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Composite Higgs Models and Extra Dimensions International School for Advanced Studies SISSA/ISAS Composite Higgs Models and Extra Dimensions Thesis submitted for the degree of Doctor of Philosophy Candidate: Supervisor: Mahmoud Safari Prof. Marco Serone Acknowledgements I would like to thank my Adviser Marco Serone for his help throughout this project, and Giuliano Panico for collaboration. I am also grateful to Alberto Parolini, Alberto Tonero, Roberto Percacci, and Alessan- dro Codello for discussions on various topics, to the professors of the Theoretical Particle Physics group for their teaching, to the students secretary Riccardo Iancer and Federica Tuniz and in general to all members of SISSA for providing a suitable environment for study and research. I would also like to thank my housemates Jian Zhao and Angelo Russomanno for their patience and sharing the responsibilities of living together. Special thanks to all my friends with whom I have had a great time and who made my stay in Trieste enjoyable, in particular Alireza Alemi neissi, Zhian Asadzadeh, Fahimeh Baftizadeh, Milad Ekram- nia, Marzieh Forough, Zhaleh Ghaemi, Ehsan Hatefi, Majid Moshtagh, Shahab Naghavi, Masoud Nahali, Khadijeh Najafi, Nader Nikbakht, Sahar Pirmoradian, Mohammad Ali Rajabpour, Houman Safaai, Shima Seyed-allaei, Shima Taallohi, Maryam Tavakoli and Mahdi Torabian. Finally I would like to express my deepest gratitude to my parents and my brother and sister for their continuous support and encouragement without which I would have never achieved this goal. Contents 1 Introduction 2 2 Composite Higgs Models 6 2.1 Introduction................................... 6 2.2 TheCCWZprescription ............................ 7 2.3 TheMinimalCompositeHiggsModel . 9 2.4 Composite Higgs and deconstructed models . ..... 13 3 5D Theories and Holography 17 3.1 Gaugefieldsin5D ............................... 17 3.2 Gauge sector of the SO(5)/SO(4)modelin5Dflatspace . 22 3.3 Fermionsin5D ................................. 41 4 Simple Composite Higgs models in Flat Extra Dimensions 47 4.1 Model I: Modified MCHM5 .......................... 47 4.2 Model II: Fermions in two fundamentals of SO(5).............. 57 4.3 Model III: Fermions in an adjoint of SO(5) ................. 69 4.4 Conclusions ................................... 79 A 1-Loop Computation of the Z¯bLbL Vertex 82 A.1 NotationandFeynmanrules. 82 A.2 Details of the 1-loop computation . 84 A.3 Proof of finiteness of F ............................. 95 B Electroweak Precision Tests 96 B.1 Introduction................................... 96 B.2 One-loop fermion contributions to the electroweak S and T parameters . 99 B.3 χ2 Fit ...................................... 105 1 Chapter 1 Introduction The Standard Model (SM) of particle physics is so far our best understanding of nature at the fundamental level. This theory which describes the constituents of matter and their interactions has stood firmly after many years of experimental scrutiny. However, despite all its success and agreement with experimental data, the SM is still filled with problems and puzzles. To name few: i) among the particles of the SM there is no candidate for the observed dark matter of the universe, ii) the amount of CP violation in the SM is not enough to explain the asymmetry between matter and antimatter, iii) neutrino masses call for physics beyond the SM, although possibly at a very high energy scale, iv) the SM does not incorporate gravity, v) an unreasonable amount of fine tuning is required to account for the Higgs mass, known as the gauge hierarchy problem. These are all motivations to develop ideas and look for physics beyond the SM. In particular the gauge hierarchy problem will be the main motivation behind the work presented here and will be explained in more detail in the following. The experimental status has finally come to a point to probe the weak scale. Re- cently both ATLAS and CMS collaborations at the Large Hadron Collider (LHC) have announced the discovery of a 125 GeV mass particle [1, 2] compatible with the long sought Higgs boson [3, 4, 5] , the last missing piece of the SM. Whether this is exactly the SM Higgs boson or not needs further investigation. It has long been known that the mass of the Higgs boson in the SM suffers from quadratic divergences. specifically the main contribution to the Higgs mass comes from the top quark loop 3 Y 2 m2 = 2µ2 | t| Λ2 h − 8π2 NP where ΛNP is the cut-off of the theory, or where new physics shows up, and √2µ is the bare mass of the Higgs. On the other hand, apart from the recent discovery of a 125 GeV Higgs, there has been indirect constraints from precision electroweak measurements 2 CHAPTER 1. INTRODUCTION 3 which put an upper bound on the mass of the Higgs roughly below 200 GeV. Although the bare parameter µ is absolutely free and one can adjust it to get the desired value for the Higgs mass, if new physics does not show up until very high energies, say the Planck energy, one must tune the bare mass to great accuracy which is highly unnatural. In other words the bare mass is extremely sensitive to the scale of new physics ΛNP . This is the statement of the hierarchy problem. The general expectation is that there must be some new dynamics, not much above the weak scale, which is responsible for stabilizing the Higgs mass. This has also been the main motivation for the construction of the LHC. There are two main classes of theories, namely weakly coupled and strongly coupled models, which aim to address this problem. Theories based on Supersymmetry, a symme- try which relates fermions and bosons, is an example of weakly coupled theories. In such theories the quadratic divergences cancel among fermion and boson loop contributions, solving in this way the hierarchy problem. A typical example of strongly coupled theories beyond the SM is Technicolor [6]. In Technicolor theories, inspired by chiral symmetry breaking in QCD, ElectroWeak Symmetry Breaking (EWSB) is triggered by strong dy- namics. Also the smallness of the scale of strong dynamics which is essentially the weak scale is naturally explained, much like ΛQCD, by renormalization group running. However these theories are highly constrained by precision electroweak data. Furthermore, incor- porating flavour into these theories is quite challenging. A variant of these theories [7] which generally perform better under ElectroWeak Precision Tests (EWPT) is based on the idea that electroweak symmetry is not broken at the strong scale where bound states are formed, but rather strong dynamics will give rise, possibly among other bound states, to a composite Pseudo Goldstone Boson (PGB) with the quantum numbers of the Higgs which in turn breaks electroweak symmetry by acquiring a Vacuum Expectation Value (VEV). The PGB nature of this Higgs particle will guarantee its lightness, as its mass will be generated only radiatively. A common obstacle before model building in such theories is that due to strong coupling explicit computation of precision electroweak observables as well as the Higgs potential is not possible and one has to rely on arguments such as naive dimensional analysis to estimate these quantities. As we extend the space dimensions, other interesting possibilities arise. One interesting idea is to identify the internal component of the gauge field in a higher dimensional theory with compact extra dimensions as the Higgs field, the so called Gauge-Higgs Unification (GHU) scenarios [8]. In such models the gauge symmetry forbids a tree level potential for the Higgs, while the loop contribution, being a non-local effect, is finite. Another proposal introduced in [9] as a solution to the gauge hierarchy problem was to take a slice of AdS5 and localize the matter fields on the IR brane, in this way the physical masses will be exponentially suppressed by the warp factor with respect to the mass parameters on the IR brane, while the Planck mass is insensitive to the compactification scale for small warp factors. It was noticed, though, that to provide a solution to the hierarchy problem it is only necessary for the Higgs to be localized close to the IR brane and embedding the CHAPTER 1. INTRODUCTION 4 matter fields in the bulk will give rise to a richer flavour structure. Inspired by the AdS/CFT correspondence one can think of these extra dimensional theories in terms of their 4D dual descriptions which are strongly coupled conformal field theories [10]. In particular, GHU models in warped space can be interpreted as weakly coupled duals of 4D strongly coupled field theories [11] in which the Higgs arises as a PGB of the strong sector [12, 13]. In fact such a correspondence does not necessarily exist in the exact AdS/CFT sense, in which case the bulk of the 5D theory along with the IR boundary can be considered as the definition of the 4D strongly interacting sector. In this work we will be interested in GHU models as they provide, due to their weak coupling nature, calculable examples of composite Higgs scenarios in which the Higgs potential and electroweak precision observables can be computed explicitly. Unfortunately model building in warped space is still a challenge and although a few 5D GHU models have been constructed so far [13, 14, 15], only in one model [16] (a modified version of a model introduced in [14] to accommodate a Dark Matter candidate) 1-loop corrections to S, T and δgb (and the Higgs potential explicitly determined) have been analyzed and the EWPT successfully passed. However, as far as we are concerned with low energy phenomenology, we do not really need to consider the technically challenging warped models. Instead, we can rely on the much simpler flat space implementations of the GHU idea. The resulting models may still be reinterpreted as calculable 5D descriptions of 4D strongly coupled composite Higgs models. This is guaranteed by the holographic interpretation, which shows that the low-energy symmetries of the theory are independent of the specific form of the 5D metric.
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