On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

More documents

Recommendations

Info

74 Chapter 2. The Randall Sundrum Model and its Holographic Interpretation Bulk Sliver Matching the solutions UV brane IR brane ɛ t 1 − η 1 η → 0 Figure 2.9: Illustration of the separation of a sliver from the bulk of the extra dimension in the context of the regularization of the delta function and the procedure of solving the coupled EOM. The red line in the bulk indicates an S-profile which solves the bulk EOM, the red line in the sliver depicts the corresponding solution of the sliver EOM, which is supposed to match the bulk solution at 1 − . equivalent, see [122], although the latter is more straightforward and will therefore be adopted here. As mentioned in the previous section, in a situation with a brane localized Higgs, it is crucial to properly regularize the delta functions and we will use the rectangular regularization introduced in (2.79). The importance of such a regularization procedure was first pointed out in [121, Sec. IV] and it was shown that the results are independent of the regularization method later in [114]. The following derivation is also outlined in the appendix of the latter reference. The regularized delta function effectively divides the bulk into two regions, t ∈ [0, 1 − η] and a small sliver t ∈ [1 − η, 1], see Figure 2.9. The solution in the sliver will generate the proper BCs for the bulk solutions at 1 − . In the sliver, only the following terms in the EOM are relevant, −∂t S Q n (t) a Q n = δ η (t − 1) ∂t S q n(t) a q n = δ η (t − 1) ∂t C Q n (t) a Q n = δ η (t − 1) −∂t C q n(t) a q n = δ η (t − 1) Using (2.79), we obtain � ∂ 2 � � � 2 Xq t − S η Q n (t) = 0 , � v √ Y q C 2MKK q n(t) a q n , (2.128) v √ Y 2MKK † q C Q n (t) a Q n , (2.129) v √ Y q S 2MKK q n(t) a q n , (2.130) v √ Y 2MKK † q S Q n (t) a Q n . (2.131) ∂ 2 t − � � � ¯Xq 2 S η q n(t) = 0 , (2.132)
in which Xq ≡ � v √ Y q Y 2MKK † q , ¯Xq ≡ 2.4. Profiles of Fermions 75 � v √ Y 2MKK † q Y q . (2.133) The solutions to these PDEs are hyperbolic functions. Integration constants are fixed by the fact that at the IR brane t = 1, the S-profiles have Dirichlet BCS, SQ,q n (1) = 0 and at 1 − η, they must be matched onto the solution of the bulk EOM at t = 1− . So that S Q � � Xq sinh (1 − t) η n (t) = sinh � � S Xq Q n (1 − ) , S q � � ¯Xq sinh (1 − t) η n(t) = sinh � � S ¯Xq q n(1 − ) . C Q � � Xq cosh (1 − t) η n (t) = cosh � � C Xq Q n (1 − ) , C q � � ¯Xq cosh (1 − t) η n(t) = cosh � � C ¯Xq q n(1 − ) , (2.134) where the solutions for the C-profiles follow from (2.128). At the boundary between sliver and bulk, the bulk solutions are therefore related by S Q n (1 − ) a Q n = −S q n(1 − ) a q n = v � � √ Y ¯Xq −1 � � q tanh ¯Xq q Cn(1 2MKK − ) a q n , (2.135) v √ Y 2MKK † � �−1 � � Q q Xq tanh Xq Cn (1 − ) a Q n , (2.136) which can be re-expressed by introducing the effective Yukawa couplings � �Y q ≡ f v � Y �q Y † � �q Y q , f(A) = A −1 tanh (A) . (2.137) √ 2MKK These correspond to the bare Yukawas plus corrections of the order O(v2 /M 2 KK ). One can therefore write (2.137) as S Q n (1 − ) a Q n = −S q n(1 − ) a q n = v √ 2MKK v √ 2MKK �Y q C q n(1 − ) a q n , (2.138) �Y † q C Q n (1 − ) a Q n . (2.139) It is straightforward to derive the eigenvalue equation and expressions for the avectors from (2.138). Since the diagonal S- and C-profiles will have only nonzero entries (otherwise the corresponding SM quark would have no kinetic term), they can be inverted and it follows S Q n (1 − ) a Q n = − v2 2M 2 KK S q n(1 − ) a q n = − v2 2M 2 KK �Y u C q n(1 − ) � S q n(1 − ) � −1 �Y † u C Q n (1 − ) a Q n , �Y † u C Q n (1 − ) � S Q n (1 − ) � −1 �Y u C q n(1 − ) a q n . (2.140)
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
Page 10 and 11:
x One of the most fascinating insig
Page 12 and 13:
2 Chapter 1. Introduction: Problems
Page 14 and 15:
Page 16 and 17:
Page 18 and 19:
Page 20 and 21:
10 Chapter 1. Introduction: Problem
Page 22 and 23:
Page 24 and 25:
Page 26 and 27:
Page 28 and 29:
Page 30 and 31:
Page 32 and 33:
Page 34 and 35: 24 Chapter 1. Introduction: Problem
Page 46 and 47: 36 Chapter 2. The Randall Sundrum M
Page 102 and 103: 92 Chapter 3. Solving the Flavor Pr
Page 110 and 111: 100 Chapter 3. Solving the Flavor P
Page 134 and 135:
124 Chapter 3. Solving the Flavor P
Page 136 and 137:
Page 138 and 139:
Page 140 and 141:
Page 142 and 143:
Page 144 and 145:
Page 146 and 147:
Page 148 and 149:
Page 150 and 151:
Page 152 and 153:
Page 154 and 155:
Page 156 and 157:
146 Chapter 4. The Asymmetry in Top
Page 158 and 159:
Page 160 and 161:
Page 162 and 163:
Page 164 and 165:
Page 166 and 167:
Page 168 and 169:
Page 170 and 171:
Page 172 and 173:
Page 174 and 175:
Page 176 and 177:
Page 178 and 179:
Page 180 and 181:
Page 182 and 183:
Page 184 and 185:
Page 186 and 187:
176 Chapter 5. Conclusions describe
Page 188 and 189:
178 Appendix A. Generating Paramete
Page 190 and 191:
180 Appendix A. Generating Paramete
Page 192 and 193:
182 Appendix B. Neutral Meson Mixin
Page 194 and 195:
184 Appendix B. Neutral Meson Mixin
Page 197 and 198:
C Input Parameters This appendix is
Page 199:
Parameter Value ± Error Reference
Page 202 and 203:
192 Appendix D. Higgs Potential for
Page 204 and 205:
194 Bibliography [13] S. Dimopoulos
Page 206 and 207:
196 Bibliography [43] G. Nordström
Page 208 and 209:
198 Bibliography [72] K. S. Babu,
Page 210 and 211:
200 Bibliography [104] R. Contino a
Page 212 and 213:
202 Bibliography [133] M. Baak, M.
Page 214 and 215:
204 Bibliography [160] J. Charles,
Page 216 and 217:
206 Bibliography [185] C. Csaki, G.
Page 218 and 219:
208 Bibliography [213] E. Thomson e
Page 220:
210 Bibliography [239] V. Ahrens, A
show all

On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

Create successful ePaper yourself

Delete template?

Save as template?