On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

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54 Chapter 2. The Randall Sundrum Model and its Holographic Interpretation presented in Section 1.1 has the symmetry breaking pattern G = SU(3) × SU(3) × U(1) F = SU(3) × U(1) H = SU(2) × SU(2) × U(1) ⇒ dim G/H = 10 dim F/I = 5 NPNGB = 5 . (2.51) The charges must again be chosen in a way which guarantees I = SU(2)L × U(1)Y . One also finds five PNGBs, which in agreement with (1.36) will form the composite Higgs doublet and a singlet. One might be alarmed by the fact that we now have a large elementary gauge group F , possibly implying that there are additional gauge bosons with masses at the electroweak scale in the theory. However, in both the composite Higgs scenarios (with and without collective breaking), the IR BCs will not break the electroweak symmetry (that is what the composite Higgs is for), but the larger global symmetry G at some scale f > v. The five additional gauge bosons in the collective breaking scenario are just what we found in (1.41) with masses in the multiple TeV range. 2.3 Profiles of Gauge Bosons In the later discussion of flavor observables, we will compute the Wilson coefficients of four fermion operators. The leading coefficients come from tree-level diagrams in the full theory, with external fermion zero modes, but with gauge boson zero modes (which are just the SM contributions), as well as their KK excitations or composite resonances propagating between the vertices. At the matching scale, the momenta in these propagators can be neglected compared to the masses. In this section, the general 5D propagator with all possible boundary conditions for a generic bulk gauge boson will be given and interpreted in the dual theory. We assume a bulk gauge field transforming as a 5D Lorentz vector AM(xµ, φ) with suppressed gauge indices (as they play no role in deriving the propagator) and add a gauge fixing term as well as an explicit mass term, to the Lagrangian. Both gauge breaking terms are kept, so that the solution can be adapted to different situations, LGauge + LGF + LMass = G KM G LN − � − 1 4 FKLFMN � + 1 rc 2ξ � � ∂µA |G| µ − ξ r2 ∂φe c −2σ Aφ 2 GMN M 2 AAMAN �2 . (2.52) Note, that the gauge fixing is specifically chosen in a way that the Aφ independent term represents the usual 4D gauge fixing and the cross terms cancel the mixing between vector and scalar components of AM which will arrange for diagonal propagators [111].
After integration by parts, one finds [112] � |G|(LGauge+LGF + LMass) = � � rc Aν ∂ 2 2 η µν � − 1 − 1 ξ +Aφ � − e−2σ r2 ∂ c 2 + ξ r4 c � ∂ µ ∂ ν − ∂φ e −2σ ∂ 2 φe−2σ − e−4σ r2 M c 2 A 2.3. Profiles of Gauge Bosons 55 e−2σ r2 ∂φη c µν + M 2 Ae −2σ � Aµ � Aφ � . (2.53) Boundary terms from partial integration vanish as long as the fields have either Dirichlet or Neumann BCs and the fields are well behaved at 4D infinity. We will once more change notation, because it is sensible for the numerics to have all expressions in dimensionless variables. Therefore, let t ≡ MKK z = MKK e σ /k with MKK = ɛk defined in (1.54), and ɛ ≡ ΛIR/ΛUV. The line element in this t-notation reads ds 2 = � ɛ �2 �ηµνdx t µ dx ν − M −2 KKdt2� , (2.54) and the positions of the branes are now intuitively expressed by ratios of scales, t = ɛ for the UV and t = 1 for the IR brane. In addition, we define the volume of the extra dimension as L = − ln ɛ = krcπ ≈ 36. Finally, we redefine the scalar component of AM, A5(xµ, t) = ɛ Aφ(xµ, φ) , (2.55) t rc in order to adjust the mass dimension and simplify the following calculations. With these conventions, equation (2.53) reads � S = d 4 � x dφ � |G| (LGauge + LGF) = 1 � d 2 4 � π x rcdφ AMK −π MN ξ AN in which ⎛ K MN ξ = ⎝ � ∂2 − M 2 KKt∂t 1 t ∂t � � η µν − = 1 2 � 1 − 1 ξ d 4 x 2π rc L � 1 ɛ dt t AMK MN ξ AN , (2.56) � ∂ µ ∂ ν + ɛ2 t 2 M 2 A ηµν 0 0 − ∂2 + ξM 2 1 ɛ2 KK∂tt∂t t − t2 M 2 A (2.57) is the inverse of the Feynman propagator in Rξ gauge. After a Fourier transformation of the coordinates of the non-compact directions, pµ = i∂µ, the following two equations ⎞ ⎠
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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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104 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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