Phenomenological Aspects of Type IIB Flux Compactifications
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Phenomenological Aspects of Type IIB Flux Compactifications Enrico Pajer Munchen¨ 2008 Phenomenological Aspects of Type IIB Flux Compactifications Enrico Pajer Dissertation an der Fakult¨at fur¨ Physik der Ludwig–Maximilians–Universit¨at Munchen¨ vorgelegt von Enrico Pajer aus Venedig, Italien Munchen,¨ den 01. Juni 2008 Erstgutachter: Dr. Michael Haack Zweitgutachter: Prof. Dr. Dieter Lust¨ Tag der mundlichen¨ Prufung:¨ 18.07.2008 Contents 1 Introduction and conclusions 1 2 Type IIB flux compactifications 9 2.1 Superstring theory in a nutshell . 10 2.2 How many string theories are there? . 13 2.3 Type IIB . 15 2.4 D-branes . 17 2.5 Flux compactifications . 18 2.5.1 The GKP setup . 19 2.6 Towards de Sitter vacua in string theory . 21 3 Inflation in string theory 27 3.1 Observations . 27 3.2 Inflation . 29 3.2.1 Slow-roll models . 32 3.3 String theory models of inflation . 35 3.3.1 Open string models . 35 3.3.2 Closed string models . 36 3.4 Discussion . 37 4 Radial brane inflation 39 4.1 Preliminaries . 40 4.2 The superpotential . 42 4.3 Warped Brane inflation . 44 4.3.1 The η-problem from volume stabilization . 44 4.3.2 F-term potential for the conifold . 46 4.4 Critical points of the potential . 47 4.4.1 Axion stabilization . 48 4.4.2 Volume stabilization . 48 4.4.3 Angular moduli stabilization . 50 4.4.4 Potential with fixed moduli . 51 4.5 Explicit examples: Ouyang vs Kuperstein embedding . 52 4.5.1 Ouyang embedding . 52 4.5.2 Kuperstein embedding . 53 4.6 Inflation . 54 vi Contents 4.6.1 The effective inflaton potential . 56 4.6.2 Inflation through an inflection point . 58 4.6.3 Damped oscillatory phase . 59 4.6.4 Uphill inflation . 61 4.7 Nice upliftings . 63 4.7.1 Nice downliftings . 64 4.8 A summary on radial brane inflation . 65 5 Angular brane inflation 67 5.1 Motivations . 67 5.2 Inflation at the tip . 69 5.3 F-term potential for a D3-brane at the tip . 71 5.4 Kuperstein embeddings . 73 5.4.1 A simple case . 74 5.4.2 Radial stability . 79 5.5 Inflationary analysis . 79 5.5.1 Slow-roll Natural Inflation . 81 5.5.2 Slow-roll hilltop inflation . 84 5.5.3 DBI Natural Inflation . 86 5.6 A no-go result and how to evade it . 92 5.6.1 Slow-roll–DBI alternation from generic embeddings . 93 5.7 A summary on angular brane inflation . 96 6 On soft terms from large volume compactifications 99 6.1 Preliminaries . 100 6.2 Review . 101 6.2.1 Back to KKLT . 101 6.2.2 Consistency of KKLT . 102 6.2.3 Large volume scenario (LVS) . 103 6.3 String loop corrections to LVS . 106 6.3.1 From toroidal orientifolds to Calabi-Yau manifolds . 106 6.3.2 LVS with loop corrections . 110 P4 6.3.3 The [1,1,1,6,9] model . 111 6.4 Gaugino masses . 114 6.4.1 Including loop corrections . 117 6.4.2 Other soft terms . 118 6.5 LVS for other classes of Calabi-Yau manifolds? . 119 6.5.1 Abundance of “Swiss cheese” Calabi-Yau manifolds . 119 6.5.2 Toroidal orientifolds . 120 6.5.3 Fibered Calabi-Yau manifolds . 122 6.6 Further corrections . 122 6.7 A summary on string compactifications . 123 Contents vii A The conifold 125 A.1 The singular conifold . 125 A.2 The deformed conifold . 126 B Technical details on radial brane inflation 129 B.1 Dependence of τcr on the uplifting and the inflaton . 129 B.1.1 On τcr and the uplifting . 130 B.1.2 On τcr and the inflaton . 131 B.2 Sign of r3/2 term . 131 B.3 Maximum and minimum of V (φ)....................... 132 B.4 Radial brane inflation close to the tip . 133 B.5 Forces on D3- and anti D3-branes. 134 C Technical details on angular brane inflation 137 C.1 Minimization of τ ................................ 137 C.2 Radial stability . 138 C.2.1 Radial stability in the C B regime (DBI inflation) . 141 C.2.2 Radial stability in the 2C = B regime (slow-roll hilltop inflation) . 141 C.3 From one to many . 142 C.3.1 K¨ahler potential . 142 C.3.2 Superpotential . 143 D Technical details on soft terms from LVS 147 D.1 Some details on LVS . 147 P4 D.1.1 LVS for [1,1,1,6,9] ............................ 147 D.1.2 Many K¨ahlermoduli . 148 P4 D.2 Loop corrected inverse K¨ahler metric for [1,1,1,6,9] .............. 150 D.3 No-scale K¨ahler potential in type II string theory . 151 D.3.1 No-scale structure in type IIA . 151 D.3.2 No-scale structure in type IIB . 153 D.3.3 Cancellation with just the volume modulus . 154 D.3.4 Cancellation with many K¨ahler moduli . 154 D.3.5 Perturbative corrections to Vnp1 and Vnp2 ............... 155 D.4 KK spectrum with fluxes . 156 D.5 The orientifold calculation . 159 D.6 Factorized approximation . 162 D.6.1 Factorized approximation of the scalar potential . 164 Bibliography 169 viii Contents Abstract ix Summary The purpose of this work is to investigate how phenomenologically successful constructions in the field of high energy physics can be embedded into a fundamental theory, and what we can learn from this procedure. In fact, the synergy of an effective and a fundamental approach might be very fruitful. On the one hand, it poses several theoretical challenges to potential candidates for a fundamental theory, which might be ruled out or restricted in their generality. On the other hand, this mutual interplay can inspire new experiments and observations or suggest correlations which are invisible from an effective point of view. Our starting point is that string theory is entitled to be a valid candidate for a fundamental theory by a series of properties such as the fact that it is a consistent, UV-finite, anomaly- free quantum theory and it describes, in appropriate limits, quantum field theory and general relativity, hence providing a unified description of the four known forces. Within string theory, we choose to work in the framework of type IIB flux compactifications where the issue of moduli stabilization can be successfully addressed. On the phenomenological side, we focus on particle physics and cosmology, and in particular on the MSSM and the mechanism of inflation, respectively. We construct and study two different models of inflation in string theory. These are models of brane inflation, i.e. models where the primordial exponential expansion of the universe is driven by a scalar field representing the position of a D3-brane (a 3 + 1-dimensional dynamical object) in a compact space. We show explicitly that, allowing for fine tuning, a prolonged stage of slow-roll inflation can be achieved during which perturbations are generated in good agreement with the CMB data. In addition, we give an example of the fruitful synergy we mentioned above. One of our two models, namely inflation at the tip, might induce also DBI inflation which produces peculiar features in the spectrum of density perturbations that might be looked for using CMB data. On the particle physics side, we consider a model, the large volume scenario (LVS), which, among other things, provides a rationale for a low scale susy-breaking. We perform some checks on the consistency of LVS as a string theory model. To this end, we formulate an educated guess for the form of string loop corrections. We then show that physical predictions such as the soft susy-breaking terms are actually surprisingly robust against the inclusion of these corrections. x Abstract 1 Introduction and conclusions To motivate string theory phenomenology, which is the subject of the present work, one could start from two different perspectives. The first is incarnated by Plato, the person on the left part of figure 1.1. From this perspective, one should start introducing and discussing the “theory of Forms”, which in the present context is to be identified with string theory1. In a second step, one should descend to our “world of phenomena”, which is populated not by the Forms themselves, but by their shadows, which in the present discussion should be identified with the physical phenomena. The second perspective is incarnated by Aristotle2, depicted on the right part of figure 1.1. Now, the starting point are experiments and observations, from which one should collect a series of empirical facts. Then, one should try to construct a simple and elegant explanation to account for these facts. The explanation might have some implications which can in principle be tested. A successful and logically consistent explanation might eventually be promoted to the status of predictive theory. We are going to embrace both perspectives. First, we are going to discover the remarkable properties of string theory, which promoted it to a very active research field. Second we are going to take a bird’s-eye view on some outstanding open phenomenological issues in high energy.