Università degli Studi di Napoli “Federico II”

Scuola Politecnica e delle Scienze di Base

Area Didattica di Scienze Matematiche Fisiche e Naturali

Dipartimento di Fisica “Ettore Pancini”

Laurea Magistrale in Fisica

Phenomenological study of W’ models: the

Topflavor scheme

Relatori: Prof. Gennaro Miele

Dott. Stefano Morisi

Candidato: Roberta Calabrese

Matricola N94000495

A.A. 2019/2020

Ai miei genitori, mia sorella ed i miei fratelli.

Introduction

The Standard Model (SM) describes strong, weak, and electromagnetic interaction and the elementary

particles; the weak and the electromagnetic interactions are uni�ed into the electroweak one. The SM is

a non-abelian gauge theory (Yang-Mills theory), whose gauge group is SU(3)c×SU(2)L×U(1)Y , and it

is based on Quantum Field Theory (QFT). The Standard Model presents many other symmetries besides

the gauge ones, for example one can mention, among others, the CPT symmetry and the Lorentz and

Poincarè symmetry.

Two of the most important characteristics of the Standard Model are the renormalizability and the

fact that it is compatible with special relativity. It is important to have a renormalizable theory because

it allows us to obtain �nite predictions. The SM owes its success to the large number of phenomenological

predictions which were proved only years later with high precision; for instance, the neutral current [1]

and the charged one. One of the most impressive results of this theory is the postulation of some particles

and their properties which were discovered after a long time, such as the W± and Z weak bosons [2];

the charm [3] and the top [4] quark; the Higgs scalar boson that was postulated in 1964 [5], but it was

discovered only in 2012 at LHC [6].

Before the establishment of this theory, there were a lot of models to describe separately the three

interactions mentioned before. The SM is compatible with all the preceding theories and is able to

overcome some (or all) of their problems. For instance, the weak interaction was seen as a point-like

interaction with a four fermions vertex; this theory works only for low energy and was not renormalizable.

The strong interaction, on the other hand, was described using a Yukawa theory; in this case, the problem

was not its renormalizability, but the fact that it was not possible to make predictions because it was

impossible to use the perturbation theory. These theories evolved during the years, including new ideas

that allowed to explain new phenomena that were discovered over time. Examples of such are the parton

model that led to the quark model years later, and the P violation in weak interaction discovered in

1956 by the Wu experiment [7]. Some ideas came from considerations made for di�erent theories, such

as the Higgs mechanism: Philip Anderson, who was working on superconductivity, suggested that the

Spontaneous Symmetry Breaking (SSB) could have been used to solve the mass problem in particles

physics [8]. Another example of an idea that was meant for another theory is the quantization of Young-

3

4

Mills theory: it was originally seen as a possible way to quantize the general relativity.

Despite all successes of the SM, it presents some problems, which can be divided into three categories:

1. real problems: experimental results that are not explained by the theory, such as:

anomalous magnetic dipole moment of the muon: the standard model prediction of this quan-

tity is signi�cantly di�erent from the measured one [9, 10, 11, 12];

matter-antimatter asymmetry : many observations show that our universe is asymmetric in its

content of matter-antimatter [13]. The mechanism that explain this di�erence in a dynamic

way (called baryogenesis) must at least satisfy the Sakarov conditions [14]. The SM satis�es

these conditions, but it is not able to explain quantitatively the observed asymmetry.

2. Conceptual problems: quantities or physical e�ects that are not explained by the standard model

because is not a complete theory, such as:

neutrinos' masses: many experiments have proven the neutrino oscillation, that requires mas-

sive neutrinos [15, 16]. At present we know olny two mass square di�erence, ∆m212 and ∆m232,

this means that at least two neutrino are massive. The problem is that in SM neutrinos are

massless particles and we have only the left-handed ones. The right-handed neutrinos are not

included in the SM because they have not been observed and because they are not subject to

any interaction described by the SM. We still don't know the origins of neutrinos' masses, so

we do not know if they are Dirac or Majorana particles.

Strong CP problem: from a theoretical point of view, we have terms that break the CP

symmetry in the strong interaction, but this violation isn't observed [17];

inconsistency with general relativity ;

Dark matter [18]: from many cosmological observations, we know that the standard model can

explain only about 5% of the universal energy. Approximately 26% of the universe is made

up of dark matter, which appears to behave like other matter, but its interactions are weakly

compared to the SM one [14];

Dark Energy [19]: the remaining part of the universe is �lled with dark energy, which isn't

included in the SM [14];

3. Aesthetical problems: features of the theory that are added ad hoc, for example:

naturalness problem (also known as the hierarchy problem):the Higgs mass's corrections are

much larger than the mass itself and for this reason, the Higgs bare mass must be �ne-tuned

to cancel these quantum corrections;

5

fermions hierarchical structure: the Yukawa couplings, except the top quark one, are small

and hierarchical ;

number of generations: we don't know why we have three generation and why the number of

them is the same for quarks and leptons;

number of free parameters, 19 to be precise;

The SM presents the aspects of an incomplete theory. In literature, there are a lot of di�erent scenarios

of beyond standard model physics (BSM), such as supersymmetry (SUSY)[20], the grand uni�ed theory

(GUT)[21], composite model [22, 23], string theory [24], and quantum gravity [25]. For each of these

theories, there are many possible realizations. Some of the previously quoted theories predict new gauge

bosons, such as the leptoquark in GUT [26], and a multi-step Spontaneous Symmetry Breaking pattern.

The bosons that arise from the extension of the electroweak interaction are called W ′ and Z ′ [27] in

analogy to the ones of the SM. The principal aim of this thesis will be the phenomenological study of W ′

models. The reasons of this work is that currently LHC is searching for the W ′ using a variety of models,

including for example the Sequential Standard Model (SSM) [28].

There are many models that predict this new boson; in this thesis we will present �ve of them:

Top�avor model [29, 30, 31, 32, 33, 34, 35, 27, 36, 37]; SU(3)C × SU(3)L ×U(1)Y model (also known as

331 model) [38, 27, 39, 40, 41, 42, 43, 44]; Little Higgs model [27, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54];

Twin Higgs model [27, 55, 56, 57, 58, 59]; L-R model [60, 61]These models have many common features,

such as the presence of both a W ′ and a Z ′ boson, and the mult-step SSB pattern .

Various W ′ models di�er from each other not only for gauge theory, but also depending on the

coupling between fermions and the W ′, the number new particles, etc. Some of these models present

other interesting features, such as the explanation of the fermion generations number, the possibility

of having Flavor-Changing-Neutral-Current (FCNC) interaction at tree level, the violation of lepton

universality, and new sources of CP violation.

The Sequential Standard Model provides only an e�ective Lagrangian that does not come from gauge

theory [21], but it is the more generic way of adding this new boson. Here it follows the e�ective

Lagrangian of the SSM:

Leff. = Vfifj

2√

2gwf̄iγµ

[αfifjR (1 + γ

5) + αfifjL (1− γ

5)]W ′µfj + h.c. (1)

Where αijL and αijR are the coupling of the left-handed and right-handed fermions respectively to the new

boson; Vfifj is the Cabibbo-Kobayashi-Maskawa matrix if fi and fj are quarks, while for the leptons it is

equal to δij ; and lastly gw = e/(sin θw). The W′ boson is seen as an heavy analogue of the W one; if the

W ′ has a mass high enough, it can have decay channels that involve the top quark. One of the problems

with this model is that it is not associated with a gauge theory for a generic set of parameters.

6

The Compact Muon Solenoid (CMS) experiment uses the SSM to make theoretical predictions for

certain cross-sections, such as W ′ → bt, for speci�c set of parameters. In these prediction they assume

αijR = αR and αijL = αL ∀i, j and they �x these quantities in three di�erent ways: 1)αR = 0 and αL = 1,

2)αR = 1 and αL = 0, 3)αR = 1 and αL = 1 [62, 63]. From the theoretical point of view, most of

the W ′ models verify the same assumptions of CMS (see table.1), except the Top�avor model, in which

αR = 0 and α11L = α

22L 6= α33L ; this is one of the reasons why we chose to dedicate an entire chapter to the

phenomenology of this particular model. All of the aforementioned models predict the W ′ with masses

greater than 1TeV, since these model must be consistent with the standard model phenomenology.

Model non-universality αijL αijR

Top�avor√ √

L-R√ √

331√

Little Higgs√

Twin Higgs√

Table 1: W ′ models classi�cation in term of the Sequential Stan-

dard Model's parameters: where the√

in the �rst column means

that there α depends on the generation; in the column αL and αR

it means that this constant is non-zero.

One of the most interesting feature of the Top�avor model is that it presents a small violation of the

lepton universality for the W and a bigger violation of lepton universality for the W ′; this is one of the

few W ′ models that has this characteristics, another example can be seen in [64].

Here it follow a short summary on the work done during this thesis. The �rst step consisted in the

study on the variousW ′ models in literature and their classi�cation; this classi�cation is made depending

on gauge group, coupling to the fermions, number of new particles, and other interesting feature.

After this initial step, we decided to study systematically the Top�avor model for the reasons stated

above. We made a complete phenomenological study for one model, because a complete study of all the

models is beyond the purpose of this thesis and will be studied elsewhere. In order to get quantitatively

the range of mass and the coupling of this new boson, we developed an algorithm using the program

"Mathematica". After this we started to consider the various constraints that we had to impose in order

to get correct prediction of already measured quantity, like for example the Fermi constant, and the ones

that comes from the perturbation theory. Since this model presents a violation of the lepton universality

both for the W and the W ′, we studied it quantitatively.

7

This thesis is divided into �ve chapters. In the �rst chapter, we describe the standard model; in

doing this we discuss the particle content, its gauge theory, the Higgs mechanism, and the Fermi coupling

constant. In the second chapter, there is a description of the previously quoted models, except the

Top�avor model. The third chapter is dedicated to the Top�avor model. In the beginning, we de�ne the

principal feature of the model and the di�erences among the di�erent types of Top�avor models. Then

we discuss the Spontaneous Symmetry Breaking pattern and we get the expression of the masses of the

new electroweak gauge bosons. Lastly, in this chapter, we discuss the various constraints that we need

to consider. In the fourth chapter, there is a collection of all our results. The last chapter is dedicated

to the conclusions of this work and ideas for future development.

Contents

Contents 8

1 The Standard Model 9

2 Models 14

2.1 Sequential Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 331 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3 L-R model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4 Little Higgs Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.5 Twin Higgs model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3 Top�avor model 32

3.1 General information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.2 Gauge masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.3 Lepton sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.4 Quark sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.5 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4 Results 41

5 Conclusions 46

8

Chapter 1

The Standard Model

The Standard model [65, 66, 67, 68, 69] is a mathematical theory that describes elementary particles and

three of the four fundamental interactions (the only one that this model isn't able to describe is gravity)

using Quantum Field Theory.

First of all, it is useful to list all the particles described in this model:

Matter, made up of quarks and leptons;

Interaction, mediated by the gauge bosons which are the gluons (eight) for the strong interaction,

W± and Z for the Weak one and the photon, γ, for the Electromagnetic (EM);

Higgs bosons, this particle is the one responsible for the mass of all the elementary particles (we

will see later why we can't add the mass term in the usual way in the standard model).

Figure 1.1: Standard model particle content.

9

CHAPTER 1. THE STANDARD MODEL 10

The particle content of the SM is summed up in �gure 1.1

The theory in exam describes the previously quoted interactions using the following gauge group:

GSM = SU(3)C × SU(2)L × U(1)Y , (1.1)

where the SU(3)C is used to describe the strong interaction while SU(2)L × U(1)Y is used to describe

the electroweak one (EW). Each �eld transforms di�erently depending on the group we are considering;

this information is summed up in table 1.1,where

Φ =

Φ+Φ0

, (1.2)

L1L =

νee

L

, L2L =

νµµ

L

, L3L =

νττ

L

(1.3)

Q1L =

ud

L

, Q2L =

cs

L

, Q3L =

bt

L

. (1.4)

.

LiL QiL `

iR u

iR d

iR Φ

SU(3)C 1 3 1 3 3 1

SU(2)L 2 2 1 1 1 2

Y − 1216 -1

23 −

13

12

Table 1.1: Standard model quantum number

From this table we can easily notice that the weak interaction is chiral, this was discovered with the

Wu experiment [7]; it is easy to show that, due to the chiral nature of weak interaction, we are not allowed

to insert the usual mass term. In order to add the mass term for fermion, we should be able to write a

term like

mΨ̄Ψ = mΨ̄(PR + PL)(PR + PL)Ψ = m(Ψ̄RΨL + Ψ̄LΨR) , (1.5)

but the left-handed fermions transform as a doublet under SU(2)L, while the right-handed ones as a

singlet, for this reason, this term would not be SU(2)L invariant. The gauge group of the weak interaction

is the reason why we can not have mass terms for the gauge bosons. For these reasons, we have to introduce

the Higgs bosons, Φ. When this �eld acquires its Vacuum Expectation Value (VEV), both fermions and

weak gauge bosons acquire their mass; this mechanism is known as the Higgs mechanism.

Standard model Lagrangian can be written as follows:

L = LF + LB + LY , (1.6)

CHAPTER 1. THE STANDARD MODEL 11

where

LF = iL̄jLγµ(∂µ + i

g

2σiWµi − i

g′

2Bµ)LjL + iQ̄

jLγµ(∂

µ + igV jkCKM

2σiWµi + i

g′

6Bµ)QkL+

− iq̄γµ(gs2λaGµa)q + iū

jRγµ(∂

µ + i2g′

3Bµ)ujR + id̄

jRγµ(∂

µ − ig′

3Bµ)djR + iē

jRγµ(∂

µ − ig′Bµ)ejR ,

LB = −1

4W iµνW iµν −

1

4BµνBµν −

1

4GaµνGaµν +

1

2(DµΦ)

†DµΦ + µ2Φ†Φ− λ(Φ†Φ)2 ,

LY = −ydijQ̄iLΦdiR − yuijQ̄iLΦ̃uiR − ylijL̄LiΦ`R + h.c.

(1.7)

where Φ̃ = iσ2Φ∗, λa are the Gell-Mann matrices, ~T = ~τ2 , τi are the Pauli matrices, and VCKM is the

Cabibbo-Kobayashi-Maskawa matrix. The VCKM matrix is given by:

V †uVd = VCKM =

Vud Vus Vub

Vcd Vcs Vcb

Vtd Vts Vtb

,

d

s

b

= Vd ×

d′

s′

b′

,

u

c

t

= Vu ×

u′

c′

t′

,(1.8)

where {q′} are the eigenstates of strong interaction, while {q} are the ones of the weak interaction, and

both Vi are unitary matrices (V†i Vi = 1).

Let's discuss the Higgs mechanism. We can parametrize the Higgs sector as follows:

Φ =

Ψ1 + iΨ2h+ iχ

, (1.9)where Ψi, h and χ are real �elds; these real �elds, except h, after the Spontaneous Symmetry Breaking

will become the longitudinal component of the phyisical gauge bosons. The �eld h is the Higgs boson.

The minimum of the potential occurs for Φ†Φ=v2, where v is the VEV of Φ. The acquirement of the

VEV will break the SU(2)L × U(1)Y down to U(1)em, which is identi�ed as the electromagnetic gauge

group. The next step consists in the evaluation near the minimum of the Yukawa and the Higgs part of

the Lagrangian; let's start with the terms that will lead to the gauge bosons masses:

Lmasses =1

4g2v2W+µW−µ +

v2

8(−gW 3µ + g′Bµ)(−gW 3µ + g′Bµ) , (1.10)

where

W± =W 1 ± iW 2√

2(1.11)

Now we have only to diagonalize the following matrix:

v2

4

g2 −gg′−gg′ g′2

, (1.12)

CHAPTER 1. THE STANDARD MODEL 12

written in the base (B,W 3). Then we get that:

Zµ = cos θWW3µ − sin θWBµ ,

Aµ = sin θWW3µ + cos θWB

µ ,(1.13)

where θW is called the Weinberg angle. The bosons masses are:

MW =gv

2,

MZ =v√g2 + g′2

2,

Mγ =0 .

(1.14)

Substituting this into the lepton part of the Lagrangian, we can see that:

Lcclepton = −√

2

2[g ¯`αLγµW

+µναL + g ¯ναLγµWµ−`αL] , (1.15)

and

Lnclepton = g′ ¯`αRγmu(cos θAµ − sin θZµ)`αR +g′

2¯`αLγµ(cos θWA

µ − sin θWZµ)`αL+

+g′

2¯ναLγµ(cos θWA

µ − sin θWZµ)ναL +g

2¯`αLγµ(sin θWA

µ + cos θWZµ)`αL+

−g2

¯ναLγµ(sin θWAµ + cos θWZ

µ)ναL .

(1.16)

Imposing that the coupling between charged leptons and the photon are equal to the electric charge, we

get:

g =e

sin θW,

g′ =e

cos θW.

(1.17)

We can rewrtite the masses as follows:

MW =ev

2 sin θW,

MZ =ev

2 sin θW cos θW,

Mγ =0 .

(1.18)

We can do the same thing for quarks:

Lccquark = −√

2

2[g ¯dαLγµW

+µV CKM†αβ uβL + g ¯uαLγµW−µV CKMiβ dβL , (1.19)

and

Lncquark =g′

3¯dαRγmu(cos θA

µ − sin θZµ)dαR −g′

6¯dαLγµ(cos θA

µ − sin θZµ)dαL+

−g′

6¯uαLγµ(cos θA

µ − sin θZµ)uαL −2g′

3¯uαRγmu(cos θA

µ − sin θZµ)uαR+

+g

2¯dαLγµ(sin θWA

µ + cos θWZµ)dαL −

g

2¯uαLγµ(sin θWA

µ + cos θWZµ)uαL .

(1.20)

CHAPTER 1. THE STANDARD MODEL 13

In the standard model there are not �avor-changing-neutral-current interactions at tree level thanks to

the unitarity of Vi.

In a similar way we get the mass term for fermions:

− Lmassfermion =v√2

(ydij d̄iLd

jR + y

uij ū

iLu

jR + y

`ij

¯̀iL`jR + h.c.) . (1.21)

As for the bosons, the leptons masses are proportional to v.

Figure 1.2: Standard model: muon decay

An interesting quantity that can be used to test the electroweak interaction is the Fermi constant.

This constant was introduced in the Fermi theory, born to explain the β-decay. The experimental value

of this constant is determined by measuring the muon life time. If we use the Standard model to compute

the muon lifetime, we get:1

τµ=P

32

g4

m4W

1 + C(me)

(1−∆r)2[70] , (1.22)

where g = e/ sin θW , ∆r is the electroweak radiative correction, C(me) cames from the radiative correction

to the V-A Fermi theory and

P = f

(m2em2µ

)[1 +

3

5

m2µm2W

]m5µ

192π3

where f(m2em2µ

)is linked to a phase space factor.

If we use the Fermi theory to the same calculation, we get

1

τµ=G2Fm

5µ

192π3

(m2em2µ

)1 + C(me)

(1−∆r)2[70] , (1.23)

Then from the two expressions of the muon lifetime, we get the following expression for the Fermi constant:

GF =

√2

8

g2

M2W

δ

1−∆r[70, 71] , (1.24)

where δ = 1 + 310m2µM2W

. The Fermi constant experimental value is: (1.1663787± 0.0000006)10−5 [72]

Chapter 2

Models

2.1 Sequential Standard Model

The Sequential Standard Model [73] provides the e�ective Lagrangian for the W ′ boson built in analogy

to the W boson one; here it follows the e�ective Lagrangian:

Leff. = Vfifj

2√

2gwf̄iγµ

[αfifjR (1 + γ

5) + αfifjL (1− γ

5)]W ′µfj + h.c. . (2.1)

Where αijL and αijR are the coupling of the left-handed and right-handed fermions respectively to the new

boson; Vfifj is the Cabibbo-Kobayashi-Maskawa matrix if fi and fj are quarks, while for fermions it is

equal to δij ; and lastly gw = e/(sin θw). It is the most general Lagrangian one can consider for the W′

boson.

The reference model is built assuming that the coupling to the new vector bosons are the same of the

SM, both to the fermions and of the vertices with three bosons, such as Z ′W+W− and W ′WZ. The

�rst assumption is justi�ed by the fact that most of the W ′ models have coupling to the fermions of the

same order of the SM ones [73]. The coupling of the vertices with three bosons can be smaller or bigger

than the SM ones.

The problem with this Lagrangian is that it is not always associated with a gauge theory; in order to

understand the last statement, it useful to make an example. Let's consider the L-R model, that will be

discussed below; the covariant derivative of this model acts on the right-handed and left-handed fermions

as follows:

DµΨL = ∂µΨL + ig

2σαWαLµΨL + ig

′(B − L)BµΨL ,

DµΨR = ∂µΨR + ig

2σαWαRµΨR + ig

′(B − L)BµΨR ,(2.2)

we can easily see that the right-handed fermions couple only to WR, while the left-handed ones only to

14

CHAPTER 2. MODELS 15

WL. The mass eigenvalues, W and W′, are related to WL and WR as follows:

W±L = cos θ′W± − sin θ′e−iωk′W ′± ,

W±R = sin θ′e+iωk′W± + cos θ′W ′± ,

(2.3)

Then we obtain the following relation for αR and αL:

αLαR

= − tan θ′e−iωk′ , (2.4)

this means that not all choices of αijL and αijR are compatible with this model.

Also the branching ratios and the decay modes of the W ′ are similar to the one of the W boson, the

principal di�erences are linked to the possibility to have decay channels that contain the top quark, since

the mass of the W ′ boson is bigger than the Top quark one.

Usually it is not possible to distinguish two types W ′′ only looking at the Sequential Standard Model

Lagrangian; for this reason, it is necessary to study the various models to �nd the di�erences among

them.

Assuming that the couplings to this boson are equal to the coupling to the W boson, the decays rate

of the W ′ has been calculated at leading order in [74]; we report these results here:

Γ(W ′) =Γ(W ′ → tq̄′) + Γ(W ′ → qq̄′) + Γ(W ′ → `ν̄) ,

Γ(W ′ → tq̄′) =|V ′tq′ |2g2(

1− m2t

m2W ′

)216πmW ′

(m2W ′ +

m2t2

),

Γ(W ′ → qq̄′) = g2

16π|Vqq′ |2mW ′ ,

Γ(W ′ → `ν̄) = g2

16π|V`ν |2

mW ′

3.

(2.5)

where g = e/ sin θW and a new Cabibbo-Kobayashi-Maskawa matrix and a mixing matrix for the leptons

have been introduced. In [73] the authors calculated the decay rates of the processes W ′ → ZW and

Z ′ → W+W− assuming the same coupling of the standard model; we report their results here:

Γ(W ′ → ZW ) = e2

192πcot2 θW

M5W ′

M2ZM2W

[(1− M

2Z −M2WM2W ′

)2− 4 M

2W

M2W ′

]3/2·

·[1 + 10

(M2W +M

2Z

M2W ′

)+M4W +M

4Z + 10M

2WM

2Z

M4W ′

],

Γ(Z ′ → W+W−) = e2

192πcot2 θWMZ′

(MZ′

MW

)4(1− 4MW

MZ′

)3/2·

·

[1 + 20

(MWMZ′

)2+ 12

(MWMZ′

)4].

(2.6)

2.2 331 Model

The 331 model [39] not only predicts new bosons, but also new fermions, and explains the fermion

generation through the request of anomaly cancellation.

CHAPTER 2. MODELS 16

Let's start with the gauge theory and the SSB pattern:

G331 = SU(3)C × SU(3)L × U(1)X , (2.7)

we have a two-step SSB:

SU(3)C × SU(3)L × U(1)X → SU(3)C × SU(2)L × U(1)Y → SU(3)C × U(1)em . (2.8)

This is possible introducing three scalar �elds that transform as triplet under SU(3)L:

η =

η0

η−1

η+2

, ρ =

ρ+

ρ0

ρ++

, χ =

χ−

χ−−

χ0

. (2.9)The �rst step is accomplished when the �eld χ acquires the VEV, while the second one is accomplished

when the other two �elds acquire the VEV.

〈η〉 = 1√2

vη

0

0

, 〈ρ〉 = 1√2

0

vρ

0

, 〈χ〉 = 1√2

0

0

vχ

. (2.10)The X charge of these �leds will be de�ned by imposing later some condition on the vacuum expectation

values.

Since there is no di�erence in the strong interaction, we will omit this part in the following calculations.

The �rst thing we can say about this model is linked to the bosons: we have eight bosons for the

"new" weak interaction, three of them are the usual one, W± and the Z0 , while the other ones are new

bosons, Y ±QY , Z0′, V ±QV . If the charge one the new bosons is equal to ±1, this boson is called W ′. Not

all the versions of this kind of model predict bosons with integer charges.

We de�ne the following �elds:

W±µ =1√2

(W 1µ ±W 2µ) ,

Y ±QYµ =1√2

(W 4µ ± iW 5µ) .

V ±QVµ =1√2

(W 6µ ± iW 7µ) .

(2.11)

The left-handed fermions transform as triplets (or anti-triplets) under SU(3)L. In order to have the

anomaly cancellation, it is required that the number of anti-triplets and the number of triplets are the

same [39]; for this reason, we need that the number of generations is a multiple of the number of colors.

All the leptons and the third generation of quark transform as anti-triplet under SU(3)L, while the �rst

two generations of quarks transform as triplets:

Q1L =

u

d

D

, Q2L =

c

s

S

, Q3L =

b

−t

T

, (2.12)

CHAPTER 2. MODELS 17

`1L =

e

−νe

Ee

, `2L =

µ

−νµ

Eµ

, `3L =

τ

−ντ

Eτ

. (2.13)In this way the condition for the anomaly cancellation is veri�ed. From these expression, we can easily

notice that this model needs at least three new quarks, D, S, and T. The third component of the anti-

triplets Eα can be new leptons or can be the right-handed neutrinos or the right-handed charged leptons

depending on the speci�c version of the model.

In order to di�erentiate the di�erent 331 models, it is useful to introduce the electric charge operator,

that is de�ned as follows:

Q = 12λ3 + β

1

2λ8 +X , (2.14)

where β is a free parameter of the model; this parameter is one of the features that di�erentiate the various

331 models. Depending on the value of β, the eigenvalue of X for each known particle is determined,

because we want to reproduce the charge the SM particles. The new particles can have even exotic

charges. Here it follows the various cases:

Triplets: QΨL = QtripletW ΨL, where

QtripletW =

12 +

β

2√

3+X 0 0

0 − 12 +β

2√

3+X 0

0 0 − β√3

+X

; (2.15)

Anti-triplets: QΨ̄L = QtripletW Ψ̄L, where

Qanti−tripletW =

− 12 −

β

2√

3+X 0 0

0 12 −β

2√

3+X 0

0 0 β√3

+X

; (2.16)

Singlets:QΨR = QsingletW ΨR, where QsingletW = X

Gauge bosons: [Q,Wµ] = QGBW Wµ, where

QGBW =

0 1 12 +

β√

32

−1 0 − 12 +β√

32

− 12 −β√

32

12 −

β√

32 0

; (2.17)and Wµ is a three-by-three matrix de�ned as follow:

Wµ = Wαµ

λα

2=

1

2

W 3µ +

1√3

√2W+µ

√2Y QYµ

√2W−µ −W 3µ + 1√3W

8µ

√2V QVµ

√2Y −QYµ

√2V −QVµ − 2√3W

8µ

. (2.18)

CHAPTER 2. MODELS 18

The principal information of this model is summarized in Table 2.1.

The choice of scalar �elds and their VEV is made in order to recover the Standard model and its

symmetry breaking. These conditions are the following :

λ1 〈χ〉 =0 ,

λ2 〈χ〉 =0 ,

λ3 〈χ〉 =0 ,

(βλ8 + 2X) 〈χ〉 =0 ,

(2.19)

Q〈ρ〉 =0 ,

Q〈η〉 =0 .(2.20)

With these conditions we are requiring that the vacuum expecation values are invariant under λ1, λ2, λ3, βλ8+

2X and Q respectively. These conditions also �x the X charge of the scalar �elds:

Xχ =β√3, Xρ =

1

2− β

2√

3, Xη = −

1

2− β

2√

3. (2.21)

The choice we made for the scalar �elds is not the only one, because it is possible to build invariant terms

for the interaction between fermions and scalar �eld also using a scalar sextet. If we impose the same

conditions for a scalar sextet, we get that we can introduce this �eld only for particular values of β.

Before the discussion of some speci�c 331 models, it is useful to de�ne the covariant derivative and

the Lagrangian; let's start with the derivative and its action on triplets and antitriplets:

Dµ = ∂µ + ig

2λαWαµ + igXXBµ , (2.22)

DµΨL = ∂µΨL + ig

2λαWαµ ΨL + igXXBµΨL , (2.23)

DµΨ∗L = ∂µΨ

∗L − i

g

2(λα)†Wαµ Ψ

∗L + igXXBµΨ

∗L . (2.24)

As usual, we can divide the Lagrangian into three part

L = LF + LB + LY , (2.25)

where

LF =iΨ̄γµDµΨ =

=iQ̄iL(∂µ + ig

2λαWαµ + igXXBµ)Q

iL + iQ̄

3L(∂µ − i

g

2(λα)†Wαµ + igXXBµ)Q

3L+

+iL̄3L(∂µ − ig

2(λα)†Wαµ + igXXBµ)L

3L + i

¯̀αR(∂µ + igXXBµ)`

αR+

+id̄αR(∂µ + igXXBµ)dαR + iūαR(∂µ + igXXBµ)u

αR + iD̄R(∂µ + igXXBµ)DR+

+iS̄R(∂µ + igXXBµ)SR + iT̄R(∂µ + igXXBµ)TR + iĒαR(∂µ + igXXBµ)E

αR ,

(2.26)

CHAPTER 2. MODELS 19

LY =ydiαQ̄iLρdαR + yd3αQ̄3Lη∗dαR + yuiαQ̄iLηuαR + yu3αQ̄3Lρ∗uαR + uJijQ̄iχJjR + yJ33Q̄3χ∗TR+

+y`αβL̄αLρ∗`βR + y

EαβL̄αχ

∗EβR + h.c. ,(2.27)

LB = (Dµχ)†(Dµχ) + (Dµρ)†(Dµρ) + (Dµη)†(Dµη)− V (ρ, η, χ)−1

4W βµνW

βµν − 14BµνB

µν , (2.28)

where the potential is de�ned as follow:

V (ρ, η, χ) = µ21(ρ†ρ) + µ22(η

†η) + µ23(χ†χ) + σ11(ρ

†ρ)2 + σ12(ρ†ρ)(η†η) + σ13(ρ

†ρ)(χ†χ)+

+ σ22(η†η)(η†η) + σ23(η

†η)(χ†χ) + σ33(χ†χ)(χ†χ) + σ̃12(ρ

†η)(η†ρ) + σ̃13(ρ†χ)(χ†ρ)+

+ σ̃23(η†χ)(χ†η) +

√2f1(�ijkρ

iηjχk + h.c.) .

(2.29)

If we evaluate the following term near the minimum

(Dµχ)†(Dµχ) + (Dµρ)

†(Dµρ) + (Dµη)†(Dµη) , (2.30)

and we diagonalize the mass matrix for the charged bosons that arises from it, we get, in the limit of

vχ � vη, vρ, the following expressions for the masses:

M2W =g2

4(v2η + v

2ρ) ,

M2V =g2

4v2χ

(1−

v2ρv2χ

),

M2Y =g2

4u2

(1−

v2ηv2χ

).

(2.31)

Is useful to introduce the following mixing angles and coupling constant:

sin θ331 =g√

g2 +g2Xβ

2

6

,

1

g2Y=

6

g2X+β2

g2,

sin θW =gY√g2 + g2Y

.

(2.32)

Using these quantities, the expression of the neutral gauge bosons masses are:

M2A = 0 ,

M2Z =g2(veta

2 + v2ρ)

4 cos θW,

MZ′ =g2v2χ cos

2 θW

3[1− (1− β) sin2 θW ]

[1 +

v2η + v2ρ

4vchi2(1 + 3β2 tan4 θW

)+

√3β

2

v2η − v2ρv2χ

tan2 θW

].

(2.33)

CHAPTER 2. MODELS 20

If we neglect the corrections in v2η/v2χ and v

2ρ/v

2χ, we get:

M2W =g2

4(v2η + v

2ρ) ,

M2V =g2

4v2χ ,

M2Y =g2

4v2χ ,

M2A =0

M2Z =M2W

cos θW,

MZ′ =g2v2χ cos

2 θW

3[1− (1− β2) sin2 θW ].

(2.34)

SU(3)C SU(3)L X

QiL 3 316 −

β

2√

3

Q3L 3 3∗ 1

6 +β

2√

3

LαL 1 3∗ − 12 +

β

2√

3

`αR 1 1 -1

uαR 3 123

dαR 3 1 − 13DR 3 1

16 −

√3β2

SR 3 116 −

β√

32

TR 3 116 +

√3β2

ER 3 1 − 12 +√

3β2

χ 1 3 β√3

ρ 1 3 12 −β

2√

3

η 1 3 − 12 −β

2√

3

Table 2.1: 331 model: particle quantum numbers

Now we will consider some particular 331 models; the �rst thing we can notice from 2.17 is that only

for some particular values of β, we have bosons with integer charge. We will consider the values of β

that predicts at least one integer charged new bosons; the principal characteristics of these models are

summarized into Table 2.2. Here it follows the discussion of the interesting cases :

β = ± 1√3, also known as ¯331: in this case, we have that the new quarks have not exotic charges and

that the new leptons are chargeless. For β = 1√3, there are even some variations, called 331RHN ,

that use the right-handed neutrinos, (νcR)α, as the third component of the anti-triplets. The problem

CHAPTER 2. MODELS 21

Qβ= 1√3Qβ=√3 Qβ=− 1√

3

D − 13 −43

23

S − 13 −43

23

T 2353 −

13

E 0 1 -1

V ±QV 0 ±1 ±1

Y ±QY ±1 ±2 0

Table 2.2: 331 model: new quarks electric charge for di�erent

choice of β

with these variations is that the mass terms for neutrinos are obtained by adding the scalar sextet

mentioned above or with dimensions �ve e�ective operators [43].

β =√

3, also known as the minimal 331 model [39]: in this case, we have exotic charges for the new

quarks, 53 and −43 , while the new leptons have the same charge as the positron; in this case, it is

possible to choose as the third component of the anti-triplets the conjugate of the charged leptons.

In the case mentioned above, the quark T has a di�erent charge than S and D; this means that we

can only have a mixing between S and D, but some models consider them as gauge and mass eigenstates

[39].

An interesting feature of the 331 model is that the �avor-changing-neutral-current interactions are

allowed; it depends on the di�erent transformation rule for the third generation of quarks with respect

to the other two. This model presents also new CP-violation sources whose dominant contributions are

given by these new FCNC interactions [39, 41, 42]

2.3 L-R model

The weak interaction is the only one in the standard model with a parity violation, for this reason, since

1970's many models that restore the parity at high energy have been studied; the left-right model [60],

whose gauge group is

SU(3)C × SU(2)L × SU(2)R × U(1)B−L , (2.35)

is one of them. In this model, the right-handed fermions transform as doublet under SU(2)R, while the

left-handed ones as a doublet under SU(2)L.

QαR,L =

uαdα

R,L

, LαR,L =

να`α

R,L

. (2.36)

CHAPTER 2. MODELS 22

It is useful to introduce the charge operator:

Q = IL + IR + (B − L) , (2.37)

where IR and IL are respectively the right-handed and the left-handed weak isospin.

As usual, we want the standard model to be restored at low energies, for this reason, we have two

steps SSB:

SU(3)C × SU(2)L × SU(2)R × U(1)B−L → SU(3)C × SU(2)L × U(1)Y → SU(3)C × U(1)EM . (2.38)

The �rst step is made introducing the following �elds

∆L =

δ+L /√2 δ++Lδ0L −δ

−L /√

2

,∆R = δ+R/√2 δ++R

δ0R −δ−R/√

2

, (2.39)this is one of the possible choice for this step, but in this way we can have Majorana masses for the

neutrinos. The second step is made introducing:

Φ =

Φ01 Φ+1Φ−2 Φ

02

. (2.40)The �rst two �elds transform as triplets under SU(2)L and SU(2)R respectively, while the third one

transform as a bi-doublet under SU(2)L × S(2)R:

Φ→ sLΦs†R ,

∆L → sL∆Ls†L ,

∆R → sR∆Rs†R ,

(2.41)

where sL,R ∈ SU(2)L,R. The principal information of the �elds is presented in Table 2.3. As usual, the

SU(2)L SU(2)R B-L

QL 2 116

QR 1 216

LL 2 1 − 12LR 1 2 − 12∆L 3 1 1

∆L 1 3 1

Φ 2 2∗ 0

Table 2.3: L-R model: particles quantum number

neutral components of the scalar �elds gain the vacuum expectation value

〈∆L〉 =

0 0vLe

iωL 0

, 〈∆R〉 = 0 0

vReiωR 0

, (2.42)

CHAPTER 2. MODELS 23

〈Φ〉 =

keiωk 00 k′eiωk′

, (2.43)where vR,L, k, and k

′ are taken real and positive. Transforming the �elds, it is possible to reabsorb two

phases; the usual choice is the following one:

〈∆L〉 =

0 0vLe

iωL 0

, 〈∆R〉 = 0 0

vR 0

, (2.44)

〈Φ〉 =

k 00 k′eiωk′

. (2.45)In order to obtain results that are in agreement with the standard model at low energies, it must turn

out to be true that vR � k , k′ � vL.

Before introducing the Lagrangian, it is useful to present the covariant derivative and its action on

the scalar �elds:

Dµ = ∂µ + ig

2σαWαLµ + i

g

2σαWαRµ + ig

′(B − L)Bµ ,

Dµ∆L = ∂µ∆L + ig

2[σαWαLµ; ∆L] + ig

′(B − L)Bµ∆L ,

Dµ∆R = ∂µ∆R + ig

2[σαWαRµ; ∆R] + ig

′(B − L)Bµ∆R ,

DµΦ = ∂µ + ig

2(σαWαLµφ− ΦσαWαRµ) + ig′(B − L)BµΦ .

(2.46)

In principle we can have two di�erent coupling gR and gL, but we will study one that presents the parity

symmetry (the Lagrangian is invariant under the exchange L↔ R ).

As we have done before, we divide the Lagrangian into three part:

L = LF + LB + LY , (2.47)

where

LF =iΨ̄γµDµΨ =

=iQ̄βL(∂µ + ig

2σαWαLµ + i

g′

6Bµ)Q

βL + iQ̄

βR(∂µ + i

g

2σαWαRµ + i

g′

6Bµ)Q

βR+

+iL̄βL(∂µ + ig

2σαWαLµ − i

g′

2Bµ)L

βL + iL̄

βR(∂µ + i

g

2σαWαRµ − i

g′

2Bµ)L

βR ,

(2.48)

LY =Q̄αL(hqαβΦ + h̃

qαβΦ̃)Q

βR + L̄

αL(h

`αβΦ + h̃

`αβΦ̃)L

βR+

+LcαR (hM )αβΣLLβL + L

cαL (hM )αβΣRL

βR + h.c. ,

(2.49)

LB =Tr[(Dµ∆L)†Dµ∆L] + Tr[(Dµ∆R)†Dµ∆L] + Tr[(DµΦ)†DµΦ]− V (∆L,∆R,Φ)

−14W βLµνW

βµνL −

1

4BµνB

µν − 14W βRµνW

βµνR ,

(2.50)

where Φ̃ = σ2Φσ∗2 , ΣL,R = iσ2∆L,R and h, h̃ , hM are three-by-three hermitian matrices. It is easy to

notice that in the Lagrangian we included both the Majorana and Dirac terms for the leptons.

CHAPTER 2. MODELS 24

From the evaluation of the Lagrangian term with the scalar �elds near the vacuum expectation values,

we get the mass matrix for the charged bosons and the one for the charged ones:

M1 =g2

4

k2 + k′2 + 2v2L −2kk′e−iωk′−2kk′e+iωk′ k2 + k′2 + 2v2R

, (2.51)where the base is ordered as W±L ,W

±R , and for the neutral ones,

M2 =1

4

g2(k2 + k′2 + 4v2L) −g2(k2 + k′2) −4gg′v2L−g2(k2 + k′2) g2(k2 + k′2 + 2v2R) −4gg′v2R−4gg′v2L −4gg′v2R 4g′2(v2L + v2R)

, (2.52)where the base is ordered as W 3L,W

3R, B. Diagonalizing these matrices, we get the following masses

eigenvalues:

M2W =g2

4

[k2 + k′2 + v2L + v

2R −

√(v2L − v2R)2 + 4k2k′2

],

M2W ′ =g2

4

[k2 + k′2 + v2L + v

2R +

√(v2L − v2R)2 + 4k2k′2

],

M2A =0 ,

M2Z =1

4

[g2(k2 + k′

2

+ 2v2R) + 2g′2v2R −

√g4(k2 + k′2)2 + 4v4R(g

2 + g′2)2 − 4g2g′2v2R(k2 + k′2)

],

M2Z′ =1

4

[g2(k2 + k′

2

+ 2v2R) + 2g′2v2R +

√g4(k2 + k′2)2 + 4v4R(g

2 + g′2)2 − 4g2g′2v2R(k2 + k′2)

].

(2.53)

In the limit vL → 0 we can write:

M2W =g2

4(k2 + k′2)

[1 +

2k2k′2

(k2 + k′2)v2R

],

M2W ′ =g2v2R

2,

M2A =0 ,

M1Z =(k2 + k′2)g2(g2 + 2g′2)

4(g2 + g′2)

[1− (k

2 + k′2)(g2 + g′2)2

4v2Rg4

],

M2Z′ =g2v2R .

(2.54)

The rotation matrices are:

R1 =

cos θ′ − sin θ′e−iωk′sin θ′eiωk′ cos θ′

. (2.55)

R2 =

sin θW sin θY cos θW cos θW sin θY

− cos θW sin θW sin θY sin θW cos θY

0 − cos θY sin θY

. (2.56)

CHAPTER 2. MODELS 25

where the angles are de�ned as follows:

sin θW =g′√

g2 + 2g′2,

cos θW =

√g2 + g′2

g2 + 2g′2,

sin θY =g′√

g2 + g′2,

cos θY =g√

g2 + g′2,

tan θ′ = −kk′

v2R.

(2.57)

We want the mixing between the charged bosons to be small, because otherwise we would not have a

parity violation weak interaction at low energies; this is automatically veri�ed for vR � k , k′.

The mass eigenstate are linked to the gauge as follows:

W±L = cos θ′W± − sin θ′e−iωk′W ′± ,

W±R = sin θ′e+iωk′W± + cos θ′W ′± ,

W 3L = sin θWA− cos θWZ ,

W 3R = cos θW sin θYA+ sin θW sin θY Z − cos θY Z ′ ,

B = cos θW cos θYA+ sin θW cos θWZ + sin θY Z′ .

(2.58)

2.4 Little Higgs Model

There are many versions of the Little Higgs model, but one of the principal feature is that the Higgs boson

mass came from the breaking of a global symmetry and this solves the naturalness problem. Basically,

we have an approximate global symmetry in the Lagrangian and the breaking of this symmetry at a

high energy scale, f , generates a pseudo-Nambu-Goldstone boson (PNGB) whose mass depends on the

asymmetry: in the limit of an exact symmetry, the mass of the PNGB goes to zero.

There are many versions of the Little Higgs model, we will report the one with the following gauge

theory [53, 51]:

SU(3)c × SU(4)L × U(1)X . (2.59)

This model predicts the existence of 15 gauge �elds for the weak interaction, three of them are the ones of

the standard model. We chose this version because, as the 331 models, it explains the generation number

and the color number requiring the anomaly cancellation: we need the same number of quadruplets and

anti-quadruplets [53].

Before the discussion of the particle content of this model, it is useful to explain brie�y why the mass

of the Higgs boson is "protected": the new fermions cancel out the one loop quadratic contribution to

the Higgs mass and the same goes for the bosons contributions [53].

CHAPTER 2. MODELS 26

The global symmetry of this theory is

[SU(4)/SU(3)]4 , (2.60)

and it is approximately veri�ed by the potential of the theory. This theory requires the following scalar

�elds in order to obtain the mass term for fermions and bosons:

Φ1 = eiHu f2f1

0

0

f1

0

,Φ2 = e−iHu f1f2

0

0

f2

0

, (2.61)

Φ3 = eiHd

f4f3

0

0

0

f3

,Φ4 = e−iHd

f3f4

0

0

0

f4

, (2.62)

where

Hu '

0 0

hu0

0 0 0

h†u 0 0

0 0 0 0

/f12 ,

Hd '

0 0 0

hd0 0 0

0 0 0 0

h†d 0 0

/f34 ,(2.63)

where fij =√f2i + f

2j , while hu,d are the the two Higgs doublets:

hu =1√2

H0uH−u

, hd = 1√2

H+dH0d

(2.64)The vacuum expectation of these doublets are:

〈hu〉 =1√2

vu0

, 〈hd〉 = 1√2

0vd

. (2.65)For simplicity, we consider f = f12 = f34. These �elds appear in the potential 2.82.

The covariant derivative is de�ned as follow:

Dµ = ∂µ + igTαAµα + igXXA

µX , (2.66)

CHAPTER 2. MODELS 27

where Tα = λα/2 and λ are the corresponding of the Gell-Mann matrices for SU(4). We can de�ne the

following matrix:

TαAα =

1√2

D01 W

+ Y 0 X ′+

W− D02 X− Y ′0

Ȳ 0 X+ D′01 W′+

X ′− Ȳ ′0 W ′− D′02

, (2.67)

where:

D01 =1√2A3 +

1√6A8 +

1√12A15 ,

D02 = −1√2A3 +

1√6A8 +

1√12A15 ,

D′01 = −2√6A8 +

1√12A15 ,

D′02 = −3√12A15 ,

(2.68)

and

Z1 = D1 −gX

g√

2AX ,

Z2 = D2 −gX

g√

2AX ,

Z ′1 = D′1 −

gX

g√

2AX ,

Z ′2 = D′2 −

gX

g√

2AX .

(2.69)

We can de�ne the electric charge operator as:

Q = a1T3 +a2√

3T8 +

a3√6T15 +X1 , (2.70)

where

T3 =1

2

1 0 0 0

0 −1 0 0

0 0 0 0

0 0 0 0

, T8 =1

2√

3

1 0 0 0

0 1 0 0

0 0 −2 0

0 0 0 0

, T15 =1

2√

6

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 −3

. (2.71)

The choice of the �rst parameter, a1, is made in order to restore the weak isospin: a1=1. The values of

the other two parameters depend on the choice of the way we build the fermions quadruplet:

ΨIL =

t

b

T

T ′

,ΨIIL =

t

b

T

B

, (2.72)

CHAPTER 2. MODELS 28

then we have the following possibilities for the electric charge operator:

Type I : Q = T3 −1√3T8 +

1√6T15 +X1 ;

Type II : Q = T3 −1√3T8 −

1√6T15 +X1 .

(2.73)

It has been proved that to cancel the quadratic divergences we have to peek the type II quadruplet [53];

in order to get an anomaly free theory, we need di�erent representation of the fermions depending on the

generation: the �rst two generation of quarks have the following representation [53]:

Q1L =

d

u

iD

iU

, Q2L =

s

c

iS

iC

, (2.74)

the third one can be built in the following way:

Q3L =

t

b

iT

iB

, (2.75)

and the leptons is built in a similar way than the third generation of quarks:

LαL =

να

eα

iNα

iEα

. (2.76)

Now we have to de�ne only the X charges for the �elds:

X(Φ1,2) = −1

2, X(Φ3,4) =

1

2, X(QL) =

1

6, X(LL) = −

1

2, X(ΨR) = Q . (2.77)

The Lagrangian is:

L = LF + LB + LY , (2.78)

LF = iΨ̄γµDµΨ , (2.79)

LB =−1

4AαµνA

αµν − V (Φ1,Φ2,Φ3,Φ4) + (DµΦ1)†DµΦ1 + (DµΦ2)†DµΦ2+

+(DµΦ3)†DµΦ3 + (DµΦ4)

†DµΦ4 ,

(2.80)

LY = (λ`α;1i`cα;1Φ†3 + λ

`α;2i`

Cα;2)Lα + (λ

t1iu

c1Φ†1 + λ

t2iu

c2Φ†2 + λ

b1id

c1Φ†3 + λ

b2id

c2Φ†4)Q

3L+

+(λuk;1iuck;1Φ

†3 + λ

uk;2iu

ck;2Φ

†4 + λ

dk;1idk;1Φ

†1 + λ

dk;2id

ck;2Φ

†2)Q

kL + h.c. ,

(2.81)

V (Φ1,Φ2,Φ3,Φ4) = k13|φ†1Φ3|+ k14|φ†1Φ4|+ k12|φ

†1Φ3|+ k23|φ

†2Φ3|+ k24|φ

†2Φ4| , (2.82)

CHAPTER 2. MODELS 29

where k=1,2, `cα1,2 are linear combinations of ecα and E

cα, u

c1,2 are linear combination of t

c and T c, dc1,2 are

linear combination of bc and Bc, uc1;1,2 are linear combination of uc and U c, uc2;1,2 are linear combination

of cc and Cc, dc1;1,2 are linear combination of dc and Dc, dc2;1,2 are linear combination of s

c and Sc.

As usual, if we evaluate the Lagrangian near the minimum and diagonalize the charged and neutral

mass matrices, we get the following masses:

M2W =1

4g2v2 ,

M2W ′ =1

4g2(4f2 − v2) ,

M2X =1

4g2(2f2 − v21 + v22) ,

M2X′ =1

4(2f2 + v21 − v22) ,

M2Y =1

2g2f2 ,

M2Y ′ =1

2g2f2 ,

M2Z =g2v2

4 cos2 θW

(1− tan

4 θW v2

4f2

),

M2Z′ = (g2 + g2X)f

2 − g2v2

4 cos2 θW

(1− tan

4 θW v2

4f2

),

M2Z′′ =1

2g2f2 ,

(2.83)

where

v21 = v2u −

v4u3f2

(f22f21

+f21f22− 1),

v22 = v2u −

v4d3f2

(f24f23

+f23f24− 1),

v2 = v21 + v22 ,

cos θW =

√g2 + g2Xg2 + 2g2X

.

(2.84)

2.5 Twin Higgs model

The Twin Higgs model [56, 57] was built to solve the naturalness problem. The gauge theory of this

model is:

SU(3)c × SU(2)L × U(1)Y × SU(3)′c × SU(2)′L × U(1)′Y . (2.85)

Basically, we have a twin copy of the standard model. We have two "electroweak" SSB:

SU(2)L × U(1)Y → U(1)EM ,

SU(2)′L × U(1)′Y → U(1)′EM .(2.86)

The mirror gauge group has the same particle content of the standard model and also the couplings are

almost the same

ĝ ' g , ĝ′ ' g′ , ĝs ' gs , ŷf ' yf . (2.87)

CHAPTER 2. MODELS 30

This model presents other symmetries, but to notice them we have to de�ne the Lagrangian:

L = L321 + L3′2′1′ − V (H,H ′) , (2.88)

where

V = −µ2(H†H+(H ′)†H ′)+λ(H†H+(H ′)†H ′)2+m2(H†H−(H ′)†H ′)+δ[(H†H)2+((H ′)†H ′)2] . (2.89)

De�ned the following quadruplet:

H =

HH ′

= e if Π

0

0

0

f + σ√2

, (2.90)

where f is an energy scale and σ is a �eld

Π =

0 0 0 iφ1

0 0 0 iφ2

0 0 0 0

−iφ∗1 −iφ∗2 0 0

, (2.91)

and φ1,2 are the element of Φ,

Φ =

φ1φ2

, 〈Φ〉 = 1√2

0v

. (2.92)We can parametrize this �eld as follows:

Φ =

0h+v√

2

(2.93)We can easily notice that the Lagrangian is almost symmetric under the interchange H ↔ H ′, the

"twin symmetry"( Z2), and that H is invariant under the global symmetry SU(4) × U(1) if the term

m2(H†H−(H ′)†H ′) goes to zero; we can takem2 smaller than the other parameter, so that the symmetry

is "softly" broken. The global symmetry is broken also by δ[(H†H)2 + ((H ′)†H ′)2], also in this case

we want δ smaller than the remaining parameters. The energy scale f is the vacuum expectation value

associated to the symmetry breaking of the global symmetry into SU(3)×U(1); this leads to the following

vacuum expectation values for the two Higgs sector H and H ′ respectively:

vSM =√

2f sin(v√2f

) ,

v′ =√

2f cos(v√2f

) .

(2.94)

For simplicity we will de�ne the following angle:

v√2f

= ω . (2.95)

CHAPTER 2. MODELS 31

This will lead to the usual symmetry breaking present in the standard model and its "twin" version; we

get the following massive bosons:

M2W =g2v2SM

4,

M2Z =(g2 + g′2)v2SM

4,

M2W ′ =g2v′2

4,

M2Z′ =(g2 + g′2)v′2

4.

(2.96)

If we calculate the expression of the potential near the minimum, we get that the mass eigenstates

h− and h+ are a linear combination of σ and h: h−h+

= cosα sinα− sinα cosα

hσ

, (2.97)where α is de�ned as follows:

sin 2α =2f2 sin 4ω

m2+ −m2−, (2.98)

being m± the mass of h±,

m2± = 2f2[λ+ δ ±

√λ2 + δ(2λ+ δ) cos2 2ω

]. (2.99)

Depending on the speci�c twin Higgs model, we have di�erent transformation rules for the fermions.

One possibility consists in the introduction of the following fermions

QL =Q11L +Q

12L +Q

21L +Q

22L ,

QR =qR + q′r ,

(2.100)

where the transformation rules of the various �elds are shown in Table 2.4.

SU(3)C SU(2)L SU(3)′C SU(2)

′L

Q11L 3 2 1

Q22L 1 1 3 2

Q12L 3 1 1 2

Q21L 1 2 3 1

qR 3̄ 1 1 1

q′R 1 1 1 3̄

Table 2.4: Twin Higgs: quarks quantum numbers example

There are even Twin Higgs model in which there is a left-right symmetry [56].

Chapter 3

Top�avor model

3.1 General information

The Top�avor model [37] is a gauge theory, whose gauge group is

G′ = SU(3)C × SU(2)12 × SU(2)3 × U(1)Y . (3.1)

The principal characteristic of this model is that we have a di�erent treatment for the fermion depending

on its generation. We have that the �rst two generations transform as a doublet under SU(2)12, while

the third one as a doublet under SU(2)3; this means that we can have a violation of lepton universality

both for the W and the W ′.

The particle content of this theory depends on the speci�c Top�avor model considered; most of the

particles are the same as the Standard model, but with di�erent quantum numbers. This theory needs

at least a scalar �eld Σ that transforms as a bidoublet under SU(2)12×SU(2)3 and a scalar Φ; the main

di�erence between the Top�avor models stands in the role of Φ, which is analogous to the Higgs sector.

Here it follows a practical classi�cation of the di�erent Top�avor

1. Φ transforms as a doublet under SU(2)12;

2. Φ transforms as a doublet under SU(2)3;

3. we have two di�erent Φ, one transforms as a doublet under SU(2)12 the other one under SU(2)3.

There is even a more general version of the Top�avor model in which the gauge group is:

G′′ = SU(3)C × SU(2)1 × SU(2)2 × SU(2)3 × U(1)Y . (3.2)

We will study the �rst case since the other two give similar results;

In this case, the SSB pattern consists of two steps: in the �rst step, we have that the group SU(2)12×

SU(2)3 breaks down to SU(2)L when the �eld Σ acquires its vacuum expectation value; then we have

the usual Spontaneous Symmetry Breaking when the �eld Φ acquires its VEV.

32

CHAPTER 3. TOPFLAVOR MODEL 33

L3 Li Q3 Qi uαR dαR `

αR Σ Φ

SU(2)12 1 2 1 2 1 1 1 2 1

SU(2)3 2 1 2 1 1 1 1 2∗ 2

Y − 12 −12

16

16

23 −

13 -1 0

12

Table 3.1: Top�avor model: particles quantum number; α=1,2,3

and i=1,2

In the more general case, we have that each generation transforms as doublets under a di�erent

SU(2)i. This model has a three steps SSB pattern: the �rst step is accomplished with the breaking of

SU(2)1 × SU(2)2 into SU(2)12 and then it has the same SSB pattern as the previously quoted versions.

This model needs three di�erent doublets Φi and three di�erent bidoublets Σij .

The quantum numbers of the various particles are summed up in Table 3.1.

It is useful discussing a little more about the scalar sector before the discussion of the Lagrangian; we

have that the transformation rules for these �elds are:

Σ→ s12Σs3 ,

Φ→ s12uY Σ ,(3.3)

where s12 ∈ SU(2)12, s3 ∈ SU(2)3 and uY ∈ U(1)Y .

We can write the bidoublet scalar �els as

Σ =

η0 −η+η−3 η̄

0

= σ + iπ3 iπ1 + π2

iπ1 − π2 σ − iπ3

, (3.4)where πi and σ are real �elds.

Φ =

Φ+Φ0

; (3.5)their vacuum expectation values are:

〈Σ〉 =

u 00 u

,〈Φ〉 =

0v

.(3.6)

The �eld Φ plays is the analogue of the Higgs sector of the standard model; the only di�erence is that

the third family does not couple to it.

Similarly to the SM, all the real �eld present in the scalar sector, except Re(Φ0) and σ, became the

longitudinal component of the bosons. The Lagrangian of this model can be divided into three part:

L = LB + LL + LY , (3.7)

CHAPTER 3. TOPFLAVOR MODEL 34

where

LB =1

2DµΦ

†DµΦ +1

4Tr(DµΣ

†DµΣ) + V(Σ; Φ)− 14

Wa12µνWaµν12 −

1

4Wa3µνW

aµν3 −

1

4BµνB

µν , (3.8)

LY = L̄1LΦ[ge11eR + ge12µR + ge13τR] + L̄2LΦ[gµ21eR + g

µ22µR + g

µ23τR]+

+1

ΛL̄3LΣ

†Φ[g31eR + g32µR + g33τR] + h.c. ,(3.9)

LF = iΨ̄γµDµΨ = iL̄jLγµ(∂µ + i

g122σiW iµ12 − i

g02Bµ)LjL + iQ̄

jLγµ(∂

µ + ig122σiW iµ12 + i

g06Bµ)QjL+

+iL̄3Lγµ(∂µ + i

g32σiW iµ3 − i

g02Bµ)L3L + iQ̄

3Lγµ(∂

µ + ig32σiW iµ3 +

g06Bµ)Q3L + iū

αRγµ(∂

µ +2g03Bµ)ujR+

+id̄αRγµ(∂µ − ig0

3Bµ)djR + iē

αRγµ(∂

µ − ig0Bµ)ejR − iq̄γµ(gs2λaGµa)q ,

(3.10)

where i,j=1,2, while α=1,2,3. The covariant derivative is de�ned as:

Dµ = ∂µ + ig12 ~T · ~Wµ12 + ig3 ~T ′ · ~Wµ3 + ig0Y Bµ (3.11)

It acts on the scalar �elds the following way:

DµΣ = ∂µΣ + ig12 ~T · ~Wµ12Σ− ig3Σ~T ′ · ~Wµ3 ,

DµΦ = ∂µΦ + ig12 ~T · ~Wµ12Φ + ig02BµΦ ,

(3.12)

where ~T = ~τ2 and τi are the Pauli matrices.

3.2 Gauge masses

In this section we will get the mass eigenvalues in term of the Vacuum Expectation Values and couplings.

The masses came from this part of the Lagrangian

1

2DµΦ

†DµΦ +1

4Tr(DµΣ

†DµΣ) , (3.13)

near the minimum. From this term, we get two mass matrices: one for the charged bosons,

M1 =

g212(u2+v2)4 − g12g3u24− g12g3u

2

4g23u

2

4

, (3.14)where the base is ordered as W±12,W

±3 , and for the neutral ones,

M2 =

g20v

2

4 −g12g0v

2

4 0

− g12g0v2

4g212(v

2+u2)4 −

g12g3u2

4

0 − g12g3u2

4g23u

2

4

, (3.15)

CHAPTER 3. TOPFLAVOR MODEL 35

in the base is B, W 312,W33 . Diagonalizing both matrices, we get the following expressions:

M2W ′ =g23u

2 + g12(u2 + v2) +

√(g23u

2 + g12(u2 + v2))2 − 4(g12g3uv)28

,

M2W =g23u

2 + g12(u2 + v2)−

√(g23u

2 + g12(u2 + v2))2 − 4(g12g3uv)28

,

M2A = 0 ,

M2Z′ =(g20 + g

212)v

2 + (g212 + g23)u

2 +√

((g20 + g212)v

2 + (g212 + g23)u

2)2 − 4u2v2(g20g23 + g212g23 + g212g20)8

,

M2Z =(g20 + g

212)v

2 + (g212 + g23)u

2 −√

((g20 + g212)v

2 + (g212 + g23)u

2)2 − 4u2v2(g20g23 + g212g23 + g212g20)8

,

(3.16)

We get the following mixing matrix for the charged bosons:

R1 =

cos θ′ − sin θ′sin θ′ cos θ′

, (3.17)And, in the limit of u�v, we get that the mixing matrix for the neutral bosons can be written as follows:

R2 '

cos θ − sin θ − v

2

u2 tan θ sinφ cos3 φ

sin θ cosφ cos θ cosφ+ v2

u2cos3 φ sin2 φ

cos θ − sinφ+v2

u2 sinφ cos4 φ

sin θ sinφ sinφ cos θ + v2

u2sinφ cos4 φ

cos θ cosφ+v2

u2 sin2 φ cos3 φ

[33] , (3.18)

R2 '

cos θ − sin θ 0

sin θ cosφ cos θ cosφ − sinφ

sin θ sinφ sinφ cos θ cosφ

. (3.19)Then we can write

Wµ±12 = Wµ± cos θ′ −W ′µ± sin θ′ , (3.20)

Wµ±3 = W′µ± cos θ′ +Wµ± sin θ′ , (3.21)

Bµ = cos θAµ − sin θZµ , (3.22)

Wµ312 = sin θ cosφAµ + cos θ cosφZµ − sinφZ ′µ , (3.23)

Wµ33 = sin θ sinφAµ + cos θ sinφZµ + cosφZ ′µ . (3.24)

3.3 Lepton sector

Now we have to substitute the previous relation into the Lagrangian; we will start doing this for the

leptons:

Lcclepton = −√

2

2[g12 ¯`iLγµ(W

µ+ cos θ′ −W ′µ+ sin θ′)νiL + g12 ¯νiLγµ(Wµ− cos θ′ −W ′µ− sin θ′)`iL+

+ g3τ̄Lγµ(Wµ+ sin θ′ +W ′µ+ cos θ′)ντL + g3 ¯ντLγµ(W

µ− sin θ′ +W ′µ− cos θ′)τL] ,

(3.25)

CHAPTER 3. TOPFLAVOR MODEL 36

Figure 3.1: Top�avor model: W ′ coupling with the leptons

and

Lnclepton = g0 ¯`αRγmu(cos θAµ − sin θZµ)`αR +g02

¯`αLγµ(cos θAµ − sin θZµ)`αL+

+g02

¯ναLγµ(cos θAµ − sin θZµ)ναL+

+g122

¯`iLγµ(sin θ cosφAµ + cos θ cosφZµ − sinφZ ′µ)`iL+

−g122

¯νiLγµ(sin θ cosφAµ + cos θ cosφZµ − sinφZ ′µ)νiL+

+g32τ̄Lγµ(sin θ sinφA

µ + cos θ sinφZµ + cosφZ ′µ)τL+

−g32

¯ντLγµ(sin θ sinφAµ + cos θ sinφZµ + cosφZ ′µ)ντL .

(3.26)

It will be useful later summarize the coupling between lepton and charged bosons (see table 3.2); the W ′

coupling with the leptons are also showed in �gure 3.1.

h h′

I,II√

22 g12 cos θ

′√

22 g12 sin θ

′

III√

22 g3 sin θ

′√

22 g3 cos θ

′

Table 3.2: Top�avor model: coupling between the �rst two gener-

ation (I, II) and the third generation (III) with the W boson (h)

and with the W ′ boson (h′).

Imposing that the coupling with the leptons and the photons is the electric charge, we get the following

CHAPTER 3. TOPFLAVOR MODEL 37

relations:

g0 =e

cos θ,

g12 =e

sin θ cosφ,

g3 =e

sin θ sinφ,

(3.27)

it is easy to verify that1

e2=

1

g20+

1

g212+

1

g23, (3.28)

It is interesting to note that these conditions are also veri�ed with the diagonalization of the neutral

matrix: in order to �nd the eigenvector of the matrix (3.15), we have to calculate:

(M2 − λ13x3)vλ = 0 , (3.29)

where λ is the eigenvalue and vλ is the relative eigenvector; in order to verify that the previous relation

are true, is su�cient to do this for λ = 0:g0x

10 − g12x20 = 0 ,

g3x30 − g12x20 = 0 ,

g12(v2 + u2)x20 − g0v2x10 − g3u2x30 = 0 .

(3.30)

We know that the eigenvectors are the columns of R2, then we can easily see that the system is auto-

matically veri�ed.

From the diagonalization of the �rst matrix, we get a relation between the mixing angle θ′ and φ:

(M1)12 = (R1 · Mdiag1 ·RT1 )12 ,

sin θ′ cos θ′ ' sinφ cosφ(

1− v2

u2g212

g23 + g212

+ 2v2

u2g212g

23

g212 + g23

).

(3.31)

In the limit u�v, we get that these angles are the same; this feature is really interesting, because thanks

to this we have automatically the lepton universality, with a little violation, for the W boson; indeed, it

must turn out to be true

g12 cos θ′ ' g3 sin θ′ , (3.32)

e

sin θ sinφsin θ′ ' e

sin θ cosφcos θ′ . (3.33)

Because this model is strictly hierarchical, we will consider only the limit u�v; for this reason, we

obtained the approximate expressions for the masses:

M2W 'v2

4

g212g23

g212 + g23

(1− g

412

(g212 + g23)

2

v2

u2

),

M2Z 'v2

4

g20e2

g212g23

g212 + g23

(1− g

412

(g212 + g23)

2

v2

u2

),

M2Z′ 'M2W ′ '1

4

(u2(g23 + g

212) + g

212v

2 − v2 g212g

23

g212 + g23

).

(3.34)

CHAPTER 3. TOPFLAVOR MODEL 38

It is easy to see some interesting features if we put these expressions in terms of the mixing angles:

M2W 'v2

4

e2

sin2 θ

(1− sin4 φv

2

u2

),

M2Z 'v2

4

e2

sin2 θ cos2 θ

(1− sin4 φv

2

u2

),

M2Z′ 'M2W ′ 'e2v2

4 sin2 θ

(tan2 φ+

u2

v2 sin2 φ cos2 φ

).

(3.35)

In this limit, we have that the new bosons W ′ and Z ′ have the same mass at �rst order in v/u. In this

limit, we have that the mass of W and Z do not depend on the new VEV and on the new coupling.

These masses depend approximately only on the usual VEV, v, and on θ angle; imposing that the masses

of the usual weak bosons are the measured ones within the error [72], we get that v ' vsm and θ ' θsm;

this is very relevant from a practical point of view, as it greatly simpli�es numerical calculations used in

the following results.

3.4 Quark sector

Let's discuss the quark sector. In the same way of lepton, we obtain

Lccquark = −√

2

2[g12 ¯dαLV

d†αi γµ(W

µ+ cos θ′ −W ′µ+ sin θ′)V uiβuβL + g12 ¯uαLVu†αi γµ(W

µ− cos θ′ −W ′µ− sin θ′)V diβdβL+

+ g3 ¯dαLVd†α3γµ(W

µ+ sin θ′ +W ′µ+ cos θ′)V u3βuβL + g3 ¯uαLVu†α3 γµ(W

µ− sin θ′ +W ′µ− cos θ′)V d3βdβL] ,

(3.36)

and

Lncquark =g03

¯dαRγmu(cos θAµ − sin θZµ)dαR −

g06

¯dαLγµ(cos θAµ − sin θZµ)dαL+

−g06

¯uαLγµ(cos θAµ − sin θZµ)uαL −

2g03

¯uαRγmu(cos θAµ − sin θZµ)uαR+

+g122

¯dαLVd†αi γµ(sin θ cosφA

µ + cos θ cosφZµ − sinφZ ′µ)V diβdβL+

−g122

¯uαLVu†αi γµ(sin θ cosφA

µ + cos θ cosφZµ − sinφZ ′µ)V uiβuβL+

+g32

¯dαLVd†α3γµ(sin θ sinφA

µ + cos θ sinφZµ + cosφZ ′µ)V d3βdβL+

−g32

¯uαLVu†α3 γµ(sin θ sinφA

µ + cos θ sinφZµ + cosφZ ′µ)V u3βuβL .

(3.37)

From the �rst expression, we can see that the Cabibbo-Kobayashi-Maskawa matrix is not unitary in this

model; indeed we have:

VCKM = V†uVd +

(1− g3 sin θ

′

g12 cos θ′

)V u†13 V

d31 V

u†13 V

d32 V

u†13 V

d33

V u†23 Vd31 V

u†23 V

d32 V

u†23 V

d32

V u†33 Vd31 V

u†33 V

d32 V

u†33 V

d33

. (3.38)We have that the �rst term is unitary and has the usual form, while the second one is not. We have to

determine two matrices. On the other hand, the second expression shows that in this model are possible

CHAPTER 3. TOPFLAVOR MODEL 39

at tree level FCNC interactions; these kinds of interactions are not present in the standard model at tree

level.

3.5 Constraints

The last step of this chapter consists of the imposition of various constraints that comes from already

measured quantities. The �rst thing we have to do is imposing that the mass prediction for the boson

W and Z are in agreement, within the error, with the experimental measurements for these quantities.

Other quantities that we have to check are listed in [30, 31], we simply updated the experimental

measurement and then the lower limit for the u vacuum expectation value. We report the corrections to

the standard model that come from its extension:

ΓZ = ΓSMZ

(1 +

1

x[−0.896 sin4 φ+ 0.588 sin2 φ]

),

Re = RSMe

(1 +

1

x[0.0794 sin4 φ+ 0.549 sin2 φ]

),

Rµ = RSMµ

(1 +

1

x

[0.0794 sin4 φ+ 0.549 sin2 φ

]),

Rτ = RSMτ

(1 +

1

x

[0.0794 sin4 φ+ 0.549 sin2 φ

]),

AeFB = (AeFB)

SM

(1 +

1

x[10.44 sin4 φ]

),

AµFB = (AµFB)

SM

(1 +

1

x[10.44 sin4 φ]

),

AτFB = (AτFB)

SM

(1 +

1

x[10.44 sin4 φ+ 12.14 sin2 φ]

),

AbFB = (AbFB)SM

(1 +

1

x[5.287 sin4 φ+ 0.157 sin2 φ]

),

Ae = ASMe

(1 +

1

x5.22 sin4 φ

),

Aτ = ASMτ

(1 +

1

x[5.22 sin4 φ+ 12.14 sin2 φ]

),

σ0h = (σ0h)SM

(1 +

1

x[−0.01 sin4 φ− 0.628 sin2 φ]

),

Rb = RSMb

(1 +

1

x[−0.015 sin4 φ+ 1.739 sin2 φ]

),

Rc = RSMc

(1 +

1

x[0.038 sin4 φ− 0.549 sin2 φ]

),

(3.39)

where x = u2/v2, Af is the asymmetry parameter, AfFB is the Forward-Back asymmetry parameter, σ

0h

is the hadronic pole cross-section, which is de�ned as

σ0h =12π

m2Z

Γ(Z → eē)Γ(Z → hadrons)Γ2Z

,

and Rf is de�ned as follows:

Rf =Γ(Z → ff̄)

Γ(Z → hadrons).

CHAPTER 3. TOPFLAVOR MODEL 40

Starting from these equations, we proceeded to calculate the lower limit for the VEV quoted before,

assuming that the other VEV, v, is approximately equal to the one of the standard model. In Table 3.3

we summarized the results we obtained. The bigger constraints came from the Rb and it will be the only

constraint we will put into �nal plots.

Observables Experimental data SM prediction u lower limit

ΓZ(GeV ) 2.4952± 0.0023 2.4942± 0.0008 4000GeV

Re 20.804± 0.050 20.737± 0.010 3800GeV

Rµ 20.785± 0.033 20.737± 0.010 3800GeV

Rτ 20.764± 0.045 20.782± 0.010 3800GeV

σohadr.(nb) 41.541± 0.037 41.481± 0.008 5800GeV

Ae 0.15138± 0.00216 0.1469± 0.0003 3200GeV

Aτ 0.136± 0.015 0.1484 3100GeV

AFBe 0.0145± 0.0025 0.01618± 0.00006 2000GeV

AFBµ 0.0169± 0.0013 0.0165 2000GeV

AFBτ 0.0188± 0.0017 0.0165 2000GeV

Rb 0.21629± 0.00066 0.21582± 0.000022 6000GeV

Rc 0.1721± 0.0030 0.17221± 0.00003 1500GeV

AFBb 0.0992± 0.0016 0.1030± 0.0002 3100GeV

Table 3.3: Top�avor model: constraints; the experimental and

theoretical values where taken from [72, 30, 31]

Figure 3.2: Top�avor model: muon decay

We have also to impose that the prediction of the Fermi constants is in agreement with the experi-

mental results. In the standard model the Fermi constant is given by:

GSMF =

√2

8

g2

M2W

δ

1−∆r[70, 71] . (3.40)

Due to the presence of the W ′ boson, the muon decay can be mediated also by it, as shown in Figure

3.2. Approximately we can say that the corrections to the standard model prediction is given by:

GF =

√2

8

[(g12 cos θ

′)2

M2W

δ

1−∆r+

(g3 sin θ′)2

M2W ′

]. (3.41)

Chapter 4

Results

In this chapter, we will describe how we developed the algorithm, using the "Mathematica" program,

necessary to study the principal characteristics of the Top�avor model. We are interested in the W ′ mass

and the coupling between fermion and this boson are allowed by this model.

Looking at the mass matrices calculated in 3.14 3.15, that are:

M1 =

g212(u2+v2)4 − g12g3u24− g12g3u

2

4g23u

2

4

,

M2 =

g20v

2

4 −g12g0v

2

4 0

− g12g0v2

4g212(v

2+u2)4 −

g12g3u2

4

0 − g12g3u2

4g23u

2

4

,we can see that they are written in term of �ve free parameter: u, v, g12, g0, and g3. Not all of these

parameter are independent, indeed, imposing the electric charge, we get the following relations:

g0 =e

cos θ,

g12 =e

sin θ cosφ,

g3 =e

sin θ sinφ.

(4.1)

Since masses lower than the TeV scale are already been excluded for the W ′ boson for coupling

constants of the same order of gSM , we are interested in the limit u � v; in this approximation the

masses expressions can be written as:

M2W 'v2

4

g212g23

g212 + g23

(1− g

412

(g212 + g23)

2

v2

u2

),

M2Z 'v2

4

g20e2

g212g23

g212 + g23

(1− g

412

(g212 + g23)

2

v2

u2

),

M2W ′ 'M2Z′ '1

4

(u2(g23 + g

212) + g

212v

2 − v2 g212g

23

g212 + g23

).

41

CHAPTER 4. RESULTS 42

In this limit, the boson W ′ and Z ′ have the same mass.

For reasons that will be clear soon, it is useful to make the following change of parameter:

g12 , g0 , g3 ←→ e , φ , θ

where e is the electric charge, which is a �xed quantity; this means that our model can be described using

four free parameters: u, v, φ, and θ. If we rewrite the previous equation in term of these parameters, we

get:

M2W 'v2

4

e2

sin2 θ

(1− sin4 φv

2

u2

),

M2Z 'v2

4

e2

sin2 θ cos2 θ

(1− sin4 φv

2

u2

),

M2W ′ 'M2Z′ 'e2v2

4 sin2 θ

(tan2 φ+

u2

v2 sin2 φ cos2 φ

).

Looking at the �rst two equations, we can easily see that if we impose that the predictions for the mass

of the W and Z bosons are in agreement, within the error, with the experimental measurement, we get

that v ' vSM and θ ' θW ; in this way we �xed them. Finally we have only two free parameters, u and

φ. We can use these expressions if v2/u2 � 1; since we need a lower limit for u in our algorithm, we

considered v2/u2 < 0.1, this means that u > 800.

Since we want both SU(2) interaction to be perturbative, for this reason, it must turn out to be true:

g212 < 4π , g23 < 4π ,

this means that in our limit 0.18 < tanφ < 5.5.

The developed algorithm generates 106 random couples (u, φ), with u > 800GeV and 0.18 < tanφ <

5.5 for the reasons mentioned above. In this way we have �xed all the parameters, this means that the

matricesM1 andM2 are numerical matrices that can be numerical diagonalized. From the diagonaliza-

tion of the �rst matrix, we obtain the rotation matrix for the charged bosons 3.17, which is linked to the

mixing angles θ′. In chapter three (see equation 3.31) we proved that θ′ and φ are almost equal if u� v;

this means that 0.18 < tan θ′ < 5.5.

We are interested in the coupling constant between fermions and the W ′ boson, but since this model

presents di�erent coupling depending on the generation, we have two di�erent coupling constants. Since

this model does not preserve the lepton universality for the new boson, it is useful to study the following

ratios: (h′

h

)I,II

=g12 sin θ

′

g12 cos θ′= tan θ′ ,(

h′

h

)III

=g3 cos θ

′

g3 sin θ′= tan−1 θ′ .

So that we can compare the coupling to the same fermions to the W ′ and the W . These quantities are

constrained by the fact that 0.18 < tanφ < 5.5 and φ ' θ′.

CHAPTER 4. RESULTS 43

We veri�ed that the predictions for the masses of W and Z bosons are within 3σ, 2σ, and 1σ that

are represented with di�erent colors, respectively blue, orange, and green, in �gure 4.1. We also required

that the theoretical prediction for the Fermi coupling constant was in agreement with the experimental

measurement.

The last step consisted of the imposition of the constraints that cames from:

Rb = RSMb

(1 +

1

x[−0.015 sin4 φ+ 1.739 sin2 φ]

).

Then we get a lower limit for u, that we call umin ' 6000GeV . The points that are allowed are the ones

on the right side of the red curve in 4.1, while the red curve is obtained for u = umin.

Since the algorithm generated a su�cient number of point, we were able to extrapolate continue band

in the plots(h′

h

)I,II

vs MW ′ and(h′

h

)III

vs MW ′ .

Figure 4.1: Results: (h′/h)I,II vs MW ′ and (h′/h)III vs MW ′ :

in the plot it the green, the orange and the blue bands are ob-

tained imposing that the correction this model to the masses of

the already measured boson, W and Z, is less than 1σ, 2σ and 3σ

respectively

The results can be seen in Figure 4.1. From these plots, we can see that the coupling constants

between fermions and the W ′ boson are of the same order of the standard model one. Another thing we

can see from these plots is that the W ′ must have a mass higher than 6-7 TeV to be compatible with

this model. We would expect to have coupling constant between fermions and the W ′ smaller in order to

recover the standard model, but too small values are not allowed due to the imposition that both SU(2)

interaction to be perturbative.

It is interesting to study the following product:

g12g3 sin θ′ cos θ′ , (4.2)

CHAPTER 4. RESULTS 44

since we have that the cross-section of process that have in the initial state the �rst (or the second)

generation and the third generation in the �nal state (or vice-versa); an example is shown in Figure 4.2.

CMS and ATLAS are interested in the cross-section of two processes: u d → t b and u d → τ ντ . The

Figure 4.2: Results: example

quantity mentioned before is approximately equal to:

g2SM =e2

sin2 θ. (4.3)

We plotted this quantity in Figure 4.3 and we can see that the variation from g2SM is less than 1.5%.

Figure 4.3: Results: The plot shows the variation from g2SM of

g12g3 sin θ′ cos θ′ depending on MW ′

Since we have di�erent coupling between fermions and the standard model W boson depending on

the generation, we decided to plot the following quantity:

2√2

|hI,II − hIII |gSM

=|g12 cos θ′ − g3 sin θ′|

gSM.

The result is shown in the right side of Figure 4.4; from this plot we can see that there is a violation from

lepton universality smaller than 1%. The following quantity:

2√2

|h′I,II − h′III |gSM

=|g12 sin θ′ − g3 cos θ′|

gSM,

CHAPTER 4. RESULTS 45

Figure 4.4: Results: Relation between the coupling constants: on

the left it is shown the plot that relates the coupling to the W

boson, while on the right it is shown the relation between the

coupling to the W ′

is not interesting because we do not expect the lepton universality for the W ′, except for particular

values of the free parameters. It is more interesting to show the relation we have between the two

coupling constant h′I,II and h′III . Since the angles θ

′ and φ are almost equal, we expect these coupling

to be one the inverse of the other because

h′I,II =e

sin θ

sin θ′

cosφ

h′III =e

sin θ

cos θ′

sinφ

h′I,IIh′III

=sin θ′

sinφ

cosφ

cos θ′

The result is shown on the right side of Figure 4.4 and we get the result we expected; an interesting

feature we can see looking at this plot: the red dot is the point in which the two couplings are equal, this

means that there is the possibility to have the lepton universality for the W ′ boson for a particular set

of parameters.

Chapter 5

Conclusions

Despite all the achievements of the Standard Model, this theory still presents some open questions, such

as the origin of neutrino masses, the dark matter and the baryon asymmetry. This means that this

model is not a complete theory and it needs to be extended. There are many extensions of the Standard

Model, but we chose to study the extensions that predict the existence of a new heavy vector boson, the

W ′. Currently the Atlas and CMS collaborations are searching for the W ′ using the Sequential Standard

Model (SSM). CMS and ATLAS produce exclusion plotss with the comparison between the theoretical

prediction of a certain cross-section and its experimental measurement; the theoretical predictions are

made using the SSM �xing its parameters; for this reason the exclusion plots are still model dependent.

The Sequential Standard Model presents two major problems: it is not associated with a certain gauge

theory for a generic set of parameters and usually it can not be used to distinguish the models. For these

reasons, it is important to make a phenomenological study of the W ′ models.

The principal aim of this thesis was to make a complete phenomenological study of the Top�avor

model. We also presented the principal features of four other W ′ models, the L-R model, the 331 model,

the Little Higgs model, and the win Higgs model.

The W ′ models we studied have many common features: all of these models predict the existence

of a new neutral boson, the Z ′, and an extended scalar sector, and a multi-step spontaneous symmetry

breaking, but di�ers for the number of new particles, the presence of FCNC interaction at tree level, the

presence of lepton-non-universality for the W ′, and sources of CP violation.

It is interesting to compare the Top�avor model with other W ′ models. For example, both the 331

model and the Top�avor treat the third generation of quark di�erently from the �rst two; in both models,

such a characteristic is responsible for the existence of FCNC interaction at tree level. However these

models predict a di�erent number of new bosons, and that the additional bosons can have larger masses

than the one of the W ′.

The L-R model and the Top�avor have the same gauge theory, but the fermions' transformation rules

46

CHAPTER 5. CONCLUSIONS 47

are di�erent. In both models, we need two steps Spontaneous symmetry breaking, but the realization is

obtained with a di�erent choice for the extension of the scalar sector. Another di�erence among these

models is that in the L-R model we have a mixing angle between W and W ′ approximately equal to zero

for the charged bosons, while in the Top�avor case such angle is bound when one requires that both SU(2)

interactions are perturbative. The coupling constant to the W ′ boson in the L-R model is approximately

equal to the coupling of the standard model, while in the Top�avor it is of the same order of the coupling

constant of the standard model.

Some of the W ′ models present a relation with αR and αL, as we have seen for the L-R model in

the second chapter. In the 331RHN and 331econ. we have that αR=αL for the neutrinos and the charged

leptons respectively, while αR=0 for the other particles. There is also a version of the Twin Higgs model

in which αR 6= 0, while αL = 0.

This thesis is focused on the phenomenological study of the Top�avor model. The main results is that

the model predicts a W ′ mass lower limit of 6-7 TeV coming from the Rb constraint. We proved that the

violation from the lepton universality for the W boson is less than 1%, while we do not expect the lepton

universality for the W ′, except for some speci�c sets of parameters. Moreover the new gauge coupling

constants with W ′ are of the same order of gSM . We obtained that the cross-sections of ud → tb and

ud→ τντ are proportional to g2SM . We also proved that the masses of Z ′ and W ′ are the same.