On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

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40 Chapter 2. The Randall Sundrum Model and its Holographic Interpretation UV brane IR brane SU(3)C × SU(2)L × U(1)Y S1/Z2 0 φ π SU(3)C × U(1)EM Figure 2.1: Illustration of the minimal RS model considered in this thesis. All fermions and gauge bosons are assumed to be bulk fields and the symmetry is broken at the IR brane by a Higgs boson. The effect of the warp factor for dimensionful parameters is indicated by gray lines. 2.2 AdS/CFT I have found that in the arena of warped compactifications, the qualitative insight gained from the AdS/CFT connection between such compactifications and strongly coupled 4D dynamics, has saved me time and time again from errors. Its like checking units as an undergraduate, in principle its not necessary, but in practice indispensable Raman Sundrum The term AdS/CFT duality originates from a conjecture of Maldacena [95] and means in its broadest sense, that a weakly coupled theory of fields defined in the bulk of a d dimensional AdS space is equivalent to a strongly coupled CFT on its d−1 dimensional boundary. This section will be a bottom-up introduction to the AdS/CFT duality, meaning that we will not go into detail about the original Maldacena conjecture, but keep the discussion close to the applications in this thesis. We will however need some of the original framework in order to motivate the conclusions in the later parts of the section. In its original formulation, the weakly coupled bulk theory is the decoupling (SUGRA) limit of a certain type of string theory on an AdS5 × S5 space and the corresponding boundary theory is a maximally supersymmetric conformal SU(N)- Yang-Mills theory in the large N limit. With R = 1/k the AdS curvature radius, ℓs the string length and gYM the coupling of the boundary Yang-Mills theory, it follows the relation R 4 ℓ 4 s = 4πNg 2 YM . (2.17)
2.2. AdS/CFT 41 It tells us, that if the string length ℓs ≪ R, the Yang-Mills gauge coupling times the number of colors N (the t’Hooft coupling) becomes large and the Yang-Mills theory is strongly coupled. In addition, nonperturbative string states with masses proportional to the inverse string coupling m ∼ 1/gs, are required to decouple as well [107, p.22]. This means gs → 0 and therefore we have necessarily a large N boundary theory, because gs ∼ 1/N. The picture becomes much more intuitive once one examines the spacetime symmetries on both sides. Consider the AdS metric in conformal coordinates (2.2). It is evident, that this metric is invariant under the rescaling x µ → λx µ , z → λz. (2.18) Fields transform under such a rescaling according to their scaling dimension ∆φ, φ(xµ, z) → λ −∆φφ(λxµ, λz). In contrast to flat space, every Lorentz invariant action will turn out to be invariant under rescalings as well due to this property of the metric. For example � SAdS = d 4 � �5 � R 1 x dz z 2 GMN M 2 ∂Mφ ∂Nφ − 2 φ2 − λ 4! φ4 � � → λd 4 � �5 � R 1 x dλz λz 2 λ−2G MN (λ∂M)λ −∆φφ (λ∂N)λ −∆φφ � M 2 − , (2.19) 2 λ−2∆φ φ 2 − λ 4! λ−4∆φ φ 4 which is invariant upon absorption of a factor λ −∆φ in each field, in contrast to flat space, where additional factors from the metric appear. This implies, that regardless of the mass dimension of whatever operator we write down in the bulk Lagrangian, the action will remain scale invariant. In the dual boundary theory the fifth coordinate is identified with the inverse of an energy scale and thus we will have a scale invariant boundary action. In general, it is still an open question whether scale invariance in 4D already implies conformal invariance (see e.g.[108]), however this is a strong hint that we are dealing with a CFT on the boundary. In fact it can be shown, that the 15 generators of the spacetime symmetry (isometry) group of AdS5 correspond to the 15 generators of the conformal group of the dual boundary theory, whereas the SO(6) isometry of the additional S5 corresponds to the R-symmetry group of the N = 4 supersymmetry [109, p.187-188]. Changing the non–AdS part of the manifold will therefore only cut down the amount of supersymmetry on the CFT side and we can consider the duality on AdS5 alone without loosing essential ingredients. At this point it should be pointed out, that there is no rigorous mathematical proof of the original conjecture as of this writing and for the stripped down versions we are considering here it could be that the corresponding CFT does not even exist. However, we will see that it is very insightful to switch to the CFT view in order to get a deeper understanding of the RS model. In a more precise formulation, the conjecture states that the 5D partition function of bulk fields φ(x µ , z) with boundary conditions at the AdS5 boundary φ(x µ , z) � � z=0 =
Page 1 and 2: On the Flavor Problem in Strongly C
Page 3 and 4: UNIVERSITY OF MAINZ ABSTRACT THEORE
Page 5 and 6: Contents Acknowledgements vii Prefa
Page 7: Acknowledgements First and foremost
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90 Chapter 2. The Randall Sundrum M
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92 Chapter 3. Solving the Flavor Pr
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100 Chapter 3. Solving the Flavor P
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146 Chapter 4. The Asymmetry in Top
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176 Chapter 5. Conclusions describe
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178 Appendix A. Generating Paramete
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180 Appendix A. Generating Paramete
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182 Appendix B. Neutral Meson Mixin
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184 Appendix B. Neutral Meson Mixin
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C Input Parameters This appendix is
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Parameter Value ± Error Reference
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192 Appendix D. Higgs Potential for
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194 Bibliography [13] S. Dimopoulos
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196 Bibliography [43] G. Nordström
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198 Bibliography [72] K. S. Babu,
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200 Bibliography [104] R. Contino a
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202 Bibliography [133] M. Baak, M.
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204 Bibliography [160] J. Charles,
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206 Bibliography [185] C. Csaki, G.
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208 Bibliography [213] E. Thomson e
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210 Bibliography [239] V. Ahrens, A
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