On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

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72 Chapter 2. The Randall Sundrum Model and its Holographic Interpretation then � S ∋ dx = � � q=u,d − e−2σ rc � π −π � π dx dφ � |G| Lmatter � rcdφ e −3σ � ¯Q i/∂ Q + ¯q c i/∂ q c −π � − e −4σ � sgn(φ) ¯Q M Q Q + ¯q c M q q c � � ¯QL ∂φ (e −2σ QR) − ¯ QR ∂φ (e −2σ QL) + ¯q c L ∂φ (e −2σ q c R) − ¯q c R ∂φ (e −2σ q c L) − δ(|φ| − π) e−4σ � rc ɛab ¯ QLa H † b + ¯ QL HY (5D) d Y (5D) u u c R + ɛab ¯ QRa H † b (5D) Y u u c L d c R + ¯ QR HY (5D) d d c L + h.c. Here, the bar on the Yukawas indicates that in principle Y (5D) q � �� . (2.122) and Y (5D) q can be chosen differently, which is suppressed in (2.4). This difference is only due to the fact, that the Higgs is a brane localized 4D field. We will in the following assume, that Y (5D) q = Y (5D) q , which can be motivated by considering the model resulting as a limit of a theory with a bulk scalar, in which the couplings would be the same, because the bulk must respect 5D Lorentz invariance. Even without introducing this limit, this assumption should not affect the physics, because we expect the 5D Yukawas to be structureless order one coefficients anyway with no effect on the hierarchies in the flavor sector, so that even being more restrictive, for example choosing Y (5D) u = Y (5D) d should not change the results. Further, the real bulk mass matrices M Q,q and the complex 5D Yukawa matrices will not be diagonal in the same basis. If not stated otherwise, we will from now on assume that we are in the bulk mass basis, in which the bulk mass matrices are diagonal. Due to the flavor degrees of freedom and the Higgs on the brane, The KK decomposition is more involved than in (2.115). It can be brought into the form uL(xµ, t) = 1 √ r uR(xµ, t) = 1 √ r u c L(xµ, t) = 1 √ r u c R(xµ, t) = 1 √ r t2 ɛ2 � u n n L(xµ) C Q n (t) a U n , t 2 ɛ 2 t 2 ɛ 2 t 2 ɛ 2 � u n R(xµ) S Q n (t) a U n , n � u n L(xµ) S u n(t) a u n , n � u n R(xµ)C u n(t) a u n , (2.123) n where the index n runs over all flavor and KK modes. So will for example m1 = mu, m2 = mc, m3 = mt give the SM quark masses, and m4, . . . , m9 the masses of the first set of six KK modes, and likewise for downtype quarks. The 3 × 3 matrices S Q,q n (t) correspond to the solutions with (DD) BCs which do not acquire a zero mode, while C Q,q n (t) denote profiles with (NN) BCs that have a zero mode. The additional a-vectors a (U,u) n quantify flavor mixing induced by the Yukawa couplings. Therefore,
2.4. Profiles of Fermions 73 while the profile functions must be the same for both components of the SU(2)L doublet, S U n = S D n ≡ S Q n and C U n = C D n ≡ C Q n , the a-vectors are not, because the two components are treated differently in the Yukawa interactions. Also, the profile functions can be chosen diagonal and real in the bulk mass basis, whereas the a-vectors are complex. Proceeding just like in the case of a free bulk gauge boson, we will first derive the orthonormality relation by matching onto the kinetic term and then fix the mass eigenvalues using the BCs. In contrast to the free bulk fermion, upon insertion of the KK decomposition, the matching of the first term in the second line of (2.122) onto the canonical kinetic term S4D = � � � q=u,d n d 4 x ¯q n (x) i/∂ q n (x) , (2.124) in which q n (x) ≡ q n L (xµ) + q n R (xµ) and q n L,R (xµ) denoting the 4D fields in the KK decomposition, will result in the more general orthonormality relation 2π Lɛ � 1 ɛ dt a (Q,q)† m C (Q,q) m (t) C (Q,q) (t)a (Q,q) n n + a (q,Q)† m S (q,Q) m (t) S (q,Q) (t)a (q,Q) = δmn . n n (2.125) In the limit of vanishing Yukawa interactions, this expression separates into two sets of three independent equations for q = u, d. Since we know, that the C- and S-profiles must reduce to the solutions fL,R in this limit, which form complete sets of functions in the bulk and fulfill separate orthonormality conditions (2.117), we can deduce that a Q† n a Q n + a q† n a q n = � . (2.126) With the result (2.125), the matching of the second to last term of (2.122) onto a canonical 4D mass term gives rise to the EOM [115, 114] � � − t∂t − cQ S Q n (t) a Q n = −xn t C Q n (t) a Q v n + δ(t − 1) √ Y q C 2MKK q n(t) a q n , � � t∂t + cq S q n(t) a q n = −xn t C q n(t) a q v n + δ(t − 1) √ Y 2MKK † q C Q n (t) a Q n , � � t∂t − cQ C Q n (t) a Q n = −xn t S Q n (t) a Q v n + δ(t − 1) √ Y q S 2MKK q n(t) a q n , � � − t∂t + cq C q n(t) a q n = −xn t S q n(t) a q v n + δ(t − 1) √ Y 2MKK † q S Q n (t) a Q n , (2.127) in which a sign has been introduced in defining cQ,q ≡ ±M Q,q/k , which will prove to be convenient, as well as the dimensionless 4D Yukawa couplings have been defined by Y (5D) q ≡ 2Y q/k. Apart from the IR brane, that is for t �= 1, these equations decouple and can be reduced to two times three sets of (2.118), because the a−vectors must be proportional to unit vectors in this limit, with the corresponding solutions (2.120). One way to find a solution to (2.127) is therefore to assume these solutions in the bulk and treat the Yukawa couplings as perturbations. Alternatively, one can directly account for the Yukawa terms in the EOM, as shown above. Both approaches are
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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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122 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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