On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

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112 Chapter 3. Solving the Flavor Problem in Strongly Coupled Theories absolute value of the Yukawa couplings from observables which are proportional to positive powers of them, typically ∆F = 1 observables, which arise at the one loop level with mass insertions. The authors of [180] find that direct CP violation from Kaon decay, measured by ɛ ′ /ɛK gives the most severe upper bound. At Yd ≈ 6, the derived bound on MKK becomes stronger than the one from |ɛK|, so that the best one can arrange for would be a KK scale of MKK ≥ 3 TeV. This problem can be avoided if instead of enhancing the Yukawa couplings, one lowers the value of the volume of the extra dimension L = − ln ɛ, which appears linearly in both the Wilson coefficients in ∆F = 1 and ∆F = 2 effective Hamiltonians [144]. This is the idea of the LRS models, which have been introduced in Section 3.1 as a means to reduce the contributions to the oblique parameters. The reason for this linearity is, that the coupling of the KK modes to a good approximation reads g √ L, which becomes apparent if one computes the contributions mode by mode instead of using the full 5D propagators. This coupling will therefore be smaller if the bulk is truncated at scales ɛLRS = ΛEW/ΛLRS, with ΛLRS ≪ MPl. In the dual theory this means, that the range of scales over which the theory is strongly coupled is smaller, i.e. asymptotic freedom sets in already at the scale ΛLRS. However, for the zero mode profiles of the light quarks holds F (c) ∼ ɛ −c−1/2 and if ɛ → ɛLRS which is orders of magnitude smaller, the localization parameters must adjust, because the values of the zero mode profiles are fixed by the mass relations (2.158). Considering the original proposal [144], ɛLRS = 10 3 (LLRS ≈ 7), this means that he c-parameters are by a factor of two larger in magnitude than in the original model. However, the RS-GIM mechanism, illustrated by the naive picture 2.11 assumes, that the integrand of the overlap integrals in (2.173) is proportional to a positive power of he bulk coordinates t and t ′ . This will only be true as long as cqi + cqj + 2 > 0 (remember, all light flavors have cQ,q < −1/2). If the c parameters become smaller, the overlap integrals (2.176) will not be dominated by the IR localization of the gluon KK modes, but by the extreme UV localization of the fermion zero modes, so that the main contributions arise from the region where t ≈ ɛ and the RS-GIM suppression is undermined. This phenomenon is called UV dominance and its implications for flavor changing as well as flavor diagonal neutral currents, for example the Zb¯b coupling, have been analyzed in [179]. One can turn this argument around and formulate a lower bound on the volume LLRS > 8.2, above which the bound from ɛK is at least not stronger, but the enhancement in (3.41) cannot be balanced out by lowering L alone. Global Symmetries and MFV The dangerous coefficients C4 and C5 appear only if flavor changing vertices with both right- and left-handed down quarks are present. This can be prevented, either by imposing a global symmetry which arranges for the singlet or doublet bulk mass parameters to be equal or by aligning them with the down-type Yukawa matrix [184]. The latter is an implication of minimal flavor violation, which was introduced in Section 1.2. Several variants have been proposed, in which the alignment is assumed only in the down sector, is extended to the up sector, or holds only for the first two
3.3. Limitations of the RS GIM Mechanism 113 generations [182, 183, 184]. In the dual theory, such an alignment means that not the mixing angles in (1.27) alone are responsible for the structure in the flavor sector, but also the fundamental Yukawa couplings denoted by λ in (1.27) in the composite sector have some structure. As in the SM, the origin of this structure is not explained, which at least partially abrogates one of the strengths of the RS model, namely the capability to explain flavor structure and FCNC suppression by the same fundamental order one parameters. If alternatively the bulk masses of all 5D fermions are chosen equal in the down-sector, cdi = cd, the couplings gR in (1.34) will be universal or equivalently, the corresponding matrices in (2.180) and (2.181), both for q = d, are diagonal in flavor space [186]. An alignment can equally well be imposed for the doublet bulk mass parameters cQi , but not for both the up- and the down singlet localizations at once without introducing additional structure in the Yukawa matrices. In essence, the reparametrization invariance (2.169) can be applied once in order to redistribute between the down-type singlets and the doublets. This ansatz is from the point of view of the dual theory, related to the idea of only electroweak singlet or doublet quarks having composite partners, as proposed by Redi and Weiler [149], which was already quoted as a possible way to reduce the tension from the Zb¯b coupling in Section 3.1. Alignment of the bulk mass parameters cdi corresponds to having an equal anomalous dimension for all composite down-type singlet partners. In the bulk, this can be assured by imposing a global symmetry, which does however not translate to a symmetry in the dual theory. Only for local bulk symmetries one knows that they are present in the composite sector as well, which for example makes it necessary to gauge the SU(2)R in order to realize the custodial protection in the dual theory. Gauged bulk flavor symmetries have therefore gained some attention, see [185] and references therein. The symmetry must also be exact, because small deviations from the alignment will already imply large corrections, because the bulk mass parameters enter the Wilson coefficients in the exponents. Such a symmetry can only be considered stable in the dual theory, if the localization of the singlet down-quarks is not only equal, but also they are confined to the UV brane, which removes the composite singlet down-quarks from the dual theory, a.k.a. left-handed compositeness (and vice versa for the left-handed partners being confined to the UV brane). Whether or not such a symmetry would be enough to balance the enhancement in (3.41) can be estimated by considering the effect of higher order operators such as ( ¯ QLYdHdR) 2 which are encoded in the O(v2 /M 2 KK ) corrections to (2.181). One finds ( � δD)mn ⊗ ( � δd)mn = mdm mdn M 2 KK ��U L† � � � L d U mi d in (� δD)ij + � U R† d � mi � � R Ud in (� δd)ij � L† � � � L Ud U mj d jn � U R† d � mj � � R Ud jn � (3.47) from the terms in (2.176) involving both even C (A) m (φ) and odd S (A) m (φ) fermion profiles. Here a summation over the indices i, j is understood. Neglecting terms suppressed by F 2 (cQi ) and F 2 (cQi )F 2 (cQj ), which are small for the light flavors, the
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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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Page 72 and 73: 62 Chapter 2. The Randall Sundrum M
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162 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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