On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

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96 Chapter 3. Solving the Flavor Problem in Strongly Coupled Theories group only, does not correspond to a theory with such a protection and therefore the T parameter turns out to be large. One can therefore consider a custodial symmetry indispensable for any extension of the SM, and in the custodially protected RS model with an corresponding extended bulk group one finds [114], S = 2πv2 M 2 � 1 − KK 1 � , T = − L π2v 2c2 wM 2 1 , (3.7) KK L as shown in Figure 3.2 by the orange band extending from the SM prediction towards larger S. One can think of other solutions, for example a version of the RS model in which the volume factor L is smaller than 36. The effect of this modification is shown in Figure 3.2 as well, in which the volume factor grows from L = 5 to L = 36 along the black arrows. The physical meaning of such a truncation is that the gauge hierarchy problem will only be cured up to some intermediate scale ΛUV = e L TeV. Based on little Higgs models, these models are called little RS (LRS) models and just like the little Higgs model they need some UV completion already at ΛUV = e L TeV. Since lowering the volume factor will affect many observables also in the flavor sector, we will come back to this option when we discuss FCNCs. Corrections to Z → b ¯ b The only observables with at least moderate deviations from the SM in 3.1 are sensitive to corrections to the Zb¯b vertex. Since the left-handed b quark is the weak isospin partner of the top, they share the same 5D mass or localization parameter cQ3 in the RS model. In order to generate the large top mass, cQ3 is shifted towards the IR compared to the light quark localizations. Consequentially, one should expect sizable corrections from the RS model. From (2.113) and the structures defined in (2.175) it follows that the Z−vertex correction for general external quark flavors can be written in the form in which L4D ∋ g cos θW × � ��gq � L g q L = � T q 3 − sin2 � θW Qq � 1 − m2 Z g q R = − sin2 θW Qq ij � 1 − m2 Z 2M 2 KK � 1 + m2 Z 2M 2 KK 4M 2 KK � 1 − 1 �� Z L 0 µ ij ¯qL,iγ µ qL,j + � g q R � ij ¯qR,iγ µ qR,j � L ∆Q − ∆ ′ � Q � − T q � 3 δQ − m2 Z � L ∆q − ∆ ′ q � � + T q � 3 δq − m2 Z 2M 2 KK � , (3.8) 2M 2 KK � L εq − ε ′ q � L εQ − ε ′ � Q � , � � , (3.9) with i, j = 1, 2, 3 denoting flavor indices and T d 1 3 = − 3 the corresponding weak isospin and charge for down type quarks. Relevant for the Zb¯b vertex are the 33 2 and Qd = − 1
3.1. Electroweak Precision Observables 97 elements of these matrices, that is the modifications to the flavor universal couplings. In general the Z coupling to light quarks is modified as well. However, because of the UV localization of the corresponding bulk fields, the SM expressions (g q L )11 = T q 3 − sinW θW Qq and (g q R )11 = − sin2 θW Qq are excellent approximations and the production can assumed to be SM like. 1With the help of the ZMA (2.180) and (2.184), we obtain g b L ≡ � g d � L 33 → � − 1 2 + sin2 � � θW 1 − 3 m2 Z 2M 2 F KK 2 (cbL ) � 5 + 2cbL L − 3 + 2cbL 2(3 + 2cbL ) �� (3.10) + m2 b 2M 2 ⎡ � ⎣ 1 1 KK 1 − 2cbR F 2 (cbR ) − 1 + F 2 (cbR ) � + 3 + 2cbR � |(Yd)3i| i=1,2 2 |(Yd)33| 2 1 1 1 − 2cdi F 2 (cbR ) ⎤ ⎦ , g b R ≡ � g d � R 33 → sin2 � θW 1 − 3 m2 Z 2M 2 F KK 2 (cbR ) � 5 + 2cbR L − 3 + 2cbR 2(3 + 2cbR ) �� (3.11) − m2 b 2M 2 ⎡ � ⎣ 1 1 KK 1 − 2cbL F 2 (cbL ) − 1 + F 2 (cbL ) � + 3 + 2cbL � |(Yd)i3| 2 |(Yd)33| 2 1 1 1 − 2cQi F 2 (cbL ) ⎤ ⎦ , in which the notation cbL ≡ cQ3 and cbR ≡ cd3 is introduced. The terms in the second lines in (3.10) and (3.11) come from the ZMA expressions of the δ matrices and are further suppressed by mb/mZ, so that the leading terms are controlled by the zero mode profiles F (cbL ) and F (cbR ). i=1,2 The ratio of the width of the Z 0 -boson decay into bottom quarks and the total hadronic width R 0 b , the bottom quark left-right asymmetry parameter Ab, and the forward- backward asymmetry for bottom quarks A 0,b FB , are given in terms of the left- and right-handed bottom quark couplings as [145] R 0 b = Ab = � 1 + 4 � � q q=u,d (gL )2 + (g q R )2� � η2ηQCD ηQED (1 − 6zb)(gb L − gb R )2 + (gb L + gb R )2� �−1 2 √ 1 − 4zb gb L + gb R gb L − gb R � gb L + g 1 − 4zb + (1 + 2zb) b R g b L − gb R �2 3 , A0,b FB = 4 Ae Ab , (3.12) where ηQCD = 0.9954 and ηQED = 0.9997 are QCD and QED radiative correction factors. The factor η2 = 0.99386 takes into account the recently published fermionic two loop contributions computed in [147], which are the only reason for the significant pull of R 0 b in Figure 3.1. The parameter zb ≡ m 2 b (mZ)/m 2 Z = 0.997·10−3 describes the effects of the non-zero bottom quark mass. As explained above, left- and right-handed couplings of the light quarks, g q L and gq R , and the asymmetry parameter of the electron, Ae can be assumed SM like and we fix these quantities to their SM values. In what follows we will employ gu L = 0.34674, gu R = −0.15470, gd L = −0.42434, gd R = 0.077345 and Ae = 0.1473 for the SM predictions [146, Table G3]. 1 For electrons as initial state the modifications are even smaller. ,
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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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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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