On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

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70 Chapter 2. The Randall Sundrum Model and its Holographic Interpretation can be found elsewhere [118, Sec. 4.2], we will concentrate on the KK decomposed theory here. As mentioned in Section 2.1, a bulk fermion is described by an irreducible Dirac spinor field, which reduces to its two Weyl components in four dimensions, due to the orbifold symmetry or equivalently, due to opposite BCs. This can be seen by considering the kinetic term (2.15), which after partial integration and in terms of the chiral components, Q = QL + QR can be written as � |G| Lmatter ∋ e −3σ Q¯ i/∂ Q − e −4σ m sgn(φ) ¯ Q Q (2.114) − e−2σ � ¯QL ∂φ (e rc −2σ QR) − ¯ QR ∂φ (e −2σ � QL) . In contrast to the vector boson, an explicit mass term does not break gauge invariance and we are going to keep it throughout the discussion. It is convenient to switch to t-notation, in which a viable KK decomposition reads QL(xµ, t) = 1 √ r QR(xµ, t) = 1 √ r t2 ɛ2 � Q n n L(xµ) f n L(t) , (2.115) t 2 ɛ 2 � Q n R(xµ) f n R(t) . (2.116) Therefore, the orthonormality relation from the 4D part of the kinetic term reads n � 1 2π dt f Lɛ ɛ n L/R (t) � f m L/R (t) �∗ = δ nm , (2.117) which implies for the matching conditions or EOM the system of coupled first order PDEs � ∂t + c � f t n R(t) = xnf n L(t) , � − ∂t + c � f t n L(t) = xnf n R(t) . (2.118) Here, xn again denotes the dimensionless ratio of the mass of the nth KK mode mn and the KK scale MKK, and c = m/k the dimensionless bulk mass parameter. This leads to the second order differential equation � − t 2 ∂ 2 � t + c(c ∓ 1) f n L/R (t) = t2x 2 nf n L/R (t) . (2.119) It is interesting to note, that in contrast to (2.91), in the limit c → 0, the left hand side reduces to a simple second derivative and one ends up with trigonometric solutions. In other words, a free fermion will not feel the AdS-curvature, if it is massless. A solution to (2.119) reads f n L/R (t) = √ t � J c∓ 1 2 � (xnt) + αnY 1 c∓ (xnt) 2 . (2.120) In deriving this solution, it was already used, that the integration constant αn is the same for fL and fR, which follows from (2.118). The value of αn is fixed by the BCs,
2.4. Profiles of Fermions 71 which simultaneously give the mass eigenvalues of the KK modes xn. It follows also from (2.118), that the BCs of fL and fR must be opposite, so that it is enough to give one of them and in the following, the notion “BCs of Q” will always be understood to be the BCs of its fL component. The following choices of BCs are possible, fL fR (NN) (DD) (DD) (NN) (ND) (DN) (DN) (ND) But only the first two lines will give rise to a zero mode for either the fL or fR component. The zero modes are massless, but introducing Yukawa couplings with a brane localized Higgs will result in zero mode masses which depend on the overlap of the corresponding Bessel function with the IR brane, and thus on the bulk mass parameter c. In this context of model building, the third and last line give still interesting solutions, because the (DN) component mimics a (NN) solution with a Higgs localized on the UV brane, which implies that a IR localization c > −1/2 results in a very light first KK mode. These ultralight KK fermions have first been recognized in [119, App A.2] and can arise in theories in which the SM top is in an extended multiplet with additional fermions, which do not have a zero mode due to (DN) BCs. Candidates are GUTs as in the original reference or higgsless models as examined in [120]. An interpretation of the different BCs in the dual theory can be given in accordance with the interpretation of a bulk gauge boson. If the BC in the UV is Neumann, there exists an elementary fermion which mixes with a composite state of the same quantum number. Only for (NN) BCs will this elementary state remain massless, Dirichlet BC in the IR result in a mass depending on the scale of the IR brane and the localization along the bulk. In the case of (DN) or (DD) BCs the dual theory has no elementary fermion and the Dirichlet BC in the IR make only for a correction of the masses of the composite meson tower. In order to model SM fermions, we will rely on the first two solutions, so that the masses and the integration constant is fixed by αn = Jc∓ 1 (xnɛ) 2 Y 1 c∓ (xnɛ) 2 , (2.121) in which the − sign holds if fL has a zero mode and and the + for the case that fR has a zero mode. In order to model the SM field content with two chiral zero modes for each quark generation, we will have to introduce three copies of Q = (u, d) transforming as a doublet under SU(2)L and a set of three downtype and three up-type SU(2)L singlets q c = u c , d c , as already hinted at in Section 2.1. The Yukawa interactions couple these bulk fermions to the Higgs, which should result in a Yukawa coupling for the zero modes and therefore implies, that the zero mode of Q corresponds to an fL solution, while the zero mode of q c is the fR solution of the bulk EOM. The ansatz (2.16) gives
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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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Page 30 and 31: 20 Chapter 1. Introduction: Problem
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120 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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