On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

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56 Chapter 2. The Randall Sundrum Model and its Holographic Interpretation define the propagator of the vector and scalar components, �� − p 2 − M 2 1 KKt∂t t ∂t + ɛ2 t2 M 2 � A η µν � + 1 − 1 � p ξ µ p ν � D ξ νρ(p, t; t ′ ) = Lt′ 2πrc � p 2 + ξM 2 KK∂t t ∂t 1 t δ µ ρ δ(t − t ′ ) , (2.58) ɛ2 − t2 M 2 � A D ξ 55 (p, t; t′ ) = Lt′ δ(t − t 2πrc ′ ) . (2.59) We further define the dimensionless quantities qµ ≡ pµ/MKK and cA ≡ ɛMA/MKK and simplify, � � − q 2 1 − t∂t t ∂t + c2 A t2 � η µν + � 1 − 1 ξ � � µ ν q q D ξ νρ(q, t; t ′ ) = � 1 ξ q2 1 + t∂t t ∂t − c2 A /ξ − 1 t2 � ξD ξ 55 (q, t; t′ ) = D ξ νρ(q, t; t ′ ) = Aξ(q, t; t ′ ) qνqρ q 2 + B(q, t; t′ ) Lt′ 2πrcM 2 δ KK µ ρ δ(t − t ′ ) , Lt′ 2πrcM 2 KK ηνρ − qνqρ q 2 (2.60) δ(t − t ′ ) . (2.61) For the vector component, one can read off an ansatz for the solution [112], � � , (2.62) which, after inserting in (2.60) yields two independent equations, � − 1 ξ q2 1 − t∂t t ∂t + c2 A t2 � � Aξ = − q 2 1 − t∂t t ∂t + c2 A t2 � B , (2.63) � � Lt B = ′ δ(t − t ′ ) . (2.64) − q 2 1 − t∂t t ∂t + c2 A t2 2πrcM 2 KK The first one tells us that Aξ(q, t; t ′ ) = B(q/ √ ξ, t; t ′ ) and therefore D ξ 55 (q, t; t′ ) = − 1 ξ Aξ(q, t; t ′ ) = − 1 ξ B(q/�ξ, t; t ′ � ) with cA → c2 A /ξ − 1 . (2.65) Solving the second equation (2.64), is sufficient to compute both propagators. Evaluated for t �= t ′ the equation can be put in the form � (qt) 2 ∂ 2 qt + (qt)∂qt + � (qt 2 ) − (c 2 A + 1) � � B(qt) = 0 , (2.66) qt which is a Bessel PDE with solutions (compare (2.27)) B>(qt>) = (qt>) � C > 1 J∆−2(qt>) + C > 2 Y∆−2(qt>) � , for t > t ′ , (2.67) B
2.3. Profiles of Gauge Bosons 57 see Table 2.1. But the A5 component will also not propagate in the unitary gauge, ξ → ∞, which is the exact behavior of a Goldstone boson and confirms therefore our the findings of Section 2.2, that A5 can be considered to be the holographic dual of a Goldstone boson. In this limit, the KK excitations of the vector fields will eat the KK modes of the A5s and become massive, compare [62]. The behavior of the zero modes will depend on the BCs. The propagator is symmetric in t and t ′ and a viable ansatz for the bulk solution is therefore B(q, t>, t, t 1 J∆−2(qt>) + C > � 2 Y∆−2(qt>) where the normalization is fixed by the jump condition, so that × � C < 1 J∆−2(qt) B(q, t>, t 1 C< 2 − C< 1 C> 2 1 Lt ′ 2πrcM 2 KK (2.70) , (2.71) . (2.72) The remaining coefficients are fixed by the boundary conditions. We will be mainly interested in gauge bosons, that is ∆ = 3. However, a mass term may be induced by a bulk Higgs, and can then be written as c 2 A → L v2 4g2 4 4M 2 , (2.73) KK in which we follow [57] in defining g5 = √ 2πrcg4, and further use v4 = ɛv5 √ √ as well as adjusting the mass dimension by 〈H〉 = v5 k/ 2. Note, that this expression is not general, but true for the SM. 9 In the case of such a spontaneous symmetry breaking in the bulk, one also has to introduce the corresponding Goldstone bosons ϕA in the gauge fixing term in (2.52), � ∂µA µ − ξ r2 ∂φe c −2σ �2 � Aφ → ∂µA µ � √k v5g5 − ξ 2 ϕA + 1 r2 ∂φe c −2σ ��2 Aφ . (2.74) Note, that in this model a linear combination of Aφ and the Goldstone boson ϕA, which has mass dimension [ϕA] = 3/2, is absorbed by the KK modes of the Aµ, while the other combination remains physical, see [50, App. A]. As a consequence, there is a tower of additional pseudo scalars, which is a distinctive signal of a bulk Higgs mechanism. This is not the case if brane Higgs fields are introduced, which can be implemented 9 Other gauge groups and symmetry breaking patterns induce different numerical factors, compare Section 3.6.
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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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4 Chapter 1. Introduction: Problems
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106 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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