On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

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14 Chapter 1. Introduction: Problems beyond the Standard Model large amount of symmetry of a conformal theory allows to extract information on the bounds on the scaling dimension ∆ ¯ QQ in dependence of ∆ ( ¯ QQ) 2 numerically. The progress made over the last years in these studies leads to the blue curve plotted in Figure 1.4, corresponding to a fit function from [30, Sec. 3.2]. The blue shaded region corresponds to the numerically excluded region. The result is, that there exists no conformal field theory in the phenomenologically preferred region, indicated by the red shaded rectangle. Additional assumptions, like minimal flavor violation, are thus necessary to find a viable CTC theory. Nevertheless, CTC may be the only way to implement strongly coupled EWSB without introducing composite fermions. Partial Compositeness The concept of partial compositeness was introduced in 1991 by Kaplan [31]. It is based on the fact that there may be a fourth kind of dimension six operators not included in (1.11), if the strongly coupled sector allows for fermionic bound states with the same quantum numbers as the SM fermions, build from three technifermions, dij qT i Q QT j Q Λ 2 ETC . (1.21) This bound state will appear as a composite Dirac-fermion and leads to a novel mechanism to generate fermion masses. The basic idea is, that the elementary SM fermions are massless and do not couple directly to the technicolor condensate, i.e. the TC condensate has a small or even a negative anomalous dimension, so that the b-terms are irrelevant. The composite fermions however may have a marginal Yukawa coupling with the TC condensate (suggestively called H in this section), because the Yukawa operator does now only contain composites and its scaling dimension is therefore not the sum of the scaling dimensions of its constituents. 13 Only through the linear mixing generated by the operator (1.21) will EWSB be communicated to the SM fermions. Kaplan first developed his idea in upscaled QCD TC and therefore he considered the techniquarks to form technibaryons Q ¯ QQ → |ψ(0)| 2 B, with |ψ(0)| 2 the wavefunctions overlap of the techniquarks in the technibaryon, measuring the coupling strength. Naive dimensional analysis tells us |ψ(0)| 2 ∼ Λ3 TC , which can be supported by comparison with the QCD analogue, see [32, eq. (2.12) and (2.13)]. Denoting the left/right-handed Lorentz-chirality of the SU(2)L-doublet composite by BL and BR and the SU(2)L-singlet with components Bc L and Bc R , one can write down an effective Lagrangian, L ∋ Λ3 TC Λ2 ( ETC ˜ d qRB c L + d qLBR) − mBBB − �mBB c B c + BL (λHB c R) + h.c. . (1.22) Here, d and ˜ d denote the mixing parameters from (1.21) and its analogue involving a right-handed SM-quark, λ is an O(1) Yukawa coupling in the composite sector, and 13 In the original paper, a coupling to a composite Higgs was not considered and only explicit mass terms for the composite fermions were introduced.
1.1. Solutions to the Gauge Hierarchy Problem 15 mB, �mB are the vector masses of the technibaryons. In walking TC, it is assumed that the technibaryons B and B c have large anomalous dimensions, bound from below only by the unitarity constraint for fermionic operators dim B ≥ 3/2. We can therefore again symbolically put in the effect of the walking in the above Lagrangian at the TC scale, so that ˜d Λ3 TC Λ2 qRB ETC c L + d Λ3 TC Λ2 ETC qLBR → d˜ Λ 3−˜γ TC Λ 2−˜γ ETC qRB c L + d Λ3−γ TC Λ 2−γ qLBR , (1.23) ETC if we denote the anomalous dimension for B by γ and for B c by ˜γ. We will further employ the redefinition γ → 3 − γ, in order to make contact with the more recent, holographically inspired literature [61]. The Lagrangian now reads L ∋ ˜ d ΛETC � ΛTC ΛETC � ˜γ q RB c L + d ΛETC � ΛTC ΛETC � γ q LBR , −mBBB − �mBB c B c + BL (λHB c R) + h.c. . (1.24) Note, that in this notation the unitarity bound requires 3 > γ > 0 and unintuitively smaller γ means stronger coupling. Upon diagonalization of the mass mixing terms, with mass eigenstates denoted by ψ and χ respectively, � qL BL � qR B c R � � � � � cos ϕL − sin ϕL ψL = , tan ϕL = sin ϕL cos ϕL χL � � � � � cos ϕR − sin ϕR ψR = sin ϕR cos ϕR χ c R d Λγ TC mB Λ γ−1 , (1.25) ETC , tan ϕR = ˜ d Λ ˜γ TC �mBΛ ˜γ−1 , (1.26) ETC the Lagrangian reads L ∋ −mχχχ − ˜mχχ c χ c + � � ψL sin ϕL + χL cos ϕL λH (ψR sin ϕR + χ c R cos ϕR) + h.c.. (1.27) Note that the right-handed component of the SU(2)L doublet vector quark does not mix, so that BR = χR and analogously Bc L = χc L . Before EWSB, the field ψ remains massless and we identify it with the SM fermions. The fields χ, χc are the New Physics mass eigenstates with masses mχ = Λ 1−γ ETC � m2 BΛ2γ−2 ETC + d2Λ 2γ TC , m˜χ = Λ 1−˜γ ETC � ˜m 2 BΛ2˜γ−2 ETC + ˜ d2Λ 2˜γ TC . (1.28) The fact that the SM fermions are admixtures of elementary and composite fermions with a composite component proportional to sin ϕL or sin ϕR motivates the name partial compositeness. After EWSB, the SM fermions gain masses through effective Yukawa couplings to the TC condensate H, whose size is also controlled by the mixing angles, and thus by the anomalous dimension of the composite technibaryons, Yψ = sin ϕLλ sin ϕR . (1.29)
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
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Page 12 and 13: 2 Chapter 1. Introduction: Problems
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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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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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210 Bibliography [239] V. Ahrens, A
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On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

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