form gsfabcAbµAcν. As a result, expanding the last term in (2.2.1), we see that gluons
can interact with each other forming triple and quartic vertices as follows:
Aa µ, p1 Ac τ, p3 Ab ν, p2 = gsfabc h gµν(p1− p2)τ+ + gµτ(p2+ p3)µ+ gτ µ(p3− p1)ν i , (2.2.5) Aa µ Adρ Ab ν Acτ = −ig2 s h feabfecd(gµτgνρ− gµρgντ)+ +feacfedb(gµρgτ ν− gµνgτ ρ)+ +feadfebc(gµνgτ ρ− gµτgνρ) i . (2.2.6)
An important aspect of QCD is known as asymptotic freedom. Opposed to, for example, quantum electrodynamics (QED) where the coupling constant α increases as one goes up in the energy scale, meaning one gets strongly coupled theory at high energies, in QCD the opposite occurs, where the coupling reduces with energy. This means that at sufficiently high energy scale the quarks began to behave more or less as free particles. This allows QCD to be dealt with in great accuracy with lattice QCD being an example of this fact [28].
On the other hand, at low energies, the coupling constant gs becomes strong and the theory
reaches a regime known as confinement. As a byproduct, in such a regime, quarks cannot exist separately and form composite states denoted as hadrons. This fact has an immediate consequence in data analysis of particle collisions. Any decay chain that ends with quarks as final states, leads to the formation of hadrons at detector level in a process denoted as hadronization. This leads to formation of tight cones of particles, known as jets.
While a detailed analytical description is not fully understood, various phenomenological mo-dels for numerical studies are readily available through Monte Carlo generators, such as PYTHIA [29], which will be used in our studies in later sections when dealing with this type of objects.
2.3
Electroweak Sector
The EW sector of the SM is a unified part of the theory that describes, in a common framework, both the electromagnetic and the weak interactions. The formal developments of the theory were designed by the theoretical physicists Sheldon Glashow, Adbus Salam and Steven Weinberg [30– 32]. It is a gauge field theory that plays a crucial role in the SM as it is responsible for the mass generation of all known particles. This follows from the Higgs mechanism which is based upon the symmetry breaking pattern SU(2)L× U(1)Y→ U(1)EM[25] as we will review below.
The EW Lagrangian, before symmetry breaking, is written as LEW =−1 4B µνB µν−1 4F µν a Fµνa + i ¯ψjDψ/ j+ (Dµφ)†(Dµφ)− V φ†, φ, (2.3.1)
where the above terms are defined as
Bµν = ∂µBν− ∂νBµ, Fa µν = ∂µAaν− ∂νAaµ− g0abcAbµAcν, Dµ= ∂µ+ ig0 Y 2Bµ+ ig σa 2 A a µ, V φ†, φ= µ2φ†φ + λ φ†φ2. (2.3.2)
When the scalar doublet φ acquires a vacuum expectation value (VEV)2,
hφi =0 vT, we can perform the shift φ = √1
2
0 v + hT, where h represent physical radial quantum fluctuations around the minimum v. Note that the spontaneous breaking of SU(2)L leads to the appearance
of Goldstone modes. However, such states are absorbed by the longitudinal polarisations of the vector bosons. We can then expand (2.3.1) to get
(Dµφ)†(Dµφ) = g2v2 4 A 1 µA µ 1 + g2v2 4 A 2 µA µ 2+ v2 4 gA 3 µ− g0Y Bµ(gAµ3− g0Y B µ) + . . . , V φ†, φ=1 2 −2µ 2h2+ . . . , (2.3.3)
where, at the minimum of the potential V φ†, φ we have µ2 < 0 and v2 = −µ2/2λ. From
this point on, we are setting Y = 1, since that is its value for the Higgs doublet hypercharge as given in Tab. 2.1. From Eq. (2.3.3) we immediately see the generation of three massive physical particles, that is, the physical Higgs boson h, with mh =
p
−2µ2, and two vector bosons with
mAi = gv/
√
2, which will give rise to the W bosons as we show below. The other two bosons, A3
and B mix, so one has to diagonalize the quadratic part of the Lagrangian involving such fields in order to determine the mass eigenstates. Doing just that, one arrives at the masses mA= 0 and
mZ = (v/√2)
p
g2+ g02, where the particlesA and Z are written in the basis
Aµ= 1 p g2+ g02 g 0A3 µ+ gBµ Zµ= p 1 g2+ g02 gA 3 µ− g0Bµ ⇔ ( Aµ = sin θwA3µ+ cos θwBµ Zµ= cos θwA3µ− sin θwBµ (2.3.4)
corresponding to the photon A and the Z boson. The angle parameterization is done in such a way thatAµ is the eigenvector of null mass, leading to tan θw= g0/g. One usually redefines the
fields A1 and A2 in terms of charge eigenstates, leading to the usual W bosons
W− µ = 1 √ 2 A 1 µ+ iA2µ W+ µ = 1 √ 2 A 1 µ− iA2µ (2.3.5)
Using these new fields, we can rewrite the covariant derivative in (2.3.2) as Dµ = ∂µ+ ig √ 2 τ −W− µ + τ+Wµ+ + ig sin θwQAµ+ ig cos θw τ3 2 − sin 2θ wQ Zµ, (2.3.6)
where we have introduced the definitions Q = (τ3+ 1)/2, τ± = (τ1
± iτ2)/2, with τa= σa/2 for
a = 1, . . . , 3. From Eq. (2.3.6) one can derive the Feynman rules of the theory. However, such calculation is performed later on, when we deal with states in the mass basis and not gauge one, as represented here.
One should note the importance of symmetry breaking in quantum field theories. Broadly speaking, a spontaneously broken symmetry is responsible for the generation of mass terms in previously massless fields, as we have seen for the Z and W±bosons, while conserved symmetries,
such as the remnant SU(3)C× U(1)EMprotect gluons and photons, respectively, from developing
mass.3
Just like in QCD, the EW sector is a non-abelian theory, meaning that W and Z bosons as well as photons can also have triple and quartic self interactions as we show below in Eqs. (2.3.7)
2Recall that φ is a SU(2) doublet, where each component is an operator acting in the Hilbert space, so a non-zero
vacuum expectation value is also two component.
3Calculations shown here are valid only at tree-level (first order in perturbation theory). However, loop
cor-rections to an arbitrary order show that photons and gluons still remain massless. This is a consequence of gauge symmetries which are preserved at all orders in perturbation theory.
and (2.3.8). The structure of these vertices is similar4to Eq. (2.2.5) and Eq. (2.2.6). Such vertices
are in fact quite relevant as they form an important background in our data analysis, as we will see in the numerical studies of the 3HDM-SHUT model.
Wβ+, p1 W− α, p3 Aµ/Zµ, p2 = igEW h gαβ(p3− p1)µ+ gβµ(p1− p2)α+ gµα(p2− p3)β i , gEW=−e for Aµ, gEW= g cos(θW) for Zµ. (2.3.7) Wβ− W+ α Aν/Zν/Wν− Aµ/Zµ/Wµ+ = − igEW h 2gαβgµν− gαµgβν− gανgβµ i gEW= g2 for Wµ+/Wν−, gEW= eg cos(θW) for Aµ/Zν, gEW=−e2 for Aµ/Aν, gEW=−g2cos2(θW) for Zµ/Zν. (2.3.8)
Moving now to a more theorist perspective, the electroweak model encapsulates the idea of unification, that is, we are able to unify electromagnetism and weak interactions into an single underlying gauge theory. This idea is fully captured in the 3HDM-SHUT model, where all forces merge into a single E8gauge group which can be seen as the unifying force.
2.4
Yukawa Sector
Explicit mass terms are not allowed by gauge invariance in the SM. One can readily observe this by looking at the gauge charges in Tab. 2.2, where one notices that terms of the type L ∝
¯
ψLψR cannot be constructed. However, couplings of fermions to the Higgs doublet are allowed
by the gauge symmetry resulting in the generation of fermion masses once the EW symmetry is spontaneously broken.
We begin by looking at leptons, the full invariant Lagrangian that obeys all the gauge symme-tries in Tab. 2.2 is simply
LY,l= (yl)ijφ · ¯L
iej
R+ H.c., (2.4.1)
where H.c. represents the Hermitian conjugate of the first term. We can expand this Lagrangian by considering the SU(2)L doublet in Eq. (2.1.2) and the Higgs doublet with the VEV shift φ = √1 2 0 v + hT, which results in Ll,SB= (yl)ij v √ 2¯e i Le j R+ (yl)ij 1 √ 2h¯e i Le j R+ H.c.. (2.4.2)
Note that there is an immediate problem here. Electrons, muons and taus gain mass via the Higgs mechanism, as we want, but the corresponding neutrinos do not, and since no explicit mass terms can be constructed, neutrinos in the SM must be massless. This is not true, as experiments have demonstrated that neutrinos do have mass, even if it is an incredibly small one. This is one of the inconsistencies that call for BSM theories incorporating right-handed neutrinos as is the case of the 3HDM-SHUT model. Also note that the generated masses are all proportional to v, the Higgs VEV, that is, they are proportional to the interaction with the Higgs given by the Yukawa
4As in, three-vector vertices are proportional to the momenta and the metric while four-vector vertices depend
couplings yltimes the Higgs VEV v = 246 GeV. Taking into account the current measured masses
for the leptons, mτ = 1776.86 MeV, mµ = 105.66 MeV and me= 0.511 MeV [33], one sees that
the third generation has the strongest interaction with the Higgs.
The first immediate issue one encounters here is the massive difference one has between different fermionic generations. Why is it that the electron is so much lighter when compared to the tau? Or even to muon? Another problem arises from higher order corrections to the Higgs mass. As it is shown latter, starting with the interaction term (yl)ij√1
2h¯e
i
Le
j
R, we compute mass corrections
far superior to the experimentally measured value. The SHUT model is constructed with the intention of addressing these issues. Let us now look at the quark Yukawa sector. The full invariant Lagrangian is LY,q= (yd)ijφ · ¯Qid j R+ (yu)ijφ · ¯˜ Qiu j R+ H.c., (2.4.3) where we define ˜φ = iσ2 2 φ
†. We can again apply the same procedure from what was done with
the leptons, resulting in the Lagrangian Lq,SB= (yd)ij√v 2 ¯ diLd j R+ (yu)ij v √ 2u¯ i Lu j R+ Terms with h + H.c., (2.4.4)
where each of the couplings yd and yu represent 3× 3 matrices in the family space. In general,
Yukawa matrices are not necessarily diagonal. While for simplicity we have assumed the lepton sector as flavour diagonal, we will consider generic quark Yukawa matrices so that one has to rotate Lq,SB to its diagonal mass basis. Since yd and yu are hermitian matrices, we diagonaize
them with the following bi-unitary transformations, that is (yu)ij= (UL∗)ik yDu kk (UR)kj, (yd)ij = (D∗L)ik y D d kk (DR)kj, (2.4.5)
where UL,R and DL,R are unitary matrices and yu,dD are diagonal matrices. With the above
transformations the physical quarks expressed in the mass eigenbasis read as ¯ d0R k = (DR)kjd j R, d¯0L k = (D∗L)kjd j L, (¯u0R)k = (UR)kju j R, (¯u0L)k = (UL∗)kju j L, (2.4.6) and the Lagrangian for the mass terms following symmetry breaking mechanism can be written as LM,q= √v 2 Y D u kk(¯u 0 L) k (¯u0R) k +√v 2 Y D d kk d¯ 0 L k ¯ d0R k , (2.4.7)
with k = 1, 2, 3. By looking at the tabulated values for quark masses in the literature and similarly to the leptons, we see that different quark generations have rather distinct interaction strengths to the Higgs field with the top quark Yukawa coupling being of the order one, the largest among all known elementary fermions.
The introduction of the unitary matrices actually results in some interesting physics in the quark sector. We begin by defining the matrix
VCKM=ULD†L, (2.4.8)
known as the Cabibbo–Kobayashi–Maskawa (CKM) matrix. This mixing then has effects on how the different quarks couple to each other via the weak interaction. Returning now to the definition of the covariant derivative in Eq. (2.3.6), the interactions vertices for quarks can be obtained by expanding
resulting in the currents
jEMµ = ¯ui
LeQγµuiL+ ¯diReQγµdiR+ ¯uiReQγµuiR,
jZµ= g cos(θW) ¯ ui Lγµguui uiL+ ¯diLγµgddi diL , jWµ+= g √ 2u¯ i LγµdiL and j µ W− = g √ 2 ¯ diLγµuiL, (2.4.10)
where guui and gddi are couplings that depend on the type of quarks in the given interaction.
This equation is written in the interaction basis. Utilising Eqs. (2.4.6) to transport it over to the mass basis, we note that both the electromagnetic current (first equation) and the neu-tral currents (second equation) are diagonal, however, for the charged current (third equation) there is a mixing between different generations of quarks, meaning that, in the quark sector, one has flavour-changing charged currents that change one quark flavour into another, whose ef-fects are encoded in the CKM matrix. Note that, in the SM, this does not occur for leptons. However, we know that they exist in nature and an analogous matrix to the CKM called Ponte-corvo–Maki–Nakagawa–Sakata, or PMNS matrix [34], that also changes flavour in charged currents between charged leptons and neutrinos.
So, in the SM, written in the mass basis, the Feynman rules are written as,
W± ν`, ¯ν` `+, `− =−i√g 2γ µP L, W± qj ¯ q0 i =−i√g 2γ µ(V CKM)ijPL, Z `−, u i, di, νi `+, ¯u i, ¯di, ¯νi =−icos(θg w)γ µ gi V − giAγ5 , Aµ `−, u i, di `+, ¯u i, ¯di =−ieQiγµ (2.4.11) with gi
V a vector-coupling and giA a axial coupling that depends on the involved fermion i. PL is
a projection operator defined as PL= (1− γ5)/2.
This finalises the chapter on the SM, however there is still something missing. Experimental observations indicate that a significant amount of the composition of the universe is made of dark matter, whose observation was detected indirectly via gravitational interactions in galaxy rotation curves. Furthermore, and as we have mentioned above, the SM lacks an explanation for neutrino masses, and in turn, the mixing in the lepton sector. A possibility to address this problem is via a seesaw mechanism as we briefly highlight. Considering, for simplicity, only one generation of leptons, the part of the Lagrangian involving right-handed neutrinos after EW symmetry breaking reads as,
LN =−yνL · ˜¯ φNR+ MNNRNR+ H.c.
=y√νv
2ν¯eNR+ MNNRNR+ H.c.,
(2.4.12) where NRis right handed neutrino, not a part of SM, and yν is a Yukawa coupling between L and
NR. Written in the flavour basis, the mass matrix corresponds to Mν = 0 y√νv 2 y∗ νv √ 2 MN . (2.4.13)
Assuming a heavy right handed neutrino, such as v MN, the eigenvalues of Mν leads to
mν ≈ 1 2 v2y2 ν MN, mN ≈ MN. (2.4.14)
To look at the potentiality of this mechanism, let us consider some numerical cases. Setting MN = 1016GeV, yν = 1, and the experimentally measured VEV of v = 246 GeV, we obtain
mν = 0.003 eV, mN = 1016GeV, (2.4.15)
in consistency with the experimental measured values for light neutrinos. The generation of neutrinos masses, at the cost of a heavy right-handed neutrino, is referred in literature as the Seesaw mechanism. Despite the lack of experimental observation of right-handed neutrinos, this mechanism is generally accepted as a potential solution to the neutrino mass problem.
Cap´ıtulo 3
3HDM-SHUT model: Theoretical
studies
In this chapter we introduce the model which will be studied in this thesis. We divide this chapter into three main sections. In Sec. 3.1 we motivate and introduce the concept of Grand Uni-fied Theories (GUTs) from where the 3HDM-SHUT model is based upon. Furthermore, provided that the high-energy limit of the 3HDM-SHUT model is supersymmetric, we will also introduce the basic ideas behind such a concept. Then, in Sec. 3.2 we make a short overview of the key properties of the unified framework. While the main purpose of this thesis is to study the pheno-menological implications at the LHC and in particular the potential observability of VLLs, it is important to explain the origin of such new physics and which parameter choices are relevant and well motivated. We then finalise this chapter in Sec. 3.3 with an effective low-energy description, by providing the full Lagrangian density, the particle’s masses as well as a brief discussion of potential benchmark scenarios for the numerical analysis.
3.1
A Motivation
One of the most attractive features of SUSY is an elegant solution to the hierarchy problem which we briefly discuss below. To see this let us discuss the effect of quantum corrections on the Higgs boson mass. At one-loop order the Higgs propagator is given by
− iΣ(p2) = H ψ H ¯ ψ +O(λ4 ψ), (3.1.1)
where ψ encodes any SM fermion. However, the dominant contribution is that of a top quark loop. Using majorant regularization, where a momentum cut-off ΛGUTis imposed, we obtain1
− iΣ(p2= 0) = −λ 2 ψ 8π2Λ 2 GUT+ λ2ψ 8π2m 2 ψln Λ2GUT+ m2ψ Λ2 GUT ! +O(λ4 ψ). (3.1.2)
Focusing on the quadratic term and considering the top contribution such that λψ ≈ 0.94 and
mψ≈ 173.0 GeV, we get − λ 2 ψ 8π2Λ 2 GUT≈ −1 × 1030(GeV) 2 . (3.1.3)
This is very problematic as such corrections to the Higgs boson mass, mH ≈ 125 GeV, are
30 orders of magnitude larger than its measured size. Of course, one can simply redefine the Lagrangian with a bare Higgs mass value that cancels this correction, however this solution is remarkably fine-tuned and is not needed when one considers fermions or gauge bosons. In fact, while the former have protection from chiral symmetry in order to forbid quadratic divergences, the latter benefit from gauge symmetry protection.
1Low momentum limit is considered, by imposing p2→ 0, meaning this result is only valid for heavy particles,
like the top quark. The relevant Lagrangian term is written as √λf 2H ¯ψψ
A possible fix is to postulate a new scalar particle s that cancels out this divergence. In fact, if we consider a scalar particle with the same mass and quantum numbers of the top quark but spin-0, the loop in Eq. (3.1.2) is written as,
− iΣ(p2) = H H
s
+ O(λ4
s), (3.1.4)
offering a new contribution to the Higgs boson mass that scales as λs/8π2Λ2GUT. Summing both
contributions, the overall correction becomes (λs − λ2ψ)/8π2ΛGUT2 .2 If λs = λ2ψ we see that
quadratic divergences in the Higgs mass cancel out and only small logarithmic divergences remain. For this to happen, a symmetry between fermions and scalars is needed which is what actually happens with SUSY. In other words, fermions and scalars are unified in a common chiral superfield, benefiting both from the protection of chiral symmetry that prevents quadratic divergences from appearing in the theory.
One of the key predictions of SUSY is that every known particle in nature has a supersymmetric partner with the same mass. However, none of the current or previous experiment have observed the existence of such particles. This means that supersymmetry cannot be a preserved symmetry in nature and should be broken in such a way to generate a larger mass contributions to the superpartners of the SM particles. The actual size of such particles is not yet known, but the current lack of observation at the LHC is indicating that the supersymmetry breaking scale should be well above the EW scale. However, this by no means excludes SUSY as a well motivated formalism to describe realistic theories. This is the case of the 3HDM-SHUT model that we analyse in this thesis. While SUSY does not manifest at low scale and we can treat it as a non-supersymmetric model, its high scale limit is indeed supersymmetric.
As we will see, the model belongs to a class of GUTs emerging from a single E8group that can
be regarded as the unifying force. One of the key properties of this framework is that flavour is promoted to a gauge symmetry that is part of E8and treated in the same footing as conventional
gauge interactions. The model addresses various inconsistencies of the Standard Model, delving into fundamental questions, such as the origin of gauge interactions and the origin of the mass hierarchies for the different matter particles, which is typically known as the flavour problem. As a byproduct of the unification picture, interesting new physics emerge such as vector-like fermions (VLFs). We will pay special attention to this sector as a potential discovery of such states at the LHC can offer important phenomenological probes of the high-energy theory.
3.2
High-Energy limit
Here, we present the high-energy scale (∼ 1016GeV) formulation of the SHUT model. A more
detailed description can be found in the papers [12–15], where I highlight [15], as the most recent and complete reference.
Throughout this thesis, we never delved deep into supersymmetric models and their description, so in this section we will be formally light, in the sense that only the main properties needed to have a basic understanding of the model will be provided. More concretely, this section will serve as a motivation for our numerical studies.
The main idea of a GUT model is to embed all SM-gauge interactions, i.e. SU(3)C× SU(2)L× U(1)Y, into a larger group. As already stated, an interesting possibility resides on the E8symmetry.
It has been presented as a possible candidate in various superstring theories [35, 36] and is, in fact, the starting point of our model.
The SHUT model was engineered to address some of the main concerns one encounters in the SM. First, it proposes a first principles explanation for a common origin of the strong and EW interactions as well as the flavour structure observed in nature. Furthermore, the Higgs and matter sectors are unified in a common superfield equipping the Higgs sector with the same
E
8
M
8E
6× SU(3)
FM
3FE
6× SU(2)
F× U(1)
FM
6[SU(3)]
3
× SU(2)
L
× U(1)
F
SU(3)
C
×
A=L,R,F
SU(2)
A×
A=L,R,F
U(1)
AM
3 p, f, ωs
iSU(3)
C× SU(2)
L× U(1)
Y ui, diSU(3)
C× U(1)
EMFigura 3.1: Symmetry breaking scheme from the original E8 gauge symmetry down to the strong
and electromagnetic gauge group (SU(3)C× U(1)EM). The various terms between the different
boxes (M6, M3, etc) represent VEVs developed by different scalars. M8is encodes details inspired
by theories with extra compact dimensions. Image is adapted from [15].
flavour structures as the fermion one. This results in a rather reduced freedom in the Yukawa sector allowing only for two free parameters,Y1 andY2, which will provide the dominant contributions
to three generations of exotic vector-like quarks (VLQs) as well as the third and second generation SM-like quark masses. All remaining fermions, including first generation quarks and charged leptons, have their masses radiatively generated and are naturally lighter. The CKM mixing is also emergent in this framework. As stated before, the starting point is the E8 gauge symmetry,
and the first symmetry breaking step reads [13, 14]
E8→ E6× SU(3)F, (3.2.1)
where the subscript F denotes the family symmetry. From this point on, the sequence of steps by which we obtain the SM gauge group is schematically illustrated in Fig. 3.1. The SM particle content and all new physics emergent at low-energy scales correspond to the states that remain light after the various breaking stages. As stated above, only second and third generation SM-like quarks and all three vector-like quark masses are tree-level generated, with their relative sizes controlled by the only two free Yukawa couplings in the theory, which are of supersymmetric origin. To see this let us consider the theory after the breaking step denoted by M3 in Fig. 3.1
whose superpotential reads [15] W =Y1εij χiqL3q j R+ `iRDL3q j R+ `iLqL3D j R+ φiD3LD j R −Y2εij χiqLjqR3 + `iRD j LqR3 + `iLq j LDR3 + φiD j LD3R +Y2εij χ3qi LqjR+ `3RDiLqjR+ `L3qLiDRj + φ3DLiDRj , (3.2.2)
couplings, and L/R denotes SU(2)L/Rdoublet superfields. Note that while χ is a SU(2)L×SU(2)R bi-doublet where the light Higgs sector resides, φ is a singlet and only carries flavour charges. Despite some allowed mixing after symmetries are sequentially broken, the left- and right-handed leptons are essentially embedded in `L and `R respectively, whereas the SM-like quark sector
belongs to both qL and qR. Note also that this model addresses neutrino masses due to the
existence of six right-handed sates, three in `Rand three in φ. Last but not least, new down-type
SU(2)L singlet VLQs and SU(2)L doublet VLLs are predicted in the SHUT model emerging from
the fermionic components of DL,R and χ, respectively. In this thesis we will study the collider
phenomenology of the latter and discuss possible implications for the high scale picture. All exotic scalars are assumed to be decoupled at the supersymmetry breaking scale beyond the reach of the LHC.
As we have mentioned above, one of the features of the SHUT model is that only second and third generation chiral quarks as well as the three VLQ generations are allowed to obtain masses at tree-level. For a better understanding of this statement let us inspect the superpotential in Eq. (3.2.2). First, after the second last breaking stage in Fig. 3.1 all six neutral scalars in ˜φiand
˜ `i
R develop VEVs which are denoted by p, f , ω and si (see [15] for details). We immediately see
that mass terms for vector-like quarks are generated from Dφ˜EDLDR and
D ˜ `R E DLqR type of terms resulting in [15] m2D/S' 1 2(f 2+ p2) Y2 2, m2S/D' ω2(f2+ p2+ ω2) 2(f2+ ω2) Y 2 2, m2B' 1 2(f 2+ ω2) Y12+ f2p2 2(f2+ ω2)Y 2 2, (3.2.3)
where, for simplicity, we have ignored the subdominant effect of the si VEVs and where we adopt
a notation where the lightest VLQ is the D-quark. Along the same lines, SM-like quark masses are generated from h ˜χi qLqR type of terms where, even for a generic setting where all six EW
doublets in ˜χ develop VEVs the up and down-quark masses are always zero. Furthermore, it was shown in [15] that a proto-realistic description of the CKM matrix requires a minimum of three light Higgs doublet VEVs where the quark masses become
mu= 0, m2c= 12Y 2 2 u21+ u22 , m2t =12Y 2 1 u21+ u22 (3.2.4) and md= 0 , m2s = 12Y 2 2 d2 2p2 (f2+ p2+ ω2), m 2 b= 12Y 2 1d22, (3.2.5)
with ui and di being i-th family EW-symmetry breaking VEVs from Higgs doublets coupling to
up-type and down-type quarks respectively. If we consider, for simplicity, that p ≈ f ≈ ω, we obtain the following ratios
Y1 Y2 = mt mc ≈ mb ms ≈ mB mD,S ∼ O(100) , (3.2.6) implying also the presence of up to two generations of VLQs at the reach of the LHC if ω and f are around 100 TeV. This relation fixes the size ofY1 andY2and implies that mass ratios in the
VLQ sector are the same as the ones found among their chiral counterparts. Note that in this thesis we will only study the VLL sector and leave a detailed numerical study of VLQs for future work.
For the case of both SM-like leptons as well as VLLs, there are no allowed terms of the form h ˜χi `L`Rand
D ˜
φEχχ respectively, which means that, at tree-level, their masses are zero just like the first generation quarks. However, below the second to last symmetry breaking stage in Fig. 3.1, such type of operators becomes allowed which means that they can be radiatively generated via loops with internal heavy scalar and fermion propagators.
3.3
Low-Energy effective limit
While directly probing the high energy limit of the SHUT model at or beyond the ω− f − p scales is far beyond the reach of the LHC, exploring new physics signatures at the TeV-scale can offer us solid indications about the structure of the model at higher scales. Furthermore, such an analysis, which is done for the first time in this thesis, will provide an important piece of information about the low-scale properties in order to match the low and the high scale theories. We consider in this section a possible low-energy scale limit of the SHUT model whose gauge symmetry is given in the second to last box of Fig. 3.1. All the quantum numbers for the gauge groups are shown in Tabs. 3.1, 3.2 and 3.3
Field SU(3)C SU(2)L U(1)Y # of generations
QL 3 2 1/3 3
L 1 2 −1 3
dR 3 1 −2/3 3
uR 3 1 4/3 3
eR 1 1 −2 3
Tabela 3.1: SM-like sector for the fermions and quarks.
Field SU(3)C SU(2)L U(1)Y # of generations
EL,R 1 2 −1 3
DL,R 3 1 −2/3 2
νR 1 1 0 6
Tabela 3.2: BSM sector for the fermions and quarks.
Field SU(3)C SU(2)L U(1)Y # of generations
φ 1 2 1 3
Tabela 3.3: Scalar sector. The SU(2)L doublets are defined as follows,
QiL= uL dL i Li= νeL eL i EL,Ri = ν0 eL,R e0L,R i (3.3.1) Let us now describe the low-scale 3HDM-SHUT model, step by step. The quantum numbers for the gauge sector are not shown3 since they are identical to those of the SM and already shown
in the first three lines of Tab. 2.1.
The matter sector, can be subdivided into two sub-sectors. The first, shown in Tab. 3.1, represents the SM-like fermions from where ordinary matter emerges. The second sector, shown in Tab. 3.2, is where new physics appears, which includes three new VLL generations denoted as EL,Rand two light VLQ generations denoted as DL,R. The BSM sector also offers a rich neutrino
sector, with six of them originating from the EL and ER SU(2)L doublets and six right-handed
Majorana neutrinos which we denote as νR. Note that the lightest of the right-handed neutrinos,
which we denote as νBSM, can be sterile enough to provide a good dark-matter (DM) candidate.
While we do not perform DM studies, we will consider this scenario in our numerical analysis by setting its mass on the keV-MeV order and the mixing to the SM-like neutrinos to zero. In such a scenario νBSM escapes the detector and is treated as missing energy.
3In fact, the model does allow for extra vector bosons, however that only becomes relevant at higher energy
While the scalar sector also offers new physics we will not further study it in this thesis. Such analysis is left for future work.
We can now write the Lagrangian of the SHUT-3HDM model by considering all renormalizable, Lorentz and gauge invariant terms. We start by writing out all kinetic terms for the fermions,
Lkin,f = i ¯QL i / D(QL)i+ i ¯L i / D(L)i+ i ¯dR i /
D(dR)i+ i(¯uR)iD(u/ R)i+ i(¯eR)iD(e/ R)i+
+ i ¯EL i / D(EL)i+ i ¯ER i / D(ER)i+ i ¯DL i / D(DL)i+ i ¯DR i / D(DR)i+ i(¯νR)iD(ν/ R)i, (3.3.2) where repeated index i represents summation over the different generations. The covariant deri-vative is defined as4 Dµ= ∂µ− ig Y 2Bµ− igw σa 2 A a µ− igs λa 2 G a µ. (3.3.3)
The kinetic terms for the bosonic sector reads
Lbos,f=−1 4B µνB µν − 1 4A µν b A b µν− 1 4G µν c Gcµν+ 1 2(Dµφa)(D µφa)†, (3.3.4)
where b and c represent SU(2)L and SU(3)Cadjoint indices, while a denotes scalar generations. For the Yukawa interactions we have
Ly=(Ya)ij Q¯L i (DR)jφa+ (Γa)ij Q¯L i (dR)jφa+ (∆a)ij Q¯L i (uR)jφ˜a+ + (Θa) ij E¯L i (eR)jφa+ (Υa)ij E¯L i (νR)jφ˜a+ (Σa)ij L¯ i (νR)jφ˜a+ + (Πa)ij L¯i(eR)jφa+ (Ωa)ij E¯R i (νR)jφ˜a+ H.c., (3.3.5)
where Γ, ∆, Θ and Π are 3× 3 Yukawa matrices, Υ, Σ, and Ω are 3 × 6 matrices whereas Y is a 3× 2 one. Note that only Y , Γ and ∆ contain entries whose leading contributions are proportional to Y1 andY2. The remaining ones are of radiative origin. Unlike what we had in SM, here, the
gauge symmetries allow for explicit construction of invariant bilinear and mass terms Lbil =(MD)ij D¯L i (DR)j+ (ME)ij E¯L i (ER)j+ 1 2(MνR)ij(¯νR)i(νR)j+ + (MLE)ij L¯ i (ER)j+ (Ξ)ij D¯L i (dR)j. (3.3.6)
These arise from the vector-like nature of the involved fields where SU(2)L transformations do
not distinguish between left and right chiralities. All such mass terms in (3.3.6) are generated at the ω-f -p scales, thus larger than the EW scale. Note that the neutrino mass matrix MνR is
generated once the p, f , ω and siVEVs are developed. However, contrary to all remaining bilinear
and Yukawa terms in the leptonic sector, its entries are generated by tree-level diagrams once the corresponding operators become allowed (see [15] for details). Therefore, small loop factors will not suppress the size of MνR, whose entries can be up to the order of the p, f and ω scales. As
a byproduct, the neutrino sector automatically contains a seesaw mechanism and an explanation for the smallness of neutrino masses as we further discuss below.
The last piece of the puzzle is the scalar sector, whose potential is that of a generic 3HDM model: V φ, φ†= (mi)2 φi 2 +m2ijφi φj † + H.c.+ λijkl φi φj†φk φl†+ H.c.. (3.3.7) The full Lagrangian density for the effective low-energy 3HDM is the sum of all previous sectors5
4We have a quite strong abuse of language here. In fact, the covariant derivate for the term i ¯Q L
i / D(QL)iis
different than, for example, i ¯EL
i /
D(EL)i. That is because quarks couple to gluons, while the leptons do not, so
the covariant derivative for leptons does not have the last term of 3.3.3. One should interpret the definition in this fashion, that is, if it interacts, it exists, if it does not, it does not exist.
5Formally speaking, this isn’t everything, as one would need to add gauge fixing terms and ghosts fields. However,
and reads as
L3HDM=Lkin,f+Lbos,f+Ly+Lbil− V φ, φ† (3.3.8)
With the model fully presented, we finalise this section by showing the fermion mass matrices in the gauge eigenbasis that are implemented in our numerical analysis. First, for the quarks, and considering the components of the of the QL SU(2)L doublets as in, (3.3.1), the new Lagrangian
is written as Lq,SB= √va 2(Y a) ij d¯L i ¯ DR j +√va 2(Γ a) ij d¯L i ¯ dR j +√va 2(∆ a) ij(¯uL)i(¯uR)j+ + (MD)ij D¯L i ¯ DR j + (Ξ)ij D¯L i ¯ dR j , (3.3.9)
with va as the VEV of the respective Higgs doublet φa. The up-type quark mass matrix, written
in the basis{u1
L,u2L,u3L} ⊗ {u1R,u2R,u3R} has the form
[Mu] = 1 √ 2 va∆ a 11 va∆a12 va∆a13 va∆a21 va∆a22 va∆a23 va∆a31 va∆a32 va∆a33 . (3.3.10)
The eigenvalues of [Mu] give the masses of the up-type quarks, that is, the up, the charm and
the top, whose leading contributions are proportional to (3.2.4). A similar strategy can be now employed for the down quark sector where, in the basis{d1
L,d2L,d3L,DL1,D2L} ⊗ {d1R,d2R,d3R,D1R,D2R}, we have [Md] = va √ 2Γ a 11 √va2Γa12 √va2Γ13a √va2Y11a √va2Y12a va √ 2Γ a 21 √va2Γa22 √va2Γ23a √va2Y21a √va2Y22a va √ 2Γ a 31 √va2Γa32 √va2Γ33a √va2Y31a √va2Y32a Ξ11 Ξ21 Ξ31 (MD)11 (MD)12 Ξ12 Ξ22 Ξ32 (MD)21 (MD)22 . (3.3.11)
Unlike in the up sector, here we have new physics. Besides the down, strange and bottom quarks, we also have two new VLQs which we name d4 and d56 defined in such a way that md5 > md4.
The leading contributions to down-type quark masses are proportional to Eqs. (3.2.3) and (3.2.5). We can now extend this analysis for the lepton sector, and write down the mass matrices for the charged leptons and neutrinos. Starting with the charged leptons, in the basis {e0i
L,eiL} ⊗ {e0jR,e j R} one gets [ML] = (ME)ij3 ×3 v a √ 2(Θ a) ij 3×3 (MLE)ij 3×3 v a √ 2(Π a) ij 3×3 , (3.3.12)
and for the neutrinos, in the basis{νi
eL,ν 0i eL,ν 0i eR,ν j R} ⊗ {νeiL,ν 0i eL,ν 0i eR,ν j R} we arrive at [Mν] = 03×3 03×3 MLE3×3 v aΣa √ 2 3×6 03×3 03×3 ME3×3 v aΥa √ 2 3×6 MLE†3×3 ME†3×3 03×3 v aΩa √ 2 3×6 v aΣa √ 2 † 6×3 v aΥa √ 2 † 6×3 v aΩa √ 2 † 6×3 MνR 6×6 . (3.3.13)
particles, and hence not relevant for our discussion.
6This rather simplistic nomenclature is used to facilitate the designation when doing numerical analysis, as this
is the name of the particle as defined in the UFO files. The designation in [15] and above in Eq. (3.2.3) corresponds to d4≡ D, d5≡ S.
For the charged lepton sector, besides SM-like leptons we also have new VLLs which we name as e4, e5 and e67, defined in such a way that me6 > me5 > me4. The neutrino sector is quite
rich in new particles, besides the 3 SM-like ones, we have a total of 12 new states. The numerical analysis will only consider the three lightest, keV-MeV scale BSM neutrinos which are denoted as ν4, ν5and ν6.
3.3.1
Physically viable benchmark scenarios for masses
Before moving to the numerical analysis, we present possible benchmark scenarios for couplings and masses in such a way to preserve the key properties emergent from the unification picture as well as complying with measured phenomenological quantities, whenever necessary to our studies. The main focus of this thesis is the study of VLLs and, in particular, understanding whether the model under consideration can be probed in such a sector at the LHC. As it was shown in [15], under certain approximations and before electroweak symmetry breaking (EWSB), the lepton mass matrix is reduced to8
[ML] = 0 0 0 0 0 0 0 0 0 0 κ7ω κ5ω 0 0 0 0 κ6ω κ8ω 0 0 0 0 κ1p κ3f 0 0 0 κ2p 0 0 0 0 0 κ4f 0 0 , (3.3.14)
where the various κi terms are radiatively generated Yukawa couplings, thus naturally small. The
VLL masses are then m2e6 = p 2κ2 2+ f2κ24, m2 e5,e4 = 1 2 ω 2Λ 1+ p2κ21+ f2κ23± h ω2Λ 1+ p2κ1+ f2κ23 2 − 4ω2 ω2Λ2− 2fpΛ3+ p2Λ4+ f2Λ5i 1/2! , (3.3.15) where we defined Λ1 = κ25 + κ26+ κ72 + κ28, Λ2 = (κ5κ6 − κ7κ6)2, Λ3 = (κ5κ7 + κ6κ8)κ1κ3,
Λ4= (κ25+ κ28)κ21and Λ5= (κ26+ κ27)κ23. The plus sign in (3.3.15) corresponds to e5and the minus
sign to e4.
Considering a scenario where ω∼ f p, Taylor expansion of (3.3.15) leads to the simplified expressions me6 ≈ pκ2, me5 ≈ pκ1, me4 ≈ ω q κ2 5+ κ28. (3.3.16)
Along the lines of what was discussed in [15], let us consider a set of possible solutions with • κ2∼ O(10−2), κ1∼ O(10−3.5− 10−2) and κ5,8 ∼ O(10−3− 10−2),
• p ∼ O(500 − 1000 TeV) and ω ∼ f ∼ O(100 TeV). This benchmark scenario leads to the following mass ranges
• me6 ∼ O(5 − 10 TeV),
7Again, in accordance with [15], we have e
4≡ E, e5≡ M , e6≡ T .
8It is important to note that this does not represent a one-to-one correspondence between (3.3.14) and (3.3.12).
One should interpret (3.3.14) as the matrix one would get by following all symmetry breaking steps as seen in Fig. 3.1, while (3.3.12) corresponds to the stage imediately after the ω, f and p VEVs.
• me5 ∼ O(0.15 − 10 TeV),
• me4 ∼ O(0.1 − 1 TeV),
which we will use as a guiding principle for our numerical analysis. In particular we see that for the model under consideration e4 can be light enough to be probed at the LHC. On another
hand, e6 will always be rather heavy and a potential observation at the LHC would likely be very
challenging. Regarding e5, we see that it can either be as heavy as e6 or as light as e4 depending
on yet unexplored model details. Based on this estimation, we will consider both possibilities in the numerical studies.
To finalise this subsection let us consider the neutrino sector. Before EWSB, the mass matrix is block diagonal, Mν= ¯M9×9 0 0 M6×6 , (3.3.17)
where ¯M represents neutral components belonging to SU(2)Ldoublets while M denotes SM singlets
corresponding to νR in tab. 3.2. Starting with the M6×6 block, which corresponds to MνR in
(3.3.6), its components offer the larger contributions to the neutrino mass matrix. In this sector, hierarchies among gauge eigenstates result from the relative sizes of the EW-preserving VEVs. On the other hand, the ¯M components are radiatively generated and share the same properties as the VLLs. Thus, after the p, f and ω VEVs one can write
¯ M = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 κ7ω κ5ω 0 0 0 0 0 0 0 κ6ω κ8ω 0 0 0 0 0 0 0 κ1p κ3f 0 0 0 0 0 0 κ2p 0 0 0 0 0 0 0 0 κ4f 0 0 0 0 0 0 κ2p κ4f 0 0 0 0 κ7ω κ6ω κ1p 0 0 0 0 0 0 κ5ω κ8ω κ3f 0 0 0 0 0 (3.3.18) with eigenvalues, m2ν1,2,3= 0, m 2 ν4,5 = m 2 e6, m 2 ν6,7 = m 2 e5, m 2 ν8,9 = m 2 e4, (3.3.19)
that is, left-handed neutrino components, at this stage, share the same masses as their charged lepton partners. In total, and before EWSB, we have 3 massless, and 12 massive neutrinos (6 from the doublets and 6 from singlets). In the corresponding mass basis, if we identify the massive states as µi (i = 1, . . . , 12), we can recast the neutrino mass matrix in a condensed notation as
mν = 03×3 vEW √ 2 (yν)3×12 vEW √ 2 y > ν 12×3 (µN)12×12 , (3.3.20)
where the contribution of EWSB VEVs was already included. Note that yν are 3× 12 Yukawa
matrices whose entries are all radiatively generated. We see that (3.3.20) has the seesaw form as we have introduced in (2.4.13). While a dedicated analysis is beyond the scope of this work, this structure can potentially offer 3 sub-eV states as well as light keV-MeV order sterile neutrinos as we will assume in our numerical studies.
3.3.2
Physically viable benchmark scenarios for couplings
In our numerical analysis we will be using MadGraph which is a tool that requires a theory written in the mass basis. Therefore, not only masses but also couplings need to be rotated to such a basis. To this end we use SARAH, which also offers a complete set of Feynman rules
among physical fields. All relevant rules for our studies are shown in appendix B. Note that all signal and background processes that we will consider involve only triple gauge self interactions as well as fermion-fermion-gauge vertices. While the gauge sector is purely SM-like with well known parameters, the Feynman rules involving fermion vertices will be sensitive to elements of the mixing matrices in the charged lepton (including VLLs) and neutrino sectors, defined as
ULe· ML· URe† = mdiage ,
Uν· mν· Uν†= mdiagν .
(3.3.21) Let us now discuss which phenomenological constraints are applied to these matrices. First, for the charged leptons, we consider the limit where the SM-like sector is flavour diagonal. Therefore, in Ue
L and URe, we add a 3× 3 identity block and consider a limiting scenario where there is no
mixing with VLLs. While this may not be the case in general, a complete study with flavour mixing is beyond the scope of this work. For the VLL block, we consider a generic mixing with the only restriction being that both Ue
L and URe are unitary. To summarize, the lepton mixing
matrices used in the numerical analysis are given by UL,Re = " 13×3 03×3 03×3 (UL,RVLL)3×3 # , (3.3.22) where UL,RVLL· UL,RVLL † = 13×3.
For the neutrino sector we also consider a limiting scenario where, for simplicity, the mixing between the three light active neutrinos and the remaining 12 BSM states is zero. Once again, a more generic case is beyond the scope of our analysis. However, mixing among light neutrinos is allowed and fixed with the PMNS matrix. For the remaining BSM 12× 12 block recall that mixing among right-handed and left-handed components is radiatively generated thus likely small. Here we will consider that those elements are always smaller than 10−3. Having said this, the full
neutrino mixing matrix reads Uν= " (UPMNS)3×3 03×12 012×3 (UνBSM)12×12 # , (3.3.23) with, UνBSM= " (U1)6×6 (D1)6×6 (D1)†6×6 (U2)6×6 # . (3.3.24)
We set the matrix elements in D1 to be of orderO(10−3− 10−8) while in U1,2 they are randomly
generated in such a way to preserve unitarity and to guarantee that e4 couples mostly to the
lightest, sterile, neutrino. With the above ingredients we have defined a possible benchmark scenario to start our collider phenomenology studies while preserving, to a certain degree, the essential features of the model under consideration.
