On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

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84 Chapter 2. The Randall Sundrum Model and its Holographic Interpretation UV brane IR brane ɛ t 1 UV brane IR brane ɛ t 1 Figure 2.11: Qualitative illustration of the RS-GIM mechanism. The dashed lines represent a t 2 term from the 5D gauge boson propagator. The solid lines show the profile of a UV and an IR localized fermion zero mode in the left panel and of to IR localized zero modes in the right panel. The blue shaded area can be interpreted as an indicator of the size of the FCNC coupling resulting from the overlap integral including the three functions. Moreover does the 5D description allow for a particularly illustrative explanation of the RS-GIM mechanism in geometric terms. In order to understand this let us analyze all possible contributions arising from tree level diagrams as shown in Figure 2.10. It describes a process in which external zero mode fermions couple to the tower of all KK modes of a gauge boson, which in the effective theory is given by the first term in the expansion of the 5D propagator for small momenta (2.82). The couplings will therefore be rescaled by an integral over the fifth dimension including the t-dependent terms of the zero modes (2.149) and the expressions (2.86) or (2.103). As an example, the Wilson coefficient of a four quark operator with four SU(2)L singlet external quarks Heff ∋ 3� i,j,k,l=1 � c cijkl ¯q i γµq c�� c j ¯q kγµq c� l , (2.172) with i, j, k, l flavor indices, gets contributions from KK photon exchange proportional to the following overlap integral cijkl ∼ α M 2 KK � 1 ɛ dt � 1 dt ɛ ′ t cq i +cq j t ′cq k +cq l � Lt 2 < − t 2 ( 1 2 − ln t) − t′2 ( 1 2 − ln t′ ) + 1 � , 2L (2.173) in which only the leading C-profiles are included and the 5D coordinates are assigned to the two vertices as shown in Figure 2.10. The integral � 1 dt t ɛ cq i +cq j × constant (2.174) will not lead to FCNCs, because in the case of the photon, the operator in (2.172) with only SU(2)L doublets gives rise to a similar integral with S-profiles which in the sum completes the orthonormality relation of the fermion profiles (2.125). FCNCs can therefore only arise if the gauge boson couplings distinguish between
2.5. Hierarchies in Quark Masses and Mixings and the RS GIM Mechanism 85 SU(2)L doublets and singlets or from propagator terms proportional to t or t ′ . Terms proportional to only one of the coordinates t or t ′ should therefore be read as t 2 → t 2 ×� in flavor space and since we will concentrate on the photon, which has vectorial couplings, can at most generate contributions to ∆F = 1 processes. Consequentially, only the term proportional to t 2 < = Min[t 2 , t ′2 ] in equation (2.173) will contribute to ∆F = 2 processes, while the second and third term will lead to flavor violation at the respective vertex and the constant is flavor diagonal. Note, that there are no negative powers of t or t ′ in the propagator. This implies, that the function becomes large in the IR and small in the UV. For the fermion zero modes, the sign of the exponent depends on the localization. The factor t cq i +cq j becomes large in the IR only if both qi and qj are IR localized, which is generically only the case if both are top quarks. If one of them or even both are UV localized, this factor drags the whole integral kernel into the UV which leads to small overlap integrals. As a rough guide one can assume that if the shared enclosed area of the function curves of the propagator term and the two zero modes is small (large), the overlap integral becomes small (large). This rule of thumb is illustrated in Figure 2.11. It works only because the propagators are IR localized. The zero mode of the graviton for example is localized in the UV, which makes the overlap integrals with SM fermion profiles even smaller, so that the conclusions drawn from Figure 2.11 are not applicable. The discussion in this section is based on the exchange of the whole KK tower, but would have led to the same qualitative results if only a single gauge boson KK mode was exchanged instead. In this case, the propagator in (2.173) is replaced by one summand of the right hand side of (2.89). The profiles of the gauge boson KK modes are given in (2.106), and are already IR localized due to the t in front of the Bessel functions. 13 Certain aspects of the full sum can however not be estimated by considering the exchange of a single KK mode. For example, the enhancement by the volume factor L ≈ 36 in front of the t 2
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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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10 Chapter 1. Introduction: Problem
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Page 44 and 45: 34 Chapter 1. Introduction: Problem
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134 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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