%
%%

�
�
�

,
,
,,

�
��

��

e
ee
@
@@

l
l
l

Q
QQ

HHPPP XXX hhhh (((( ��� IFT Instituto de F́ısica Teórica

Universidade Estadual Paulista

DISSERTAÇÃO DE MESTRADO IFT-D.003/18

Enhancement in the double Higgs boson production by e+e− annihilation and

physics Beyond the Standard Model.

Presented by: Andrés Felipe Vásquez Tocora

Advisor: Rogerio Rosenfeld

Co-Advisor: Alberto Tonero

October 3, 2018



 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  
               Vásquez Tocora, Andrés Felipe 
 V335e          Enhancement in the double Higgs boson production by e

+
e­ annihilation     

               and physics beyond the standard model/ Andrés Felipe Vásquez Tocora. –  
               São Paulo, 2018  
                     120 f. : il. 

 
                     Dissertação (mestrado) - Universidade Estadual Paulista (Unesp), 
               Instituto de Física Teórica (IFT), São Paulo 

Orientador: Rogerio Rosenfeld 
Coorientador: Alberto Tonero 

 
                     1. Modelo padrão (Física nuclear). 2. Higgs, Bosons de. 3. Teoria 
                de campos (Física)  I. Título. 
 

 

  
Sistema de geração automática de fichas catalográficas da Unesp. Biblioteca do Instituto de 

Física Teórica (IFT), São Paulo. Dados fornecidos pelo autor(a). 

 



Abstract

The main goal of this dissertation is to show the enhancement of the cross-section for the double Higgs produc-

tion through pair annihilation by including effective interactions and through the non-perturbative Sommerfeld

effect. Bounds to some Wilson coefficients were obtained from such enhancement, this in the scenarios of the

future e+e−-colliders (FCC-ee, ILC, CLIC). In order to achieve this, some computational tools were imple-

mented: FeynRules, FeynArts, FormCalc, and LoopTools. It is also shown the enhancement of the double

Higgs production in 2HDM and MSSM, discussing the general framework of these two models. In addition,

it is studied the threshold behavior of the cross-section for the double Higgs production when a hidden sector

couples to the Higgs boson, yielding resonances below the threshold energy due to non-perturbative effects.

We study the Sommerfeld effect in the double Higgs production in the scenario of e+e−-colliders. The enhance-

ment is discussed as generated from a hidden sector coupled to the Higgs boson. Below and above threshold

enhancements are presented. Such analysis is of importance in the ILC project, which will operate up to the

threshold energy
√
s = 250 GeV. The results has been achieved by the use of computational tools like FeynArts,

FormCalc, and LoopTools.

Keywords— Beyond the Standard Model, Effective Field Theory, Higgs boson production, Sommerfeld effect.

Resumo

O objetivo principal dessa dissertação é, mostrar o aprimoramento da seção de choque para a produção em

dobro dos bósons de Higgs, por meio de aniquilação de pares, incluindo interações efetivas e através do efeito

não perturbativo de Sommerfeld. De tais aprimoramentos, os limites para alguns coeficientes de Wilson foram

obtidos, isso nos cenários de futuros aceleradores de e+e− (FCC-ee, ILC, CLIC). Para atingir estes resulta-

dos, algumas ferramentas computacionais foram implementadas: FeynRules, FeynArts, FormCalc e LoopTools.

Também, é mostrado o aprimoramento da produção em dobro de bósons de Higgs no 2HDM e MSSM, discutindo

o marco geral desses dois modelos. Além disso, foi estudado o comportamento, perto do limite de produção, da

seção de choque da produção em dobro dos bósons de Higgs, quando um setor escondido é acoplado ao Higgs,

produzindo ressonâncias abaixo da energia limite de produção, devido à efeitos não perturbativos.





To Luz Stella and Gabrielly.



Contents

Abstract ii

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

I The Standard Model of particle physics 1

I.1 Strong Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

I.2 Higgs sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

I.3 Electroweak Bosonic Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

I.4 Ghosts and gauge fixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

I.5 Electroweak Fermion sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

I.6 Yukawa Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

I.7 Entire Standard Model Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

II New physics 15

II.1 Why physics beyond the Standard Model? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

II.1.1 Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

II.1.2 Cosmological Constant Problem and Dark Matter . . . . . . . . . . . . . . . . . . . . . . 16

II.1.3 Neutrino Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

II.1.4 Additional open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

II.2 Physics BSM in an effective approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

II.3 Naturalness and Hierarchy problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

v



vi CONTENTS

IIITwo-Higgs-Doublet Model (2HDM) 30

III.1 Construction of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

III.2 Types of 2HDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

III.3 The Alignment Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

III.4 Constraints to the Parameters of the 2HDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

IVMinimal Supersymmetric Standard Model 42

IV.1 Basics of Supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

IV.2 Construction of the MSSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

IV.3 Spontaneous Symmetry Breaking in the MSSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

IV.4 Supersymmetry Breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

IV.4.1 Soft Supersymmetry Breaking terms and full MSSM Lagrangian . . . . . . . . . . . . . . 56

IV.4.2 Spontaneous Supersymmetry Breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

IV.5 Mass spectrum of the MSSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

IV.6 Phenomenological constraints on the MSSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

V Higgs physics 70

V.1 Channels of production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

V.2 Software implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

V.2.1 e+e− → µ+µ− . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

V.3 h→ γγ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

V.4 h→ gg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

V.5 Branching ratios and decay widths of the Higgs boson . . . . . . . . . . . . . . . . . . . . . . . . 82

VIDouble Higgs production via electron-positron annihilation 89

VI.1 Double Higgs production in the SM (e+e− → hh) . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

VI.2 Enhancement of e+e− → hh via Effective operators . . . . . . . . . . . . . . . . . . . . . . . . . . 93

VI.3 e+e− → h0A0 (2HDM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

VI.4 e+e− → {h0A0, H0A0} (MSSM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103



VI.5 Sommerfeld enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

VI.5.1 Remarks on Sommerfeld effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

VI.5.2 Sommerfeld effect from ladder approximation applied to the e+e− → hh process . . . . . 107

VI.5.3 Solution to the Schrödinger equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

VI.5.4 Sommerfeld enhancement of e+e− → hh . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

VIIConclusion 115

Bibliography 115

vii



viii



List of Tables

I.1 Quark properties.[15] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

I.2 Lepton properties.[15] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

I.3 Properties of bosons. [15] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

I.4 Quantum numbers Q, τ3 and Y for the fermion and higgs fields. . . . . . . . . . . . . . . . . . . 5

II.1 Dimension-6 independent effective operators. The top table shows 34 operators that take into

account interactions involving bosons, while in the bottom table there are 29 operators describing

four-fermion interactions. Without the 4 B-violating operators, remain 59 independent operators

described in Ref. [62] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

III.1 Higgs doublets that couples to an up-, down- and charged-lepton-like fermions in different Two-

Higgs Doublet Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

III.2 Higgs doublets that couples to an up-, down- and charged-lepton-like fermions in different Two-

Higgs Doublet Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

IV.1 Conventions of the R-charge belonging to an U(1)R symmetry for some objects. . . . . . . . . . . 48

IV.2 Particle spectrum of the MSSM that are arranged in chiral superfields, together with their rep-

resentations under the gauge symmetry of the SM. The index i stands for the generation. . . . . 49

IV.3 Particle spectrum of the MSSM that are arranged in vector superfields, together with their

representations under the gauge symmetry of the SM. The indices run as A = 1, · · · , 8 and

I = 1, 2, 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

V.1 The Branching Ratios for a Higgs boson with mh = 125 GeV.[15] . . . . . . . . . . . . . . . . . . 88

VI.1 Benchmark luminosities and energies for different future experiments.[65] . . . . . . . . . . . . . 101

ix



x



List of Figures

I.1 Triple and Quartic Higgs interactions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

I.2 Triple interactions between weak massive bosons and the Higgs . . . . . . . . . . . . . . . . . . . 6

I.3 Quartic interactions between weak massive bosons and the Higgs . . . . . . . . . . . . . . . . . . 7

I.4 Yukawa interaction between Higgs boson and fermions. . . . . . . . . . . . . . . . . . . . . . . . . 12

I.5 Electroweak interactions for fermions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

IV.1 Corrections to the Higgs mass from scalars and fermions. . . . . . . . . . . . . . . . . . . . . . . 51

V.1 Main production processes for the Higgs boson in hadronic collisions. . . . . . . . . . . . . . . . . 71

V.2 Main production processes for the Higgs boson in e+e− colliders. . . . . . . . . . . . . . . . . . . 72

V.3 Cross-section for the process e+e− → µ+µ− (Compare to Fig. 5.1 in [54]) a) in QED, b) in the

Glashow-Weinberg-Salam model with the full Z-boson propagar (Compare to Fig. 1 in [66]). . . . 75

V.4 Feynman diagrams in the unitary gauge for the Higgs decay into two photons, with εµ2 = εµ(p2)

and εν3 = εν(p3), and where the crossed diagrams have been omitted. . . . . . . . . . . . . . . . . 75

V.5 Width for the decay H → γγ computed theoretically (Black) and by FormCalc implementation

(Magenta). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

V.6 Feynman diagrams for the Higgs decay into two gluons. . . . . . . . . . . . . . . . . . . . . . . . 81

V.7 Width for the decay H → gg given by FormCalc (Black) and theoretical computations from [15]

(Magenta). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

V.8 Branching ratios and their uncertainties of the Higgs boson as a function of its mass.[67] . . . . . 86

VI.1 Feynman diagrams at tree-level for e+e− → hh. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

VI.2 (a) Triangle and (b) Box topologies at 1-loop for e+e− → hh. . . . . . . . . . . . . . . . . . . . . 90

xi



xii LIST OF FIGURES

VI.3 Bubble topologies at 1-loop for e+e− → hh. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

VI.4 Total cross-section and box contribution for the process e+e− → hh in the SM, where it is

shown that only the box topologies do contribute substantially. Here σ(0+1) stands for the total

cross-section at NLO and σ(Box) for the cross-section coming just from the Box diagrams. . . . . 91

VI.5 Dependence on the trilinear Higgs coupling of the cross-section for e+e− → hh just for the 1-loop

contribution (Left), and of the total cross section at NLO (Right). . . . . . . . . . . . . . . . . . 92

VI.6 Example of the helicity structure for diagrams involving the trilinear Higgs interaction. . . . . . . 92

VI.7 Vanishing amplitude for the process e+e− → hh near threshold, where it has been fixed randomly

two collision angles in order to present that the suppression is not due to such fixing. . . . . . . . 93

VI.8 Feynman Diagram of an effective contribution that reproduces a Fermi-like interaction. . . . . . 94

VI.9 Sensitivity (Left) and enhancement (Right) of the cross-section σ (e+e− → hh) due to variations

of the Wilson coefficient C1 that appears in Eq. [VI.1]. . . . . . . . . . . . . . . . . . . . . . . . 95

VI.10Sensitivity (Left) and enhancement (Right) of the cross-section σ (e+e− → hh) due to variations

of the Wilson coefficient Cet introduced in the Lagrangian of Eq. [VI.15]. . . . . . . . . . . . . . 97

VI.11Chosen Feynman diagrams that introduces the effective interactions effects from Eq. [VI.15] into

the cross-section σ (e+e− → hh). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

VI.12χ2-distribution for the Wilson coefficients CeH (left) and Cet (right) from the cross-section of the

double Higgs production via electron-positron annihilation. Here the horizontal lines stand for

the limit of 95% of confidence level from a χ2 distribution with 1 degree of freedom. The bounds

a)-b) are expected for L = 2.6 ab−1 at 350 GeV, benchmark values of the FCC-ee Project. The

bounds c)-d) are expected for L = 4 ab−1 at 500 GeV, benchmark values of the ILC Project. The

bounds e)-f) are expected for L = 1.5 ab−1 at 1.5 TeV, benchmark values of the CLIC Project. . 100

VI.13χ2-distribution for the Wilson coefficients CeH and Cet from the cross-section of the double Higgs

production via electron-positron annihilation. It is presented, from inner to outer, the 68.5%,

95% and 99% confidence regions for 1 degree of freedom for the mentioned Wilson coefficients.

These bounds are expected for (a) L = 500 fb−1 at 380 GeV, (b) L = 1.5 ab−1 at 1.5 TeV, (c)

L = 3 ab−1 at 3 TeV, benchmark values of the CLIC project; and (d) L = 4 ab−1 at 500 GeV,

benchmark values of the ILC projects (arXiv: 1704.02333v2). . . . . . . . . . . . . . . . . . . . . 101

VI.14Total cross-section of the process e+e− → A0h0 in the 2HDM for tanβ = 1 and sinα = 1, at tree

level and at 1-loop with different values for λ5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

VI.15Some examples of the Feynman diagrams that enter in the the process e+e− → A0h0 in the

2HDM and in the MSSM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103



VI.16Cross-sections at tree-level for the Higgs boson production e+e− → H0A0 in the MSSM a) as a

function of the energy (mA0 = 200 GeV), and b) as a function of the mass of the CP-even neutral

Higgs boson H0 (
√
s = 3 TeV). Compare the above results with [70]. . . . . . . . . . . . . . . . . 104

VI.17Sommerfeld Factor (S) as a function of εφ with the fixed value εv = 10−3. Here the black solid

curve represent the numerical solution of the Schrödinger equation while the red dashed curve

stands for analytical solution with the approximation taking the Hulthen potential. . . . . . . . 106

VI.18Relation that defines the amplitude of the process e+e− → e+e− mediated by Higgs bosons in

terms of Γ (q, p). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

VI.19Relation that must be satisfied by Γ in the ladder approximation. . . . . . . . . . . . . . . . . . . 109

VI.20Behavior around threshold of the cross-sections for the double Higgs production when the Som-

merfeld effects of a hidden scalars φ with mφ = 10 GeV are taken into account. The left plot

corresponds to the Higgs boson with a width of Γh = 0.005 GeV as in the SM, while the left plot

corresponds to Γh = 0.01 GeV. The black solid curve represents the SM cross-section σ0. The

vertical lines in magenta represent the energies
√
s = 227.65 GeV and

√
s = 240.35 GeV of the

` = 0 bound-state from the formulas [VI.78]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

xiii



xiv



Introduction

Huge advances have been achieved in the understanding of the basic components of matter and its dynamics.

From the Fermi theory of the β-decay, many attempts were made in order to unveil the fundamental nature

of the weak interactions. Thus it was introduced the Glashow-Winberg-Salam theory (GWS) of electroweak

interactions, that explains the dynamics of electroweak interactions up to smaller distances than those that Fermi

theory could. The GWS theory, together with Quantum Chromodynamics (QCD), compose the Standard Model

(SM) of particles, a theory that has shown incredible predictive power. This theory was completely confirmed

July 4 of 2012, when the resonance at 125 GeV was identified as the Higgs boson.

Despite all the success that the Standard Model represents by itself, from its creation it is known that it cannot

be the full story. An ever deeper theory is needed in order to explain the known physical phenomena. The

simple fact that gravity cannot be explained in the SM framework is a strong hint to say that it is required

a theory that explains the physics at enough short distances, where gravity becomes comparable in strength

to the other interactions, i.e. at the Planck scale. Like this, there are many other hints in the way to find

theories beyond the SM: massive neutrinos, dark matter, dark energy, cosmological problem, CP violation and

matter-antimatter asymmetry are some observational clues.

Theoretical arguments also can guide us in the way of looking for UV - completions of the SM. How is it

possible that between the electroweak and the Planck scales there is not new physics? This is like 16 orders

of magnitude in which basically nothing interesting happens. Thus the hierarchy problem guide us to seek the

right underlying theory beyond the SM. This motivates the study of supersymmetric theories, compositeness

models, extra-dimensions, among others.

The list of extensions of the SM model is huge, for this reason a method that can describe beyond SM effects in

a model-independent way is very useful. Following the Fermi theory, an effective extension of the SM that let us

parametrize the unknown physics was built. Precisely, through this effective extension, in this dissertation, the

Higgs sector is studied. It is very important the full characterization of the Higgs sector. So far, we know the

predictions for the parameters in the scalar potential of the SM, but the trilinear and quartic self-interactions

1



2 LIST OF FIGURES

of the Higgs boson has not been directly measured. Thus, some deviations are expected from the predictions,

in such a way that this sheds light on physics beyond the SM.

It could also happen that there exist some scalars that, from the particle spectrum of the SM, only couple to

the Higgs boson. In this case, it could happen that such interactions bind Higgs bosons in order to compose

a bound-state. This situation is studied in a non-relativistic way, through the Sommerfeld effect, where the

effects of the hidden scalars could be relevant.

In this dissertation, the Standard Model of particle physics is discussed in chapter [I] and the motivations for

going beyond the Standard Model in chapter [II]. In the same chapter, the EFT approach for physics beyond

the Standard Model is presented. Chapters [III] and [IV] give the general aspects of the Two Higgs Doublet

Model and the Minimal Supersymmetric Standard Model, respectively. Then, in the chapter [V], some general

aspects of the Higgs boson physics are shown . Finally, in the chapter [VI], the cross-sections for the Double

Higgs production by e+e− annihilation in the SM, 2HDM, and MSSM, are studied, including the way that

some effective operators can enhance this process, then imposing bounds to their respective Wilson coefficients.

Moreover, the non-perturbative effects in the double Higgs boson production due to a hidden sector are presented

in the same chapter.



Chapter I

The Standard Model of particle physics

This chapter is dedicated to one of the most successful theories conceived so far: the so-called Standard Model

of particles (SM). The unprecedented accuracy between the predictions yielded by the SM and the experiments

have located it as the current paradigm for high energy processes.

The SM takes some of the outstanding results in particle physics of the 60’s and 70’s. The SM is composed

by the Electroweak theory of Glashow[1], Weinberg[2] and Salam[3] (GWS theory), and by the Quantum

Chromodynamics (QCD). The former gives a consistent unification of the weak and electromagnetic interactions

between quark and leptons, making use of the Goldstone’s theorem through the Higgs mechanism[4]-[6]. The

latter describes the strong interaction between gluons and quarks, and got its theoretical structure from the

works of Gell-Mann, Fritzsch, and Leutwyler in ref. [8], and Gross, Wilczek and Politzer in ref. [10][11]. The

renormalizability of the theory was proven in 1972 by ’t Hooft and Veltman[9].

All these ingredients have received great experimental confirmations, from the prediction of neutral currents

and the baryon and meson spectra, up to the last discovery in 2012 at the LHC: the Higgs boson[13][12]. This

theory, that has been tested up to 13 TeV, is the best description for the physics in particle accelerators, where

the gravitational interactions are negligible.

Flavor Mass (GeV) Charge (e units)

Up quark Sector
u (2.2)

+0.6
−0.4 × 10−3

2
3c 1.27± 0.03

t 173.21± 0.51

Down quark Sector
d 4.7+0.5

−0.4 × 10−3

− 1
3s 96+8

−4 × 10−3

b (4.18)
+0.04
−0.03

Table I.1: Quark properties.[15]

1



2 Chapter I. The Standard Model of particle physics

Flavor Mass (GeV) Charge (e units)

e− (0.5109989461± 0.000000031)× 10−3

-1µ− (105.6583745± 0.0000024)× 10−3

τ− 1.77686± 0.00012
νe, νµ, ντ < 2 eV (Tritium decay) 0

Table I.2: Lepton properties.[15]

Particle Mass (GeV) Charge (e units)

γ < 1× 10−21 < 10−35

g 0 (Theoretical) 0
W± 80.385± 0.015 ±1
Z 91.1876± 0.0021 0
h 125.09± 0.24 0

Table I.3: Properties of bosons. [15]

The SM, as a quantum field theory, is determined by its degrees of freedom, that is, the particle content, and

the symmetries, both global and gauge symmetries. The particle content of the theory is given by, first, six

quark types and six type of leptons, with masses and charges as shown in tables [I.1] and [I.2]. Second, spin-1

fields, where it is noted that photons (long-range Coulomb interaction) and gluons (confined due to strong

interaction and so a short-range interaction) are massless, and that the weak bosons are so heavy that leads to

a short-range weak interaction. At last but not least, we have the Higgs boson, a spin-0 field.

It is observed that the quark masses span a range of (10−3−102) GeV. As it is well known, the measure of such

quark masses is tricky, since the confinement phenomenon. Thus, bound states of quarks are measured, such as

pions, instead of free quarks. These bound states are set by the QCD scale at ΛQCD ∼ 200 MeV. Furthermore,

in the fermionic sector, we have the hierarchical span of the charged leptons masses and the particular values

for the neutrinos. The latter species is very special, since the unknown absolute scale for these masses, i.e. we

only can measure differences between masses. In addition, there exists the possibility that just two of the three

neutrinos are massive.

On the other hand, the gauge symmetry that the Lagrangian of the theory manifests is SU (3)C × SU (2)L ×

U (1)Y , thus, in general, the covariant derivative is

Dµ = ∂µ + igsG
a
µT

a + igW a
µ τ

a + ig′Bµ
Y
2 , (I.1)

where the Gaµ correspond to the eight gauge fields of the SU(3)C , i.e. the eight gluons, the W a
µ are the three

gauge fields of the SU(2)L and Bµ is the gauge field of the U(1)Y . Then, we see that the interaction particles

are known once the gauge group is set. Now, to endow with mass some particles of the SM it is implemented

the Higgs Mechanism, thus the spontaneous symmetry breaking (SSB) of the theory must be specified. Such

SSB is realized just in the electroweak sector (EW-SSB), in a way that strong interaction remains untouched.



I.1. Strong Sector 3

Specifically,

SU (2)L × U (1)Y SSB−−−→ U (1)em , (I.2)

i.e., three of the four generators of SU (2)L ×U (1)Y are broken, then the Higgs field should have at least three

real scalars to play the role of the Nambu-Goldstone bosons (NGB), which are ”swallowed” by the three gauge

fields, becoming the longitudinal degree of freedom of the new massive bosons. The complex field H, that is in

the smallest representation of the gauge group and satisfies the previous requirements, is made up of four real

scalars and is a singlet of the color group, a doublet of SU (2)L with Y = +1. Hence, an unbroken generator

and a real scalar field remain, which are the photon and the Higgs, respectively.

I.1 Strong Sector

Let us set

Gaµν = ∂µG
a
ν − ∂νGaµ + gsf

abcGbµG
c
ν , (I.3)

with a = 1, ..., 8, gs the gauge coupling for the QCD and fabc the group structure constant, which are related

to the generators T a of the group through [T a, T b] = ifabcT c. In QCD, the quarks are in the fundamental

representation, and the generators can be chosen to be T a = λa/2, with λa the Gell-Mann matrices, thus in

this sector we have

DµQ = (∂µ − igsGµ)Q. (I.4)

The gauge transformations are given by operators of the form U = e−iT
aαa , so that the SM Lagrangian remains

invariant under the field transformations

Q → e−iT
aαaQ, δQ = −iT aαaQ, (I.5)

Gµ → UGµU
−1 − i

gs
(∂µU)U−1, δGaµ = − 1

gs
∂µα

a + fabcαbGcµ. (I.6)

The gauge field Lagrangian for the gluons is written as

LYMQCD = −1

4
GaµνG

aµν . (I.7)

The Lagrangian including the matter fields will be introduced in a few sections.



4 Chapter I. The Standard Model of particle physics

H

H

H

H

H

H

H

= -i6 Λ

= -i6 Λ v

Figure I.1: Triple and Quartic Higgs interactions.

I.2 Higgs sector

The Higgs doublet used to get the EW-SSB is

H =

 H+

H0

 = 1√
2

 φ1 + iφ2

φ3 + iφ4

 . (I.8)

As it was already said, H transforms as a singlet of SU(3)C , a doublet of SU(2)L and with an hypercharge of

+1/2 under U(1)Y (which is noted as H ∼ (1, 2,+1/2)). Hence, we have the covariant derivative over the Higgs

field

DµH = ∂µH + igWµH + ig′
YH
2
BµH, (I.9)

where g and g′ are the gauge couplings for SU (2)L and U (1)Y , respectively, and Wµ = W a
µ τ

a, with τa = σa

2 .

The Lagrangian for this sector is

LHiggs = (DµH)
†

(DµH)− V (H) , (I.10)

with the potential

V (H) = λ

(
H†H − 1

2
v2

)2

, (I.11)

where the mass dimensions of the parameters in the potential are [λ] = 0 and [v] = 1, with v > 0. Now, let



I.3. Electroweak Bosonic Sector 5

Field `L `R νL uL dL uR dR H+ H0

T 3 − 1
2 0 1

2
1
2 − 1

2 0 0 1
2 − 1

2
Y −1 −2 −1 1

3
1
3

4
3 − 2

3 1 1
Q −1 −1 0 2

3 − 1
3

2
3 − 1

3 1 0

Table I.4: Quantum numbers Q, τ3 and Y for the fermion and higgs fields.

us focus on the vacuum expectation value (VEV) of the Higgs field. In general, it must be 〈H〉 =

 a

b

, but

without loss of generality, it can be set to

〈H〉 =

 0

v√
2

 , (I.12)

this by using transformations of the type φ→ exp(−2iθ2,3τ2,3)φ. From the Goldstone theorem, it is known that,

to get an unbroken generator, this must annihilate the vacuum. With the generators in the form presented so

far, naively, we can say that all the four generators are broken, but with the linear combinations Q = τ3 + 1
212×2

we find an unbroken generator. Here Q is identified as the electric charge.

We want to identify the would-be NGB’s, then the exponential parametrization is the more suitable realization.

In general:

H =
1√
2

(h+ v) e−iχ
aτa

 0

1

 , (I.13)

where the χa are the NGB’s (a = 1, 2, 3) and the radial mode h is the Higgs boson. After spontaneous symmetry

breaking, we have that the potential of the scalar sector is

V (H) = λ

(
H†H − 1

2
v

)2

=
λ

4

(
2vh+ h2

)2
(I.14)

=
1

2
mhh

2 + λvh3 +
λ

4
h4, (I.15)

where the Higgs mass comes to be m2
h = 2λv2. From this, the Feynman rules of the figure [I.1] are derived.

I.3 Electroweak Bosonic Sector

The Yang-Mills Lagrangian for this sector is

LEW = −1

4
W a
µνW

aµν − 1

4
BµνB

µν + |DµH|2. (I.16)



6 Chapter I. The Standard Model of particle physics

For the electroweak bosons, it would be useful to see the interactions with the Higgs boson. For this sake, we

H

Z Μ

ZΝ

H

W
+

Μ

W
-

Ν

= ig
Z

MZ Η ΜΝ

= ig MW Η ΜΝ

Figure I.2: Triple interactions between weak massive bosons and the Higgs

take into account the covariant derivative of the latter in the following analysis. So far, we know that the SSB

is in the form of Eq. [I.2], thus, in the end, we must be able to recover the electromagnetic field. With this in

mind, we change the gauge field basis. With τ± = τ1 ± iτ2, the following commutation relations are obtained

[
Q, τ±

]
=

[
τ3, τ±

]
= ±τ±, (I.17)

which suggests that the gauge bosons corresponding to τ+ and τ are electrically positive and negative charged,

respectively. We can define W±µ = 1√
2

(
W 1
µ ∓ iW 2

µ

)
, so that

g
(
W 1
µτ

1 +W 2
ν τ

2
)

=
g√
2

(
W+
µ τ

+ +W−µ τ
−) . (I.18)

In the same way, we rotate the other gauge fields by doing

Zµ = cWW
3
µ − sWBµ, Aµ = sWW

3
µ + cWBµ, (I.19)

where cW ≡ cos θW = g√
g2+g′2

, sW ≡ sin θW = g′√
g2+g′2

, so that tan θW = g′

g . In the canonical normalization

of the kinetic terms, we have the equivalence

−1

4

∑
a=1,2

(
∂µW

a
ν − ∂νW a

µ

)
(∂µW νa − ∂νWµa) = −1

2

(
∂µW

+
ν − ∂νW+

µ

) (
∂µW ν− − ∂νWµ−) . (I.20)



I.3. Electroweak Bosonic Sector 7

We can also check that

gW 2
µτ

3 +
Y

2
g′Bµ = g (cWZµ + sWAµ) τ3 + g′ (cWAµ − sWZµ)

Y

2
(I.21)

= gsWAµ

(
τ3 +

Y

2

)
+

g

cW
Zµ

(
c2wτ

3 − s2
W

Y

2

)
. (I.22)

There are two remarks from the last equation:

• Aµ is coupled to Q = τ3 + Y
2 (This relation can be tested with the values of the table [I.2]).

• It can be identified the electromagnetic constant by doing e = g sW = g′cW .

With this rotation on the fields, we can see the way the Higgs mechanism endows with mass the weak bosons:

|DµH|2 =

∣∣∣∣∣∣∣
1√
2

 0

∂µh

+
i

2
√

2
(v + h)

 √2gW+
µ

−gZZµ


∣∣∣∣∣∣∣
2

(I.23)

=
1

2
∂µh∂

µh+M2
WW

+
µ W

−µ +
1

2
M2
ZZµZ

µ +
1

8

(
2vh+ h2

) (
2g2W+

µ W
−µ + g2

ZZµZ
µ
)
, (I.24)

where it has been identified

MZ =
1

2

√
g2 + g′2v, MW =

1

2
gv, (I.25)

which allows us to define ρ ≡ MW

MZcW
, in such a way that at tree-level ρ = 1. After the SSB, which led us to Eq.

[I.24], we see that the photon remains massless. The last term in Eq. [I.24] yields the interactions shown in the

Figures [I.2] and [I.3].

H

H

Z Μ

ZΝ

H

H

W
+

Μ

W
-

Ν
= 2 i

MW
2

v 2

ΗΜΝ

= 2 i
MZ

2

v 2

ΗΜΝ

Figure I.3: Quartic interactions between weak massive bosons and the Higgs



8 Chapter I. The Standard Model of particle physics

I.4 Ghosts and gauge fixing

In the previous section we saw that just the physical Higgs field h appeared. In general, it is known that all

the four fields introduced in the Higgs doublet are able to be in our Lagrangian. So far, we have worked in the

unitary gauge. Now, we discuss a generalization of the class of Lorenz gauges: the Rχ-gauge. We can use the

following parametrization for the Higgs field:

H =

 ϕ+

1√
2

(v + h+ iϕZ)

 , H̃ = iσ2H
∗ =

 1√
2

(v + h− iϕZ)

−ϕ−

 , (I.26)

where ϕ± = χ1 ± χ2 and ϕZ = χ3 . The Lagrangian in this gauge is written as

LRξ = − 1

2ξ
F 2
G −

1

2ξ
F 2
A −

1

2ξ
F 2
Z −

1

ξ
F−F+, (I.27)

with

F aG = ∂µGaµ, FA = ∂µAµ, FZ = ∂µZµ − ξMZϕZ , (I.28)

F+ = ∂µW+
µ − iξMWϕ

+, (I.29)

F− = ∂µW−µ + iξMWϕ
−. (I.30)

Then we must specify the Lagrangian for the ghosts. With the Faddeev-Popov technique, we have

LGhost =

4∑
i=1

[
c̄+
∂ (δF+)

∂αi
+ c̄−

∂ (δF+)

∂αi
+ c̄Z

∂ (δFZ)

∂αi
+ c̄A

∂ (δFA)

∂αi

]
ci

+

8∑
a,b=1

ω̄a
∂ (δF aG)

∂βb
ωb, (I.31)

where the ghosts related to the QCD sector are denoted with ωa, transforming as U = e−iT
aβa (a = 1, ..., 8),

and the ghosts associated with the electroweak sector are denoted with c±, cA, cZ , transforming as U = e−iT
aαa

(a = 1, 2, 3) and U = eiY α
4

. To be more explicit, we redefine the parameters of the electroweak sector as

α± =
α1 ∓ α2

√
2

,αZ = α3 cos θW + α4 sin θW , αA = −α3 sin θW + α4 cos θW . (I.32)

Then, we have the set of transformations

δF aG = −∂µβa + gsf
abcβbGcµ, (I.33)

δFA = −∂µαA, (I.34)



I.5. Electroweak Fermion sector 9

δFZ = ∂µ (δZµ)−MZδϕZ , (I.35)

δF+ = ∂µ
(
δW+

µ

)
− iMW δϕ

+, (I.36)

δF− = ∂µ
(
δW−µ

)
− iMW δϕ

−. (I.37)

where, for the electroweak bosons,

δZµ = −∂µαZ + igcW
(
W+
µ α
− −W−µ α+

)
, (I.38)

δW±µ = −∂µα± ± ig
[
α± (ZµcW −AµsW )− (αZcW − αAsW )W±µ

]
, (I.39)

while, for the unphysical scalars,

δϕZ = −1

2
g
(
α−ϕ+ + α+ϕ−

)
+

g

2cW
αZ (v + h) , (I.40)

δϕ± = ∓ig
2

(v + h± iϕZ)α± ∓ igc2W
2cW

ϕ±αZ ± ieϕ+αA. (I.41)

I.5 Electroweak Fermion sector

Now, we focus on the fermions and the SU(2)L×U(1)Y part of the Standard Model. The GWS theory is known

to be a chiral gauge theory, i.e. left- and right-handed components of the fermions can have different quantum

numbers under the gauge group. Specifically, just the left-handed components of the fermions are sensitive to

the weak interaction, while the right-handed do not. Hence, the sensitive components to SU(2)L transform in

the fundamental representation, thus we define the quark and lepton doublets

Q1 =

 uL

dL

 , Q2 =

 cL

sL

 , Q3 =

 tL

bL

 , (I.42)

L1 =

 νeL

eL

 , L2 =

 νµL

µL

 , L3 =

 ντL

τL

 . (I.43)

The right-handed components are then SU(2)L-singlets: U1 = uR, U2 = cR, U3 = tR, D1 = dR, D2 = sR, D3 =

bR, E1 = eR, E2 = µR, E3 = τR. With these definitions, we can specify the couplings of fermions to the gauge

bosons of the electroweak sector. We can use now the hypercharges that we have already set on the table [I.2].

The covariant derivatives explicitly are

DµQi =

(
∂µ + igW a

µ τ
a + ig′Bµ

YQ
2

)
Qi, (I.44)

DµLi =

(
∂µ + igW a

µ τ
a + ig′Bµ

YL
2

)
Li, (I.45)



10 Chapter I. The Standard Model of particle physics

DµUi =

(
∂µ + ig′Bµ

YU
2

)
Ui, (I.46)

DµDi =

(
∂µ + ig′Bµ

YD
2

)
Di, (I.47)

DµEi =

(
∂µ + ig′Bµ

YE
2

)
Ei. (I.48)

With these relations, the Lagrangian for the kinetic terms in the electroweak sector of the fermions is written

as

LkinFermion = iQ̄iD/Qi + iŪiD/Ui + iD̄iD/Di + iL̄iD/Li + iĒiD/Ei, (I.49)

where a sum over the generation index i is implicit. Here, we must note that the hypercharges of the two com-

ponents of any doublet are the same, because of the commutation between SU(2)L and U(1)Y . A fundamental

difference can be noticed between leptons and quarks: the former are singlets of the SU(3)C while the latter

are in the fundamental representation. Thus, the covariant derivatives used for the quarks in this section must

include the term with the gluon fields that appears in Eq. [I.4]. Now, we can write the Lagrangian in Eq. [I.49]

explicitly as

LkinFermion =iQ̄i∂/Qi + iŪi∂/Ui + iD̄i∂/Di + iL̄i∂/Li + iĒi∂/Ei

− g√
2
W+
µ J

µ+ − g√
2
W−µ J

µ− − eAµJµQ −
e

sW cW
ZµJ

µ
Z , (I.50)

with the Noether currents given by

Jµ+ =Ψ̄τ+γµΨ =
∑
i

(
ŪiLγ

µDiL + N̄iLγ
µEiL

)
, (I.51)

Jµ− =Ψ̄τ−γµΨ =
∑
i

(
D̄iLγ

µUiL + ĒiLγ
µNiL

)
, (I.52)

JµQ =Ψ̄QγµΨ =
∑
i

(
2

3
Ūiγ

µUi −
1

3
D̄iγ

µDi − ĒiγµEi
)
, (I.53)

JµZ =Ψ̄
(
τ3 − s2

WQ
)
γµΨ

=
∑
i

[
ŪiL

(
1

2
− 2

3
s2
W

)
γµUiL + D̄iL

(
−1

2
+

1

3
s2
W

)
γµDiL

+ ŪiR

(
−2

3
s2
W

)
γµUiR + D̄iR

(
1

3
s2
W

)
γµDiR + ĒiL

(
−1

2
+ s2

W

)
γµEiL

+ĒiR
(
s2
W

)
γµEiR + +N̄iL

(
1

2

)
γµNiL

]
, (I.54)

where NiL stands for the neutrinos of the SM, i.e. left-handed neutrinos. Thus, the interactions of the figure

[I.5] are generated. The interactions between the fermions and Z-boson can be written in a suitable way. For



I.6. Yukawa Interactions 11

this, we take into account that the left- and right-handed terms can be combined by making

(
τ3 − s2

WQ
)
PL − s2

WQPR = τ3PL − s2
WQ

=
1

2

[(
τ3 − 2s2

WQ
)
− τ3γ5

]
=

1

2

(
giV − giA

)
, (I.55)

with giV = τ3
i −2s2

WQi and giA = τ3
i γ5. It is observed that there are two kinds of interactions involving fermions

in the weak sector: the flavor-changing charged currents, generated by the W± bosons, and the flavor-preserving

neutral currents, mediated by the Z and the A bosons.

I.6 Yukawa Interactions

The usual mass terms mψ̄ψ spoil the SU(2)L × U(1)Y symmetry. In the SM, the fermions get their masses

through EW-SSB. We are able to write

LY uk = −λuijQ̄iH̃Uj − λdijQ̄iHDj − λeijL̄iHEj + h.c. (I.56)

Now, for simplicity let us work in the unitary gauge, thus after SSB

LY uk = − 1√
2

(v + h)
[
λuijŪiLUjR + λdijD̄iLDjR + λeijĒiLEjR + h.c.

]
. (I.57)

Then, there appear the fermion-mass matrices Mu = v√
2
λu, Md = v√

2
λd and Me = v√

2
λe. We can rotate the

fermionic field basis such that these matrices are diagonalized, with the eigenvalues being the real and positive

masses mf . Therefore:

LY uk =

(
1 +

h

v

)[
muūu+mcc̄c+mtt̄t+mdd̄d+mus̄s+mbb̄b

meēe+mµµ̄µ+mτ τ̄ τ ] . (I.58)

With this, it is readily visible the Yukawa interactions between the Higgs and different fermions, being the

coupling proportional to the mass of the given fermion, as is shown in the figure [I.4]. There is another important

remark: the flavor-changing interaction terms, appearing in the Lagrangian of Eq. [I.50], are manifestly modified

by the field-basis change used to diagonalize the mass matrices. To see this, we take the change of the field basis

to be U ′L = LuUL and D′L = LdDL, then ŪLγ
µDL = Ū ′LL

uγµLd†D′L = Ū ′Lγ
µVCKMD

′
L, where VCKM ≡ LuLd†



12 Chapter I. The Standard Model of particle physics

H

Ψ

Ψ

= -i
mΨ

v

Figure I.4: Yukawa interaction between Higgs boson and fermions.

is the Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix. Hence,

LkinFermion ⊂ −
g√
2
W+
µ

(
ū′, c̄′, t̄′

)
L


Vud Vus Vub

Vcd Vcs Vcb

Vtd Vts Vtb

 γµ


d′

s′

b′


L

+ h.c. (I.59)

The CKM matrix can be parametrized through three angles and one complex phase that takes into account CP

violation.

I.7 Entire Standard Model Lagrangian

The full Lagrangian of the SM is then given by adding up the different Lagrangians listed in the previous

sections:

LSM = LYMQCD + LHiggs + LEW + LRξ + LGhost + LkinFermion + LY uk. (I.60)

The fermionic fields contained in this Lagrangian transform under SU(3)C × SU(2)L × U(1)Y as

Qi ∼
(

3, 2,
1

3

)
, Ui ∼

(
3, 1, 4

3

)
, Di ∼

(
3, 1,−2

3

)
, (I.61)

Li ∼ (1, 2,−1) , Ei ∼ (1, 1,−2) , H ∼ (1, 2, 1) . (I.62)

The Standard Model is determined by a total of 18 parameters: there are 3 gauge couplings, 2 constants from

the Higgs potential, 3 lepton masses, 6 quark masses and 4 CKM parameters. Thus, we must choose the best

18 measured quantities that allow us to determine the full set of parameters. So far, in the context of the SM,

all the fermion masses are known, as well as the CKM parameters, obtained from flavor physics experiments.

The parameters of the Higgs sector have been measured through the Higgs mass and the VEV, the latter known

through EW relations.



I.7. Entire Standard Model Lagrangian 13

W ±
ΜΝ

Ψd , u , Ν e

Ψu , d , e

=
- ig

2

Γ Μ
P L

A Μ

Ψ f

Ψ f

= - ieQ
f

Γ Μ

Z Μ

Ψ f

Ψ f

= - i
e

SW CW

I g
f

V
- g

f

A
Γ 5 M Γ Μ

Figure I.5: Electroweak interactions for fermions.

The best-measured quantities in the SM are

Electron magnetic moment : ge−2
2 = (1159.65218091± 0.00000026)× 10−6,

Muon lifetime : τ−1
µ = 2.99588(1)× 10−19 GeV,

Z pole mass : mZ = 91.1876± 0.0021 GeV,

where the first two are measured at rest. The Z pole mass is measured at the electroweak scale,
√
s = mZ . We

also have

Z decay width : ΓZ = 2.4952± 0.0023 GeV,

Z asymmetry : Ae =
σ
(
e+
Le
−
L → Z

)
− σ

(
e+
Re
−
R → Z

)
σ
(
e+
Le
−
L → Z

)
+ σ

(
e+
Re
−
R → Z

) = 0.1515± 0.0019,

W pole mass : mW = 80.385± 0.015 GeV,

which are measured at the electroweak scale. The electric magnetic moment gives the fine-structure constant

through the 1-loop relation ge − 2 = α(0)
π , so that α−1(0) = 137.035999139(31). The muon lifetime enables the

determination of the Fermi constant (GF ) by using the formula Γµ =
G2
Fm

5
µ

192π3 , yielding 1.1663787(6)×10−5 GeV−2.

The last result unveils the value of the VEV since GF√
2

= g2

8m2
W

= 1
2v2 , hence v = 246 GeV. The weak mixing angle

is measured through asymmetries in some processes, for example e+e− → µ+µ− forward-backward asymmetry

and neutrino-electron processes, so that s2
W = 0.23129(5) at

√
s = mZ . Once some SM parameters are known,

we can make some predictions about other parameters. For instance, we can predict through the formula



14 Chapter I. The Standard Model of particle physics

mW = ev
2sW

that at tree-level mW = 79.794 GeV, and similarly, at tree-level Ae = 0.1252. These predictions

are close to the measured values, but can be improved by including radiative corrections. This motivates the

important high precision program of electroweak observables[18][19].



Chapter II

New physics

In this chapter, it is briefly discussed why the Standard Model (SM), presented in the Chapter [I], is incomplete,

and why it is needed a new underlying theory. Also, a discussion of effective extensions of the SM is made.

II.1 Why physics beyond the Standard Model?

Despite the fact that the SM of particles has been broadly tested, showing an unprecedented accuracy on its

predictions, there are several experimental facts that escape to its scope, which have led us to the conclusion

that this theory should be extended in order to take account for those experimental results. In the following a

brief discussion of some of such phenomena is done.

II.1.1 Gravity

So far, it is well known that in our universe there are four fundamental interactions that dictate its dynamics.

Three of them are described within the frame of the SM, they being the electromagnetic, the strong and the

weak interactions. The remaining interaction, gravity, is not contained in the description given by the SM. For

the predictions made in the SM, it is not so relevant the inclusion of gravity due to its weakness at the scales

where experiments are made in particle colliders, which explains the fact that excellent predictions can be done

without taking into account gravity.

Our knowledge of gravitational interactions comes from the theory of general relativity, which is not a quantum

theory. With this in mind, it is expected that new physics should appear at scales where a quantum description

15



16 Chapter II. New physics

of gravity should be needed, and at scales where the strength of this interaction can have effects on our

measurements, i.e. at the Planck scale MP =
√

~c
GN

= 1.22093(7)× 1019GeV/c2.

Below the Planck scale, the effects of gravity are completely negligible. This fact has been shown through an

effective quantum version of gravity, which is valid up to the Planck scale. In such description, the leading

corrections to general relativity come from very weak couplings with a typical strength of ∼ 10−40, at a distance

of one fermi[20].

II.1.2 Cosmological Constant Problem and Dark Matter

Through the different collaborations in cosmological observations, it has been observed that the SM gives a

description of roughly 5% of the energy present in our universe. Actually, it is well known that, the present-day

density parameter of the vacuum is Ωv = 0.68± 0.02, and the present-day density parameter of the pressureless

matter is Ωm = 0.32 ± 0.01, for a universe spatially flat[15] (The updated measurements for the cosmological

parameters given by the Planck collaboration can be found in [21]). The energy stored in the vacuum has been

called as dark energy, and as is shown by Ωv. Our present-day universe is dominated by such vacuum energy.

This constituent of our universe represents one of the current greatest puzzles in physics, since the measurements

of the cosmological constant conclude that it is many orders of magnitude smaller than the expected value from

the SM[22].

In addition to this, about of 26% of the composition of the universe corresponds the so-called dark matter, i.e.

non-luminous and non-absorbing matter, which is also not described by the SM. The existence of dark matter

is well established from the measurement of galactic rotation curves, gravitational lensing and anisotropy of the

cosmic microwave background (CMB)[23]. On the other hand, the actual nature of the dark matter remains to

be unveiled, even though, we already have much information about its features. For instance, from analysis of

structure formation in the universe, it is known that the most of the dark matter should be cold, i.e. should be

non-relativistic.

In order to characterize the nature of the dark matter, it should be taken into account that the candidates to

compose non-baryonic dark matter must be stable on cosmological time scales, as well as very weakly interacting

with electromagnetic radiation, and must reproduce the known relic density. Among the candidates we can find

sterile neutrinos (massive neutrinos that are also SU(2)L×U(1)Y singlets), gravitinos, axions, weakly interacting

massive particles (so-called WIMPs) and primordial black holes.



II.1. Why physics beyond the Standard Model? 17

II.1.3 Neutrino Masses

The observation of neutrino oscillations is understood as the mixing phenomenon between flavor eigenstates

(Ne, Nµ, Nτ ) due to the fact that it can be chosen other basis Nα =
∑
i UαiNi, that describes the mass states

for the neutrinos, with α running in the flavors and i = 1, 2, 3 the mass eigenstates. Thus, in the simplest case

of two flavors, the probabilities for a flavor change in a neutrino’s journey through a distance x is

Pchange (x) = sin2 2θ sin2

(
|∆m2|

4E
x

)
, (II.1)

with |∆m2| = |m2
1−m2

2|, and where θ parametrizes the change of the basis. From this, it is remarkable that we

can only measure the difference between masses, not absolute values. This last feature yields different possible

hierarchies for the neutrino masses. Usually, we refer as normal and inverted hierarchies when ∆m2 > 0 and

∆m2 < 0, respectively.

In analogy to the CKM matrix in the quark sector, the PMNS (Pontecorvo-Maki-Nakagawa-Sakata) matrix is

defined by the elements Uαi, which arise in the unitary transformation for the change of basis of the neutrinos.

Following this, the neutrinos coupled to charged leptons of a given flavor through the W bosons are not mass

eigenstates, but a coherent superposition of mass eigenstates[25]

LCC =
g√
2
W−µ L̄αγ

µNα + h.c. =
g√
2
W−µ L̄αγ

µUαiNi + h.c., (II.2)

where sum in the indices α and i is understood.

So far, there are three experimental sectors to measure the important parameters in neutrino oscillations:

1. Solar neutrinos and the reactor experiment KamLAND, sensitive mainly to ∆m2
21 and θ12. (Solar sector)

2. Atmospheric neutrino studies, K2K and MINOS accelerator experiments are sensitive to ∆m2
23 and θ23.

(Atmospheric sector)

3. The CHOOZ experiment yields bounds to θ13 as a function of ∆m2
31

From such experiments it is known that the measured values for the parameters in neutrino oscillations are

(Normal hierarchy)[15]

∆m2
21 ≡ ∆m2

� = 7.54+0.26
−0.22 × 10−5 eV2, sin2 θ13 = 0.0234+0.0020

−0.0019 (II.3)

∆m2
32 ≡ ∆m2

A = (2.51± 0.10)× 10−3eV2, sin2 θ23 = 0.437+0.033
−0.023 (II.4)

δ/π = 1.39+0.38
−0.27, sin2 θ12 = 0.308± 0.017 (II.5)



18 Chapter II. New physics

These results have established the necessity of an extension for the SM where it could be possible to describe

massive neutrinos. In order to have massive neutrinos, it is needed the inclusion of right-handed neutrinos. In

this way, it is where seesaw models arise[24][26][27]. The seesaw models are minimal extensions of the SM that

add one heavy right-handed neutrino per family so that NL and NR can form a mass term that is similar to

those for charged leptons and quarks. Since the right-handed neutrinos are colorless and singlets under SU(2)L

with hypercharge zero, their kinetic terms are of the form N̄R∂/NR. Therefore, the mass terms of the neutrinos

(with one flavor) can be obtained after spontaneous symmetry breaking from

Lseesaw = YDL̄eH̃NR +
MR

2
NT
RCNR + h.c., (II.6)

without a spoiling of the gauge symmetries of the usual SM. From this Lagrangian, it is not difficult to see that

the smaller the mass of the light neutrino is, the bigger the mass of the heavy neutrino becomes. The latter

feature gives the name for this mechanism.

The main goals in neutrino physics are

1. To determine the individual values of the neutrino masses.

2. To determine whether they are Majorana-like.

3. To discover Leptonic CP-violation.

II.1.4 Additional open problems

In addition to the above mentioned open problems, currently, there are other issues of big interest in particle

physics. The strong CP poblem and the matter-antimatter asymmetry are of remarkable importance.

II.2 Physics BSM in an effective approach

In order to discover physics beyond the Standard Model, one could go in two ways: first, seek new matter

particles, or second, looking for unknown interactions between the already known particles. The Standard Model

extended through effective interactions lies in the latter situation, and can be seen as a way to parametrize the

unknown physics, through the known matter fields.

Following the discussion in the previous sections, we know that there should be some new physics at energies

above the scale of the electroweak spontaneous symmetry breaking. Whichever be the theory that successes

in solving the current problems in high energy physics, we can consider that it is possible to integrate out the



II.2. Physics BSM in an effective approach 19

unknown degrees of freedom, so that the respective effective field theory (EFT) is left only with the Standard

Model particle content. The quantum field theory obtained through this procedure stands up to a scale (Λ), at

which we are able to see the new particles.

In this construction, it is important the fact that the SU(2)L × U(1)Y symmetry is spontaneously broken for

energies below the Higgs mass, while holds for energies above such mass. Thus, the interactions in the effective

field theory must respect the full gauge symmetry of the SM.

As well, another condition that can be taken into account is the fact that if we remove the cutoff by doing

Λ → ∞, the remaining theory should be the Standard Model. The SM does only contain renormalizable

operators. In the scheme of an EFT, the absence of non-renormalizable operators in the SM is understood from

the fact that the new interactions are proportional to factors of 1
Λ , being suppressed as the scale for new physics

is sufficiently above.

The dominant effective interactions are expected to be of order O( 1
Λ ). For one generation of fermions, there is

only one operator, which leads to mass terms for the neutrinos, and corresponds to an effective term coming from

underlying theories. All the operators of dimension five violate lepton and(or) baryon number, and actually,

any odd-dimensional operator does not respect the conservation of lepton and baryon number[28].

When we go on, the dimension-six operators follow, which areO( 1
Λ2 ). At this level, there are 5 different operators

for one generation that violate lepton and baryon number, while there are 59 independent interactions that

preserve them. The interactions that involve L and(or) B violation are highly constrained for light fermions,

mainly from proton decay experiments. The interactions that do preserve such numbers are less restricted. At

this point, it is clear the simplicity of the SM, which just has 14 operators for one generation.

A classic example for an EFT is the V-A model postulated by Fermi, which remains as a valid QFT up to

energies close to the W boson mass. In this model, weak interactions were postulated as touch interactions

between two currents, where the corresponding term in the action is of order six and the coupling constant is

given by GF√
2

= g2

8m2
W

. Here, it is remarkable that, in this model, the action describes the effective interactions,

but the Fermi constant does not give information about the new energy scale, just the ratio between the strength

of the underlying interaction (g) and the new energy scale (mW ), which individuals values just can be known

through the underlying theory.

The above facts for the Fermi model are general facts and serve as a guide to build an effective extension of the

SM. Thus, we start by expanding the possible new interactions in terms of SM degrees of freedom, as

LeffSM = LSM +
∑
i

ci
Λ
O(5)
i +

∑
i

ci
Λ2
O(6)
i + . . . (II.7)



20 Chapter II. New physics

In the above equation we can have effective operators of dimension 5, 6,... and so on, multiplied by adimensional

coefficients, named Wilson coefficients, and divided by powers of the cutoff energy in a consistent way, so that

the mass dimension of the Lagrangian remains to be 4. In this fashion, the ratio between Wilson coefficients

and the new scale energy is restricted by experimental values, as in the case of the Fermi constant. When we

are interested in an analysis preserving lepton and baryon number, some Wilson coefficients are set to zero. For

instance, the only dimension-five operator that is allowed by the SM symmetry is the operator

Qνν = (H̃†Lp)
TC(H̃†Lr), (II.8)

known as the Weinberg operator, but by imposing lepton number conservation its contribution vanishes. Above,

r and p stand for generation indices. It also can be observed that by integrating out the right-handed degrees

of freedom in Lseesaw in Eq. [II.6], the effective operator Qνν is yielded.

Now, let us think about the possibility of having a fundamental interaction between fermions of the Standard

Model mediated by some new vectorial boson X. We are not able to see this boson at energies below its mass,

but we can see some of its effects indirectly as 4-fermion interactions. Thus, in the effective Lagrangian, such

effects of the X boson, will appear through dimension-6 operators with four fermionic fields. We can also see

that this dimension-6 operator comes from a process between four fermions mediated by the X boson, then

from its propagator arises a factor 1
m2
X

at energies much below the boson mass. Hence, it is identified the factor

1
m2
X
∼ 1

Λ2 .

The main advantage of this approach is the model-independence of the procedure. The new physics implies

new interactions between the already known particles. With this, we are able to study these effects without

a full fundamental description in this EFT approach. Currently, there is no evidence for the existence of a

dimension-6 operator for the Standard Model. Instead, some bounds for combinations of the coefficients in Eq.

[II.7] are known.

With the fields of the Standard Model is possible to construct a total of 80 dimension-6 operators, as was

shown by Buchmüller and Wyler in 1986[29]. After that, some works presented equivalences between some

operators[30], i.e. not all of the 80 operators are linearly independent. In the end, taking away all the redun-

dancies, stands a set of 59 independent operators, mentioned before. Another possible dimension-6 operator

can be obtained by means of the equations of motion, integration by parts, Fierz identities or by a suitable

redefinition of the fields. The first work presenting the full reduced list of operators was Ref. [62], where

Grzadkowski et al published an updated list of the effective operators.

There are different ways of choosing a given basis that allows the parametrization of new physics. Currently,

there is no convention for a preferred basis, so that we can find different options, such that, according to



II.2. Physics BSM in an effective approach 21

X3 ϕ6 and ϕ4D2 ψ2ϕ3

QG fABCGAνµ GBρν GCµρ Qϕ
(
H†H

)3
Qeϕ

(
H†H

) (
L̄pErH

)
QG̃ fABCG̃Aνµ GBρν GCµρ Qϕ�

(
H†H

)
�
(
H†H

)
Quϕ

(
H†H

) (
Q̄pUrH̃

)
QW εIJKW Iν

µ W Jρ
ν WKµ

ρ QϕD
(
H†DµH

)∗ (
H†DµH

)
Qdϕ

(
H†H

) (
Q̄pDrH

)
QW̃ εIJKW̃ Iν

µ W Jρ
ν WKµ

ρ

X2ϕ2 ψ2Xϕ ψ2ϕ2D

QϕG H†HGAµνG
Aµν QeW

(
L̄pσ

µνEr
)
τ IHW I

µν Q
(1)
ϕl

(
H†i

↔
DµH

)(
L̄pγ

µLr
)

QϕG̃ H†HG̃AµνG
Aµν QeB

(
L̄pσ

µνEr
)
HBµν Q

(3)
ϕl

(
H†i

↔
D
I

µH

)(
L̄pτ

IγµLr
)

QϕW H†HW I
µνW

Iµν QuG
(
Q̄pσ

µνTAUr
)
H̃GAµν Qϕe

(
H†i

↔
DµH

)(
Ēpγ

µEr
)

QϕW̃ H†HW̃ I
µνW

Iµν QuW
(
Q̄pσ

µνUr
)
τ IH̃W I

µν Q
(1)
ϕq

(
H†i

↔
DµH

)(
Q̄pγ

µQr
)

QϕB H†HBµνB
µν QuB

(
Q̄pσ

µνUr
)
H̃Bµν Q

(3)
ϕq

(
H†i

↔
D
I

µH

)(
Q̄pτ

IγµQr
)

QϕB̃ H†HB̃µνB
µν QdG

(
Q̄pσ

µνTADr
)
HGAµν Qϕu

(
H†i

↔
DµH

)(
Ūpγ

µUr
)

QϕWB H†τ IHW I
µνB

µν QdW
(
Q̄pσ

µνDr
)
τ IHW I

µν Qϕd

(
H†i

↔
DµH

)(
D̄pγ

µDr
)

QϕW̃B H†τ IHW̃ I
µνB

µν QdB
(
Q̄pσ

µνDr
)
HBµν Qϕud

(
H̃†DµH

) (
Ūpγ

µDr
)

(
L̄L
) (
L̄L
) (

R̄R
) (
R̄R
) (

L̄L
) (
R̄R
)

Qll
(
L̄pγµLr

) (
L̄sγ

µLt
)

Qee
(
ĒpγµEr

) (
Ēsγ

µEt
)

Qle
(
L̄pγµLr

) (
Ēsγ

µEt
)

Q
(1)
qq

(
Q̄pγµQr

) (
Q̄sγ

µQt
)

Quu
(
ŪpγµUr

) (
Ūsγ

µUt
)

Qlu
(
L̄pγµLr

) (
Ūsγ

µUt
)

Q
(3)
qq

(
Q̄pγµτ

IQr
) (
Q̄sγ

µτ IQt
)

Qdd
(
D̄pγµDr

) (
D̄sγ

µDt
)

Qld
(
L̄pγµLr

) (
D̄sγ

µDt
)

Q
(1)
lq

(
L̄pγµLr

) (
Q̄sγ

µQt
)

Qeu
(
ĒpγµEr

) (
Ūsγ

µUt
)

Qqe
(
Q̄pγµQr

) (
Ēsγ

µEt
)

Q
(3)
lq

(
L̄pγµτ

I lr
) (
Q̄sγ

µτ IQt
)

Qed
(
ĒpγµEr

) (
D̄sγ

µDt
)

Q
(1)
qu

(
Q̄pγµQr

) (
Ūsγ

µUt
)

Q
(1)
ud

(
ŪpγµUr

) (
D̄sγ

µDt
)

Q
(8)
qu

(
Q̄pγµT

AQr
) (
Ūsγ

µTAUt
)

Q
(8)
ud

(
ŪpγµT

AUr
) (
D̄sγ

µTADt
)

Q
(1)
qd

(
Q̄pγµQr

) (
D̄sγ

µDt
)

Q
(8)
qd

(
Q̄pγµT

AQr
) (
D̄sγ

µTADt
)(

L̄R
) (
R̄L
)

and
(
L̄R
) (
L̄R
)

B-violating

Qledq
(
L̄jpEr

) (
D̄sQ

j
t

)
Qduq εαβγεjk

[(
Dα
p

)T
CUβr

] [(
Qγjs

)T
CLkt

]
Q

(1)
quqd

(
Q̄jpUr

)
εjk
(
Q̄ksDt

)
Qqqu εαβγεjk

[(
Qαjp

)T
CQβkr

] [
(Uγs )T CEt

]
Q

(8)
quqd

(
Q̄jpT

AUr
)
εjk
(
Q̄ksT

ADt
)

Qqqq εαβγεjnεkm
[(
Qαjp

)T
CQβkr

] [
(Qγms )T CLnt

]
Q

(1)
lequ

(
L̄jpEr

)
εjk
(
Q̄ksUt

)
Qduu εαβγ

[(
Dα
p

)T
CUβr

] [
(Uγs )T CEt

]
Q

(3)
lequ

(
L̄jpσµνEr

)
εjk
(
Q̄ksσ

µνUt
)

Table II.1: Dimension-6 independent effective operators. The top table shows 34 operators that take into
account interactions involving bosons, while in the bottom table there are 29 operators describing four-fermion
interactions. Without the 4 B-violating operators, remain 59 independent operators described in Ref. [62]



22 Chapter II. New physics

our necessities, we can choose one of those. The reduced list mentioned above is usually referred as Warsaw

basis[62]. The full set of operators in the Warsaw basis is listed in table [II.2]. There, the generation and color

indices are denoted by p = 1, 2, 3 and α = 1, 2, 3, respectively, while the SU(2)L index appears as j = 1, 2. The

generators of the SU(3)c and SU(2)L groups are TA = 1
2λ

A and SI = 1
2τ

I , respectively, with λA the Gell-Mann

matrices and τ I the Pauli matrices. Also, in this part, the convention is such that Qr (r being the generation

index) stands for the isospin doublets, which are made up of left-handed quarks, and Lr the isospin doublets

of left-handed leptons; the Ur, Dr and Er correspond to right-handed components. The duals of the strength

field tensors are defined as X̃µν = 1
2εµνρσX

ρσ. The hermitian derivatives are defined as:

H†i
↔
DµH ≡ H†i

(
Dµ −

←
Dµ

)
H, H†i

↔
D
I

µH ≡ H†i
(
τ IDµ −

←
Dµτ

I

)
H. (II.9)

In the table [II.1], the purely bosonic operators are all hermitian. In the same table, the operators containing

X̃µν , with X being the dual to a strength field tensor, are the only CP-odd operators involving bosons. For the

operators involving currents of fermions (which are written in brackets), with fields of the same chirality, the

hermitian conjugate corresponds to exchanging the generation indices. For the other operators the hermitian

conjugates were not written.

Among the other possible bases, we find a basis that maximizes the number of operators describing interactions

between electroweak bosons and the Higgs field, then, in this way, the works in Ref. [31] are useful. After the

discovery of the Higgs boson, some modified bases were published, giving an updated lists, as for instance Ref.

[32]. The latter basis presents an interesting feature: it is possible to identify if the new physics is of strong

character or not.

In the following section, we will see the hierarchy paradox in particle physics, but now we anticipate that

supersymmetric theories and compositeness are among the possible scenarios that would provide a solution. The

advantage of the approach presented in Ref. [32] is that, it is possible to identify if the new physics correspond

to a compositeness model (where the Higgs boson turns out to be a composite state in the fundamental theory)

or not.

In the development of the basis in Ref. [32], it was very important the work done in Ref. [33]. In the latter, it

is studied the situation in which the Higgs field arises as a pseudo-Goldstone boson from a strongly-interacting

sector. Also, it was shown an effective approach for the scenario of a light composite Higgs in analogy to the

Chiral effective theory. Through such analysis, the SILH (Strongly-Interacting Light Higgs) Lagrangian can be



II.2. Physics BSM in an effective approach 23

reached, which looks like

∆LSILH =
c̄H
2v2

∂µ
(
H†H

)
∂µ
(
H†H

)
+

c̄T
2v2

(
H†

↔
DµH

)(
H†
↔
DµH

)
− c̄6λ

v2

(
H†H

)3
+
(( c̄u

v2
yuH

†HQ̄HcU +
c̄d
v2
ydH

†HQ̄HD +
c̄l
v2
ylH

†HL̄HE
)

+ h.c.
)

+
ic̄W g

2m2
W

(
H†σi

↔
DµH

)
(DνWµν)

i
+
ic̄Bg

′

2m2
W

(
H†

↔
DµH

)
(∂νBµν)

+
ic̄HW g

m2
W

(DµH)
†
σi (DνH)W i

µν +
ic̄HBg

′

m2
W

(DµH)
†

(DνH)Bµν

+
c̄γg
′2

m2
W

H†HBµνB
µν +

c̄gg
2
S

m2
W

H†HGaµνG
aµν . (II.10)

Here, some operators readily equivalent to those shown in the top of the table [II.2] can be seen. Precisely are

these operators that enter in the basis defined in Ref. [32]. This last basis is suitable for Higgs physics. In such

basis, the effective Lagrangian reads,

L = LSM +
∑
i

c̄iOi ≡ LSM + ∆LSILH + ∆LF1
+ ∆LF2

, (II.11)

where ∆LSILH was already defined, and with

∆LF1
=

ic̄Hq
v2

(
Q̄γµQ

)(
H†
↔
DµH

)
+
ic̄′Hq
v2

(
Q̄γµσiQ

)(
H†σi

↔
DµH

)
+
ic̄Hu
v2

(
ŪγµU

)(
H†
↔
DµH

)
+
ic̄Hd
v2

(
D̄γµD

)(
H†
↔
DµH

)
+

(
ic̄Hud
v2

(
ŪγµD

)(
Hc†↔DµH

)
+ h.c

)
+
ic̄HL
v2

(
L̄γµL

)(
H†
↔
DµH

)
+
ic̄′HL
v2

(
L̄γµσiL

)(
H†σi

↔
DµH

)
+
ic̄Hl
v2

(
ĒγµE

)(
H†
↔
DµH

)
, (II.12)

∆LF2
=
c̄uBg

′

m2
W

yuQ̄H
cσµνUBµν +

c̄uW g

m2
W

yuQ̄σ
iHcσµνUW i

µν +
c̄uGgS
m2
W

yuQ̄H
cσµνλaUGaµν

+
c̄dBg

′

m2
W

ydQ̄Hσ
µνDBµν +

c̄dW g

m2
W

ydQ̄σ
iHσµνDW i

µν +
c̄dGgS
m2
W

ydQ̄Hσ
µνλaDGaµν

+
c̄lBg

′

m2
W

ylL̄Hσ
µνEBµν +

c̄lW g

m2
W

ylL̄σ
iHσµνLW i

µν + h.c. (II.13)

By counting, we note that the leading operators in Higgs physics are given by a set of 12+ 8+ 8 = 28 operators.

When this basis is compared to the Warsaw one, it is noted that in order to have a complete set of operators,

there must be added the 25 four-fermion operators, and 2 X3-like operators for the strength tensors of the

SU(3)c and SU(2)L groups. It must be noted that, in the SILH basis, a CP-even Higgs is assumed, so that

CP-odd terms do not appear here, thus, by relaxing the CP-condition, 6 operators can be added: QG̃, Q
W̃

,



24 Chapter II. New physics

Q
ϕW̃

B and QϕX̃ , with X = G,W,B. Hence, we would have a set of 61 operators, but actually, there are two

redundant operators in the ∆LF1 Lagrangian, so that the minimal set of 59 independent operators appears.

To conclude this section, we have seen the usefulness of this EFT approach in order to get hints of new physics.

It is interesting the fact that a non-renormalizable theory can be so helpful. The important issue is that we

are conscious that this is a theory valid at low energies, so that the contributions of the effective operators are

suppressed by positive powers of the ratio between the energy and the scale Λ (This can be seen by simple

dimensional analysis), then at sufficiently low energies we do not have to worry about the full expansion in the

Lagrangian, and thus the epistemological problem of dealing with a non-renormalizable theory is of no matter.

II.3 Naturalness and Hierarchy problem

Now, we have seen that the SM must be extended, and we have a model-independent approach for studying

new physics. Then, we focus now on the search for the appropriate underlying and fundamental description of

nature that extends the SM. In this section, a guidance in such search is discussed: the naturalness principle.

Let us first recall the main classes of symmetries relevant in quantum field theory:

• Gauge symmetries: in this class we found local symmetries, which relate different non-observable properties

of the fields. This symmetries tells us that there are non-physical degrees of freedom in our descriptions.

• Global symmetries: This class of symmetries encloses accidental symmetries or accidental approximate

symmetries. An example of accidental symmetries is Parity in QED, since actually Parity is violated in

the electroweak model. Even though, global symmetries are normally broken they take an important role

in order to understand the dynamics of the system.

We can analyze the situation where a physical system presents an underlying global symmetry, which is broken

by a small interaction controlled by a given parameter. Such controlled breaking of symmetries can be reflected in

the observables, modifying them. Then, these modifications are completely understood in terms of symmetries.

Now, an unnatural situation happens when the experiments go in contradiction to our understanding in terms

of symmetries. What could we expect in an unnatural situation? the answer is that there should be additional

symmetries that were out of the starting analysis.

We can move on to unnatural situations in QFT, where it can be seen natural and unnatural mass hierarchies

from the point of view of RG-flow. It can be thought that, we have an infrared physical description at some

scale ΛIR, and that there is new physics above some ultraviolet scale ΛUV . With this, by definition, we are

saying that there is a separation of scales and that nothing interesting is happening in between, i.e. there is not



II.3. Naturalness and Hierarchy problem 25

any dramatical change in the dynamics. Thus, we are able to state that, in a situation where a separation of

scale has been stated, the theory is almost scale-invariant. Now, we can ask, under which conditions is such a

separation of scales natural?

To see this, first, let us recall the meaning of the β-function of a given coupling. It can be defined

t ≡ log
ΛUV
µ

,
d

dt
λ ≡ µ d

dµ
λ = β(λ),

thus, it is understood t as an RG-time, and the β-function of the coupling λ turns out to be a kind of velocity.

Then, we can see that the β-function gives information of how fast some scale is reached along the RG evolution.

In this sense, the naturalness of the hierarchy is just the question of the slowness of the motion in the parameter

space as we move along the RG evolution. In the limit of β → 0, the couplings do not change, or equivalently,

the theory does not ”move”, what corresponds to a scale-invariant theory (SFT), this happens in the so-called

fixed points. The smallness of the β-function in an RG-flow sense means the stability of the fixed point.

Hence, the notion of naturalness is related to the features of the β-function around a fixed point, or equivalently

with the deformations of the fixed point. In this way, an unnatural case arises when the fixed point admits

relevant deformations, coming from relevant operators (d < 4), such as a scalar mass term. In the latter case,

simple dimensional analysis tells us that the corrections for the mass should be proportional to some energy

scale, m2 ∼ cΛ2, and the coefficient c should be understood in terms of symmetries. In the situation where the

difference between the two scales is huge, the coefficient must be extremely small, here the problem arises when

there is no explanation in terms of symmetries about such smallness.

The standard definition for the principle of naturalness, broadly used in particle physics, was given by Gerardus

’t Hooft as

” any energy scale µ, a physical parameter or a set of physical parameters αi(µ) is allowed to be very small

only if the replacement αi(µ) = 0 would increase the symmetry of the system”.[14]

There are some well-known situations, where the truthfulness of this principle can be seen. In QED, for instance,

an enhancement in the symmetries of the theory arises when the electron is taken to be massless. The chiral

symmetry ψ → eiωγ5ψ, in QED, is broken just by the mass term of the electron. Hence, following the ’t Hooft

naturalness principle, we have that the electron mass is allowed to be small. Moreover, any quantum correction

to the electron mass must respect the chiral symmetry, then, it is expected that these quantum corrections

vanish in the limit me → 0. Precisely, this is what we get by doing the complete computation:

me = m(0)
e

(
1− 6α

4π
log

(
me

µ

))
. (II.14)



26 Chapter II. New physics

Another interesting situation happens in QCD. We can think about the contributions to the proton mass,

mp = 0.9382 GeV, coming from the Planck scale, so that m2
p ∼ εM2

P . Then, it is addressed the question of

the smallness of the parameter ε because of the huge difference between the scales. In this situation, we recall

that, the masses of the quarks from which the proton is made up are very small compared to mp, so that the

mass terms of the QCD Lagrangian are negligible. Is in this situation where we have classically conformal

symmetry. This symmetry is broken by quantum effects, and, is in this way, that the proton acquires the most

of its mass. Therefore, the smallness of ε is understood in terms of the quantum breaking of the conformal

symmetry. Actually, from the renormalization group of QCD, we can write for the strong coupling, αs,

dαs
dt

= −2b0α
2
s, (II.15)

with t = log(MP /µ), where b0 is a constant that could depend on the number of colors, among other factors,

particularly b0 = 7 in QCD. Therefore, by solving this equation, and at the scale µ = mp

mp = e
− 2π
b0αs(MP )MP , (II.16)

then, with gs(Mp) = 0.5, the exponential factor is extremely small. Particularly, it is from the renormalization

group analysis that the conformal symmetry is broken.

Now, we want to see the hierarchy problems in the SM. For this, we start by analyzing at which scale new

physics should come.

It was defined, at the end of Section 1.3, the quantity ρ, which gives information of the similarity between the

masses of the electroweak bosons. We can rewrite at tree-level the relation between these masses as M2
W =

(1 − s2
W )M2

Z , thus we see that the difference between them is related to sW , or equivalently to g′, so that

for sW = 0 then g′ = 0. This situation correspond to that when we are not taking into account the U(1)Y

interaction, hence all of the three W bosons are equivalent and the term W a
µW

aµ is SO(3) invariant. We can

go even deeper, to note that the four degrees of freedom of the scalar Higgs sector can be rearranged as

H =

 H+

H0

 =
1√
2

 φ1 + iφ2

φ3 + iφ4

⇒ Φ =
1√
2

 φ3 − iφ4 φ1 + iφ2

−φ1 + iφ2 φ3 + iφ4

 , (II.17)

where there is an underlying invariant φ2
1 + φ2

2 + φ2
3 + φ2

4, which leads to a SO(4) ' SU(2)GaugeL × SU(2)GlobalR

symmetry. The bi-doublet can be taken to transform as Φ(x) → UL(x)ΦU†R. After symmetry breaking, it

is seen SU(2)L × SU(2)R → SU(2)diag. This remnant symmetry is the so-called custodial symmetry, that is

equivalent to the SO(3) symmetry mentioned before. When we do g′ 6= 0, the global SU(2)R symmetry is

broken, remaining U(1)Y . Therefore, the custodial symmetry protects the relation ρ ∼ 1. A more detailed



II.3. Naturalness and Hierarchy problem 27

analysis can be found in Ref. [34].

Such protection in the ρ parameter puts strong bounds on new physics. This symmetry can be broken by the

irrelevant term

1

Λ2

(
H†DµH

) (
H†DµH

)
, (II.18)

from which δρ =
m2
W

Λ2 . In order to control such violations in the custodial symmetry, we must set a sufficient

big scale Λ.

On the other hand, as was discussed before, the term that leads to neutrino masses, by using the degrees

of freedom of the Standard Model, is the dimension-5 Weinberg operator. Hence, the separation of scales

generically implies that the charged leptons (with mass terms coming from marginal terms) are much heavier

than the neutrinos. It also occurs that the accidental lepton number global symmetry could be violated. Such

new term can be written as

Yν
Λ

(LT iσ2H)C(HT iσ2L), (II.19)

which after electroweak symmetry breaking generates the mass-terms for the neutrinos.

Now, from the bounds set by the measurements of ρ and neutrino masses, it is concluded from our breaking

of the approximate symmetries that, at least, ΛUV � TeV. Actually, there are more bounds measured, from

the K − K̄ mixing and the proton decay, that constraint other irrelevant operators, such as the four fermions

contact interactions.

There exist a marginal operator that is not already contained in the Standard Model Lagrangian:

θG̃µνG
µν , (II.20)

where Gµν is the QCD strength tensor and, as usual, Gµν = 1
2εµναβG

αβ is its dual. This term is known to be a

total derivative, thus it does not appear in the equations of motion. Despite this fact, it induces an electric dipole

moment (dn) in the neutron, a quantity that has been searched in the experiments. Current measurements yield

the maximal value dn < 3× 10−26e · cm, but by using current algebra we can find the dependence

dn = 5.2× 10−16 · θ cm, (II.21)

thus, it turns out that θ < 10−10. The term in Eq. [II.20] also violates CP, although, other sources of CP

violations are known even in the Standard Model, coming from the Yukawa sector.

So far, in our analysis, unnaturalness has not come. The paradoxical situations arise when we start to analyze



28 Chapter II. New physics

the relevant operators. We consider the Higgs mass, written as

cΛ2
UVH

†H, (II.22)

which is a dimension 2 operator. Here, through selection rules, we can set the coefficient of the Higgs mass from

dilatations (the selection rule dictated by dimensional analysis), so that a given fundamental scale Λ2
UV should

appear multiplied by some coefficient constrained by symmetries. In this sense, there is a shift symmetry of the

Higgs Goldstone boson that is broken by the Yukawa interaction. Then, the coefficient should be controlled by

the same parameter that breaks the shift symmetry. Thus c ∼ λ2
t

16π2 , since the biggest Yukawa coupling is that

corresponding to the heaviest fermion, i.e. the top quark. Actually, the corrections to the Higgs mass are given

by

δm2
h =

3λ2
t

4π2
Λ2
t −

9g2

32π2
Λ2
g −

3g′2

32π2
Λ2
g′ −

3λ

8π2
Λ2
h + . . . . (II.23)

It is already known that the Higgs mass is of order ∼ 102GeV, and from the irrelevant operators, we have seen

that ΛUV ∼ 1015GeV, so c is suppressed by more than 20 orders of magnitude. At this stage, we start to see a

huge difference between selection rules and the results of the experiments. This is the naturalness problem and

the hierarchy paradox in the Standard Model: we do not have a whole understanding of the parameter structure

in terms of symmetries. The situation could be even worst: there could be a relevant term of null dimension:

Λ4
UV

√
g, (II.24)

which is related to the Cosmological constant term. Here the problem could exist even if ΛUV were the electron

mass.

Under this light, the θ parameter in Eq. [II.20] can be seen as an unnatural quantity, since its smallness cannot

be explained in terms of some symmetry. It could be argued that the CP violation could lead to such smallness,

but we know that CP is already violated in other sectors.

With all this, we see that some paradoxical situations happen in the Standard Model, and, if we want to solve

them, we have to propose new models that give explanations for the smallness of some parameters in terms of

new symmetries. Is in this fashion that it is preferred the kind of models that solve the Hierarchy problem.

There are several models that do not have this SM problem, among those the supersymmetric and compositeness

scenarios are found.

For the supersymmetric scenario, a new fundamental symmetry is introduced: the physics at fundamental scales

is invariant under the exchange between fermion and bosons. Under supersymmetry (SUSY), the Higgs mass



II.3. Naturalness and Hierarchy problem 29

parameters acquire a new symmetry, so that their smallness can be understood in terms of SUSY, thus the

Higgs mass is protected. Hence, it is said that, such smallness is explained by the fact that SUSY is broken by

small effects that can be even non-perturbative. With this the t’ Hooft criterion is satisfied.

On the other scenario, in compositeness, the Higgs is viewed as a composite field in the fundamental theory. The

basic idea in this approach is that, above the weak scale, the Higgs field does not have dimension 1 anymore,

instead, such dimension is bigger than 1, in such a way that the mass operator of the Higgs has dimension

bigger than 2. This diminishes its relevance, then the RG-evolution of the parameters is slower, explaining the

separation between scales.

Another option to solve the Hierarchy problem is the introduction of large extra-dimensions, which solve the

problem by changing the fundamental scale, bringing it closer to the weak scale.



Chapter III

Two-Higgs-Doublet Model (2HDM)

In the chapter [II], it was shown why there should be physics beyond the Standard Model. Also, it was presented

a model-independent approach to study the effects of such new physics. Now, in this chapter, we want to see

a simple extension of the SM, where an additional copy of the SU(2)L Higgs doublet is introduced. This is

the two Higgs doublet model (2HDM), where the Higgs boson discovered in 2012 is assumed to be one of the

physical fields contained in the two scalar SU(2) doublets, which appear in the scalar potential of this new

model.

When it is intended to extend the SM, an important parameter that allows measuring the phenomenological

validity of such extensions is the ρ parameter discussed in the previous chapter. The new models cannot spoil

the custodial symmetry. This imposes that, the deviations (∆ρ) from the tree-level value of ρ in the SM must

be |∆ρ| . 10−3. It can be shown that for SU(2) singlets (T = 0) and doublets (T = 1/2), with hypercharges

Y = 0 and Y = ±1, respectively, it must be satisfied ρ = 1 at tree level. The simplest extension of the SM that

fits in the latter option is the 2HDM.

This chapter is dedicated to the review of the basic features of this SM extension. There is a vast literature

concerning the theory and phenomenology of the 2HDM, thus, here, we are focused on the construction of the

model, with the current constraints on the parameter space after the detection of the Higgs boson, which usually

come from unitarity and stability of the potential.

30



III.1. Construction of the Model 31

III.1 Construction of the Model

The model starts by introducing a new SU(2)L doublet of scalars. In this fashion, our new scalar sector will be

made up of the two doublets

Φ1 =

 Φ+
1

Φ0
1

 (Y = +1) , Φ2 =

 Φ+
2

Φ0
2

 (Y = +1) . (III.1)

With this, the kinetic terms of the scalars, the Yukawa, and the potential terms are modified. We can write

Lscalar = LKin + LY uk − V (Φ1,Φ2) , (III.2)

where, naturally,

LKin = (DµΦi)
†

(DµΦi) , (i = 1, 2) and with Dµ = ∂µ + igW a
µ τ

a + ig′
Y

2
Bµ. (III.3)

The Yukawa interactions in L can be chosen in different ways that will be studied in the next section. Now,

we can write the most general Higgs potential so that the conditions of CP-conservation, gauge invariance and

renormalizability are satisfied, and even more, we can ask for a discrete Z2 symmetry to be present in a way

that the potential is symmetric under transformations

Φi → (−1)
i
Φi, i = 1, 2. (III.4)

This Z2 symmetry forbids the appearance of big contributions to Flavor-Changing-Neutral-Current (FCNC)

effects

V (Φ1,Φ2) = λ1

(
Φ†1Φ1 −

v2
1

2

)2

+ λ2

(
Φ†2Φ2 −

v2
2

2

)2

+ λ3

[(
Φ†1Φ1 −

v2
1

2

)
+

(
Φ†2Φ2 −

v2
2

2

)]2

+ λ4

[(
Φ†1Φ1

)(
Φ†2Φ2

)
−
(

Φ†1Φ2

)(
Φ†2Φ1

)]
+ λ5

[
Re
(

Φ†1Φ2

)
− v1v2

2

]2
+ λ6

[
Im
(

Φ†1Φ2

)]2
,

(III.5)

with the λi being dimensionless parameters and where the vacuum expectation values of the neutral part of the

scalar fields correspond to 〈Φ0
i 〉 = vi/

√
2. In addition, in order to get the usual electroweak symmetry breaking



32 Chapter III. Two-Higgs-Doublet Model (2HDM)

SU(2)L × U(1)Y → U(1)em we choose the minimum of the potential to be1

〈Φ1〉 =
1√
2

 0

v1

 , 〈Φ2〉 =
1√
2

 0

v2

 . (III.6)

In general, the physical scalars may be a linear combination of the different degrees of freedom contained among

the complex components of the Φi’s. By writing

Φ1 =

 φ+
1

v1+η1+iχ1√
2

 , Φ2 =

 φ+
2

v2+η2+iχ2√
2

 , (III.7)

we can extract the mass terms from the potential in terms of the field components. We start by recognizing

Φ†iΦi = φ−i φ
+
i +

1

2
(vi + ηi − iχi) (vi + ηi + iχi)

= φ−i φ
+
i +

1

2
(vi + ηi)

2
+

1

2
χ2
i , (III.8)

so that it is readily visible that

(
Φ†iΦi −

v2
i

2

)2

= v2
i η

2
i +

(
φ−i φ

+
i

)2
+

1

4
η4
i −

1

4
χ4
i + φ−i φ

+
i η

2
i + 2viφ

−
i φ

+
i ηi

− φ−i φ
+
i χ

2
i + viη

3
i − viηiχ2

i −
1

2
χ2
i η

2
i , (III.9)

where just the first term will contribute to the mass expressions for each i = 1, 2. By taking advantage of the

relation above, we also can extract mass terms from the expression

(
Φ†1Φ1 + Φ†2Φ2 −

v2
1 + v2

2

2

)2

⊃ v2
1η

2
1 + v2

2η
2
2 + 2v1v2η1η2. (III.10)

In the potential of the 2HDM appears the expression Φ†1Φ2, as well as its conjugate, and its real and imaginary

part, thus, it is useful to fully expand it:

Φ†1Φ2 = φ−1 φ
+
2 +

1

2
η1η2 +

1

2
v2η1 +

1

2
v1η2 +

v1v2

2
+

1

2
χ1χ2

+
i

2
(η1χ2 + v1χ2 − χ1η2 − v2χ1) . (III.11)

1Here, we are using a specific parametrization of the scalar potential. In [39] it is discussed vacuum stability in 2HDMs with a
different parametrization. It is possible to obtain the potential in eq. [III.5] from that in [39], but not vice-versa. In addition, there
can be seen that the minimization of the scalar potential could lead to different local minima, and that the form of the vev’s in eq.
[III.6] is subtle. In some cases, there could be a non-zero upper component in the vev’s, which corresponds to the so-called charged
vacuum, leading to a U(1)em symmetry spontaneously broken. In this work, we decide to take the most common vev’s in 2HDMs.



III.1. Construction of the Model 33

Hence, we extract the mass terms,

(
Φ†2Φ1

)(
Φ†1Φ2

)
⊃ 1

2
v1v2φ

−
1 φ

+
2 +

1

2
v1v2φ

+
1 φ
−
2 +

1

4
v2

1χ
2
2 +

1

4
v2

2χ
2
1

+
1

4
v2

2η
2
1 +

1

4
v2

1η
2
2 + v1v2η1η2, (III.12)

but this contribution is subtracted from the expression

(
Φ†1Φ1

)(
Φ†2Φ2

)
⊃ 1

2
v2

2φ
−
1 φ

+
1 +

1

2
v2

1φ
−
2 φ

+
2 +

1

4
v2

1χ
2
2 +

1

4
v2

2χ
2
1

+
1

4
v2

2η
2
1 + v1v2η1η2 +

1

4
v2

1η
2
2 , (III.13)

where from Eq.[III.12-III.13] we can see that almost all the mass terms cancel out, except those related to the

charged scalars. The mass terms coming from the real and imaginary part of Φ†1Φ2 are easy to get from Eq.

[III.11]. Thus, for the neutral scalars:

Lmassneutral = λ1v
2
1η

2
1 + λ2v

2
2η

2
2 + λ3

(
v2

1η
2
1 + v2

2η
2
2 + 2v1v2η1η2

)
+λ5

(
1

4
v2

2η
2
1 +

1

4
v2

1η
2
2 +

1

2
v1v2η1η2

)

=
1

2
(η1, η2)M2

neutral

 η1

η2

 , (III.14)

where the mass matrix is given by

M2
neutral =

 m11 m12

m12 m22

 , (III.15)

with

m11 = 2 (λ1 + λ3) v2
1 +

1

2
λ5v

2
2 , m12 =

(
2λ3 +

1

2
λ5

)
v1v2, m22 = 2 (λ2 + λ3) v2

2 +
1

2
λ5v

2
1 . (III.16)

Analogously, we have for the pseudo-scalars

Lmassp−s =
1

4
λ6

(
v2

1χ
2
2 + v2

2χ
2
1 − 2v1v2χ1χ2

)
=

1

2
(χ1, χ2)M2

p−s

 χ1

χ2

 , (III.17)



34 Chapter III. Two-Higgs-Doublet Model (2HDM)

with the mass matrix in this case given by

M2
p−s =

λ6

2

 v2
2 −v1v2

−v1v2 v2
1

 . (III.18)

Finally, for the charged scalars it is yielded

Lmasscharged = λ4

(
1

2
v2

2φ
−
1 φ

+
1 +

1

2
v2

1φ
−
2 φ

+
2 −

1

2
v1v2φ

−
1 φ

+
2 −

1

2
v1v2φ

+
1 φ
−
2

)

=
(
φ+

1 , φ
+
2

)
M2
charged

 φ−1

φ−2

 , (III.19)

where

M2
charged =

λ4

2

 v2
2 −v1v2

−v1v2 v2
1

 . (III.20)

With these mass expressions, we are able to write the physical scalars states by diagonalizing the mass matrices.

For this task, we note the similitude betweenM2
charged andM2

p−s, then both of these matrices will be diagonalized

by the same rotation in the field basis. Then, we change our field basis by applying the rotations,

 H0

h0

 = R1

 η1

η2

 ,

 G0

A0

 = R2

 χ1

χ2

 ,

 G±

H±

 = R2

 φ±1

φ±2

 , (III.21)

with

R1 =

 cosα sinα

− sinα cosα

 , R2 =

 cosβ sinβ

− sinβ cosβ

 . (III.22)

Then, the diagonalization of the mass matrix in Eq. [III.15] for the scalar bosons is

R1

(
M2
neutral

)
RT1 =

 m11 cos2 α+m12 sin 2α+m22 sin2 α m12 cos 2α+ 1
2 (−m11 +m22) sin 2α

m12 cos 2α+ 1
2 (−m11 +m22) sin 2α m11 sin2 α−m12 sin 2α+m22 cos2 α


=

 m2
h 0

0 m2
H

 , (III.23)

where in order to get a diagonal matrix we do:

m12 cos 2α+
1

2
(−m11 +m12) sin 2α = 0 ⇒ tan 2α =

2m12

m11 −m22
, (III.24)



III.1. Construction of the Model 35

but, by recalling that the components mij are given in the relations [III.16], it turns out that the rotation angle

α is given by

tan 2α =
2
(
λ3 + 1

4λ5

)
v1v2

λ1v2
1 − λ2v2

2 +
(
λ3 + 1

4λ5

)
(v2

1 − v2
2)
. (III.25)

The masses of the CP-even scalars can be obtained by solving the characteristic equation of the matrix, so that

m2
h =

1

2

[
(m11 +m22)−

√
(m11 −m22)

2
+m2

12

]
,

m2
H =

1

2

[
(m11 +m22) +

√
(m11 −m22)

2
+m2

12

]
. (III.26)

Similarly, it can be done for the other mass matrices. The matrix where the mass of the pseudo-scalar is

contained (Eq. [III.18]) can be diagonalized by:

R2

(
M2
p−s
)
RT2 =

λ6

2

 (v2 cosβ − v1 sinβ)
2

v1v2

(
sin2 β − cos2 β

)
−
(
−v2

1 + v2
2

)
sinβ cosβ

v1v2

(
sin2 β − cos2 β

)
−
(
−v2

1 + v2
2

)
sinβ cosβ (v2 cosβ + v1 sinβ)

2

 ,

(III.27)

so that in order to get a diagonal matrix, the β-angle must be

v1v2

(
sin2 β − cos2 β

)
−
(
−v2

1 + v2
2

)
sinβ cosβ = 0 ⇒ tanβ =

v2

v1
. (III.28)

This is equivalent to v2 cosβ = v1 sinβ, then, the diagonal entries are

(v2 cosβ − v1 sinβ)
2

= 0, and (v2 cosβ + v1 sinβ)
2

= v2
1 + v2

2 ≡ v2, (III.29)

where it was made the identification v =
√
v2

1 + v2
2 = 246 GeV. We note that the fact of having a diagonal zero

value in the matrix corresponds to a Goldstone boson, G0, while the mass of the pseudo-scalar is given by

m2
A =

1

2
λ6v

2. (III.30)

The same analysis stands for the matrix of the charged bosons, as was mentioned before. Then, we also have a

couple of charged Goldstone bosons G±, while the physical charged Higgses have mass

m2
H+ =

1

2
λ4v

2. (III.31)

Let us summarize. Through diagonalization it was yielded the scalar spectrum made up of eight real fields h0,



36 Chapter III. Two-Higgs-Doublet Model (2HDM)

H0, H±, A0, G0 and G±, from which three of them, the massless one, would be eaten by the electroweak vector

fields in order to get their longitudinal degrees of freedom, so that in the end just the first five scalars compose

the physical particle spectrum from the scalar sector.

By looking at the potential in Eq. [III.5], we note that there are eight parameters in it, where six of them are

couplings and two are the vev’s. The vev’s are related through their relation to the vev in the SM, thus we

are introducing seven new parameters in the 2HDM. Usually, those seven parameters are chosen to be the four

masses of the physical scalars, the parameter tanβ, the mixing angle α and the coupling λ5:

(mh,mH ,mA,mH+ , tanβ, α, λ5) . (III.32)

Then, the couplings must be given in terms of the input parameters. This is not a difficult task. By inverting

the relations [III.26], [III.30, [III.31] and with the definitions of the mixing angles, we obtain:

λ1 =
1

2v2 cos2 β

[
m2
H cos2 α+m2

h sin2 α− sinα cosα

tanβ

(
m2
H −m2

h

)]
− λ5

4

(
tan2 β − 1

)
,

λ2 =
1

2v2 sin2 β

[
m2
h cos2 α+m2

H sin2 α− sinα cosα tanβ
(
m2
H −m2

h

)]
− λ5

4

(
cot2 β − 1

)
,

λ3 =
1

2v2

sinα cosα

sinβ cosβ

(
m2
H −m2

h

)
− λ5

4
,

λ4 =
2

v2
m2
H+ ,

λ5 =
2

v2
m2
A. (III.33)

We note that from the eight parameters introduced in the potential of the 2HDM, two of them are already known.

One is given by the vev already known in the SM, v = 246 GeV, and the other is given by mh = 125 GeV. This

assuming that the detected scalar in the LHC is the lightest CP-even scalar of the 2HDM. The other parameters

must be bounded by theoretical and experimental constraints.

III.2 Types of 2HDM

So far, we have established the physical spectrum of the 2HDM. In this section, we want to see the way the new

physical states interact with the SM particles. This issue is basically what defines the different possible types

of 2HDM. Then, we discuss here the possible forms of LY uk in Eq. [III.2]. In general, we can have couplings of

the form

−yaijΨ̄i
LΦaΨj

R + h.c. (III.34)



III.2. Types of 2HDM 37

Model Type I Type II Lepton-specific Flipped

U Φ2 Φ2 Φ2 Φ2

D Φ2 Φ1 Φ2 Φ1

E Φ2 Φ1 Φ1 Φ2

Table III.1: Higgs doublets that couples to an up-, down- and charged-lepton-like fermions in different Two-Higgs
Doublet Models.

Here, we can have two different Yukawa couplings for each fermionic field. This does not allow in the 2HDM to

ensure that the mass matrices can be diagonalized together with the Yukawa coupling matrices, as in the SM.

This leads to a Higgs neutral boson to mediate flavor-changing interactions among fermion mass eigenstates.

Such situation is very constrained from the Kaon, D and B meson phenomenology. To avoid this situation

it is taken into account the Glashow-Weinberg criterion2, which states the necessary conditions to have flavor

conservation in the neutral current interactions[35][36]. By following this criterion it is enough that each charged

fermionic field couples to just one of the Higgs doublet.

The Yukawa interactions in different Two-Higgs Doublet models that satisfy the Glashow-Weinberg criterion are

shown in the table [III.1]. In type-I models just one Higgs doublet couples to the up, down and charged-lepton

fermions, while the other does not couple to any of them. Thus, the Yukawa Lagrangian of type-I 2HDM is

LY ukType−I = −ydijQ̄iΦ2Dj − yuijQ̄iΦ̃2Uj − yeijL̄iΦ2Ej + h.c. (III.35)

The type-II 2HDM Yukawa interactions are chosen such that one of the Higgs doublets couples to the down-like

fermions and to the charged-leptons, while the other doublet couples to the up-like fermions. To get vanishing

the dangerous FCNC, the right-handed components should transform under the discrete symmetry as Di → −Di

and Ui → Ui. The Lagrangian for this type of models is

LY ukType−II = −ydijQ̄iΦ1Dj − yuijQ̄iΦ̃2Uj − yeijL̄iΦ1Dj + h.c. (III.36)

In the next chapter, we shall see that the Higgs sector of the MSSM corresponds to a type-II 2HDM.

The Yukawa interactions can be written explicitly in terms of the mass eigenstates. Let us see the way this

happens for the type-I 2HDM. In this model, all the mass matrices are defined as mij = v2√
2
yij , so that they can

be diagonalized by rotations in the fermionic basis, as it was done for the SM (see Section [I.6]). The relations

2In order to avoid FCNC at tree-level it is necessary and sufficient that all fermions of the same charge and helicity transform
under the same irreducible representation of SU(2), have the same T3 eigenvalue and that a basis exists in which they receive their
contributions in the mass matrix from a single source[37].



38 Chapter III. Two-Higgs-Doublet Model (2HDM)

Model Type I Type II Lepton-specific Flipped

ξuh cosα/ sinβ cosα/ sinβ cosα/ sinβ cosα/ sinβ
ξdh cosα/ sinβ − sinα/ cosβ cosα/ sinβ − sinα/ cosβ
ξ`h cosα/ sinβ − sinα/ cosβ − sinα/ cosβ cosα/ sinβ
ξuH sinα/ sinβ sinα/ sinβ sinα/ sinβ sinα/ sinβ
ξdH sinα/ sinβ cosα/ cosβ sinα/ sinβ cosα/ cosβ
ξ`H sinα/ sinβ cosα/ cosβ cosα/ cosβ sinα/ sinβ
ξuA cotβ cotβ cotβ cotβ
ξdA − cotβ tanβ − cotβ tanβ
ξ`A − cotβ tanβ tanβ − cotβ

Table III.2: Higgs doublets that couples to an up-, down- and charged-lepton-like fermions in different Two-Higgs
Doublet Models.

v2 = v sinβ and ev = 2MW sin θW allow us to write

LY ukType−I = − 1√
2
ydij (v2 + η2 + iχ2) d̄LidRj − ydijφ+

2 ūLidRj −
1√
2
yuij (v2 + η2 + iχ2) ūLiuRj + yuijφ

−
2 d̄LiuRj

− 1√
2
yeij (v2 + η2 + iχ2) ēLieRj − yeijφ+

2 ν̄LieRj + h.c.

⊃ −m(i)
d d̄idi −

e

2MW sin θW sinβ
m

(i)
d η2d̄idi −

ie

2MW sin θW sinβ
m

(i)
d χ2

(
d̄iγ5di

)
−
{

e√
2MW sin θW sinβ

φ+
2 V

ij
CKM ūi

(
m

(j)
d PR −m(i)

u PL

)
dj + h.c.

}
= −m(i)

d d̄idi −
e

2MW sin θW sinβ
m

(i)
d d̄idi

[
H0 sinα+ h0 cosα

]
− ie cotβ

2MW sin θW
m

(i)
d d̄iγ5diA0

−
{

e cotβ√
2MW sin θW

V ijCKM ūi

(
m

(j)
d PR −m(i)

u PL

)
djH

+ + h.c.

}
, (III.37)

where the Cabibbo-Kobayashi-Maskawa matrix arises after diagonalization of the mass matrices, and where the

expressions for the up-like quarks and leptons are very similar. It is also understood that the computations are

made in the unitary gauge, where the ghosts and the would-be Nambu-Goldstone bosons do not appear. In

fact, the Yukawa interactions can be written in a very general way, following the notation in [43]

LY uk =−
∑

f=u,d,`

e

2MW sin θW
mf

(
ξfh f̄fh

0 + ξfH f̄fH
0 − iξfAf̄γ5fA

0
)

+

{
e√

2MW sin θW
VCKM ū

(
muξ

u
APL +mdξ

d
APR

)
d H+ +

e√
2MW sin θW

m`ξ
`
Aν̄L`RH

+ + h.c.

}
,

(III.38)

with the generation indices omitted. The parameters ξfh , ξfH and ξfA are enlisted for various 2HDM in the table

[III.2]. The way to reach such couplings is the same as the shown above. The difference in the couplings come

from the vev that defines the masses of the fermions and the rotation between the scalar bases. For instance,

in type-II 2HDM the masses are defined as mu ij = v2√
2
yuij , md ij = v1√

2
ydij and me ij = v1√

2
yeij , so that for

down-like quarks and leptons instead of v2 = v sinβ, it enters v1 = v cosβ. In a phenomenological sense, it is

worth mentioning that the Yukawa couplings can be enhanced or suppressed depending on the type of 2HDM.



III.3. The Alignment Limit 39

For instance, in the type I the couplings of the pseudoscalar and the charged Higgs bosons get suppressed for

tanβ > 1.

III.3 The Alignment Limit

In the 2HDM it is possible to recover a physical CP-even scalar with the same gauge, Yukawa and self-coupling

interactions at tree level as those in the SM for the Higgs boson. This situation in the 2HDM is known as the

alignment limit. To see this limit we start by noting that the trilinear coupling of the W-bosons with the Higgs

is of the form

LKin ⊃ 1

8
g2
(
W 1
µ + iW 2

µ

) (
W 1
µ − iW 2

µ

)
(v1 + η1)

2
+

1

8
g2
(
W 1
µ + iW 2

µ

) (
W 1
µ − iW 2

µ

)
(v2 + η2)

2

⊃ 1

2
g2W−µ W

+
µ (v1η1 + v2η2) . (III.39)

Now, by comparing with the SM model situation (see Eq. [I.24]), the identification

H ′1 =
1

v
(v1h1 + v2h2) , (III.40)

leads to an interaction between a scalar and the charged weak bosons like that in the SM. The orthogonal

combination (H ′2) of h1 and h2 does not present a trilinear coupling with the weak bosons. Then, let us write

 H ′1

H ′2

 =

 cosβ sinβ

− sinβ cosβ


 h1

h2

 . (III.41)

In general, the fields H ′i not need to be physical. The alignment limit states the situation under which H ′1 is a

CP-even physical scalar, but these physical states are given by the transformation

h0 = sin (β − α)H ′1 + cos (β − α)H ′2, (III.42)

H0 = cos (β − α)H ′1 − sin (β − α)H ′2. (III.43)

Then, the alignment limit is at sin (β − α) = 1. The measurements at the LHC regarding the Higgs boson

decays are in a good agreement with the SM, thus, it is expected the parameters to be near the alignment limit.

Let us also note that the alignment condition fixes one parameter. It is also remarkable that in the alignment

limit the trilinear Higgs coupling takes the same form as in the SM.



40 Chapter III. Two-Higgs-Doublet Model (2HDM)

III.4 Constraints to the Parameters of the 2HDM

After the measurement of the resonance identified with the Higgs boson, the parameters of the 2HDMs got

stronger constraints. Bounds to these parameters come from experiments and from theoretical arguments. In

the former, we can find direct bounds to the masses of the non-standard scalars and measurements on FCNC;

in the latter enter unitary and stability constraints, as well as small deviations from ρ = 1.

Measurements in the LHC set that at 95% of C.L. the masses of the non-standard Higgs bosons should be[15]

mH± > 80 GeV, and mA0 > 93.4 GeV for tanβ > 0.4. (III.44)

Other constraints on the mass of the charged Higgs bosons can be found from quantum corrections to the FCNC

through the charged-current interactions. Measurements on the B-meson decays [38] claim that mH± > 295 GeV

for tanβ > 1.

Constraints on tanβ can also be established from measurements of Z0 → b̄b and B − B̄ mixing processes. This

parameter can take values from a wide range of values. The mentioned experiments pointed out