On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

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60 Chapter 2. The Randall Sundrum Model and its Holographic Interpretation qi qk qk × qi × qi qi A × µ Oqi Oqi A µ Oµ Oqi × qj qj × Oqi Oµ × × qi qi , qi qi qi , × qi × Oqi Oqi Oµ × × Aµ A µ + the same diagrams with the replacement: qj qj Oµ × Aµ qj qj qj qj OH OH Oµ Oµ Oµ ∆F = 0 ∆F = 1 ∆F = 2 Figure 2.7: Collection of the contributions to the full four fermion operator in the strongly coupled dual theory with at most one Oµ propagator, constructed from the building blocks given in Figures 2.5 and 2.4. Thick lines represent composites and thin lines elementary fields. dependence has been kept at this point. It is also puzzling on first sight, why the coefficient c1 would be exactly one, regardless of the BCs, which seems not motivated at all from the 5D theory. This will however become clearer by considering the strongly coupled dual theory. Before we discuss specific limits, let us therefore collect the different contributions we would expect in the dual theory. In terms of diagrams, they are given in Figure 2.7. In principle, fermion lines are not directly relevant for the following discussion (we only compute the boson propagator), but they are included to illustrate the different contributions. If there is an elementary gauge boson, that is the propagator has Neumann BCs in the UV, there exist contributions which can only be flavor diagonal, shown in the first line of Figure 2.7. They consist of a pure gauge boson contribution and a contribution from mixing in the composite state. Following the discussion of section 2.2, composite states are always in the theory. Boundary conditions will only tell us whether there is a gauge boson to mix into or not, and if the composites get mass corrections from a composite Higgs coupling (depicted in the last line of Figure 2.7). Also, in our model, fermions are bulk fields and therefore the elementary quarks mix into corresponding composite states. This allows for flavor violation at either one vertex, see the second line of Figure 2.7, or ∆F = 2 contributions from
2.3. Profiles of Gauge Bosons 61 pure composite exchange, which can be found in the third line of Figure 2.7. Keep in mind, that these diagrams are flavor-diagonal in Figure 2.7, but not flavor universal, which will lead to the described flavor violation after rotation into the basis in which the Yukawa matrices are diagonal. Also, the diagrams in the second and third line of Figure 2.7 can give flavor conserving and ∆F = 1 contributions as well, the right hand side only denotes the maximal number of vertices at which flavor violation can occur simultaneously. Further diagrams can be constructed through additional mixing from composites into the elementary states and back. However, each additional Oµ propagator carries another factor M −2 KK and will the corresponding terms will thus be suppressed. An interesting result of this thesis is, that we are able to assign diagrams in the dual theory to the different contributions of (2.82), by systematically considering different BCs. In the following, the respective BCs and the contributing diagrams are shown on the left hand side. Also, after performing the limits for the respective BCs, ɛ-suppressed factors will be omitted and the results are given in Feynman-’t Hooft gauge, ξ = 1. Let us start with the “cleanest” scenario. (DN) Dirichlet BCs in the UV imply, that there is no elementary gauge boson in the theory. Consequentially, there is no mixing as in (2.41). If there were no composite fermions, the elementary and composite sector would not talk to each other at all. There is also no symmetry breaking in the composite sector, because all fields (for all possible gauge indices) have Neumann BCs in the IR. Referring to Figure (2.7), that means, that we do only expect the exchange of composite states, i.e. only the diagram in the third line. Interestingly, for these BCs, (2.82) reduces to D ξ=1 µν (q, t; t ′ ) = ηµν L 4πrc M 2 t KK 2 < , (2.84) which represents the term which we found to appear in (2.82) with always the same coefficient regardless of the BCs. In the dual theory, this corresponds to the exchange of exclusively composites, which is only controlled by the bulk. That means, one can check that a different localization, ∆ �= 3, will in general change this factor. This term will however always be present as long as ∆ = 3 and a change of BCs will only generate additional diagrams.
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On the Flavor Problem in Strongly C
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UNIVERSITY OF MAINZ ABSTRACT THEORE
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Contents Acknowledgements vii Prefa
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Acknowledgements First and foremost
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x One of the most fascinating insig
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2 Chapter 1. Introduction: Problems
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Page 20 and 21: 10 Chapter 1. Introduction: Problem
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110 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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On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

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