On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

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106 Chapter 3. Solving the Flavor Problem in Strongly Coupled Theories b quark as well as being constrained by very precise measurements. The branching ratios of both the Bd and Bs mesons to muons meet these criteria. Not only is the experimental limit very close to the SM prediction, which excludes large new physics effects and cuts deep into the parameter space of many extensions of the SM 6 , but also did CDF observe an excess of events in B(Bs → µ + µ − ), which allows for the first time to establish a two sided bound on the branching ratio with a central value slightly above the SM prediction. In the RS model, there are the following operators contributing to the effective Hamiltonian for b → sℓ + ℓ − transitions H b→sℓ+ ℓ − eff = Cℓ1 (¯sLγ µ bL) � ( ¯ ℓLγµℓL) + Cℓ2 (¯sLγ µ bL) � ( ¯ ℓRγµℓR) ℓ + ˜ Cℓ1 (¯sRγ µ bR) � ( ¯ ℓRγµℓR) + ˜ Cl2 (¯sRγ µ bR) � ( ¯ ℓLγµℓL) l + Cℓ3 (¯sLbR) � ( ¯ ℓℓ) + ˜ Cℓ3 (¯sRbL) � ( ¯ ℓℓ) . (3.33) ℓ The Wilson coefficients can only get contributions from the Z and its KK excitations, the photon KK modes and from the Higgs, see Section 2.6. They depend on the overlap integrals (2.175) and (2.184), which are described in the dual theory by the diagrams in the second line of Figure 2.7. Note, that we do not implement leptons as bulk fields. However, non-universal contributions at the lepton vertex are suppressed by m 2 µ/(msmb) relative to the leading contributions and can be neglected to very good approximation and for the same reason the tensor structures which are proportional to msmbm 2 µ/v 4 give only subleading corrections. One obtains in which ΣQ,q ≡ ω Z LLL C RS ℓ1 C RS ℓ2 ˜C RS ℓ1 ˜C RS ℓ2 C RS ℓ3 ˜C RS ℓ3 4πα = − 3M 2 (∆ KK ′ D)23 − 2πα (1 − 2s2w) s2 wc2 w M 2 (ΣD)23 , KK 4πα = − 3M 2 (∆ KK ′ D)23 + 4πα c2 w M 2 (ΣD)23 , KK 4πα = − 3M 2 (∆ KK ′ d )23 − 4πα c2 w M 2 (Σ KK ′ d )23 , 4πα = − 3M 2 (∆ KK ′ d )23 + 2πα (1 − 2s2w) s2 wc2 w M 2 (Σ KK ′ d )23 , 2mℓ = − m2 hv � ms v (δd)23 + mb v (δD)23 + (∆g d h )23 2mℓ = − m2 hv � mb v (δd)23 + ms v (δD)23 + (∆g d h )23 � 1 2 − s2 � w ∆Q,q + 3 M 2 KK m2 δQ,q Z 6 For example in the MSSM, the branching ratio (3.36) grows with tan β 6 . ℓ ℓ ℓ � , � , (3.34) Σ ′ Q,q ≡ ω Z RRL s2w 3 ∆Q,q + M 2 KK m2 δQ,q , Z (3.35)
3.2. RS GIM Working 107 with ωZ LL = ωZ RR = 1 in the minimal model and if the electroweak gauge group is extended in order to comprise a custodial symmetry, ωZ LL = 0 and ωZ RR = 3c2w s2 , as in w (3.15) and (3.16), compare also [167, App. B]. Corrections to b → sℓ + ℓ− arising from CRS ℓ3 and ˜ CRS ℓ3 are either O(mµmb/M 2 KK O(v4 /M 4 KK ) and will consequentially also be ignored in the numerical analysis. The branching ratios for the Bq → µ + µ − decays can be expressed as B(Bq → µ + µ − ) = G2 F α2 m3 Bqf 2 BqτBq 64π3s4 w � � (qb) � λ � t 2 � 1 − 4m2 µ m2 Bq � 4m2 µ × m2 � �CA − C Bq ′ � � A 2 + m 2 � Bq 1 − 4m2 µ m 2 Bq � �� mb CS − mq C ′ S mb + mq � � � � 2 � ) or , (3.36) where mBq, fBq, and τBq are the mass, decay constant, and lifetime of the Bq meson and λ (pr) q ≡ V ∗ qpVqr. The electromagnetic coupling α entering the branching ratios should be evaluated at mZ. The expressions for the coefficients CA,S and C ′ A,S read CA = cA − s4wc 2 wm2 Z α2λ (qb) � C t RS ℓ1 � − CRS ℓ2 , C ′ A = s4wc 2 wm2 Z α2λ (qb) � ˜C RS ℓ1 t − ˜ C RS � ℓ2 , CS = 2s4wc 2 wm2 Z α2mbλ (qb) C t RS l3 , C′ S = 2s4wc 2 wm2 Z α2mqλ (qb) t ˜C RS ℓ3 , (3.37) where cA = 0.96 ± 0.02 denotes the SM contribution to the Wilson coefficient of the axial-vector current [169, 170], and the coefficients CRS contain the 13 ℓ1−3 and ˜ CRS ℓ1−3 or 23 elements of the mixing matrices in the case of Bd → µ + µ − and Bs → µ + µ − , respectively. 7 The SM branching ratios of the Bq → µ + µ − decay channels evaluate to [171, 172] B(Bd → µ + µ − )SM = (1.0 ± 0.1) · 10 −10 , (3.38) B(Bs → µ + µ − )SM = (3.2 ± 0.2) · 10 −9 . (3.39) These predictions are obtained by normalizing the decay rates to the well-measured meson mass differences (∆mq)exp. This eliminates the dependence on CKM parameters and the bulk of the hadronic uncertainties by trading the decay constants for less uncertain hadronic parameters. The dominant source of error is nevertheless still provided by the hadronic input. The result is plotted in Figure 3.5 and shows again very good agreement with the current bounds, which however start to cut into the parameter space. Experimental bounds refer to the very precise limit from LHCb measurements, obtained including the full 2011 dataset [166], and the two-sided bound from CDF [168] respectively. It is 7 While CA and C ′ A are scale independent, the coefficients CS and C ′ S have a non-trivial RG evolution. We can ignore that because they are numerically insignificant.
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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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36 Chapter 2. The Randall Sundrum M
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156 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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