Supersymmetry vis-à-vis Observation
Weak-scale supersymmetry is one of the most favoured theories beyond the Standard Model of particle physics that elegantly solves various theoretical and observational problems in both particle physics and cosmology. In this thesis, I describe the theoretical foundations of supersymmetry, issues that it can address and concrete supersymmetric models that are widely used in phenomenological studies. I discuss how the predictions of supersymmetric models may be compared with observational data from both colliders and cosmology. I show why constraints on supersymmetric parameters by direct and indirect searches of particle dark matter are of particular interest in this respect. Gamma-ray observations of astrophysical sources, in particular dwarf spheroidal galaxies, by the Fermi satellite, and recording nuclear recoil events and energies by future ton-scale direct detection experiments are shown to provide powerful tools in searches for supersymmetric dark matter and estimating supersymmetric parameters. I discuss some major statistical issues in supersymmetric global fits to experimental data. In particular, I further demonstrate that existing advanced scanning techniques may fail in correctly mapping the statistical properties of the parameter spaces even for the simplest supersymmetric models. Complementary scanning methods based on Genetic Algorithms are proposed.
Key words: supersymmetry, cosmology of theories beyond the Standard Model, dark matter, gamma rays, dwarf galaxies, direct detection, statistical techniques, scanning algorithms, genetic algorithms, statistical coverage
Supersymmetry vis-à-vis Observation
Dark Matter Constraints, Global Fits and Statistical Issues
Doctoral Thesis in Theoretical Physics
|Department of Physics|
Doctoral Thesis in Theoretical Physics
Supersymmetry vis-à-vis Observation
Dark Matter Constraints, Global Fits and Statistical Issues
Oskar Klein Centre for Cosmoparticle Physics and Cosmology, Particle Astrophysics and String Theory Department of Physics Stockholm University SE-106 91 Stockholm Stockholm, Sweden 2011
Cover image: Artist’s impression of ‘supersymmetry vis-à-vis observation’. The image is composed of: (1) A real photograph from the dome of Hāfezieh (the tomb of the persian poet Hāfez) in Shiraz, Iran. The roof is decorated by enamelled mosaic tiles. Credit: The author’s father. (2) The cosmic microwave background (CMB) temperature fluctuations as observed by the Wilkinson Microwave Anisotropy Probe (WMAP). Credit: NASA/WMAP Science Team.
ISBN 978-91-7447-312-4 (pp. i–xx, 1–142)
pp. i–xx, 1–142 © Yashar Akrami, 2011
Printed by Universitetsservice US-AB, Stockholm, Sweden, 2011.
Typeset in pdfLaTeX
In eternity without beginning, the splendor-ray of Thy beauty boasted
Revealed became love; and, upon of the world, fire dashed.
From that torch, reason wanted to kindle its lamp
Jealousy’s lightning flashed; and in chaos, the world dashed.
The Persian Poet, Hāfez (1325/26-1389/90)
Svensk sammanfattning\@mkbothSvensk sammanfattningSvensk sammanfattning
Supersymmetri är en av de mest välstuderade teorierna för fysik bortom standardmodellen för partikelfysik. Den löser på ett elegant sätt flera teoretiska och observationella problem inom både partikelfysik och kosmologi. I denna avhandling kommer jag att beskriva de teoretiska fundamenten för supersymmetri, de problem den kan lösa och konkreta supersymmetriska modeller som används i fenomenologiska studier. Jag kommer att diskutera hur förutsägelser från supersymmetriska modeller kan jämföras med observationella data från både partikelkolliderare och kosmologi. Jag visar också varför resultat från direkta och indirekta sökanden efter mörk materia är särskilt intressanta. Observationer av gammastrålning från astrofysikaliska källor, i synnerhet dvärggalaxer med Fermi-satelliten, samt kollisioner med atomkärnor i kommande storskaliga direktdetektionsexperiment är kraftfulla verktyg i letandet efter supersymmetrisk mörk materia och för att bestämma de supersymmetriska parametrarna. Jag diskuterar några statistiska frågeställningar när man gör globala anpassningar till experimentella data och visar att nuvarande avancerade tekniker för att skanna parameterrymden ibland misslyckas med att korrekt kartlägga de statistiska egenskaperna, även för de enklaste supersymmetriska modellerna. Alternativa skanningsmetoder baserade på genetiska algoritmer föreslås.
List of Accompanying Papers
First and foremost, I would like to thank my supervisor Joakim Edsjö for his excellent guidance, encouragement and enthusiastic supervision especially during the completion of this thesis. Thanks also to my secondary supervisor Lars Bergström for his valuable advices, generous support and providing the opportunity of pursuing my academic interests and goals. Thanks to both of them also for understanding my situation as a foreigner here in Sweden and for their pivotal helps in resolving intricate life-related problems. Many thanks also to Jan Conrad whose various helps and guidance have been crucial for the successful completion of this work. I am also grateful to him for invaluable non-physics advices that will certainly have indisputable influence on my future career. Jan, I do not forget the nice discussions we had during the visit to CERN.
Many thanks to all other professors and senior researchers at Fysikum, Department of Astronomy and KTH for sharing their invaluable knowledge and expertise with me. Thank you Marcus Berg, Claes-Ingvar Björnsson, Claes Fransson, Ariel Goobar, Fawad Hassan, Garrelt Mellema, Edvard Mörtsell, Kjell Rosquist, Felix Ryde, Bo Sundborg, Christian Walck and Göran Östlin. Special thanks to Marcus Berg for bringing to our group a new and highly enthusiastic ambiance to learn and discuss interesting aspects of high energy physics and cosmology. My warmest thanks to Fawad Hassan for being an excellent teacher and a good friend, and for his great willingness and patience in answering my endless questions. I am grateful to Ulf Danielsson and Stefan Hofmann for broadening my knowledge in theoretical physics with exciting discussions and ideas that made me think about ‘other’ possibilities. I also thank Hector Rubinstein for all the nice conversations I had with him. Although he is no longer with us, he will always be in my mind.
My thanks also to the CoPS, HEAC and guest students Karl Andersson, Michael Blomqvist, Jonas Enander, Michael Gustafsson, Marianne Johansen, Joel Johansson, Jakob Jönsson, Natallia Karpenko, Maja Llena Garde, Erik Lundström, David Marsh, Jakob Nordin, Narit Pidokrajt, Anders Pinzke, Sara Rydbeck, Angnis Schmidt-May, Pat Scott, Sofia Sivertsson, Alexander Sellerholm, Stefan Sjörs, Mikael von Strauss, Tomi Ylinen, Stephan Zimmer and Linda Östman, the CoPS and OKC postdocs Rahman Amanullah, Torsten Bringmann, Alessandro Cuoco, Tomas Dahlen, Hugh Dickinson, Malcolm Fairbairn, Gabriele Garavini, Christine Meurer, Serena Nobili, Kerstin Paech, Antje Putze, Are Raklev, Joachim Ripken, Rachel Rosen, Martin Sahlen, Chris Savage, Vallery Stanishev and Gabrijela Zaharijas, and all other current or former students and postdocs that I may have forgotten to enumerate here. I have definitely benefited from all the conversations and discussions I have had with them and enjoyed every second I have spent with them. Special thanks to Pat and Chris for good times in the office and for all I have learned from collaborating with them.
I would also like to thank Ove Appelblad, Stefan Csillag, Kjell Fransson, Mona Holgerstrand, Marieanne Holmberg, Elisabet Oppenheimer and all other people in administration for their valuable helps over the last few years.
Thanks also to the Swedish Research Council (VR) for making it possible for me and all my colleagues at the Oskar Klein Centre for Cosmoparticle Physics to work in such a work-class and highly prestigious institution.
Thanks to my parents and sister for all their continous encouragement and unconditional support. ‘Baba’ & ‘Maman’ thank you for all troubles you endured stoically over the years. What I learned from you was all eagerness for truth, integrity and wisdom. Thanks to you Athena for being such a kind and supportive sister.
And last but not least, thanks to you Mahshid for all the confidence, independence and strength you have shown in me, for all your support and encouragement and for all great moments we shared over the last four and a half years of my life.
This thesis deals with the phenomenology of weak-scale supersymmetry and strategies for comparing predictions of supersymmetric models with different types of observational data, in particular the ones related to the identification of dark matter particles. Currently, various experiments, either terrestrial, such as colliders and instruments for direct detection of dark matter, or celestial, such as cosmological space telescopes and dark matter indirect detection experiments, are providing an incredibly large amount of precise data that can be used as valuable sources of information about the fundamental laws and building blocks of Nature. Analysing these data in statistically consistent and numerically feasible ways is now one of the crucial tasks of cosmologists and particle physics phenomenologists. There are several issues and subtleties that should be addressed in this respect, and dealing with those form the bulk of the present work.
The papers included in this thesis can be divided into two general categories: Some (Paper I and Paper III) mostly aim to illustrate how real data can be used in constraining supersymmetric and/or other fundamental theories, and others (Paper II and Paper IV) are more about whether existing statistical and numerical tools and algorithms are powerful enough for correctly comparing theoretical predictions with observations.
This thesis is organised as follows. It is divided into three major parts: Part I is an introduction to the theoretical and statistical backgrounds relevant to my work, Part II summarises the main results we have obtained in our investigations and Part III presents the included papers. Part I is itself divided into 7 chapters: Chapter 1 is a short and non-technical introduction to the field and the main motivations for investigating models of physics beyond the Standard Model of particle physics in particular supersymmetry, Chapters 2 and 3 discuss the motivations for considering supersymmetry as a possible underlying theory of Nature in more detail and in demand for explaining both the dark matter problem in cosmology and theoretical issues with the Standard Model, Chapter 4 introduces supersymmetry and its theoretical foundations in a top-down approach and in a rather technical language, Chapter 5 details the most interesting supersymmetric models that are being used in current phenomenological studies, Chapter 6 provides a review of different observational sources of information that can constrain supersymmetric models and parameters, and Chapter 7 describes statistical frameworks and techniques for analysing supersymmetry.
Almost all the included papers are written in rather comprehensive, self-contained and self-explanatory manners. Therefore, in order to avoid any unnecessary repetitions, I have written the introductory chapters such that they provide in a rather consistent and coherent way a more general and detailed description of the field to which the papers contribute. This also provides some additional background material that may not have been discussed in detail in the papers. The reader is therefore strongly recommended to consult the papers for more advanced and technical discussions.
Contribution to papers
Paper I focuses on potential experimental constraints one may place upon supersymmetric models from indirect searches of dark matter (this has been done for the particular case of the Constrained Minimal Supersymmetric Standard Model (CMSSM) as the model, and gamma-ray observations of the dwarf galaxy Segue 1 as the data). We have assumed that the lightest neutralino is the dark matter particle that annihilates into gamma rays observable by our detectors. The instrument for observations is the Large Area Telescope (LAT) aboard the Fermi satellite. Conventional state-of-the-art Bayesian techniques are employed for the exploration of the CMSSM parameter space and the model is constrained using the LAT data alone and also together with other experimental data in a global fit setup. In preparing and writing the paper, I was mostly involved in general discussions and edition of the manuscript. I also helped Pat Scott in setting up SuperBayeS for the numerical calculations.
Paper II deals with the issue of efficiently scanning highly complex and poorly-understood parameter spaces of supersymmetric models. It attempts to introduce a new scanning algorithm based on Genetic Algorithms (GAs) that is optimised for frequentist profile likelihood analyses of such models. In addition to comparing its performance with that of the conventional (Bayesian) methods and illustrating how our results can affect the entire statistical inference, some physical consequences of the results (in terms of the implications for the Large Hadron Collider (LHC) and dark matter searches) are also presented and discussed. The analyses are done for a global fit of the CMSSM to the existing cosmological and collider data. I have been the main author of the paper. The use of Genetic Algorithms for exploration of supersymmetric parameter spaces was to a great extent my own initiative. I modified SupeBayeS and added GA routines to it. I did the numerical calculations, analysed the results and produced the tables and figures. I wrote most of the text.
Paper III aims to predict how far one can go in constraining supersymmetric models with future dark matter direct detection experiments. The methodology and the main strategy of the paper are very similar to the analysis of Paper I: The studied supersymmetric model is the CMSSM and nested sampling is used as the scanning technique. Both profile likelihoods and marginal posteriors are presented. I have been the main author of the paper, performed the numerical scans, analysed the results and produced tables and plots. Christopher Savage also significantly contributed to the work by providing the background material for direct detection theory and experiments, as well as preparing the likelihood functions for the experiments that I used in the analysis.
Paper IV studies a rather technical issue in the statistical investigations of supersymmetric models, namely the coverage problem. The analysis of this paper was computationally very demanding and required a substantial amount of computational power; this made the project a rather lengthy and challanging one. I have been the main author for this paper as well. I wrote most of the text and produced the results and all plots and tables. The numerical likelihood function for the analysis was provided by Christopher Savage, but I performed all the scans and interpreted the results.
Stockholm, April 2011
- Svensk sammanfattning\@mkbothSvensk sammanfattningSvensk sammanfattning
- List of Accompanying Papers
- 1 Why dark matter and why go beyond the Standard Model?
- 2 The cosmological dark matter problem
- 3 Theoretical issues with the Standard Model
- 4 Theoretical foundations of supersymmetry
5 Supersymmetry in real life
- 5.1 The Minimal Supersymmetric Standard Model
- 5.2 … and beyond
- 5.3 Shrinking the parameter space
6 Observational constraints on supersymmetry
- 6.1 Supersymmetric WIMPs
- 6.2 Collider constraints
- 7 Statistical subtleties
- II Summary and outlook
- III Papers
Part I Introduction
Chapter 1 Why dark matter and why go beyond the Standard Model?
The visible Universe that we know and love is made up of planets, stars, galaxies and clusters of galaxies. We know that these objects exist mostly because they emit light or other types of electromagnetic radiation which we detect either by eye or by various telescopes. In addition, the celestial objects substantiate their existence through their gravitational effects which impact the motions of other objects in their vicinity. For most nearby astrophysical objects the two sources of information fairly agree and are therefore used as complementary ways in studying interesting properties of their sources. A problem emerges however when we look at scales of the order of galaxies or larger, where the gravitational effects imply the presence of massive bodies that are not detected electromagnetically. These objects that exhibit all the gravitational properties of normal matter but do not emit electromagnetic radiation (and are therefore invisible) are referred to as ‘dark matter’ (DM).
Almost every attempt at explaining the nature of DM with the known types of matter has so far failed. This is mainly because DM seems to be required in order to consistently explain very different astrophysical phenomena that have been observed by completely different methods. This inevitably leads us to the assumption that DM is composed of new types of matter that are beyond our current understanding of the elementary particles and their interactions.
Our present knowledge of the fundamental building blocks of the Universe is summarised in the so-called Standard Model (SM) of particle physics (for an introduction, see e.g. ref. ). The SM provides a mathematically consistent (though rather sophisticated) framework for describing different phenomena in a relatively large range of energy scales. At low energies the model describes the everyday life processes in terms of normal atoms, molecules and chemical interactions between them, and at high energies it has been capable of explaining various processes observed in nuclear reactors, particle colliders and high-energy astrophysical processes with remarkably high precision. The SM is a quantum-mechanical description of particles (or fields) and is based on a particular theoretical framework called quantum field theory.
The SM is now extensively tested at colliders and is in excellent agreement with the current data. However, as we stated earlier, the SM does not contain any type of matter with properties similar to the ones we need for DM. This simply implies that if DM exists, the SM has to be appropriately modified or extended so as to include DM particles with required properties. The need for DM is therefore one of the strongest motivations for going ‘beyond’ the SM.
Apart from the lack of any DM candidates in the SM, there are additional reasons in support of the existence of new physics beyond this framework. These reasons are mainly motivated by some theoretically irritating characteristics of the model that cannot be explained otherwise. Perhaps the most notorious one is that the SM does not contain gravity. Currently four different type of force have been known in Nature: the gravitational force between massive objects, the electromagnetic force between charged particles, the strong force that put together neutrons and protons inside atomic nuclei, and the weak force which is responsible for radioactive processes. While three of these forces, i.e. electromagnetic, strong and weak are well described quantum mechanically by the SM, the gravitational interactions do not fit consistently into the model. The reason is that when one attempts to quantise gravity with the known mathematical methods of quantum field theory, the resulting theory contains some infinities that cannot be removed in an acceptable manner. This is done for the other interactions through the so-called ‘renormalisation’ procedure, a method that breaks down for gravitational interactions. We are therefore forced to treat gravity as a classical field which is best described by Einstein’s theory of general relativity. This distinction between gravity and the other forces does not lead to serious problems provided that we do not want to describe gravitational processes at high energies where the quantum effects become important. There are however interesting high-energy cases where one needs to have a quantum-mechanical description of gravity so as to be able to study the physical systems. Two important examples are (1) extreme objects such as black holes and (2) the physics of the very early Universe. It is therefore commonly accepted that the SM must be modified at least at those high energies where gravity needs to be quantised.
In addition, the SM possesses a very special mathematical structure that is based on particular types of fields and symmetries. This structure, although being crucial for the model to successfully describe different phenomena in particle physics, does not find any explanation within the theoretical principles of the model. The model also contains some free parameters, such as masses and couplings whose values have been determined experimentally. Some of these parameters take on values that require extensive fine-tuning. All these aesthetically vexatious issues and a few more give us strong hints that the SM is not the fundamental description of Nature and has to be appropriately extended.
Fortunately, several interesting extensions for the SM exist, the best of which are those that address all or most of the aforementioned issues simultaneously. One of these proposals is weak-scale supersymmetry. It is a very powerful framework in which the SM is conjectured to be modified by some new physics that kicks in at energies just above the electroweak scale, i.e. the energy scale at which the electromagnetic and weak forces are assumed to be unified into one single electroweak force. This new physics assumes that all particles of the SM are accompanied by some partner particles that are more massive than the original ones. The existence of these so-called superpartners provides elegant solutions to many of the problems listed above, and paves the way for the resolution of many others in some broader theoretical framework. An important example is the inclusion of new matter fields with properties similar to what we need for a viable DM candidate.
Supersymmetric models, like any other theories in physics, need to be tested experimentally. Indeed, there have been many theoretically fascinating ideas in the history of physics that were abandoned only because they have not been consistent with particular experimental data. Fortunately, there are various sources of information from both man-made experiments, such as particle colliders, and astrophysical/cosmological observations that can be used for testing the supersymmetric models. Ideally, all these different types of data should be combined appropriately so as to give the most reliable answers to our questions about the validity of particular models and frameworks. This is however not a trivial task, because there are usually various sources of complication and uncertainty that enter the game and, if not addressed properly, can make any interpretations completely unreliable. This is exactly where the main objectives of the present thesis stand. We would like to examine how a class of interesting supersymmetric models can be compared with observations in the presence of different experimental (and theoretical) uncertainties and statistical/numerical complications.
First, in the following two chapters we give a more thorough (and more technical) description of the problems with the SM, including the need for DM. In each case, we describe in rather general terms how the problem finds appropriate solutions in supersymmetry. The detailed resolutions of some of the problems will be discussed later when supersymmetry is defined and concrete supersymmetric models are presented in chapters 4 and 5, respectively. In chapter 6 we review important observational constraints we have employed in our analyses and describe different uncertainties in each case. Chapter 7 will be devoted to a discussion of the main statistical and numerical issues that we have dealt with in our endeavour. In the last chapter, i.e. chapter 8, we will briefly review our major results and present an outlook for future work.
Chapter 2 The cosmological dark matter problem
2.1 The standard cosmological model
The standard model of cosmology (for an introduction, see e.g. refs. [2, 3]) is a mathematical framework for studying the largest-scale structures of the Universe and their dynamics. In other words, cosmologists attempt to answer various fundamental questions about the origin and evolution of the cosmos using the fundamental laws of physics. The model is based on Einstein’s theory of general relativity as the currently best description of gravity at the classical level, as well as two important assumptions about the distribution of matter and energy in the Universe that are usually called together cosmological principles: the homogeneity and isotropy on large scales. The cosmological principles immediately imply that the correct metric for the Universe has to be of a particular form that is known as Friedmann-Lemaître-Robertson-Walker (FLRW) metric and has the following form:
Here , and denote the spherical coordinates and is time. as a function of time, is called the scale factor of the Universe and is an unknown function that can be determined by solving the Einstein field equations
Here is the Einstein tensor which contains all geometric properties of spacetime and is the stress-energy-momentum tensor (or simply stress-energy tensor) that includes the information about various sources of matter and energy on that spacetime. is Newton’s gravitational constant. The time evolution of therefore depends upon the assumptions we make for the matter and energy content of the Universe. is called the curvature parameter and depending on its value, the Universe may be closed, open or flat (corresponding to , and , respectively).
The assumption for the stress-energy tensor on the right-hand side of Eq. 2.1 is that the matter and energy of the Universe can be well described by a perfect fluid that is characterised by two quantities (its energy density) and (its pressure). By inserting the stress-energy tensor for such a fluid, , into Eq. 2.1 we end up with the following simple equations:
By solving these so-called Friedmann equations, one can obtain the dynamics of the Universe in terms of the time evolution of the scale factor . The quantity on the left-hand side of the first equation that gives the expansion rate is called Hubble parameter . In order to solve Eqs. 2.0, it is essential to also know how and are related, i.e. what the equation of state (EoS) is for the perfect fluid. For normal non-relativistic matter, the energy density is much larger than the pressure and one can therefore reasonably assume that the EoS is simply . For relativistic matter (or radiation) on the other hand , and for the vacuum energy (vacuum energy can be effectively written in terms of a cosmological constant in which case ).
In cosmology it is useful to write the various energy density contributions to the total density (at present time) in terms of the so-called density parameters , , and for matter, radiation and vacuum, respectively. The same is usually done for the curvature term in the first Friedmann equation by defining in an analogous way. These density parameters are defined as the ratio of a density at present time () to a specific quantity called the critical density . (defined as , where is the present value of the Hubble parameter) is the density for which the Universe has an exact flat curvature:
The first Friedmann equation in Eqs. 2.0 can be written in the following simple form in terms of the density parameters:
where is the redshift with being the present value of the scale factor usually taken to be . There are various ways to measure the Hubble parameter as a function of time from which one can determine the values for different density parameters and therefore the energy budget of the Universe.
Thanks to different high-precision cosmological observations, we have now been able to not only confirm the relative validity of our standard cosmological model, but also determine the values of different parameters that enter the mathematical formulation of the model to a high degree of accuracy. We now know that (see e.g. Fig. 2.1) the Universe started from an extremely hot and dense state about billion years ago (a state that we call the Big Bang) and then expanded, cooled down and became structured by galaxies, stars and other astrophysical objects. We also know that the curvature of the Universe is, to a good approximation, flat and also that it has recently entered an accelerated expansion phase. Although we still need a quantum theory of gravity to understand what exactly happened in the very early moments of the cosmic evolution, we have been able to infer some properties of the Universe at those times. For example there are various reasons to believe that shortly after its birth the Universe has seen a short inflationary phase during which its size has grown exponentially: (1) The Universe is (at least approximately) flat. (2) The observed cosmic microwave background radiation (i.e. the relic radiation from the recombination epoch at which photons that were originally in thermal equilibrium with matter could escape the equilibrium and freely travel in the Universe) is to a great degree isotropic. (3) The Universe is not perfectly homogeneous and structures exist. All these features can be gracefully explained by inflation. The underlying mechanism for inflation is yet to be understood, but the evidence for its occurrence is so strong that it has now become one of the main paradigms of modern cosmology.
2.2 The need for dark components
Perhaps the best confirmation of our cosmological picture to date has been from observations of the cosmic microwave background (CMB). It is extremely difficult (if not impossible) to explain the black-body spectrum of the CMB with alternative cosmological models. The measurements performed by the NASA satellite Wilkinson Microwave Anisotropy Probe (WMAP) have played a central role in this direction . Not only have such measurements confirmed the fact that the Big Bang theory is a successful description of the Universe, they have also determined the actual values of the density parameters we introduced in the previous section. By fitting the model to the so-called angular power spectrum of the CMB for the tiny temperature fluctuations observed on the 7-year WMAP sky map (see e.g. Fig. 2.2), it is now known that, for example, and .
The first surprising observation is that the vacuum energy (or the cosmological constant) is non-zero and even constitutes about of the total energy budget of the Universe. A similar number was for the first time reported in 1998 by two different measurements of the so-called luminosity distance (a quantity that is defined in terms of the relationship between the absolute magnitude and apparent magnitude of an astronomical object and can be calculated theoretically for a cosmological model in terms of the Hubble parameter for an object with a specific redshift) using Type Ia supernovae (SNe) [5, 6]. The first explanation for this energy component that implies a recent transition of the Universe to an accelerated expansion epoch was that it is just a cosmological constant. From a particle physics point of view, however, the vacuum energy density of the SM contributes to the cosmological constant and hence affects the expansion history of the Universe. But the value estimated in this way is much larger than the observed one and this poses a serious problem that cannot be explained within the SM . It was then proposed that perhaps some new physics has made such contributions from the vacuum energy small (or zero) and what we observe cosmologically is not the cosmological constant but rather a new energy source (with an EoS parameter that is not identically equal to ) that can be detected only gravitationally (hence the name dark energy). There are numerous suggestions for the nature of the dark energy, most of which come from particle physics theories beyond the SM (for a review, see e.g. ref. ).
Although the WMAP results imply that normal matter (with the EoS of ) forms about of the total energy density, the surprise comes from the value it has measured for the energy density of baryonic matter in the Universe. This is the matter that is composed mainly of baryons and includes all types of atoms we know. The baryons’ energy density can be measured because the CMB angular power spectrum is sensitive directly to the amount of baryonic matter: While the location of the first peak (see Fig. 2.2) gives us information about the total amount of matter, i.e. , the second peak tells us about the total amount of baryonic matter . Estimations then determine to be . Comparing the values for and indicates that the usual baryonic matter constitutes only about of the energy content of the Universe and about is non-baryonic (see Fig. 2.3). All baryons interact with photons and can be detected also through non-gravitational effects whereas the non-baryonic component has been detected only gravitationally and is therefore named dark matter. In order to agree with observations of large-scale structure of the Universe, this non-baryonic dark matter must be dominantly cold (i.e. almost non-relativistic). This cold dark matter (CDM) together with the assumption that dark energy is nothing but the cosmological constant , a hypothesis that is in excellent agreement with all existing observations, contrives the foundations of our current standard model of cosmology that is accordingly called CDM.
The left panel of Fig. 2.4 shows the currently best constraints on the energy densities of matter and dark energy from three important types of cosmological observations, i.e. the CMB, Type Ia SNe and baryon acoustic oscillations (BAO) . The latter refers to an overdensity of baryonic matter at certain length scales due to acoustic waves that propagated in the early Universe. BAO can be predicted from the CDM model and compared with what we have observed from the distribution of galaxies on large scales. The right panel of Fig. 2.4 depicts constraints from the same set of data but in terms of versus the equation of state parameter for dark energy ( is for dark energy being the cosmological constant). By looking at both plots, it is quite interesting to see that the constraints from all these three sources of information are in perfect agreement with each other and also consistent with our theoretical model. This model is also in harmony with many other observations (such as constraints from Big Bang Nucleosynthesis (BBN) on the baryon density , gravitational lensing  and X-ray data from galaxy clusters ), and is accordingly called the concordance model of cosmology.
The argument for the existence of dark matter, i.e. the mass density that is not luminous and cannot be seen in telescopes, is actually very old. Zwicky back in 1933 already reported the “missing mass” in the Coma cluster of galaxies by studying the motion of galaxies in the cluster and using the virial theorem . A classic strong evidence for dark matter existing in the scale of galaxies comes from the study of rotation curves in spiral galaxies by Rubin [14, 15, 16, 17]. The observed rotation curves are not consistent with the standard theoretical assumptions unless one assumes the existence of dark matter halos surrounding all known contents of the galaxies, i.e. stars and gas (for an example, see e.g. Fig. 2.5).
We should note here that some alternative explanations have been put forward that claim the anomalous observational data do not necessarily lead to the conclusion that dark matter exists. Some of these alternative proposals, such as the ones in the context of modified Newtonian dynamics (MOND) [19, 20], have been successful in for example explaining the rotation curves of spiral galaxies (although in a rather ad hoc way). As we saw, the dark matter problem is not limited to astrophysical phenomena on particular scales and shows up in different observations from the scale of a galaxy to cosmological scales. It is in fact extremely difficult to explain all those observations without dark matter.
Perhaps the best direct evidence for the existence of dark matter is the so-called Bullet Cluster (see Fig. 2.6). The Bullet Cluster consists of two galaxy clusters that have recently collided. Fig. 2.6 is a composite picture that shows (apart from the optical image) two types of observations of the cluster: gravitational lensing (in blue) and X-ray observations (in pink). Comparing these two cases evidently show that the baryonic gas component, which emits X-ray radiation, does not form the total mass of the cluster. Most of the mass, mapped by the lensing measurement, seem to come from a component that, in contract with the baryons, is collisionless: it does not interact with either baryonic gas or itself. These properties are all consistent with the assumption of dark matter.
2.3 Weakly Interacting Massive Particles
The astrophysical/cosmological observations we discussed in the previous section all imply that dark matter probably exist. The next question we need to answer is what is the nature of dark matter, i.e. what are the basic constituents of it. We have already inferred some of the properties the dark matter components should possess: (1) They must be massive otherwise we would not have seen their gravitational effects. (2) They must be dark, i.e. they should not emit or absorb electromagnetic radiation (at least not noticeably), otherwise they would have already been detected by our telescopes. (3) They must be non-baryonic (confirmed by e.g. the observations of CMB anisotropies and BBN). (4) They must be effectively collisionless with respect to both normal matter and themselves, otherwise they would loose energy through electromagnetic (or stronger) interactions and form dark matter disks (which contradicts the observations of galactic rotation curves). Observations of astrophysical systems like the Bullet Cluster could also not be explained in this case. (5) Dark matter must be cold(ish) (i.e. almost non-relativistic), otherwise it would have not given rise to proper structure formation as we observe on cosmological scales. (6) It must be stable or at least very long-lived (compared to the age of the Universe); this is required because dark matter comprises a significant fraction of the total energy of the Universe at the present time (this fraction is given in terms of the dark matter relic abundance ).
Unfortunately, all attempts at finding a suitable dark matter candidate in the framework of the SM of particle physics have so far failed. This is because there are no standard particles that can satisfy all the requirements we listed above, and this means that cosmology requires new particles. This takes us to the realm of particle dark matter, namely that dark matter is composed of some new particles that have not been discovered yet. The need for particle dark matter is one of the main motivations for us to go beyond the SM (for detailed introductions to particle dark matter, see e.g. refs. [23, 24, 25]).
Fortunately, several viable dark matter candidates have been proposed in the literature (for a review, see e.g. ref. ) and most of the interesting ones fall into the class of Weakly Interacting Massive Particles (WIMPs). WIMPs are particles that couple to the SM particles only through interactions that are of the order of the weak nuclear force (or weaker). This immediately tells us that WIMPs are electrically neutral, dark, effectively collisionless and non-baryonic. They are also massive, usually with masses within a few orders of magnitude of the electroweak scale. Having high enough masses also means that they are cold. WIMPs are also stable on cosmological timescales and this characteristic comes from a (usually imposed) discrete symmetry of the theory that gives WIMPs some conserved quantum number. This quantum number then prevents WIMPs from decaying into other particles and therefore makes them stable. In most scenarios, WIMPs are produced thermally in the early Universe [27, 28, 29, 30]. A generic (and highly interesting) feature of thermally-produced WIMPs is that they naturally provide the correct relic density of dark matter (), i.e. a value that is in excellent agreement with observations. We explain this intriguing feature in more detail below.
In the early Universe, right after the Big Bang, all the created particles (including WIMPs) are in both chemical and thermal equilibrium. Here chemical equilibrium refers to the situation where the primordial particles are created and destructed with almost equal rates and no net changes in their abundances with time. On the other hand, by thermal equilibrium (which is also called kinetic equilibrium) we mean that the particles are in thermal contact with each other without a net exchange of energy. In this latter case the temperatures associated with the particles follow the global temperature of the Universe.
Suppose that the number density associated with our hypothetical WIMP particles is , their relative velocity is and they annihilate into lighter particles with the total annihilation cross-section . The equation governing the evolution of the WIMP density is the Boltzmann equation 
where is the equilibrium number density of the WIMPs, is the Hubble parameter and the brackets denote thermal average. For WIMPs with the mass , the equilibrium number density (in the non-relativistic limit) at the temperature reads
where is the number of degrees of freedom associated with the species .
A direct implication of Eq. 2.0 is that as long as the creation and annihilation of the WIMPs is larger than (or comparable with) the expansion rate of the Universe (specified by the Hubble parameter), the particles remain in chemical equilibrium. However, the Universe expands and cools, and this means that at some time and temperature, the interaction rate drops below the expansion rate and the equilibrium can no longer be maintained. This process during which the WIMPs decouple from the other particles is called chemical ‘freeze-out’. The number density of such thermally-produced WIMPs at the end of chemical freeze-out determines the relic density of dark matter today. Obviously, the abundance of WIMPs at freeze-out (and consequently the dark matter relic density) depends on how large the annihilation cross-section is: Larger cross-sections cause the WIMPs to remain in chemical equilibrium for a longer period and therefore generate a lower relic density (see Fig. 2.7).
Chemical freeze-out happens at a temperature that for WIMPs with weak-scale masses is given approximately as . After chemical freeze-out, WIMPs still remain in thermal contact with the other particles for some time and kinetic freeze-out (or decoupling) happens later. The temperature of the WIMPs before this time is the same as the equilibrium temperature, and becomes fixed by kinetic decoupling afterwards. This means that the WIMPs will have a temperature lower than after kinetic freeze-out and this makes the WIMPs move non-relativistically up to the present moment. This characteristic is crucial for WIMPs to be ‘cold’ dark matter.
In order to obtain the relic density of WIMPs , one needs to solve Eq. 2.0 numerically. However, to a first-order approximation, it can be shown that under very general assumptions does not depend explicitly on the WIMP mass and only depends on its annihilation cross-section [23, 31] in the following way:
where . For weakly-interacting particles with reasonable masses (i.e. with values close to the scale of the electroweak symmetry breaking), the quantity can be estimated as , where is the fine structure constant. Assuming a typical value of GeV for the WIMP mass, we obtain . By inserting this value into Eq. 2.0, we obtain an approximate value for with the right order of magnitude. This interesting ‘coincidence’, also often referred to as ‘the WIMP miracle’, means that, under the assumption of chemical freeze-out as the actual dark matter production mechanism occurred in the early Universe, any particles with generic properties of WIMPs can provide a dark matter relic density of the correct order. This particular characteristic of WIMPs makes them amongst the most interesting and popular dark matter candidates.
There are a large number of WIMP dark matter candidates on the market proposed in different contexts , amongst which the lightest neutralino in supersymmetry [31, 32, 33], the lightest Kaluza-Klein particle in models of Universal Extra Dimension (UED)  and the lightest inert scalar in the Inert Doublet Model (IDM) [35, 36] are the most widely-studied ones. The first one, i.e. the lightest neutralino provides arguably the leading dark matter candidate with almost all desired properties. A substantial part of this thesis is devoted to the phenomenological aspects of the neutralino with particular emphasis on its implications for constraining models of weak-scale supersymmetry.
Before we end this section, let us emphasize that although WIMP dark matter proves to be an extremely powerful idea that provides extensive scope for phenomenological studies of particle dark matter, there are a number of other viable dark matter candidates that are either entirely non-WIMP or only WIMP-inspired. We do not intend to go through any of them here and just provide a list of the most interesting ones and refer the reader to the given references for detailed discussions (see also ref.  for a comprehensive review): axions [37, 38], gravitinos , axinos , sterile neutrinos , WIMPzillas , Minimal Dark Matter [43, 44], Inelastic Dark Matter (iDM) [45, 46], eXciting Dark Matter (XDM) , WIMPless dark matter [48, 49] and models with Sommerfeld enhancement [50, 51].
Chapter 3 Theoretical issues with the Standard Model
As stated earlier in chapter 1, the Standard Model of particle physics is currently the minimal mathematical description of all known matter particles and their interactions that consistently explains various experimental observations, and holds over a wide range of energies. This includes phenomena that we observe in our everyday experiments (i.e. energies of the order of a few eV), as well as the ones that can be observed only at high-energy colliders and astrophysical processes (i.e. energies of GeV). The only key ingredient of this mathematical framework that still needs to be confirmed experimentally is the Higgs boson which is thought to be responsible for giving masses to the other particles. There are however alternative proposals for making the particles massive that although not excluded yet, are arguably less motivated (see e.g. refs. [52, 53] for one of the most competitive ones). Having said that, it became relatively manifest soon after its establishment in the 1970s that for purely theoretical reasons the SM is incomplete and probably not the end of the story. It therefore has to be modified or extended beyond certain energies (which are argued to be energies higher than TeV scales).
As we discussed in the previous chapter, the need for a viable dark matter candidate is one pivotal reason for thinking about extensions of the SM. We advertised supersymmetry as one of the leading theories beyond the SM that provides such candidates. However, the nice thing with supersymmetry is that it also helps us circumvent many of the theoretical issues with the SM that are not related to the dark matter problem.
Before we introduce supersymmetry and review supersymmetric models, their properties and phenomenological implications in chapters 4, 5 and 6, we remind ourselves in this chapter of some of the most notable theoretical problems in the SM and corresponding arguments in support of the physics beyond the SM, in particular supersymmetry. Clearly without describing its mathematical foundations and concrete realisations in particle physics, we cannot discuss in detail how supersymmetry helps us address these problems. We will therefore come back to some of the issues raised here in chapter 5 and explain how they can be gracefully resolved in some interesting supersymmetric models.
3.1 The gauge hierarchy problem
In any quantum field theory, including the SM, all present parameters (such as masses and coupling constants) are affected by quantum radiative corrections. The amount of the corrections is generically a function of the cut-off scale that is used in the process of renormalising the theory or removing the divergences arising from various loop integrals. For the case of fermions (i.e. particles with half-integer spin) interacting with photons, the radiative corrections to the fermion masses have a logarithmic dependence on the cut-off scale (here we use for a Lorentz-invariant cut-off): (see e.g. refs. [54, 55, 56] for a detailed discussion). For gauge bosons (i.e. particles with spin in the SM, such as photons), by using a gauge-invariant regulator (as is for example used in dimensional regularisation), one can show that the radiative corrections to the masses vanish. The reason for the absence of linear, quadratic, or higher-order corrections to the masses of fermions and gauge bosons is known and attributed to the presence of some particular symmetries of the theory: chiral symmetry in the former case and gauge invariance in the latter. Such symmetries are said to protect the particle masses from large radiative corrections.
The situation is however different for the scalar fields present in the theory, such as the Higgs boson of the SM. Restricting the discussion to the SM Higgs mass, the radiative correction to its mass from the self-interaction term in the SM Lagrangian reads
which is quadratically divergent (i.e. when increases to infinity, the term quadratic in dominates over the others and becomes infinitely large). It should be noted that this is not the only quadratically divergent contribution to the radiative mass corrections for the Higgs boson: others come from gauge boson loops and fermion loops. An interesting feature of field theory is that the quadratically divergent contributions from the fermion loops have opposite signs relative to the contributions from the boson loops, an observation that, as we will argue below, plays an important role in one of our strongest motivations for extending the SM to its supersymmetric version.
Since the SM is a renormalisable theory, there is in principle no problem with the divergent radiative corrections to exist, because they can be absorbed into the so-called bare mass parameter. However, in an ‘effective field theory’ interpretation of the SM (for an introduction, see e.g. ref. ), it is believed that the model is a valid description of particle physics up to some particular energy scale which is characterised by the cut-off scale . At energies beyond , the SM may be modified by adding new degrees of freedom (i.e. new fields) that are associated with some heavy particles whose effects are neglected at low energies. One example of such modifications is the assumption that the gauge group of the SM (i.e. ) is generalised to a larger grand unification group such as or . A rather trivial value for beyond which we expect new degrees of freedom to become important is the Planck scale GeV, but can certainly be as small as TeV scales beyond which the SM has not been tested yet. In this effective field theory framework, quadratically divergent corrections pose a theoretical problem.
There are several reasons which indicate that the ‘physical’ Higgs mass (the mass that is measured experimentally) has to be no larger than a few hundred GeV. This is the total value after adding the correction given in Eq. 3.0 to the bare mass parameter of the theory. If becomes very large, the quadratic term in Eq. 3.0 will dominate over the other terms and this effectively means that the physical mass is determined by the bare mass and the quadratic term. If one now assumes that the SM is valid below the scale of grand unification theories (GUTs) GeV (see section 3.3), the required cancellation of the two large values implies that the bare Higgs mass parameter will have to be “fine-tuned” to part in . This becomes even worse if is as large as the obvious cut-off scale of . This fine-tuning problem is often referred to as the ‘gauge hierarchy problem’ of the SM [58, 59, 60, 61]. In other words, the large quadratic corrections imply that if TeV, any predictions we make for physics at TeV energies are highly sensitive to the structure of the underlying high-energy theory with the SM being its effective incarnation at low energies.
Although such a fine-tuning of the SM structure is mathematically allowed, it has been taken as a strong hint (although not necessarily111Examples of the alternative approaches include: (1) Simply accepting that Nature is actually fine-tuned. (2) Leaving the assumption that elementary scalar fields exist in Nature, in models with composite states of bound fermions such as the idea of technicolor [52, 53]. (3) Assuming that the Higgs bosons interact strongly (rather than perturbatively) with themselves, gauge fields or fermions at the cut-off scale [62, 63]. (4) Making gravitational effects strong at energies close to TeV scales by for example assuming the existence of additional compact spatial dimensions [64, 65]. (5) Assuming that the quadratic divergences only show up at multi-loop level and not necessarily at the lowest order in models such as the Little Higgs .) that some new degrees of freedom must exist above the electroweak scale that ‘naturally’ cancel the problematic quadratic corrections in Eq. 3.0. These new degrees of freedom should then be soon revealed by TeV-energy experiments and observations both at colliders and in high-energy astrophysical phenomena.
Weak-scale supersymmetry is arguably the leading proposal that provides the required new degrees of freedom and solves the hierarchy problem in a simple and elegant way. We mentioned earlier in this section that the fermion and boson loops contribute to the dangerous quadratic divergences with opposite signs. This immediately suggests that in a theory with equal numbers of fermionic and bosonic degrees of freedom, the quadratic divergences will be cancelled. In order for this idea to work at any loop level, the couplings of fermions and bosons are additionally required to be related due to some symmetry. As we will see in the following chapters, both of these requirements are fulfilled in supersymmetry as a symmetry that transforms fermions to bosons and vice versa (see e.g. Fig. 3.1).
As we will argue in section 4.5, even if supersymmetry is a correct extension of the SM, it has to be broken at least spontaneously (i.e. through a mechanism similar to the Higgs mechanism of electroweak symmetry breaking). One can show that in a supersymmetric theory where supersymmetry is appropriately broken, the scalar masses all remain stabilised against radiative corrections and the hierarchy problem is still resolved . This observation is so remarkable that it essentially served as a watershed in the history of supersymmetry and provided one of the strongest motivations for it.
3.2 Electroweak symmetry breaking
Electroweak symmetry breaking (EWSB) is an essential ingredient in the SM. Through this process all the particles of the model acquire mass, a feature that is obviously a crucial requirement for the model to successfully describe the real world. EWSB is realised in the SM through the Higgs mechanism: The Higgs boson of the theory is believed to have acquired a vacuum expectation value (VEV) which results in the breaking of electroweak gauge symmetry. This is a ‘spontaneous’ symmetry breaking, because the fundamental Lagrangian of the theory (i.e. the SM Lagrangian) still remains symmetric while the ground state is no longer invariant under the symmetry. In order for the Higgs boson to develop an appropriate VEV, a so-called scalar potential of the theory should be minimised properly. This requires some particular parameters of the potential to acquire specific values. Strictly speaking, in order for the EWSB mechanism to work, some squared mass parameter for the Higgs boson has to be negative and this can be achieved only if some parameters of the model possess certain values. Although these values have been set experimentally, there is no explanation for such choices and again some fine-tuning seems to be necessary.
As we will discuss in sections 5.1.3 and 5.3.2 for particular supersymmetric models, supersymmetry can naturally lead to EWSB and provide a deeper understanding of why it happens. This is mainly because in supersymmetric models one usually does not have to tune the EWSB parameters directly: The conditions of EWSB can be satisfied by setting the model parameters to some typical values that are motivated for other reasons. In models for supersymmetry with parameters that are set at some high-energy scales (such as the models of sections 5.3.1 and 5.3.2), starting from a few parameters and evolving them with energy by means of the so-called renormalisation group equations (RGEs; see e.g. section 5.1.6) can give rise to EWSB at the electroweak scale. This process is often referred to as ‘radiative electroweak symmetry breaking’ (REWSB).
3.3 Gauge coupling unification
The SM is constructed based on the gauge group and all particles are different representations of this particular symmetry group. But why is this group special? It certainly looks peculiar and there is no theoretical explanation within the framework of the SM for this particular choice.
The three subgroups of the above gauge group (i.e. , and ), correspond to three forces of Nature, i.e. strong, weak and electromagnetic forces, respectively. Each group has a coupling constant that determines the strength of its associated force. Experimental measurements over a wide range of energies tell us that the three forces are very different in strength and this is related to the fact that the three corresponding coupling constants have very different values. Like any other quantity in quantum field theory that in general runs with energy, the couplings are also scale-dependent. However, experiments indicate that even at energies slightly higher than the electroweak scale where the spontaneously broken (sub-)symmetry becomes restored, the two associated coupling constants do not unify (see e.g. the left panel of Fig. 5.1 in section 5.1.6).
The peculiar gauge structure of the SM has however important implications. For example, it prevents the occurrence of some unwanted phenomena such as proton decay and large flavour-changing neutral currents (FCNCs). Although these characteristics are crucial for the success of the model, the way they are achieved in the SM is highly non-trivial and seems to be pure luck. In addition, the SM contains many free parameters whose values are constrained by experiments. There is however no theoretical explanation for such experimentally favoured values. All these types of tuning problems, as well as the question about different values of gauge coupling constants find reasonable explanations through the intriguing idea of ‘unification’.
In unified theories, the gauge symmetries of the SM are assumed to be extended to larger symmetries. For example in the so-called grand unified theories (GUTs), that are of particular interest in this respect, the SM symmetry group is extended to some simple Lie groups such as  or [69, 70, 71]. This extension is largely motivated by the fact that the SM field content perfectly fits into multiplets (or representations) of these groups, i.e. these larger groups include the SM group as their subgroup . This can therefore potentially explain the reason for the particular assignment of quantum numbers (such as hypercharges) in the SM (which seem to be randomly assigned). This consequently illuminates why dangerous experimental processes are forbidden in the SM.
One requirement for unification to occur is that all gauge couplings of the theory unify to a single quantity. As we mentioned above, this is not the case for the SM. We however know that these couplings, as well as all other parameters of the model, generally evolve with energy through the RGEs. This then gives the hope that although the gauge couplings have different values at low energies, they may unify at some high energy scale where the underlying larger symmetry group manifests itself. If this scenario is true, it provides an appropriate answer to the question why different forces of Nature have different strength: this is only a natural consequence of running of parameters with energy in quantum field theory. In addition, as a bonus, unification usually provides extra relations between various parameters of the theory and therefore gives rise to a (sometimes dramatical) reduction in the number of free parameters of the model. This alleviates the problem with the large number of free parameters in the SM. Finally, promoting the peculiar gauge group of the SM to a simple group such as or is on its own an interesting feature.
The problem manifests itself if we now solve the RGEs for the SM gauge couplings up to very high energies: the result is that the couplings do not unify at any scale (again see e.g. the left panel of Fig. 5.1 in section 5.1.6) and the idea of unification seems to be excluded. However, the unification scale (if exists) cannot be chosen arbitrarily and is determined by the particle content of the theory and measured values of different parameters at some energy scale (e.g. the weak scale). Although for the SM, with the known particle content and experimental constraints on its free parameters, the gauge couplings do not unify at any scale, a way out is to modify the particle content appropriately by adding new degrees of freedom to the model. Clearly these new particles should be heavy enough so as to remain hidden at low energies.
This is exactly where supersymmetry enters the game and turns out to be quite helpful. In most interesting versions of weak-scale supersymmetric models (as we will see in section 5.1.6) the SM field content is modified such that the gauge coupling unification can be elegantly achieved. In the case of minimal supersymmetric extensions of the SM, the unification is obtained typically at a GUT scale of GeV with a unified gauge coupling of (see e.g. the right panel of Fig. 5.1 in section 5.1.6) [94, 74, 75, 76, 77]. A detailed and technical discussion of why gauge unification is achieved in concrete realisations of supersymmetry can be found in section 5.1.6 of this thesis.
Before we stop our discussion here, let us note that although unification can be obtained this way, it is however highly non-trivial from a theoretical point of view. For example, no firm theoretical explanations exist for why should remain in perturbative regime, or why the GUT scale resides in a narrow energy range that is required for both suppression of proton decay and prevention of possible quantum gravitational effects. These characteristics therefore remain as accidental properties of the theory.
3.4 Experimental bounds on the Higgs boson mass
In the SM, the Higgs mass is set by the quartic Higgs coupling which is fairly unrestricted. Consequently, there are no strong limits on the Higgs mass and while the lower limit is set by experiments such as the Large Electron-Positron (LEP) collider to be about GeV , the mass can be as large as about GeV.
On the other hand, as we will see in section 5.1.3, in the most widely-studied supersymmetric extensions of the SM, the Higgs mass is not a free parameter and is actually a prediction of the theory. In these models, the lightest Higgs scalar222As we will see later, consistency conditions require supersymmetric models to have more than one Higgs boson. is required to be lighter than about GeV and this much narrower range for the Higgs mass makes the theory more falsifiable and therefore phenomenologically more interesting.
On the other hand, by fitting the SM parameters to the available electroweak precision data, the favoured value for the Higgs mass (i.e. the minimum- point) is fairly low and well below the experimental direct limit from the LEP (see Fig. 3.2). Although this discrepancy is not statistically very significant, it definitely shows some tension. For comparison, Fig. 3.2 shows also the result of a typical supersymmetric fit using one of the simplest supersymmetric extensions of the SM called the CMSSM (see section 5.3.2) . It can be seen from this example that it is possible to reconcile theoretical predictions for the Higgs mass with experimental data within the supersymmetric extensions of the SM.
3.5 The need for quantum gravity
As stated in chapter 1, the SM of particle physics as a framework for describing the matter components of the Universe and their interactions, has been able to provide such a description in a mathematically consistent way only for three fundamental forces (out of four). The SM as a quantum-field theoretical framework is renormalisable only if gravitation is not included. Arguably, string theory (for an introcuction, see e.g. ref. ) has so far been the most favoured candidate for a consistent quantum theory of gravitation which is expected to include the SM as its effective field theory valid at low energies. It is however highly difficult to build a phenomenologically successful string theory that does not require supersymmetry, and this means that supersymmetry is an essential ingredient of the best quantum description of gravitation so far. This makes supersymmetry particularly interesting.
Even if we do not believe in string theory as a valid description of high-energy phenomena, there is yet another intriguing connection between supersymmetry and gravity. As will be seen in the following chapters, the phenomenologically interesting versions of supersymmetry that we will consider are all based on a ‘global’ symmetry. In the language of chapter 4 this means that the generators of supersymmetric transformations are not functions of space and time. It is however entirely justified to promote the global symmetry to a local one, in a way analogous to the gauge symmetries of the SM. Such a localisation process is shown to inevitably lead to the existence of a new spin- massless gauge field together with its supersymmetric partner, a spin- particle (see e.g. ref. ). The former is exactly the particle that is assumed to be responsible for gravitational interactions, and is accordingly called the graviton. The interesting characteristic of the supersymmetric graviton is that its dynamics, which is entirely fixed by local supersymmetry, contains Einstein’s general relativity as our currently best classical theory of gravity. Regarding this connection with gravity, local supersymmetry is often called ‘supergravity’ (or SUGRA). Although such a supergravity theory is not renormalisable333Strictly speaking, this statement may not be correct. There is a particular version of supergravity, called ‘ supergravity’ (see chapter 4 for the terminology), which is conjectured to be renormalisable . As we will point out in the next chapter, these versions of supersymmetry are however phenomenologically not very interesting., its natural connection to gravity should not be ignored.
3.6 Other issues
In addition to the issues with the SM we enumerated in the previous sections, there are a few other reasons to believe that the SM is not the complete theory of Nature. Most of these arguments are again purely theoretical (or aesthetic) in nature, but are still highly intriguing so that one cannot simply ignore them. We will not attempt to detail these other problems here and only list (or briefly introduce) a few interesting ones with some references for further reading.
The first problem comes from the observations of neutrino oscillations. These indicate that neutrinos have small but non-zero masses. In the SM, neutrinos are however massless and this directly implies that the model must be extended so as to accommodate massive neutrinos. In order to avoid a ‘fine-tuning’ problem, this is usually done through the so-called ‘seesaw’ mechanism that is generally implemented within the framework of grand unified theories discussed in section 3.3 (for a review of neutrino masses and mixing, see e.g. ref.  and references thein).
The other problem that is again related to a fine-tuning within the SM, is the ‘strong CP problem’. This deals with the fact that the quantum chromodynamics (QCD) sector of the SM, contrary to the electroweak sector, respects the CP-symmetry. This leads to a highly fine-tuned value for a parameter called ‘vacuum angle’ and denoted by . The strong CP problem finds natural resolutions in models of physics beyond the SM, in particular through the introduction of new particles called axions (these are the same particles as the axions we mentioned in section 2.3 in our list of viable dark matter candidates) .
Let us end this chapter by adding to our list of issues two other theoretical speculations on the structure of the SM: (1) Why are there only three generations for matter particles, i.e. for leptons and quarks? (2) What is the origin of fermion masses? These two may also find appropriate answers in theories beyond the SM.
Chapter 4 Theoretical foundations of supersymmetry
In the previous chapters, we attempted to review some answers to the question why we are interested in physics beyond the Standard Model of particle physics and in particular its supersymmetric extensions. Our discussions so far have been based on a very vague understanding of supersymmetry. Before we enter the world of concrete supersymmetric models in the next chapter and investigate various observational constraints on these models in chapter 5, we briefly introduce supersymmetry in this chapter and review some of its fundamental properties. In addition, some formalisms are discussed and basics of supersymmetric model building are presented. This chapter is a rather technical one and the reader who is only interested in phenomenological aspects of the field can simply skip it and continue directly from chapter 5.
4.1 Supersymmetry is a symmetry
All the known elementary particles are either bosons or fermions. Bosons are those particles (or fields) that obey Bose-Einstein statistics and this means that they can occupy the same quantum state at any given time. Fermions, on the other hand, obey Fermi-Dirac statistics and, consequently, only one fermion can occupy a particular quantum state at a time. Although the quantum mechanical distinction between matter and force is not a clear cut, fermions are often associated with matter whereas bosons are considered as carriers of forces and interactions between the fermions. According to the so-called spin-statistics theorem in quantum field theory, bosons have integer spin while fermions possess half-integer spin (for an introduction to quantum field theory, see e.g. ref. ).
Elementary fermions that are known to exist in Nature, according to the Standard Model of particle physics, are categorised as quarks (6 particles (up), (down), (charm), (strange), (top), (bottom) and 6 corresponding antiparticles , , , , , ) or leptons (3 charged particles (electron), (muon), (tau), 3 neutrinos , , and 6 corresponding antiparticles (positron), , , , , ). The SM also contains 7 elementary bosons in total (if we include the graviton), some of which, such as the gauge bosons (photon), (gluon), and have already been discovered, while the other two, i.e. (Higgs boson) and (graviton) are to be observed experimentally (see e.g. ref.  for an introduction to the SM).
As we pointed out in the previous chapters, the mathematical structure of the SM that describes its field content and various interactions between the fields, is constructed based on some particular symmetries, some of which are thought to be fundamental.
The first symmetry from the latter category is called Pioncaré symmetry and is a ‘spacetime’ symmetry. The Poincaré group (for an introduction to group theory and its applications in particle physics, see e.g. ref. ) is the full symmetry of special relativity and correspondingly any relativistic field theory; the SM is no exception. This is a 10-dimensional noncompact Lie group and is the group of isometries of Minkowski spacetime. The Poincaré group includes the Lorentz group as a subgroup and is a semi-direct product of translations in spacetime and Lorentz transformations (i.e. , where stands for the former and denotes the latter). Mathematically speaking, all elementary particles (or fields) are different ‘irreducible representations’ of the group and are specified by two quantities: mass (or four-momentum) and spin (an intrinsic quantum number). The Poincaré symmetry is considered as a fundamental symmetry which every quantum-field theoretical framework that describes particles and their interactions should possess (including the SM and its potential extensions).
The other symmetry that is implemented in the SM, and has been used as a guiding principle in constructing its theoretical structure, is an ‘internal’ symmetry, i.e. a symmetry which is not obviously related to space and time. This determines how different components of a theory (e.g. different fields in the SM) transform into each other. The SM is called a ‘gauge theory’, and this is because the fundamental Lagrangian of the theory is invariant (or symmetric) under a particular non-abelian gauge symmetry: . This gauge symmetry is an example of internal symmetries.
While spacetime symmetries of a quantum field theory dictate the properties of the field components and classify them into various categories of scalars, vectors, tensors and spinors, internal symmetries rather determine how different terms in the Lagrangian must be written. The two symmetries are entirely independent in the SM.
Supersymmetry or SUSY (for an introduction, see e.g. refs. [54, 55, 56, 86, 87]) is a symmetry that transforms fermionic degrees of freedom into bosonic ones and vice versa. In a supersymmetric theory, every fermion has a bosonic ‘superpartner’ and every boson has a fermionic superpartner. With this definition, supersymmetry can be considered as a new internal symmetry because it gives a new way to transform some components of the theory, say fermions, to some other ones, i.e. bosons. This is however not entirely true. Supersymmetry is actually also an extension of the Poincaré group in the sense that it extends the ‘Poincaré algebra’ (and therefore special relativity) through the introduction of four anticommuting ‘spinor’ generators. In other words, although in supersymmetry, fermions are transformed into bosons and vice versa, these transformations only modify the particles’ spin and this is essentially a spacetime property.
4.2 The supersymmetry algebra
Supersymmetry, as any other continuous symmetry, is characterised by a symmetry algebra, and as we mentioned in the previous section, this algebra is obtained by extending the Poincaré algebra such that it relates two types of fields (bosons and fermions) in a single algebra. The resultant algebra is called a Lie ‘superalgebra’.
Let us first look at the Poincaré algebra: Since the Poincaré group is a semi-direct product of the Lorentz group and the group of spacetime translations, a general Poincaré transformation contains both Lorentz transformations and translations. The Lorentz group has 6 generators: (3 rotations) and (3 boosts). One often denotes the generators of translations as . In a more covariant looking form, the Lorentz generators are usually written as , where and . In this notation, the full Poincaré algebra can be written as 
In supersymmetry, the Poincaré algebra is enlarged by generators that are ‘spinorial’, i.e. transform as spinors in contrast to the original ‘tensorial’ Poincaré generators which transform as tensors. Such generators are often denoted by dotted and undotted spinors and . These are objects that transform under the group as
where and . Here, the extra index labels distinct SUSY generators in case there are more than one pair. The number of such generator pairs is usually denoted by (i.e. ). The simplest case with only one pair is accordingly called ‘ supersymmetry’.
Mathematically, there is no limit on , but with increasing the theories contain particles of increasing spin. Since no consistent quantum field theory with spins larger than two exists, this then leads to the condition 111This is however the case when gravity is part of the theory, otherwise, spins cannot be larger than one and this leads to . supersymmetry (that is also called ‘unextended’ supersymmetry) is of particular interest since it is the only case which permits chiral fermions. We know that chiral fermions exist, therefore from a phenomenological point of view, any supersymmetric theory of particle physics has to be of type, at least at low energies. We therefore restrict our discussions to supersymmetry.
The extended Poincaré algebra (i.e. the superalgebra) includes the following new commutation and anti-commutation relations:
Here, are the so-called ‘central charges’ of the group, and they are the members that commute with all generators of the algebra. The (or unextended) SUSY algebra, is the simplest supersymmetry algebra which has no central charges. One important property of any supersymmetric theory that can be inferred from the above SUSY algebra is that the energy is always positive.
Any irreducible representation of the Poincaré algebra is associated with a particle. Since the Poincaré algebra is a subalgebra of the superalgebra, any representation of the latter is also a representation of the former. However in general, an irreducible representation of the superalgebra corresponds to a reducible representation of the Poincaré algebra, and this means that it corresponds to several particles. The particles of each SUSY representation are related to each other by the SUSY generators and . This means that these particles have spins that differ by units of one half, i.e. some are bosons and some are fermions. The spin-statistics theorem then implies that the generators and transform fermions to bosons and vice versa. The particles that are obtained via supersymmetric transformations of other particles, are called supersymmetric partners or simply ‘superpartners’ of the original particles.
An irreducible representation of supersymmetry that is equivalent to a set of supersymmetrically-related particle states is called a ‘supermultiplet’. All particles (or states) belonging to a supermultiplet have equal masses, and any supermultiplet contains an equal number of fermionic and bosonic degrees of freedom. One can show that for (i.e. unextended) supersymmetry (with gravity), only three types of massless multiplets exist: chiral multiplets (consisting of a Weyl fermion with spin and a complex scalar with spin ), vector multiplets (consisting of a gauge boson with spin and a Weyl fermion) and graviton multiplets (consisting of a graviton with spin and a gravitino with spin ). It is tempting to also add gravitino multiplets (consisting of a gravitino and a gauge boson) to the above set of multiplets, but such a multiplet can only happen in an extended (i.e. ) SUSY.
4.3 The Wess-Zumino model
In this section, we briefly describe how a supersymmetric field theory can be constructed at the ‘action’ level. Although the described model is too simple compared to the more sophisticated ones thought to be implemented in Nature (see chapter 5), it shows the main ingredients of any supersymmetric theories including the ones that are of phenomenological interest (see also refs. [54, 55, 56, 86]).
By looking again at the possible SUSY multiplets we enumerated in the previous section, we see that the simplest representation of supersymmetric transformations which includes a chiral fermion (with spin ) is a chiral multiplet. In addition to a left-handed two-component Weyl fermion, that we denote by , this multiplet includes a complex scalar field, say .
Let us now try to write down a four-dimensional SUSY-invariant action that is composed of chiral multiplets with scalar fields and Weyl fermions (). We demand the action to be supersymmetric ‘off-shell’. This means that the action is invariant under supersymmetry even if the classical equations of motion are not satisfied. The latter requirement leads to the addition of a set of ‘auxiliary’ complex scalar fields (fields without kinetic terms) to the field content of the theory. The resultant Lagrangian density for our SUSY theory reads
This is a theory for massless and free fields (i.e. includes no interaction terms) and was first derived by Wess and Zumino . The next step is obviously to add non-gauge interaction terms to the Lagrangian such that they preserve the supersymmetric property of the action. If we only retain the renormalisable interactions (i.e. ones with mass dimension ), it can be shown that the most general Lagrangian with non-gauge interactions for chiral multiplets has to have the following form:
Here and are defined as derivatives of the so-called ‘superpotential’ that is a function of the scalar fields :
Here, is a symmetric mass matrix for the fermions, is a Yukawa coupling of a scalar and two fermions, and are some additional parameters that influence only the scalar potential of the Lagrangian . Auxiliary fields are eliminated from the expression using their classical equations of motion. The term in Eq. 4.0 is only a function of the scalar fields and and is essentially the ‘scalar potential’ of the theory (usually denoted by ). The model introduced in Eq. 4.0 is called the Wess-Zumino model .
4.4 Supersymmetric gauge theories
In the previous section, we showed how a simple supersymmetric theory looks like for a chiral supermultiplet. This can be used in constructing a model that describes particle physics fermions (i.e. leptons and quarks) and scalars (such as the Higgs boson). We however know that in reality, at least at low energies, there are other types of fields which should also be described in any supersymmetric extension of the SM: gauge fields.
As pointed out in section 4.2, any SUSY theory for gauge fields should include vector (or gauge) multiplets as basic ingredients. These multiplets have massless gauge bosons (that we denote by ), as well as Weyl fermions . Here the index can take on different integer values depending on the particular gauge group of the theory (e.g. for , for and for ). As for the chiral multiplet case, one has to also add an auxiliary field to the field content of the theory. Such a field, traditionally named , is real and bosonic, and is required for the action to be SUSY-invariant off-shell. The Lagrangian density for the gauge multiplet is shown to have the following form:
where is the Yang-Mills field strength for the gauge fields and is the covariant derivative of the field .
As the final step towards constructing a general supersymmetric Largangian, one needs to consider both contributions from the chiral and gauge supermultiplets, as well as any additional interaction terms that are allowed by gauge invariance and keep the theory supersymmetric. Adding the requirement that the interaction terms should be renormalisable (i.e. of mass dimension in four dimensions), our general Lagrangian density will have the following form:
where and are defined in Eqs. 4.0 and 4.0, respectively. The only difference is that the ordinary derivatives in Eq. 4.0 for the chiral supermultiplet Lagrangian are now replaced by gauge-covariant derivatives . are the generators of the gauge group that satisfy . Here are the structure constants that define the gauge group and is the ‘gauge coupling’.
The complete scalar potential of the theory in this case is shown to be expressible purely in terms of the auxiliary fields and (which are in turn expressible only in terms of the scalar fields ):
The first and second terms in Eq. 4.0 are called -terms and -terms, respectively. The former are entirely fixed by Yukawa couplings and fermion mass terms, while the latter are fixed by the gauge interactions. In addition, the scalar potential can be shown to be bounded from below, i.e. it is always greater than or equal to zero.
Finally, one should notice here that the theory defined in Eq. 4.0 is invariant under ‘global supersymmetry’. This means that the parameters of supersymmetric transformations do not depend on the spacetime co-ordinates. We however know that local symmetries also exist and some of them have played critical roles in our current description of particle physics: The best example is the gauge symmetries of the SM. ‘Local supersymmetry’ also exists and as we pointed out in section 3.5 makes an interesting connection between supersymmetry and gravity (in the context of ‘supergravity’). The theory of supergravity is highly technical and we do not detail it in this thesis. We only briefly describe in the next chapter (mainly section 5.3) some phenomenologically interesting models that have been constructed based on supergravity assumptions. We refer the interested reader to the literature for detailed discussions (see e.g. refs. [54, 56] and references therein).
4.5 Spontaneous supersymmetry breaking
As we will argue in the next chapter, supersymmetry cannot be implemented in Nature as an exact symmetry and is required to be broken appropriately. From a theoretical point of view, arguably the most interesting way of breaking a symmetry in any quantum field theory is via a ‘Higgs-like’ mechanism, where the symmetry is broken ‘spontaneously’. This idea has seemingly worked very well in the SM when the electroweak gauge symmetry is broken at TeV scales. It is therefore quite interesting to see how the same idea could work for supersymmetry (see also refs. [54, 55, 56, 86]).
Spontaneous supersymmetry breaking means that while the Lagrangian of the theory is SUSY-invariant, the vacuum state is not, i.e. and . In an unbroken supersymmetry, the vacuum has zero energy (since , where is the Hamiltonian operator), while in a spontaneously-broken supersymmetry the vacuum has positive energy (i.e. ). It can be shown from this that if the vacuum expectation value (or VEV) of and/or (the auxiliary fields introduced in sections 4.3 and 4.4) become non-zero (i.e. and/or ), supersymmetry will be spontaneously broken.
In SUSY-breaking models in which the vacuum state we live in is assumed to be the true ground state of the theory, the structure of the models usually imply that the equations and cannot be satisfied simultaneously and this breaks SUSY spontaneously. Other models exist in which we are not assumed to live in the true ground state and instead live in a metastable SUSY-breaking state with sufficiently long lifetime (comparable to the current age of the Universe) (see e.g. ref. ). This metastable state might have been chosen by some finite temperature effects in the early Universe.
In the -term SUSY-breaking mechanism, the gauge symmetry group needs to contain a subgroup with a non-zero -term VEV. Supersymmetry is then broken by introducing the following little extra piece to the SUSY Lagrangian:
which is a term proportional to ( being a constant).
In the -term mechanism, SUSY breaking occurs due to the existence of a non-vanishing -term VEV that comes from a particular property of the superpotential , namely that there is no simultaneous solutions for the equations
with defined in Eq. 4.3 (i.e. ).
One property of all types of spontaneous global SUSY breaking models (with stable or metastable vacuum states), is the existence of a massless neutral Weyl fermion as the Nambu-Goldstone mode. This fermion is called goldstino, is denoted by and possesses the same quantum numbers as the broken symmetry generator (which in our case is the fermionic charge ). The can be shown to have the form , i.e. its components are proportional to the VEVs of the auxiliary fields and .
With this brief introduction to the two aforementioned SUSY-breaking mechanisms, we stop our discussion here. We will instead come back to the discussion of supersymmetry breaking in section 5.3.1 of the next chapter where we discuss various concrete scenarios in connection with phenomenologically interesting SUSY models. We will see how some of the general strategies described here can be used in constructing real-world theories.
4.6 Superfield formalism
In order to construct more complex supersymmetric Lagrangians, with larger numbers of fields and more complicated interaction terms, one needs to develop a rather general procedure that generates SUSY-invariant interactions in a systematic way. A compact and convenient way is to use ‘superspace’ and ‘superfield’ formalism. In supersymmetry, the superspace is the usual four-dimensional spacetime (labelled by the four coordinates ) enlarged by adding four anticommuting ‘Grassmannian coordinates’ and . These new coordinates are fermionic and transform as a two-component spinor and its conjugate. In general, for an extended supersymmetry with SUSY generator pairs, there are extra fermionic coordinates. Superfields are quantum fields that differ from the usual ones in that they are defined on the superspace rather than the spacetime. Superfields are defined as single objects with components being all the different fields (fermionic, bosonic and auxiliary) that belong to a supermultiplet.
The main advantage of using superfield formalism is that the invariance under SUSY transformations remain manifest during the Lagrangian construction; this is because the Lagrangian is defined in terms of integrals over the superspace. Working with superfield formalism has also the advantage that the spacetime nature of supersymmetric transformations is more manifest. Despite all the definite benefits of working with superfields, the formalism is fairly complicated and we do not detail it here. We refer the interested reader to e.g. ref.  for a detailed introduction.
Chapter 5 Supersymmetry in real life
In the previous chapter, we described supersymmetry in general and discussed various properties of a supersymmetric field theory. This was done mainly through the presentation of the simplest possible SUSY models with a minimal field content, i.e. the Wess-Zumino model and its gauge extension. However, these models are obviously too simple to describe the real world. It is the goal of the present chapter to discuss viable SUSY models and scenarios that may describe reality. We also argued in chapters 1, 2 and 3 why a supersymmetric extension of the SM is helpful, although our discussions were limited to rather general arguments. We may therefore want to see in a more explicit way how suprsymmetric models could address the issues discussed there. We will detail in this chapter ‘some’ of those issues in terms of definite SUSY models. Finally, in order to examine how observations could enhance our knowledge about supersymmetry, its validity and possible implementations in Nature (which has been the primary objective of this thesis), we need to have concrete theoretical frameworks to work in. The present chapter also provides these frameworks.
5.1 The Minimal Supersymmetric Standard Model
There are various strategies in building a SUSY model that has to do with reality. In a top-down approach, one looks at some fundamentally motivated theories that accommodate supersymmetry, such as string theory. These theories are usually defined at very high energies that are not accessible by experiments. Phenomenological studies can then be carried out by extracting an effective supersymmetric field theory valid at low energies.
In an alternative bottom-up approach, one starts with the SM itself and adds all the ingredients that are required for it so as to become supersymmetric. It is important that in the latter approach one takes into account all phenomenological considerations and constraints in such a way that the emergent theory is consistent with observations as well as theoretical conditions. It must also give the SM as an effective theory valid up to certain energies since we know that the SM is an excellent description of particle physics below those energies.
The simplest phenomenologically-constructed SUSY model (i.e. obtained through a bottom-up approach) is the so-called ‘Minimal Supersymmetric Standard Model (MSSM)’  (see also refs. [54, 55, 56] for comprehensive introductions to the MSSM). It is minimal in the sense that it contains the smallest number of new particles (or fields) that can be added to the SM in order to make it supersymmetric, and the theory still remains consistent with all phenomenological requirements. In this section, we describe the MSSM and its properties that are of most interest for phenomenological studies of supersymmetry.
5.1.1 Field content and superpotential
In every supersymmetric model, including the MSSM, the number of degrees of freedom for particles and their corresponding supersymmetric partners (or superpartners) match. This particularly implies that some SM particles have more than one superpartner. For example, the elementary fermionic spin- particles with two degrees of freedom (such as leptons and quarks) need two scalar superpartners with one degree of freedom each. The superpartners of the SM fermions are called ‘sfermions’ (sleptons for leptons and squarks for quarks) and the superpartners of the bosons are called ‘bosinos’ (gauginos for gauge bosons and Higgsinos for Higgs bosons). We also often refer to the superpartners of the SM particles simply as ‘sparticles’.
In the MSSM, every known (i.e. SM) particle has a spin , or and must therefore reside, together with its superpartners, in either a chiral or gauge supermultiplet (see the previous chapter). We summarise in Tab. 5.1 all the particles and spartners in the MSSM. As we see, they are divided into two categories of chiral and gauge supermultiplets. Tab. 5.1 also shows different hypercharges associated with the particles. These correspond to the three SM gauge groups.
One interesting feature of the MSSM is that, contrary to the SM, it contains ‘two’ Higgs doublets (shown as and in Tab. 5.1) that consequently require two chiral supermultiplets. There are two main reasons for this: (1) Only one Higgs chiral supermultiplet would introduce a gauge anomaly in the electroweak gauge symmetry that would make the theory quantum-mechanically inconsistent. (2) The Higgs chiral supermultiplet that has the Yukawa couplings necessary for giving masses to the up-type quarks, has a hypercharge that is different from the hypercharge of the Higgs chiral supermultiplet that has the Yukawa couplings necessary for giving masses to the down-type quarks and the charged leptons (see e.g. ref.  for more details).
The existence of two Higgs doublets in the MSSM and the fact that every bosonic degree of freedom has a corresponding fermionic degree of freedom and vice versa, together imply that the MSSM particle content is slightly more than a doubling of the SM particle content. It is also important to note that the sparticles presented in Tab. 5.0 are the ‘interaction’ (or gauge) eigenstates of the theory and the ‘mass’ eigenstates are in general linear combinations of the gauge eigenstates (we will come back to this in section 5.1.4).
|Name||Symbol||spin 0||spin 1/2|
|Name||spin 1/2||spin 1|
|winos, W bosons|
|bino, B boson|
Like any other supersymmetric theory, the SUSY part of the MSSM Lagrangian is determined by a superpotential that is defined in terms of the chiral supermultiplets (see the previous chapter). The MSSM superpotential is 
where , , , , , , and denote chiral superfields of the theory. It is important to notice that there are three generations (i.e. three families) for quarks/squarks and lepton/sleptons and although not written explicitly in Eq. 5.0, the summation over the generations is understood. Similarly, all gauge indices and summations are suppressed. The presence of three generations implies that the Yukawa couplings , and in Eq. 5.0 (which are exactly the same Yukawa couplings as those that enter the SM Lagrangian) are matrices in the family space.
The superpotential defined in Eq. 5.0 completely determines the structure of the MSSM if SUSY is not broken (see the next section). This means that we have now obtained an exactly supersymmetrised version of the SM albeit at the cost of introducing one new parameter, i.e. .
5.1.2 SUSY breaking and soft terms
Simple phenomenological considerations imply that supersymmetry cannot be an exact symmetry of Nature (at least at low energies), and if implemented in Nature, must be broken spontaneously. In other words although the fundamental Lagrangian might be SUSY invariant, the vacuum state that Nature has chosen need not be (see section 4.5 in the previous chapter). In a fully supersymmetric theory, masses of particles and their corresponding superpartners are equal. This immediately puts the theory in trouble if it is to describe reality. For example masses of selectrons (i.e. the superpartners of electrons) should be as low as the electron mass, i.e. about MeV. A particle with such a low mass should be easily detected experimentally, as electron is, and should essentially show up in our everyday life. This all means that SUSY is a broken symmetry. In addition, all sparticle masses should be much higher than the SM masses (in order not to have been observed in low-energy experiments).
As we stated earlier, the MSSM is a phenomenological model in the sense that its general structure is not set by any fundamental high-energy theory. This clearly means that phenomenological considerations should also fix the structure of any SUSY-breaking terms that we may add to the MSSM Lagrangian (see e.g. refs. [54, 55, 56, 86, 95, 96]).
One important guiding principle in determining the SUSY-breaking interactions in the MSSM comes from one of the strongest theoretical motivations for extending the SM to its SUSY version, i.e. providing a solution to the gauge hierarchy problem. We argued in section 3.1 that the quadratic divergences from the radiative corrections to the scalar masses can be cancelled out in a supersymmetric theory if fermionic fields and their bosonic partners have equal masses. This is clearly not the case in a SUSY-broken theory. It can however be shown that if the sparticles have masses not much larger than TeV scales, the cancellation of the different loop contributions does not require huge fine-tuning and therefore SUSY can still provide a solution to the hierarchy problem .
Supersymmetry is broken in the MSSM by adding the so-called ‘soft SUSY-breaking terms’ to the exact supersymmetric Lagrangian described in the previous section. These are the terms that while breaking supersymmetry, satisfy four conditions: (1) They do not reintroduce quadratic divergences to the Higgs mass (i.e. the gauge hierarchy remains stabilised). (2) They preserve the gauge invariance of the SM (and correspondingly the SUSY-unbroken MSSM). (3) They do not violate the renormalisability of the theory (which can be achieved by adding only mass terms and coupling parameters with positive mass dimensions). (4) They respect baryon and lepton symmetries of the SM and therefore conserve the corresponding quantum numbers and . The most general soft supersymmetry-breaking Lagrangian then reads
Here, , and are bino, wino and gluino mass terms, respectively. Trilinear couplings , and are complex matrices in the family space and are in one-to-one correspondence to the Yukawa couplings , and in the superpotential (see Eq. 5.0). Squark and slepton mass terms , , , and are also (Hermitian) matrices in the family space with potentially complex entries. and are explicit real mass terms in the Higgs sector and is a complex bilinear coupling.
5.1.3 Electroweak symmetry breaking and Higgs sector
In order for the Higgs mechanism to work, the scalar potential for the Higgs scalar fields needs be minimised and the minimum then breaks electroweak symmetry. It can be shown that at the minimum, both and can be set to , a property that is satisfactory. The reason for this satisfaction is that electromagnetism is not spontaneously broken at the minimum. Ignoring the terms in the potential that involve or , one obtains the following expression for the Higgs scalar potential that only contains the neutral Higgs fields and :
Here and are the and gauge coupling constants, respectively, and is the parameter defined in Eq. 5.0.
Now, in order to break electroweak symmetry, is required to be minimised (with a stable minimum) and the fields and acquire real and non-zero vacuum expectation values (VEVs). We denote these VEVs by and , i.e.
One can show that in order for the scalar potential to develop a well-defined local minimum such that electroweak symmetry is appropriately broken, the following two conditions must be satisfied :
The existence of two Higgs doublets in the MSSM implies that the Higgs sector of the theory consists of eight degrees of freedom (two per each field for , , and ). As in the SM, when electroweak symmetry is broken, three of these degrees of freedom are eaten so as to make and bosons massive. This means that five degrees of freedom remain intact and form five physical Higgs scalars. They are usually shown as
where and are CP-even and neutral (with lighter than ), is CP-odd and neutral, and are charged (with charges ).
The quantities and in Eq. 5.0 are related to the masses of the -boson and -boson ( and ) and the gauge couplings and in the following way: