On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

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128 Chapter 3. Solving the Flavor Problem in Strongly Coupled Theories The covariant derivative in (3.73) reads � � �r (DµS) αa = δabδαβ∂µ − igD5δαβ GD µ T r � � �r � � r ∗δab ab + igS5 GS T µ αβ Sbβ , (3.74) in which the color indices of the SU(3)D (SU(3)S) are denoted by greek (latin) letters and the index r = 1 . . . , 8 runs over the number of generators of the respective group. If the vev of S is supposed to break the bulk SU(3)D × SU(3)S down to its diagonal subgroup, it must read Inserting the linear combinations (3.54) and using cos θ = 〈Saα〉 = vS √ δaα . (3.75) 2NC gS5 � g 2 S5 + g2 D5 and (3.56), it follows from the kinetic term ��DµS Tr � †� � µ D S � ∋ 1 2 , sin θ = g 2 s5 v2 S 2NC c 2 θ s2 θ gD5 � g 2 S5 + g2 D5 , (3.76) A r µA r µ . (3.77) Comparing this with the expressions (2.93) and (2.102), one can infer that for the axigluon vIR = L m2 A M 2 KK = L g 2 s v 2 S 2NC c2 θ s2 θ M 2 KK , (3.78) which we already anticipated in (3.64). In SM implementations of chiral color, vS is directly proportional to the axigluon mass. This is also true for an axigluon implemented in an RS model, if the IR brane provides the only source of SU(3)D × SU(3)S breaking. However, we could rule out this choice of BCs for the model at hand in Section 3.4, because it leads to a too light first KK mode. This is rooted in the fact that the IR brane is connected to the electroweak breaking scale. The vev of every scalar confined to the IR is naturally bounded from above by the cutoff vS ≤ MKK. However, if the axigluon has Dirichlet BCs in the UV, the mass of its first excitation will be in the TeV range even for vS well below MKK, as discussed in Section 2.3. Already in the SM it is questionable to assume such a minimal extension of the Higgs sector, even if the vev of S can in principle be chosen arbitrarily large. The problem arises not from the axigluon mass, but from the top quark mass. If the Yukawa interactions are dimension five operators as in (3.59), the ratio 〈S〉/Λ should be of order one if the top mass is to be generated with a perturbative Yukawa coupling. That makes it problematic to neglect higher dimensional operators with additional S insertions. In the extension of the RS model it brings about another drawback. If vS/MKK ∼ 1, the contributions to the mixed chirality Wilson coefficients in (3.66) do not have an additional suppression and become equally large as the original terms in (3.20) so that
3.6. The Extension of the Scalar Sector 129 the RS flavor problem is not solved. The scalar sector introduced in the next section will allow for dimension four Yukawa couplings, so that the vev is necessarily connected to the electroweak scale and a sufficient suppression of the dangerous FCNCs is achieved. A Realistic Higgs Sector In order to evade the tension between a realistic top Yukawa coupling and the suppression of contributions to ɛK, we consider a scalar sector in which the Higgs which is responsible for EWSB also carries color charges. Such an extension is shortly mentioned for a chiral color extension of the SM in [195]. The corresponding Yukawa couplings have already been given in (3.60), and the additional terms in the Lagrangian read � δ(|φ| − π) ��DµHu � †� � µ LS = Tr D Hu rc � ��DµHd � †� � µ + T r D Hd � + � Dµh � � †� � µ D h − V (h, Hu, Hd) , (3.79) where the potential will again be given in Appendix D. The three scalar fields carry the quantum numbers SU(3)D SU(3)S SU(2)L U(1)Y Hu 3 ¯3 2 − 1 2 Hd 3 ¯3 2 1 2 h 1 1 2 1 2 The Higgs with SM quantum numbers will in this section be denoted by h for a better differentiation, and the corresponding covariant derivative given by (2.96) does not include couplings to the color group gauge bosons. Notice also, that the Dirichlet BCs in the UV can be modeled by putting the Lagrangian (3.73) on the UV brane and consequentially introducing a vev 〈S〉 ≈ MPl. 10 There need to be different color charged Higgs fields for up and down quarks, because the complex conjugated Hd transforms as H † d ∼ (¯3, 3) under SU(3)D × SU(3)S and therefore if it is inserted in the up-Yukawa interactions it will not saturate the quark color charges11 . The SM Higgs h will give Yukawa interactions for the leptons. In principle one could write down effective lepton Yukawas with only the two color charged fields, because h ∼ HdHdHu since 3 ⊗ 3 ⊗ 3 = (6 ⊕ ¯3) ⊗ 3 ∋ 1. However, the potential V (Hu, Hd) has a global U(3) × U(3) symmetry, while the vacuum is invariant under SU(3)C × U(1). This will lead to nine Goldstone bosons from which only eight will be absorbed by the axigluons in order to become massive. It is therefore necessary 10 A scalar sector on the UV brane may not include an electroweak doublet, because this would introduce Yukawa interactions on the UV brane. 11 Notice, that for a Higgs bidoublet as in the custodial model, there is no problem in writing down both Yukawa terms with one field, because for SU(2) the fundamental and anti-fundamental representation are isomorphic.
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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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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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