On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

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20 Chapter 1. Introduction: Problems beyond the Standard Model Here {κ = (κ − , κ 0 ), �κ} and {h = (h − , h 0 ), η} denote each five Goldstone bosons from the breaking G/H. Now assume that the diagonal subgroup F = SU(3)diag of G is gauged, so that the kinetic part of (1.35) becomes Lkin = (Dµφ1) † (D µ φ1) + (Dµφ2) † (D µ φ2) (1.37) with Dµ = ∂µ − ig T a A a µ, a = 1, . . . , 8. This is an explicit breaking of the subgroup H, so that only I = H ∩ F = SU(2)diag (generated by the T a s with the Pauli matrices in the upper left corner) remains unbroken. The interaction terms from (1.37) can be written as � i � |Dµφi| ∋ Tr g 2 (A a µT a ) 2 (φ1φ † † 1 + φ2φ 2 ) � , (1.38) where g is the weak coupling constant in this model and (ignoring the singlet η) φ1φ † † 1 + φ2φ 2 = � h † h 0 0 f 2 − h † � . h (1.39) And, if we denote the new gauge bosons by X and the SM ones as usual, we find in the mass eigenbasis so that A a µT a = 1 √ 2 � Tr g 2 (A a µT a ) 2 (φ1φ † † 1 + φ2φ 2 ) � ⎛ B ⎜ ⎝ 0 µ + X8 µ √3 W − µ X 0 µ W + µ X 8 µ √3 − B 0 µ X + µ X 0 µ X − µ −2 √ 3 X 8 µ = g2 � 2 8 f 2 3 X8 µX µ8 + X 0 µX µ0 + X + µ X µ− � ⎞ ⎟ ⎠ , (1.40) + g2 2 h2 � W + µ W µ− + 2B 0 µB µ0 − X + µ X µ− − 2X 8 µX µ8� + 2g2 √ 3 h 2 X 8 µB µ0 . (1.41) Two things become evident from this result. First, the five gauge bosons corresponding to the broken T a s, a = 1, . . . , 5 have eaten the κ fields from (1.36) and gain masses proportional to the scale f, while the remaining three gauge bosons will become massive if the Higgs field h acquires a vev. Second, the couplings to the Higgs in the second line of (1.41) are of the same magnitude but of opposite sign between the new gauge fields and the weak gauge fields. This leads to a cancellation (between equal spin fields), which will assure that no quadratic divergent loop diagrams appear in the theory. The reason is, that if one covariant derivative in (1.37) was replaced by Dµ → ∂µ, all GBs of the corresponding scalar would be exact. Thus, only gauge interactions including both scalars φ1 and φ2 break the shift symmetry and can give
1.1. Solutions to the Gauge Hierarchy Problem 21 a contribution to the Higgs mass. A corresponding diagram is given by φ2 φ1 : g4 log 16π2 � Λ 2 µ 2 � |φ † 1 φ2| 2 = g4 � Λ2 log 16π2 µ 2 � (f 2 − 2h † h) 2 , (1.42) and a similar mass is generated for the real singlet η. Such a cancellation must also be installed in the fermion sector, so that these models at least have an additional toppartner. Light fermions may give quadratic corrections up to a larger scale because their Yukawa couplings are small (the same reason which allows for a splitting in the MSSM sfermion mass spectrum). Thus, the hierarchy between the scale of compositeness and the electroweak scale is explained and one naturally achieves the desired sin 2 θ ∼ 1/(16π 2 ) ≪ 1. For completeness, it should be mentioned that it is also possible to construct little Higgs models with one symmetry breaking scalar, but two different gauge groups with a common factor group. In these scenarios, collective symmetry breaking is achieved by multiple gauged subgroups of the global symmetry (if only one gauge group is active, the residue global symmetry protects the Higgs mass). In the minimal model, one can get along with exactly four gauge partners for the four electroweak SM gauge bosons [40]. Extra Dimensions Flat Extra Dimensions Additional compact spatial dimensions may provide an explanation for the extreme weakness of gravity, if it is the only force which feels the extra volume and is thus diluted by propagating in the extra dimension. Historically, additional compact dimensions were first proposed by Nordström [43] and subsequently by Kaluza and Klein [45, 44] almost 100 years ago, originally introduced with the purpose to find a unified theory of gravity and electrodynamics. In the context of the hierarchy problem however, the significance of extra dimensions was only realized by Arkani-Hamed, Dvali and Dimopoulos (ADD) in 1998 [48]. Their idea was motivated by the observation, that gravity, in contrast to the SM, is only tested at distances of order 0.1 mm, corresponding to an energy scale of 10 3 eV [46]. 18 At these energies, for small masses, Newtons law is a good approximation of general relativity. A modification of the inverse square law might therefore have gone undetected if it only comes to the fore at distances smaller than a tenth of a millimeter. To be 18 This bound is so weak, that there are serious attempts to explain the cosmological fine-tuning problem –the question why the cosmological constant is so small despite quartic radiative corrections– by a modification of gravity close to this scale [47].
Page 1 and 2: On the Flavor Problem in Strongly C
Page 3 and 4: UNIVERSITY OF MAINZ ABSTRACT THEORE
Page 5 and 6: Contents Acknowledgements vii Prefa
Page 7: Acknowledgements First and foremost
Page 10 and 11: x One of the most fascinating insig
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92 Chapter 3. Solving the Flavor Pr
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100 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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