Search for electroweak production of charginos and sleptons decaying into final states with two leptons and missing transverse momentum in √
s
= 13 TeV proton-proton collisions using the ATLAS detector
Helén Persson
Thesis submitted for the degree of Master in Nuclear and Particle Physics
60 credits
Department of Physics
Faculty of mathematics and natural sciences
UNIVERSITY OF OSLO
Search for electroweak production of charginos and
sleptons decaying into final states with two leptons and missing transverse momentum in √
s = 13 TeV proton-proton collisions using the ATLAS
detector
Helén Persson
© 2020 Helén Persson
Search for electroweak production of charginos and sleptons decaying into final states with two leptons and missing transverse momentum in √
s = 13 TeV proton-proton collisions using the ATLAS detector
http://www.duo.uio.no/
Printed: Reprosentralen, University of Oslo
Abstract
The Standard Model (SM) accounts for about 5 % of the known univers, and can not explain gravity, the hierarchy problem or the matter-antimatter asymmetry in the univers. Supersymmetry is a new symmetry which aim to bridge the gap between the SM and the unanswered questions.
This analysis studies production of Supersymmetric particles using data from the ATLAS detector at the Large Hadron Collider at CERN taken at
√s = 13 TeV. In particular, three decay models of electroweak production of charginos and sleptons decaying into final states with two leptons and missing trans- verse momentum are studied. The effect of different isolation point combinations are tested and the expected significance calculated.
It is found that the different isolation point combination has little effect on
the total amount of background in the signal regions, but have a large effect
on the fake and non-prompt sample, with one of the combination tested in
this analysis being better than the others.
Acknowledgements
First of all I would like to thank my supervisor Heidi Sandarker for being incredible supportive and motivating to work with. Thank you for always being there for advice, listen to my frustrations and helping me find solutions to every situation. I will also like to thank you for giving med the opportunity to work with one of the most exiting topics I could think off.
I would also like to thank my co-supervisor Eirik Gramstad for helping me find my path during the work with this thesis. You where always ready to help, and have guided me through the every day life problems that comes with embarking on a master thesis. Without you my comprehension of high- energy physics would not have been the same.
The HEP group at the University of Oslo have been my home for some time now, and it is with both sadness and joy the next chapter will start.
We have had so many good discussions, laughs and quizzes. I will miss it.
To my fellow master student, thank you for always being there for a dis- cussion, help, and a great atmosphere in the master room. Oda Langrekken, you may have finished before me, but I will always remember our time at UiO. I would also like to thank Mona Anderssen for sitting right behind me and always keeping my spirit up. It would not have been the same in you weren’t there. Also Eli Bæverfjord Rye deserves a mention. A happy, smil- ing addition to may days at UiO with great knowledge I have benefited from more than once.
Finally I want to thank my friends and family for always having my back,
and supporting me what ever choices I make.
Contents
Introduction 6
1 The Standard Model 8
1.1 Fermions . . . . 9
1.2 Bosons . . . . 11
1.3 Symmetries and conservation . . . . 12
1.3.1 Noether’s Theorem . . . . 13
1.4 Quantum field theory . . . . 13
1.4.1 Lagrangian formalism and interpretation of the La- grangian . . . . 14
1.4.2 Group description and gauge theory . . . . 15
1.4.3 Quantum Electrodynamics . . . . 16
1.4.4 Quantum Chromodynamics . . . . 17
1.4.5 Glashow-Weinberg-Salam theory . . . . 19
1.5 Spontaneous Symmetry Breaking . . . . 22
1.5.1 Real scalar field . . . . 22
1.5.2 Complex scalar field . . . . 24
1.6 Brout-Englert-Higgs mechanism . . . . 25
1.6.1 Local U (1) Y symmetry breaking . . . . 25
1.6.2 Local SU (2) L × U (1) Y symmetry breaking . . . . 27
1.6.3 Fermion masses . . . . 28
1.7 Full GWS Lagrangian . . . . 29
1.8 Beyond the Standard Model . . . . 29
2 Supersymmetry 32
2.1 Introduction to SUSY . . . . 32
2.2 MSSM theory . . . . 33
2.2.1 Symmetry breaking in SUSY . . . . 35
3 CERN and the ATLAS experiment 36 3.1 The CERN of today . . . . 36
3.2 The LHC . . . . 38
3.3 The ATLAS Detector . . . . 38
3.3.1 Detector overview - coordinate system and nomenclature 40 3.3.2 The inner detector . . . . 42
3.3.3 Calorimeters . . . . 43
3.3.4 Magnets . . . . 45
3.3.5 Muon system . . . . 45
3.3.6 Triggers and data acquesition . . . . 46
4 Proton-proton Collisions 49 4.1 Kinematics of proton-proton collisions . . . . 49
4.1.1 Particles in collisions . . . . 50
4.1.2 Proton-Proton interactions . . . . 51
4.1.3 Parton distribution function . . . . 53
4.1.4 Hadronization . . . . 54
4.2 Important parameters . . . . 56
5 Search strategy 58 5.1 Search algorithm on 1-2-3 . . . . 58
5.2 Supersymmetric decay models . . . . 59
5.2.1 Chargino-chargino with W- boson mediator decay . . . 59
5.2.2 Chargino-chargino with slepton mediator decay . . . . 60
5.2.3 Direct slepton production . . . . 61
5.3 Kinematics of supersymmetric decay models and searches . . . 62
5.4 Object reconstruction . . . . 64
5.4.1 Electrons . . . . 64
5.4.2 Muons . . . . 65
5.4.3 Jets . . . . 66
5.4.4 Missing transverse energy . . . . 67
5.5 Object definition and Event selection . . . . 67
5.5.1 Terminology . . . . 68
5.5.2 Leptons . . . . 68
5.5.3 Jet . . . . 69
5.5.4 Missing transverse energy . . . . 69
5.5.5 Overlap removal . . . . 70
5.6 Search regions . . . . 70
5.7 Isolation points . . . . 73
6 Data and simulated event samples 74 6.1 Data set . . . . 75
6.2 Signal samples . . . . 75
6.3 Standard Model background estimations . . . . 76
6.3.1 Z + jets . . . . 76
6.3.2 Diboson . . . . 77
6.3.3 V + γ . . . . 77
6.3.4 Drell-Yan . . . . 78
6.3.5 t ¯ t . . . . 78
6.3.6 W + jets . . . . 78
6.3.7 Single top . . . . 78
6.3.8 Top other . . . . 79
6.3.9 Higgs . . . . 79
6.3.10 Triboson . . . . 79
6.3.11 Fake and non-prompt leptons . . . . 79
7 Analysis framework 81 7.1 Programming tools needed . . . . 81
7.1.1 Python . . . . 82
7.1.2 C++ . . . . 82
7.1.3 ROOT . . . . 83
7.2 Program workflow . . . . 84
7.2.1 Selecting interesting events . . . . 86
7.2.2 Making histograms and tables . . . . 86
7.3 Validation of code . . . . 86
8 Results 88 8.1 Discovery, exclusion and significance . . . . 89
8.1.1 Discovery . . . . 89
8.1.2 Exclusion . . . . 90
8.1.3 Significance calculations . . . . 91
8.2 Distributions . . . . 91
8.3 Significance plots . . . . 96
8.4 Cutflow tables background simulations . . . . 106
Conclusion 109
Introduction
The Standard model is the current theory used to describe the particles that build up matter (fermions) and the interaction between (bosons) them by the use of a few building blocks known as the fundamental particles. Covering only around 5% og the observable universe the Standard Model is a robust theory, unmatched in its success to outlive other theories trying to take its place. In 2012 the last pice of the puzzle, the Higgs boson, was discovered at CERN by the ATLAS[16] and the CMS [21] experiments. Still, by leav- ing around 95 % of the universe unexplained, physicists believe that there are some other expansion or replacement for the Standard Model giving us further answers. Although the standard model is very successful in describ- ing and predicting outcome from the experiments at CERN, there are som fundamental problems. One of the problem is the lack of an explanation for gravity, and another is the matter-antimatter asymmetry in the universe.
Supersymmetry (SUSY) is a common name containing multiple theories as a sort of expansion to the Standard Model. Common for them is that they contain all of the particles in the Standard Model and a corresponding supersymmetric partner particle to those. In this analysis three decay models of electroweak production of charginos and sleptons decaying into final states with two leptons and missing transverse momentum are studied. The goal is to reproduce the results from the newest paper on the topic [18] published early 2020, and study the effect of different isolation point criteria for leptons (e and µ).
Chapter 1 starts at the beginning with the Standard Model and build a bridge to chapter 2 giving a brief introduction to Supersymmetry. In chapter 3 an introduction to CERN, the LHC accelerator and the ATLAS detector is given. Chapter 4 lays the foundation for understanding how a proton-proton collision happen, before starting on the analysis part of the thesis.
In chapter 5 the process of a SUSY search is explained, with details about
object reconstruction in the detector, and object definition used in this anal-
ysis is presented. Chapter 6 covers the data and simulated events used, and
chapter 7 explain the necessary tools and the framework is explained. In
chapter 8 the results from this analysis is presented and discussed.
Chapter 1
The Standard Model
Physics aims to describe the univers around us, and particle physics focusses on the fundamental constituents of the universe: the elementary particles and their interactions. Today our best model describing the univers is the Standard Model (SM). The Standard Model is a unified picture of parti- cles and the forces acting between them, where forces also are described as particles. Clustering particles with similar properties together, two particle groups are created: fermions (leptons and quarks) and bosons, and a visual representation of the Standard Model can be given as in figure 1.1. The fermions are matter constituents and make up matter while bosons carries the forces holding matter together. In 2012 the last missing particle in the Standard Model was found by the ATLAS [16] and CMS [21] experiments at CERN, a Higgs boson with a mass of 125 GeV, named after Peter Higgs who in 1964 predicted the existence of such a particle.
In this chapter an introduction to the Standard Model and the individual theories behind it will be given based upon the sources [36], [23], [32] and [37].
For a more phenomenological insight to the Standard Model the sources [36], [23] are recommended, while the sources [32] and [37] can shed light upon the more theoretical parts of the Standard model and this chapter.
The Standard Model turns out to be a very robust model withstanding the years and years of searches for new physics, still lacking explanations to sev- eral large questions physicist are wondering about. One of the questions you may have about the Standard Model is why there is exactly three generations of fundamental particles as shown in figure 1.1?
The answer is, put simply, three fits. A more complex answer is that
studies done on the Z 0 -boson and its lifetime[23] shows that tree generations
will describe the predominance of matter over antimatter in the universe.
Figure 1.1: The Standard Model. The figure is reprinted from [7]
But this is still not the ultimate answer, because it could be there exists a fourth generation of fundamental particles, much heavier that the three first generations so that the Z 0 -boson can not decay into them. But there are no good reasons for why there suddenly should be a huge jump in particle mass, forcing the fourth generation neutrino mass to be over 45 GeV/ c 2 .
Not content with this answer, or wondering about why we still use the Standard Model or what other questions there might be connected to it, this will hopefully become more clear during this chapter, specifically in section 1.8.
In the first two sections a non-mathematical overview of the building blocks of the Standard Model will be given, before diving in to symmetries and conservation laws, and the Quantum Field Theories (QFTs) building up the Standard Model.
1.1 Fermions
The fermions are spin- 1 2 particles split into two subcategories, 6 quarks (up,
down, charm, strange, top, bottom) and 6 leptons (electron, electron neu-
trino, muon, muon neutrino, tau, tau neutrino), divided into three genera-
tions. Fermions obey the Pauli exclusion principle meaning that only one
fermion can occupy a given quantum state at any given quantum number.
Each of the 12 Standard Model fermions have an antiparticle, with the same quantum numbers except from opposite charge. This comes from the Dirac equation The antiparticles are noted by their charge (e + ) or by a bar over ( u ¯ ).
Not a topic for this thesis, but the interesting nature of the neutrinos must be mentioned whenever discussing fermions. They where long assumed to be massless particles but from experiments showing neutrino oscillation occurring over large distances we now know that ν e , ν µ and ν τ are quantum- mechanical mixtures of the three fundamental neutrino eigenstates ν 1 , ν 2 and ν 3 [36]. This means that they must have mass since they are able to oscillate, but they are much lighter than the the other fermions.
The leptons may exits on their own, but quarks are never observed as free particles, they are always grouped together to form particles we call hadrons.
Hadrons can either be baryons/anti-baryons, consisting of three quarks/anti- quarks ( qqq / q ¯ q¯ ¯ q ) or mesons/anti-meson consisting of a quark and an anti- quark ( q¯ q ). Why this specific structure you may ask. The answer comes from the fact that in addition to carry the quantum numbers of spin, charge etc., quark carries a colour charge, either red, blue or green 1 . Hadrons exists in a colour neutral state, meaning you either need one of each colour, or a colour and a anti-colour. Hadrons and leptons have different properties, and hence ”feel” different forces acting on them. The forces felt by the different fermions can be found in table 1.1.
Table 1.1: Table of which fermion feels which force
Strong Electroweak Weak
Quarks up-type u c t
X X X
down-type d s b
Leptons charged e − µ − τ − X X
neutrinos ν e ν µ µ τ X
1 This is of course not the colour red, blue and green as we normally think of colour,
but only a naming convention for a quantum number.
1.2 Bosons
In modern particle physics forces between fundamental particles are de- scribed by Quantum Field Theory (QFT), and a force carrying particle can be thought of as an excitement of these fields. For now we will just think of them as particles. The particles on the right hand side of figure 1.1 shows the integer spin particles known as (gauge) bosons. Bosons follow Bose-Einstein statistics and contain the force carriers for the electromagnetic force (photon (γ), the weak force (W ± -,Z 0 -bosons) and the strong force (gluons (g)). If gravity was included in the Standard Model, the graviton ( G ) would be the associated gauge boson.
The Higgs boson is a bit different than the other bosons, as it is a spin-0 scalar particle in the SM. It does not mediator of a force, but rather provides the mechanism that gives mass to the other particles as they interact with the Higgs field. More about the Higgs mechanism in section 1.6.
As can be seen from the table 1.1 different fermions interact with different forces. The photon, carrier for the electromagnetic force connects to all charged particles. For the weak force the W ± - and the Z 0 -bosons connect with different particles. The W ± have a charge of +e/ − e and are each others anti-particles. They couple together pairs of fermions that differ by one unit of electric charge. For leptons this means the coupling are limited to the pairs
ν e e
,
ν µ µ
,
ν τ τ
,
not changing the generation of the leptons. For quarks the change is not limited to generations, but the strength of these weak charged current cou- pling are strongest among quarks in the same generation as long as there is a change of flavour. This means that all the coupling pairs below are allowed by an exchange of either a W + - or W − - boson
u d
,
u s
,
u b
,
c d
,
c s
,
c b
,
t d
,
t s
,
t b
.
The Z 0 -boson is electrically neutral, it is its own anti-particle. The weak neutral current, Z 0 never changes the type of particle or the flavour of the quark.
From the figure 1.1 it may not be clear that there are in fact eight massless
gluons as carriers for the strong interaction. Gluons must carry both colour
and anti-colour and can be viewed in colour representation as
r¯ g, g¯ r, r ¯ b, b¯ r, g ¯ b, b¯ g, 1
√ 2 (r¯ r − g¯ g) , 1
√ 2 r¯ r + g¯ g − 2b ¯ b .
1.3 Symmetries and conservation
The rules governing a model can be described by symmetries and conserva- tion laws for that particular model. In the Standard Model, in the same way as in Newtonian physics, energy, E , three momentum, ~ p , and total angular momentum, J is conserved. From Special Relativity (SR) we know that the (rest) mass, m, is not conserved, leading to a possibility of production of heav- ier particles than the once involved in the process. Quantum Field Theory states that all physical processes are symmetric under CPT-transformation.
The C stands for charge conjugation, meaning that you replace the parti- cle with it’s anti-particle (particle → anti − particle) , while the P stands for parity (~ x → −~ x) , and finally T stands for time-reversal (t → −t) . The strong interaction also follows C-, P-, T-, and CP-symmetries, while the weak interaction violates these.
For each interaction there is conservation of a quantum number that comes from the gauge symmetries. For the electromagnetic interaction this quan- tum number is the electric charge. For the weak interaction it is the weak isospin, and for the strong interaction the quantum number conserved is colour charge.
Besides these quantum numbers, there are two more that are worth men- tioning here. The first one is baryon number, B, which is always conserved in the Standard Model. All baryons are given a baryon number of B = 1 , while anti-baryons are assigned a baryon number of B = −1 . This means that quarks have a baryon number of 1 3 , while anit-quarks have B = − 1 3 . All other particles have baryon number B = 0 , including mesons.
The lepton number, L x , where x denotes the flavour, is almost always
conserved except in the oscillations of neutrinos. e − and ν e have a lepton
number of L e = 1, while e + and ν ¯ e have L e = −1. All other particles have
L e = 0 . The same system applies for L µ and L τ . When neutrinos travel large
distances they may oscillate between the different flavours, breaking lepton
number conservation. This only happens when they travel large distances,
and is not something we need to take into account when working with particle
interactions as the once studied in this thesis.
1.3.1 Noether’s Theorem
To be able to describe physics and have rules that are applicable for multiple scenarios, a basic principle in physics is that the laws of physics must be the same for all times and for all locations. Such a statement can be though of as a symmetry: the laws of physics are invariant when the viewpoint is changed.
In 1915 Emmy Noether proved a connection between these symmetries and the conservation of physical quantities [29], [30]. This theorem stats that that every differentiable symmetry of the action of a physical system has a corresponding conservation law.
1.4 Quantum field theory
As stated earlier forces in modern particle physics are described by fields in stead of particles, where each particle is an excited state of their underlying fields. This moves us away from the thought of particles being point-like and also from the idea of having an empty vacuum. In this section an introduction to Quantum Field Theory and its connection to high-energy particle physics will be given. Quantum Field Theory is not an easy subject, and many steps and calculations will not be included in this thesis, just stated as results.
The reader is referred to material such as An Introduction to Quantum Field Theory by Michael E. Peskin, Daniel V. Schroeder [32] for further material.
To (almost) describe Quantum Field Theory in (physics) layman terms it is a combination of Quantum Mechanics (QM) - describing small things, and Special Relativity (SR) - describes things moving fast.
The problem is that there is no way to simply combine Quantum Mechanics and Special Relativity because the Schrödinger equation (the most important building block in QM) is not Lorentz invariant, meaning it is not equal for two observers in different reference frames. This leads to problems with causality, negativ energy states and no possibility to create other particles.
In stead of the Schrödinger equation Quantum Field Theory includes the Dirac equation (for spin- 1 2 particles) and the Klein-Gordon equation (for scalar (spin-1) particles).
To work efficiently with Quantum Field Theory the Lagrangian formalism,
explained in the next section, is used. The Lagrangian formalism is suited to
relativistic dynamics because all expressions are explicitly Lorentz invariant
[32].
1.4.1 Lagrangian formalism and interpretation of the Lagrangian
The Lagrangian is used to describe fields and their interactions in a concise manner. From classical mechanics the Lagrangian is given as L = T − U , where T is the kinetic energy and U is the potential energy and this is used as a baseline also in Quantum Field Theory. From the Lagrangian of a system via the Euler-Lagrange equation the equation of motion (e.o.m) can be found, giving insights into the rules of the system in question.
The Euler-Lagrange equation can be found by minimizing the action, S, defines as
S = Z
L dt, (1.1)
giving for a general coordinate q the Euler-Lagrange equation d
dt ∂L
∂ q ˙
− ∂L
∂q = 0 (1.2)
leading to the respective equation of motion when solved for L .
In Quantum Field Theory fields are used in stead of kinetic and potential energy so the Lagrangian L is changed in to the Lagrangian density L which is a function of fields (Φ) and their derivatives (∂ µ Φ). L is the spatial integral over L . For simplicity the Lagrangian density will herby be referred to as only the Lagrangian. By changing to L the action now becomes
S = Z
L(Φ, ∂ µ Φ) d 4 x (1.3) and the Euler-Lagrange equation becomes
∂ µ
L
∂ (∂ µ Φ)
− ∂L
∂Φ = 0. (1.4)
For a free fermion field ψ = ψ(x) the Lagrangian is
L = ¯ ψ (iγ µ ∂ µ − m) ψ (1.5)
where ψ ¯ = ψ † γ 0 is the adjoint spinor, and the first term is a kinematic term, while the second term is a mass term. By inserting this into the Euler- Lagrange equation 1.4 the Dirac equation pops out:
(iγ µ ∂ µ − m) ψ = 0. (1.6)
For a free theory the Lagrangian is L = 1
2
(∂ Φ) 2 − m 2 Φ 2
, (1.7)
and by inserting this into the Euler-Lagrange eq. 1.4 the Klein-Gordon equa- tion is found:
∂ 2 + m 2
Φ = 0. (1.8)
The Lagrangian can be used to learn more about the equations of motion governing a system. It can also, in combination with perturbation theory, give Feynman diagrams describing interaction vertices, and lead to the tran- sition amplitude M for a certain process, which can be used to calculate cross-sections and decay rates. The Lagrangian also shown which interac- tions the theory allows and how the fields of the theory are coupled (interact) to each other. In general the kinematic terms of a Lagrangian are quadratic involving both ψ and ∂ψ while the mass terms are quadratic without the derivative of the field involved. The constant in front of the mass term is interpreted as the mass of the field.
1.4.2 Group description and gauge theory
An efficient way to describe symmetris of a physical system is to use groups.
In mathematics a group G is a set of elements g i combined with a binary operation • , satisfying the group axionoms
1. g i • g j ∈ G (closure)
2. (g i • g j ) • g k = g i • (g j • g k ) (associativity)
3. ∃e ∈ G such that g i • e = e • g i = g i (identity element) 4. ∃g −1 i ∈ G such that g i • g i −1 = g −1 • g i = e (inverse element).
Not much time will be spent diving into groups and group theory, but it lays the foundation for the following sections. A symmetry group is a set of all transformations that leaves a given object invariant, meaning it does not change the object. For Special Relativity the symmetry group is called the Poincaré group, assumed to be the fundamental symmetry group for spacetime (including rotations and boosts).
In Quantum Field Theory there may also be invariance under the trans-
formation of the field it self, not only in spacetime. This is called internal
symmetris and describe particle interactions. The special unitary group of degree n SU(n) is such a symmetry, referred to as a Lie group of n×n unitary matrices with determinant 1. A Lie group describes continuous symmetries.
In physics, a gauge theory is a type of field theory in which the Lagrangian is invariant under local transformations from certain Lie groups. The term gauge refers to the redundant degrees of freedom in the Lagrangian, meaning they have no observable consequence. To describe a gauge theory, and the interactions it covers, we start by identifying the global gauge symmetries 2 of the theory we want to study. These symmetries must leave the Lagrangian L invariant. From this we identify local gauge transformations 3 , and require that they to keep the Lagrangian L invariant. This is referred to as the gauge principle. The transformations we identify are described by symmetries, and the symmetry group of a given theory is called the gauge group of that theory.
The last two terms needed to know is Abelian and non-Abelian groups.
In an Abelian group all the elements commute, meaning that the order of the elements in an operation is not important. For a non-Abelian group (you might have guessed it) there is at least one pair of elements that do not commute, meaning that the order of the elements is important under an operation.
The Standard Model is a combination of the local gauge symmetries SU(3) C × SU (2) L × U (1) Y . In the following sections we will build up the gauge group and the Lagrangian of the Standard Model, staring with the U (1) group, the simplest of these gauge symmetries, the gauge group of Quantum Electro- dynamics (QED). After that we will go through Quantum Chromodynamics and Glashow-Weinberg- Salam (GWS) theory.
1.4.3 Quantum Electrodynamics
The gauge gruop U (1) is an Abelian group, with the global gauge transfor- mation ψ(x) → e iα ψ(x), while the local gauge transformation is given by ψ(x) → e iα(x) ψ(x) .
To build the Lagrangian of Quantum Electrodynamics we do the following:
We start the Lagrangian of a free fermion field L = ¯ ψ (iγ µ ∂ µ − m) ψ , which is invariant under the global U(1) transformation, but not under the local transformation due to the derivative of the kinematic term. To make the Lagrangian invariant we start by replacing the derivative with the covariant derivative
2 A global symmetry is independent of the coordinate x
3 A local symmetry is dependent on the coordinate x
∂ µ → D µ = ∂ µ − ieA µ (x), (1.9) where A µ (x) is a vector field, often called a gauge field. For this change to be invariant, we must require that the field A µ (x) transforms as A µ (x) → A µ (x) + 1 e ∂ µ α(x).
For the vector field A µ to be a true propagating field it needs a kinetic term in the Lagrangian, still keeping the Lagrangian gauge invariant. This term is constructed from the field strength tensor F µν = ∂ µ A ν − ∂ ν A µ . The full Lagrangian for QED then becomes
L QED = ¯ ψ(iγ µ ∂ µ − m)ψ + e ψγ ¯ µ ψA µ − 1
4 F µν F µν . (1.10) In the Lagrangian shown in equation eq. 1.10 we se that there is a term with both ψ and A µ . This therm indicates an interaction between the fermion and the gauge field, and from this term we can extract a Feynman diagram.
By associating the gauge field to the massless photon, we get the well known interaction vertex of QED, shown in figure 1.2
f
f ¯
γ
Figure 1.2: QED vertex
1.4.4 Quantum Chromodynamics
Quantum Chromodynamics (QCD) describes the strong interaction, which relates to the colour charge of fundamental particles (quarks). As mentioned earlier there are three possible colour charges (red, green, blue) so quantum chromodynamics must work in a three dimension colour space. The sym- metry group for quantum chromodynamics is the SU(3) C gauge symmetry group. As for quantum electrodynamics this section will focus on finding the Lagrangian of quantum chromodynamics.
For a field ψ = ψ(x) the local SU (3) C gauge transformation is given as ψ → exp h
ig s ~ α(x)˙ˆ T i
ψ (1.11)
where g s is the strong coupling constant, T ˆ = {T a } are eight generators, a = 1, 2, 3, . . . , 8 of SU(3) given by T a = 1 2 λ a , where λ a is the Gell-Mann matrices.
For quantum chromodynamics the covariant derivative is
D µ = ∂ µ + ig s G a µ (x)T a (1.12) where G a µ is a representation of eight gauge fields that we require to transform as
G k µ → G k0 µ = G k µ − ∂a k − g s f ijk α i G j µ (1.13) and in this equation f ijk is the structure constant of SU (3) defined from the commutation relation
[λ i , λ j ] = 2if ijk λ k . (1.14) This equation is a non-zero relation, meaning that SU (3) is a non-Abelian group. Being a non-Abelian group leads to a property of self-interactions between the gauge fields in SU(3) , which is also apparent from the field strength tensor
G i µν = ∂ µ G i ν − ∂ ν G i µ − g s f ijk G j µ G k ν. (1.15) The kinematic term − 1 4 G µν G µν contains both three and four gauge field vertices.
q
¯ q
g
g
g
g
g g
g g
Figure 1.3: QCD vertex
The full Lagrangian of quantum chromodynamics become L QCD = ¯ ψ (iγ µ ∂ µ − m) ψ − g s ψT ¯ a ψG a µ − 1
4 G µν G µν (1.16)
where the ψ now is a colour charge carrying fermion, i.e. a quark. As can be seen from the Lagrangian there is noe mass term, meaning that in the SM the gluons are predicted to be massless.
1.4.5 Glashow-Weinberg-Salam theory
The Glashow-Weinberg-Salam (GWS) theory is a bit more complicated than the previous two sections, as it is a combination of the weak and the electro- magnetic interactions often known as the electroweak theory. Developed by Glashow [22], Weinberg and Salam in the 1960 and earned them the Nobel prize in physics in 1979 [12]. The GWS theory is one of the last steps towards have a full Lagrangian for the Standard Model, and all fields in the Stan- dard Model is believed to have a vacuum expectation value of zero except the Higgs field which is believed to have a non-zero value. The Glashow- Weinberg-Salam theory calls for at least one Higgs particle in the Standard Model.
The quarks in GWS theory are not (completely) the same quarks as the ones in quantum chromodynamics. The strongly interacting quarks are mass eigenstates, while the weak interacting quarks of this theory are superposi- tions of the mass eigenstates. A common practice is to chose a basis such that the up-type quarks (u, c, t) are both weak and strong eigenstates, while the down-type quarks ( d, s, b ) are mixed states. The weak states are denoted by d 0 , s 0 , b 0 and the mixing is described by the Cabibbo-Kobayashi-Maskawa (CKM) matrix
d 0 s 0 b 0
=
V ud V us V ub V cd V cs V cb V td V ts V tb
d s b
(1.17)
where V xy describes the probability of a transition by emission or absorption of a charge W-boson from quark type x to quark type y .
Before we can start to look at the Lagrangian we need to look at chirality and gauge group of the GWS theory.
Chirality and gauge group
The first thing that makes the GWS theory more complicated than quantum
electrodynamics or quantum chromodynamics is the fact that in this GWS
theory fermions exists as left- and right-handed chirality states, ψ L and ψ R
given by
ψ L = 1 − γ 5
2 ψ, and ψ R = 1 + γ 5
2 ψ (1.18)
where ψ is a fermionic field.
What makes it complicated is that only left-handed fermions couple to the W -boson and not right-handed fermions. To take this into account the quantum number: weak isospin I W is introduced. Left-handed fermions have I W = ± 1 2 ( + for neutrions and up-type quarks, − for charged leptons and down-type quarks) and appears in isospin doublets
` L = ν l
l −
or q L = u
d
(1.19)
while right-handed fermions have I W = 0 and are found in isospin singlets
` R , u R or d R . (1.20)
The gauge gruop for GWS theory is SU (2) L × U(1) Y , where the L denotes the left-handedness of the interaction, and Y stands for the hypercharge.
Hypercharge is connected to electric charge (Q) and the third component of the weak isospin ( I W 3 ) by the following relation
Y = 2(Q − I W 3 ). (1.21)
Particles in weak isospin doulets have a hypercharge Y = 1 for leptons and Y = 1 3 for quarks.
The Lagrangian
The local gauge transformation for SU (2) L × U (1) Y for a field ψ = ψ(x) is ψ → exp
i Y
2 α(x) + iI W β(x)σ
ψ (1.22)
where β(x) is a three dimensional function and σ are the Pauli matrices.
As stated earlier, we can not work out a Lagrangian in the same way as for quantum electrodynamics or quantum chromodynamics due to the fact that the mass term would mix left- and right-handed states, destroying the gauge invariance. Still, let us continue and see how far we can go with this method.
The adding of a mass term that does not destroy gauge variance can only happen if there is some symmetry breaking, and a mechanism giving rise to the masses. These topics will be covered in the sections 1.5 and 1.6 leading up to the complete Lagrangian for GWS theory in section 1.7.
Let us see how far we can come without these new tools. The transforma-
tion of the left- and right-handed fields become
ψ L → ψ 0 L = exp
i Y
2 α(x) + iI W β(x)σ
ψ L , (1.23)
ψ R → ψ 0 R = exp
i Y 2 α(x)
ψ R . (1.24)
The covariant derivative takes the form D µ = ∂ µ + ig 0 Y
2 B µ (x) + igI W σW µ (x) (1.25) where B µ is a gauge field for the U (1) Y symmetry, W µ is three gauge fields for SU (2) L symmetry, and g and g 0 are related to the elementary charge by e = g sin θ W = g 0 cos θ W (1.26) where θ W is the weak mixing angle. In the Standard Model this angle is not explained by theory, and the value can only be found as experimental results.
The covariant derivative offers up no physical fields, and the fields we find in the Standard Model are mixtures of the fields found in eq. 1.25 by the formulas
For γ: A µ = B µ cos θ W + W µ 3 sin θ W For Z 0 : Z µ = −B µ sin θ W + W µ 3 cos θ W For W ± : W µ ± = 1
√ 2 W µ 1 ∓ W µ 2 The field strength tensor for these fields become
B µν = ∂ µ B ν − ∂ ν B µ (1.27) and
W µν = ∂ µ W ν − ∂ ν W µ − gW µ × W ν , (1.28) and the (not complete) electroweak Lagrangian becomes
L GWS = ¯ ψ L γ µ
i∂ µ − g
2 σW µ − g 0 Y 2 B µ
ψ L + ¯ ψ R γ µ
i∂ µ − g 0 Y 2 B µ
ψ R − 1
4 B µν B µν − 1
4 W µν W µν
(1.29)
f L,R
f ¯ L,R
Z, γ
` − L , d L
¯ ν L , u ¯ L
W −
ν L , u L
` + L , d ¯ L
W +
W +
W −
Z, γ
W + W −
Z γ
Figure 1.4: Elemental GWS vertices
1.5 Spontaneous Symmetry Breaking
Before moving on to the full Lagrangian for the GWS theory, symmetry breaking must be touched upon. The understanding developed here will be necessary to have both for the Brout-Englert-Higgs mechanism giving mass to particles, but also for Supersymmetry 2 A presentation for spontaneous symmetry breaking for both a real scalar field and a complex scalar field will be given.
1.5.1 Real scalar field
For a real scalar field φ with the Lagrangian L = 1
2 (∂ µ φ) (∂ µ φ) − µ 2 φ 2 − 1
4 λφ 4 , (1.30)
and if we choose λ > 0 , the Lagrangian is invariant under the transformation
φ → −φ, but the case of µ 2 > 0 and µ 2 < 0 must be studied separately.
µ 2 > 0 :
If µ 2 > 0 the mass term is √
2µ 4 . From the Lagrangian we see that there is a quartic self-coupling term with the coupling constant λ .
Remembering that the Lagrangian is given as L = T − U , eq. 1.30 has the potential
V (φ) = µ 2 φ 2 + 1
4 λφ 4 , (1.31)
with a vacuum expectation value (a minimum) of 0 when φ = 0 . This upholds the symmetry of the Lagrangian.
µ 2 < 0 :
If µ 2 < 0 the Lagrangian and the potential will still be symmetric under the transformation φ → −φ , but it will lead to two changes: The quadratic term can no longer be considered a mass term, and the vacuum expectation value is now found at
φ = ± µ
λ , (1.32)
where one of these solutions can be chosen as the ground state.
Perturbation theory can be used to move further, but then the field must be expressed as derivatives of the ground state. This is achieved by introducing a field η
η ≡ φ ± µ
λ (1.33)
and the Lagrangian (ignoring the constant term involving only µ and λ , since it never would contribute to the equation of motion) becomes
L = 1
2 (∂ µ η) (∂ µ η) − µ 2 η 2 ± µλη 3 − 1
4 λφ 4 . (1.34) The Lagrangian now contains two self-coupling terms, and a mass term
√ 2µ . It also contains a cubic term, meaning that the Lagrangian is no longer symmetric under η → −η, and we have spontaneous 5 symmetry breaking.
4 Mass terms of a scalar field is accompanied by a factor of 1 2 in the Lagrangian (not so for fermions). If this term is not included in the Lagrangian it must be added when writing down the mass of the scalar field
5 Spontaneous is used to indicate that there are no external influence that leads to the
symmetry breaking
1.5.2 Complex scalar field
For a complex scalar field φ = √ 1
2 (φ 1 + iφ 2 ) with the Lagrangian
L = (∂ µ φ ∗ ) (∂ µ φ) − µ 2 (φ ∗ φ) + λ (φ ∗ φ) 2 (1.35) again with λ > 0 . This Lagrangian is symmetric under U (1) transformations, and as for the real scalar field the case of µ 2 > 0 and µ 2 < 0 must be studied separately.
µ 2 > 0 :
If µ 2 > 0 the potential V (φ) = µ 2 (φ ∗ φ) − λ (φ ∗ φ) 2 has a vacuum expectation value when φ = 0 , and will uphold the symmetry of the Lagrangian. From the real scalar field we saw that µ 2 < 0 lead to symmetry braking. Will it be the same here?
µ 2 < 0 :
For the case of µ 2 < 0 the potential V (φ) = µ 2 (φ ∗ φ) − λ (φ ∗ φ) 2 has a minima at
φ 2 1 + φ 2 2 = − µ 2
λ 2 = v 2 (1.36)
which forms a circle in the φ 1 , φ 2 -plane, referred to as the ”Mexican hat”- potential, shown in figure 1.5, and the corresponding ground state break the U (1) symmetry. By choseing a ground state with φ 1 = v and φ 2 = 0 , two new fields η and ξ are introduces as
φ 1 (x) = η(x) + v (1.37)
and
φ 2 (x) = ξ(x) (1.38)
which let us describe the original field as φ = 1
√ 2 (η + v + iξ) . (1.39)
The Lagrangian can be rewritten as L = 1
2 (∂ µ η) (∂ µ η) + 1
2 (∂ µ ξ) (∂ µ ξ) − λv 2 η 2 + V int (η, ξ) (1.40) where V int (η, ξ) represent the interaction term for the two fields. This La- grangian has a massive field η with mass m η = √
2λv , and a massless field ξ .
Figure 1.5: Mexican hat potential
This massless field can be explained by the Goldstone theorem which states that a massless field will appear whenever a continuous symmetry is broken.
These fields are referred to as Goldstone bosons.
1.6 Brout-Englert-Higgs mechanism
In the section about the GWS theory or development of the Lagrangian came to a halt since there was no symmetry breaking and no way to give particles mass. In this section the Brout-Englert-Higgs mechanism, which said in easy terms gives masses to the different particles in the Standard Model will be introduces”. First only symmetry breaking in the local U (1) Y transformation will be discussed, before moving on to the full GWS gauge group of SU (2) L × U (1) Y .
1.6.1 Local U (1) Y symmetry breaking
We start out with the same complex scalar field we used in section 1.5, φ ,
which has the potential V (φ) = µ 2 φ 2 + λφ 4 . If we require the Lagrangian
to be invariant under the U(1) transformation φ → e igχ(x) φ the covariant
derivative is given as
D µ = ∂ µ + igB µ (1.41)
which gives the Lagrangian
L = (D µ φ) ∗ (D µ φ) − µ 2 φ 2 − λφ 4 − 1
4 F µν F µν . (1.42) So far this looks very similar to the Lagrangian found for QED. For µ 2 > 0 section 1.5 showed that nothing exiting happens, so only the the µ 2 < 0 case which leads to spontaneous symmetry breaking will be presented. From section 1.5 we learned that for a complex field with φ 1 = v and φ 2 = 0 two new fields η and ξ could be introduced, rewriting the field to
φ = 1
√ 2 (v + η + iξ) . (1.43)
Combining the Lagrangian for this new field with eq. 1.42 the Lagrangian become
L = 1
2 (∂ µ η) (∂ µ η) + 1
2 (∂ µ ξ) (∂ µ ξ) − λv 2 η 2 + 1
2 g 2 B µ B µ − 1
4 F µν F µν (1.44)
+ V int (η, ξ, B) + gvB µ (∂ µ ξ) (1.45)
where there is one massiv field η , one Goldstone boson ξ and one massive gauge field B µ . The Lagrangian also shows the interaction term for η , ξ and B V int (η, ξ, B). This Lagrangian brings two new problems: i) the gauge field has mass which means that we have gained one degree of freedom, and ii) the last term in the Lagrangian links a spin-1 boson directly to a scalar field, which is not possible.
By choosing the gauge to be the unitary gauge group the number of scalar degrees of freedom becomes minimal from transforming the scalar fields re- sponsible for the BEH mechanism into a basis in which their Goldstone boson components are set to zero. The Goldstone bosons are often said to have been
”eaten” by the gauge boson. The unitary gauge introduces a new field χ(x) = − ξ
gv . (1.46)
The field φ can then be written as an approximation φ ≡ 1
√ 2 (v + η)e i
vξ(1.47)
that under U (1) transformation gives
φ → 1
√ 2 (v + η) ≡ 1
√ 2 (v + h). (1.48)
By also making the gauge transformation B µ → B µ + 1
gv ∂ µ ξ (1.49)
the Lagrangian becomes L = 1
2 (∂ µ h) (∂ µ h) − λv 2 h 2 + 1
2 g 2 B µ B µ − 1
4 F µν F µν + V int (h, B) (1.50) that contains a massiv scalar field h , a massive gauge field B µ , and the interaction term between the two fields h and B µ V int , which also includes h self-interactions.
1.6.2 Local SU (2) L × U (1) Y symmetry breaking
Since we only include left-handed particles in the SU (2) L gauge group, the nest step is to include two complex scalar fields in a weak isospin doublet, where upper and lower components differ by one unit of charge,
φ = φ +
φ 0
= 1 2
φ 1 + iφ 2 φ 2 + iφ 3
(1.51)
with the Lagrangian
L = (D µ φ) † (D µ φ) − µ 2 φ † φ − λ φ † φ 2
. (1.52)
The potential for this Lagrangian is called the Higgs potential. As previ- ously seen the choice of µ 2 < 0 must be made for the ground state to break the symmetry. In unitary gauge the field φ is expressed as
φ = 1
√ 2 0
v + h
(1.53)
where v 2 = − µ λ
2.
By replacing the derivative with the covariant derivative the Lagrangian has three massive gauge fields (W ± , Z 0 ), one massless gauge field (A µ for A) and a massive scalar field h . The masses of the fields are
m w = 1
2 gv, m Z = 1 2 v p
g 2 + g 02 , m H = √ 2λv.
The Lagrangian now contains all previously mention interactions, including
the new coupling between the Higgs-boson and the massive gauge bosons,
and the Higgs self-coupling.The coupling between the Higgs-boson and the gauge bosons are proportional to the square of the mass of the gauge boson.
H
W + , Z W − , Z
H H
W + , Z W − , Z
H H H
H H
H H
Figure 1.6: Elemental GWS vertices
H
f f ¯
Figure 1.7: Higgs to fermion (Yukawa) coupling
1.6.3 Fermion masses
So far the masses of the gauge bosons have been explained but what about the fermion masses? For the down type quarks, the lower element of the isospin dublet, the Lagrangian becomes
L = − g f v
√ 2
ψ ¯ L ψ R + ¯ ψ R ψ L
− g f h
√ 2
ψ ¯ L ψ R + ¯ ψ R ψ L (1.54)
which describes fermions with a mass of m f = g √
f2 v and the coupling shown in 1.7. The factor g f is called the Yukawa coupling constant. Both the Yukawa and the Higgs coupling constant contains the mass of the heavy gauge group meaning that the Higgs-boson favours coupling to the heavy particles.
To create mass for the up-type fermions the scalar isospin doublet is re- placed with φ c = −iσφ ∗ , which transforms in the same way as φ .
1.7 Full GWS Lagrangian
By now all the necessary tool to derive the full GWS Lagrangian has been developed, and it is time to put it all together. The full Lagrangian for the GWS theory is
L GWS = ¯ ψ L γ µ
i∂ m u − g
2 σW µ − g 0 Y 2 B µ
ψ L + ¯ ψ R γ µ
i∂ m u − g 0 Y 2 B µ
ψ R
− 1
4 B µν B µν − 1
4 W µν W µν +
i∂ m u − g
2 σW µ − g 0 Y 2 B µ
φ
2
− V (φ) − g f ψ ¯ L φψ R + G 0 f ψ ¯ L φψ R + h.c.
(1.55) where the first line contains couplings between the fermions and the gauge field and kinetic terms for fermions, the second line contains kinetic terms for the gauge fields and the BEH field, the coupling between the gauge field and the BEH field and the coupling between the gauge fields, while the third line contains the scalar potential, the Yukawa coupling and the fermion mass terms 6 .
1.8 Beyond the Standard Model
The Standard Model is the best model describing the physical universe we have, yet it only covers ≈ 5 % of the universe 7 , and have a lot of unanswered questions attached to it. Containing over 20 fee parameters, such as the masses of the quarks and leptons, that can only be extracted from experi- mental data, it is hard to accept it as a ”final” theory that will describe it all. In [23] an argument is made that a mature theory would presumably
6 Remember that ψ L is a isospin dublet while ψ R is a isospin singlet.
7 The rest is believed to be dark matter and dark energy.
explain these numbers, not only give them to us by experimental data. Still it withstands the test of time in searches for a new and better model.
Having many free parameters in the model is one problem, but there are also many others. For instance the Standard Model does not include grav- ity, it can not explain the hierarchy problem 8 or the fact that neutrinos oscillate between different states 9 The contribution from the CP-symmetry violation of the weak interaction is not enough to explain the matter-anti- matter asymmetry in the universe, and only counting for around 5% it can not alone account for the movements of stars in galaxies.
f
f
H H
W, Z
W, Z
H H
H
H
H H
Figure 1.8: Loop corrections
Grand Unification Theories (GUTs) predict that there should be unifica- tion of the forces at some energy, but the coupling constants in the Standard Model, called ”running constants”, depend on the energy scale of where the interaction takes place. This makes a unification of forces impossible as seen in figure 1.9.
8 Due to effects from loop corrections diverging for the Higgs boson in higher order perturbation theory it is hard to keep the Higgs boson mass on the electroweak scale of 10 2 GeV.
9 In the Standard Model neutrinos are massless and hence can not oscillate.
Figure 1.9: Coupling constants of Standard Model from [31]
There are many different theories trying to take the place of the Standard
Model, but yet non of them have succeeded. Some examples are string the-
ory, extra-dimensions, and Supersymmetry which will be more explained in
chapter ??.
Chapter 2
Supersymmetry
Most of this chapter is based on the source , while a more detailed insight to Supersymmetry can be found in .
2.1 Introduction to SUSY
One of the proposed expansions to the Standard Model is Supersymme- try (SUSY). Supersymmetry allows for a connection between fermions and bosons by introducing a new set for particles. Each SM-particle is given a new superpartner, a ”sparticle”, equal in all quantum numbers except a half unit difference in spin, i.e. the SM-fermions get a bosonic superpartner (denoted by an ”s” in front of the name, e.g. electron → selectron) while the SM-bosons get a fermionic superpartner (denoted by ”-ino” added to the end of the name, e.g. W → wino). All sparticles are denoted by a tilde above their symbol, e.g. selctron ( e ˜ ). The Higgs boson is accompanied by superpartners called higgsinos.
There are a few reasons to why SUSY is one of the contenders to be the needed expansion of the SM, and one attractive property is that we do not need to include any new gauge groups, hence no new fundamental forces. All that is needed is a supersymmetry operator Q that alters spin by 1/2, and commute with the gauge transformations of the Standard Model.
Supersymmetry could then be described as
Q|f ermions >= |bosons > Q|bosons >= |f ermions > (2.1)
Supersymmetry offers a solution to the hierarchy problem, and when im-
Figure 2.1: This figure shows the Feynman diagram of the process pp → ˜ l ˜ l → ll χ ˜ 1 0 χ ˜ 1 0 , also known as direct slepton production, that will be studied in this exercise.
posed as a local symmetry, gravity, in form of Einstein’s theory of general relativity, is included automatically.
If we assume R-parity to be conserved SUSY particles are produced in pairs and the lightest supersymmetric particle (LSP) arises as a stable dark-matter candidate.
SUSY can not be an exact symmetry, since the sparticles must have differ- ent mass then the particles of the SM. Or else, we would have been able to discover them at experiments all ready. This means that SUSY is a broken theory. If SUSY is a valid extension of the Standard Model, we know that it must be a broken symmetry where the particles and sparticles must have different masses although all the quantum numbers are equal. The sparticles must be heavier than their partner, or else we would already have been able to detect them in particle detectors, such as the ATLAS experiment.
The fundamental representation of the SM particles is gauge groups (colour triplets and weak isospin dublets) together with their superpartner is called supermultiplets. Left- and right-handed ferimons belong to different super- multiplets from the SM organisation in dublets and singlets, and they have distinct superpartners even though their scalar superpartners do not have handedness.
There exists many different Supersymmetric models.The potential that describes the dynamics of a SUSY model is referred to as a superpotensial.
2.2 MSSM theory
The Minimal Sypersymmetric Standard Model (MSSM) is the Supersymmet-
ric model used in this thesis. It introduces a minimal amount of new particles,
basically one new particle for each SM particle, and is pretty straight forward
until the Higgs sector . In the GWS theory explained in section we had 4 de-
grees of freedom (the W ± , the Z and the Higgs mass). A charged conjugated
field, φ C was used to give mass to the up-type . This can not be done in
SUYS, since the charge conjugate would destroy the invariance under gauge
and SUSY transformation . To give up-type mass in SUSY two new scalar
doublets must be introduced. This gives 8 degrees of freedom, which gives
W ± , Z, and 5 Higgs bosons. These 5 turns out to be two charged bosons, H ± , one heavy CP-even boson, H 0 , one heavy CP-odd boson, A 0 , and one light Higgs boson, h 0 .
In MSSM it is assumed that there is some spontaneous electroweak sym- metry breaking and in figure . The neutral gauge bosons (bino and neau- tral wino) mix with the neutral higgsino ( H ˜ 0 , h ˜ 0 ) to form four neutrali- nos ( χ ˜ 0 1 , χ ˜ 0 2 , χ ˜ 0 3 , χ ˜ 0 4 ). The charged winos and the charged higgsinos form the charginos ( χ ˜ ± 1 , χ ˜ ± 2 ). The left- and right-handed sleptons and squarks mix to form mass eigenstates different from the weak eigenstates, but the mixing is negligible for the first two generations.
In MSSM the superpotensial contains terms that looks to violate the con- servation of lepton and baryon number. We will not go into further details about the potensial, but this is a big problem. If this was the case, the proton can decay at an unusual rate which has been ruled out by experiments. To fix this dilemma another quantum number have been introduced, the R-parity.
This is a multiplicativ quantum number given as
P R = (−1) 3(B−L)+2s (2.2)
where B is the baryon number, L is the lepton number and s is spin. From this it is found that all particles have P R = +1 while all sparticles have P R = −1.
By requiring that R-parity must be conserved this leads to that all SUSY particles must be produced in pairs in particle collisions and that the LSP must be stable . If the LSP is massive and weakly interacting particle (WIMP ), this is a good candidate for the dark matter particle. MSSM can hence (maybe) solve the mystery of dark matter.
With running couplings MSSM have a unification of the fundamental forces. This comes from various SUSY particles showing up in loop-corrections and alternating the relative evolution of the three fundamental coupling con- stants. MSSM solves the hierarchy problem in a perfectly natural way. . The particle and sparticle loop-corrections nearly (in a mathematical sence) cancles each other out, meaning there is no more diverging Higgs mass loop- corrections.
MSSM solves many of the problems in the SM, is still not a ”ultimate”
theory of nature. In SM there is over 20 free parameters, but in MSSM there is a 124 in total, and it does not include gravity. How can this be better?
Well, SUSY may be an important step towards an ultimate theory. Proposed
quantum theories of gravity, such as string theory, needs SUSY.
2.2.1 Symmetry breaking in SUSY
In this section I will try to explain the why there is a difference in the mass of the SM particles and their superpartners. We know that if SUSY/MSSM is to solve the hierarchy problem sparticles can not be to heavy compared with particles. The mass of sparticles should not be much more than 1 TeV.
This is referred to as soft SUSY breaking.
Some unknown mechanism is believed to cause a spontaneous symmetry breaking, and it is expected that this mechanism will reduce the number of free parameters to only a few at some higher energy scale, i.e GUT scale .
Possible models for soft symmetry braking is supergravity (SUGRA). This model introduce a gravitational interaction between the MSSM field and fields in a hidden sector. The simplest of these models is minimal supergrav- ity (mSUGRA), which at GUT scale reduces the number of free parameters to only five:
m 0 , m
12
, A 0 , tan β, sgn(µ) (2.3) where m 0 is the common mass of scalar particles, m
12
