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Domain-Wall Brane Models of an Infinite Extra Dimension

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Domain-Wall Brane Models of an Infinite Extra Dimension Domain-wall brane models of an infinite extra dimension Damien P. George Submitted in total fulfilment of the requirements of the degree of Doctor of Philosophy March 2009 School of Physics The University of Melbourne Produced on archival quality paper Abstract In this thesis we consider aspects related to the construction and phe- nomenology of domain-wall brane models of a single, infinite, extra spatial dimension. We discuss how gravity, gauge- and matter-fields become four- dimensional at low energies due to the dynamical formation of a topological defect known as a brane. These ideas are amongst the leading candidates for extensions to the standard models of particle physics and cosmology. We first examine a toy model where a pair of scalar fields, charged under a U(1) U(1) gauge symmetry, form a background domain-wall configura- ⊗ tion. This analysis demonstrates the general ideas of domain-wall formation and stability, dimensional reduction and semi-confinement of gauge fields. Gravity is incorporated into this toy model, in the form of a regularised version of the Randall-Sundrum warped metric. For the case of a single real scalar field, we show in detail how a domain wall can be obtained, and determine its relationship to the fundamental brane in the infinitely thin wall limit. Explicit expressions for the modes of such a domain wall are found, as well as the modes associated with cou- pled fermions and scalars. The symmetric modified P¨oschl-Teller potential arises in this context, and the analysis elucidates the reduction of a five- dimensional field to a tower of four-dimensional modes. Adding gravity to the model alters the spectrum of four-dimensional modes trapped to the wall. What were once discrete modes become res- onances within a continuum, and low-energy modes are coupled to these continuum bulk modes. We show explicitly how this comes about, and use a toy model to demonstrate that this coupling can be made small enough to avoid experimental constraints. The Dvali-Shifman mechanism for gauge field localisation is introduced, and we use it, along with the previously discussed techniques for construct- ing domain walls, to write down an SU(5) grand unified, single generation iii iv version of the standard model. Fermion mass relations and Higgs induced proton decay are significantly improved from usual SU(5) models due to an inherent split fermion set-up. An extension of the gauge group to E6 is considered, naturally including the clash-of-symmetries mechanism, and it is shown that the field content can be simplified. Cosmological implications of domain-wall branes are investigated, and we attempt to reproduce an effective four-dimensional Friedmann, Lemaˆıtre, Robertson-Walker metric for brane-localised matter. We find that for a do- main wall of finite thickness it is not possible to define a common spacetime for all localised species of matter. As a consequence, different species ex- perience a different effective four-dimensional scale factor, an unusual effect that is suppressed when the domain wall is made sufficiently thin. We conclude that domain-wall brane models of an infinite extra dimen- sion are viable extensions of the standard models of particle physics and cosmology. These models provide a rich set of phenomenology, and allow new ways to tackle existing theoretical problems. Acknowledgements Available only in printed version. v Contents 1 Introduction 1 1.1 KaluzaandKlein......................... 3 1.2 Thestandardmodelofparticlephysics. 6 1.2.1 Gaugetheories ...................... 6 1.2.2 Symmetry breaking and the electroweak force . 9 1.2.3 Quarks,gluonsandstrings . 13 1.2.4 Thestandardmodel . 15 1.2.5 Shortcomings of the standard model . 18 1.2.6 Grandunification. 20 1.3 The early years of brane world models . 26 1.3.1 Field theoretic domain walls . 26 1.3.2 Superstring theory and D-branes . 31 1.3.3 The Dvali-Shifman mechanism . 33 1.3.4 Low-energy string theory and brane worlds . 37 1.4 Basiccosmology ......................... 40 1.4.1 Expansion and the FLRW metric . 41 1.4.2 Cosmological domain walls . 43 1.5 TheRandall-Sundrumwarpedmetric. 44 1.5.1 The two Randall-Sundrum scenarios . 45 1.5.2 Beyond Randall-Sundrum . 52 1.5.3 SummaryofRandall-Sundrum . 59 1.6 Overviewofthethesis . 59 2 The U(1) U(1) toy model 63 ⊗ 2.1 Themodelanditskinksolutions . 65 2.2 Stability of the background configuration . 71 2.3 Includinggravity ......................... 76 vii viii Contents 2.4 Conclusion ............................ 81 3 Kink modes and confined matter fields 83 3.1 Thekinkanditslimits. 86 3.1.1 Kink modes and the effective model . 88 3.1.2 Limiting behaviour of the kink . 91 3.1.3 Is the kink zero mode really frozen out? . 93 3.2 Addingascalarfield . 103 3.2.1 Scalarmodes ....................... 103 3.2.2 The thin kink with a scalar field . 106 3.3 Addingafermionfield . 109 3.3.1 Fermionmodes . 109 3.3.2 Kink limits with a coupled fermion field . 114 3.3.3 Four- and five-dimensional interacting fields . 115 3.4 Conclusion ............................116 4 Warped gravity and matter spectra 119 4.1 Randall-Sundrum and the volcano potential . 122 4.2 Fermions in the presence of gravity . 127 4.3 Localised scalar fields with gravity . 132 4.4 Toy model calculation . 134 4.5 Conclusion ............................139 5 The SU(5) model 141 5.1 The Dvali-Shifman mechanism . 144 5.2 Themodel.............................148 5.3 Aspectsofthemodel. 154 5.3.1 The fermion and Higgs fields . 155 5.3.2 Effective Higgs sector and fermion masses . 160 5.3.3 Includinggravity . 163 5.4 Relationships among the scales . 165 5.5 Conclusion ............................166 6 E6 invariance and the clash of symmetries 169 6.1 Theclashofsymmetries . 172 6.2 Attempt at an SO(10) model . 180 6.3 Structure of the E6 invariants . 184 Contents ix 6.3.1 Truncated analysis of I6 .................190 6.3.2 Thefullanalysis . 194 6.4 An E6 domain-wallmodel . 196 6.4.1 Domain-wall solutions . 198 6.4.2 Localisation of fermion zero modes . 205 6.5 Conclusion ............................209 7 Cosmology of domain-wall branes 211 7.1 Fundamental-brane cosmology . 213 7.2 The extension to a domain-wall brane . 217 7.2.1 A localised scalar field . 220 7.2.2 Localised fermions . 226 7.2.3 The effective Newton’s constant . 229 7.3 Effective scale factor for a thin domain wall . 231 7.4 Conclusion ............................234 8 Conclusions and outlook 237 List of publications 245 Bibliography 247 A Conventions, definitions and identities 269 A.1 Spinors ..............................269 A.2 Some useful integrals and limits . 270 A.3 Sign conventions for general relativity . 271 A.4 Thevielbeinformalism. 273 B Numerical techniques 275 B.1 Relaxationonamesh . 276 B.2 Twofold fourth-order Runge-Kutta . 277 C Symmetric modified P¨oschl-Teller potential 281 List of Figures 1.1 Weight diagram of the fermions of the standard model. 23 1.2 (a) Plot of the quartic potential that induces a domain-wall configuration. (b) The domain-wall kink solution and a lo- calisedfermionprofile. 29 1.3 (a) A na¨ıve approach to confining a U(1) gauge field that fails. (b) A schematic representation of the Dvali-Shifman mechanismatwork. ....................... 34 2.1 Time-independent domain-wall solutions for the two scalar and two gauge fields in the U(1) U(1) model. 70 ⊗ 2.2 The function U(w) used to determine the eigenvalues of the ri perturbation in the light- and space-like cases. 75 2.3 The minimum of the function U(w) in the light-like case plot- ted against λ2. .......................... 75 2.4 The minimum of the function U(w) in the space-like case. 75 3.1 Plot of the potential well corresponding to modes of the kink. 89 3.2 Typical extra-dimensional profiles for the lowest, discrete, bound states arising from a scalar field coupled to a kink. 105 3.3 Extra-dimensional profiles for the massless, left-handed four- dimensional fermion, and two massive fermions, which arise when a five-dimensional fermion is coupled to a kink. 112 4.1 Plot of the volcano-shaped graviton trapping potential Ugrav(z) and the zero mode profile E˜0(z) for the Randall-Sundrum warped metric with an infinitely thin brane. 124 xi xii List of Figures 4.2 Plot of the volcano-shaped graviton trapping potential Ugrav(z) and the zero mode profile E˜0(z) for domain-wall brane model withasmoothwarpedmetric. 126 ˜ eff 4.3 An example of the effective Schr¨odinger potential, WL , which traps a left-handed fermion field for no gravity, weak gravity andstronggravity. 136 4.4 Fourier decomposition of the extra-dimensional profile h˜typ(z)= l/2 cosh−1(lz) in terms of the scalar modes h˜n(z). 137 (4) 4.5p The “interaction per continuum mode”, λn /λ√l, for contin- uum fermion modes interacting with a typical bound mode onthebrane. ...........................138 4.6 The extra-dimensional profiles of the resonant mode at (E/l)2 ≃ 2.3, and a mode off-resonance by 2.0 10−4 in units of (E/l)2. 139 × 5.1 Plot of the average plaquette as a function of the inverse lattice temperature β for 4 + 1-dimensional SU(2) and SU(5) Yang-Mills. ............................147 5.2 Typical extra-dimensional profiles fnY (w) for the fermion com- ponents contained in the 5∗ and the 10. ...........158 5.3 Example potential profiles WY (w) which trap the Higgs dou- blet Φw and coloured scalar Φc. ................159 6.1 The vacuum manifold of a G Z H model and some ⊗ 2 → typical choices for domain-wall configurations.
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