On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

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88 Chapter 2. The Randall Sundrum Model and its Holographic Interpretation as well as ∆ ′ A ≈ ∆A, both for A, B ∈ {Q, q}. For completeness, the matrices (2.179) are given δQ → xq U q† R diag � � 1 1 1 − 2cqi F 2 (cqi ) − 1 + F 2 (cqi ) �� U 3 + 2cqi q R xq , (2.184) δq → xq U q† L diag � � 1 1 1 − 2cQi F 2 (cQi ) − 1 + F 2 (cQi ) �� U 3 + 2cQi q L xq , ∆g q h → √ 2 v2 U q† L diag [F (cQi )] Y q Y † q q Y q diag [F (cqi )] U R , 3M 2 KK where xq denotes a vector in flavor space with xq = mq/MKK as its entries. 2.6 Four Fermion Interactions Given the flavor mixing matrices introduced in the last section and the expressions for the SM gauge boson propagators (2.103) and (2.86), it is straightforward to derive effective Hamiltonians featuring all dimension six operators with four fermion legs. In matching the full with the effective theory, one has to evaluate all diagrams in the full theory, and take into account possible symmetry factors for the effective diagrams, which are process dependent. Therefore, a common prefactor S(¯qi, qj; ¯q ′ k , q′ l ) depending on the flavor indices i, j, k, l of all external quarks in a given process, is introduced. In order not to clutter up the notation we will write the external quarks q and q ′ as vectors in flavor space and the sum implicitly assumes a sum over all KK modes as well. For the photon one finds, H (γ) eff 2πα = M 2 � KK q,q ′ S(¯qi, qj; ¯q ′ k , q′ l ) Qq Qq ′ � 1 2L (¯qγµ q) � ¯q ′ γµq ′� − 2 � ¯qLγ µ ∆ ′ QqL + ¯qRγ µ ∆ ′ � � ′ qqR ¯q γµq ′� + 2L � ¯qLγ µ � ∆QqL + ¯qRγ µ � ∆qqR (2.185) � � ′ ⊗ ¯q Lγµ � ∆Q ′q′ L + ¯q ′ Rγµ � ∆q ′q′ � R � . The same expression holds for the gluon, with α → αs and the electromagnetic charges replaced by the color matrices T a ⊗ T a inserted in the respective quark bilinears. The
2.6. Four Fermion Interactions 89 induced interactions arising from the exchange of a Z-boson read H (Z) 4πα eff = s2 wc2 w m2 � 1 + Z m2 Z 2M 2 � 1 − KK 1 � m4 Z + O 2L M 4 �� (2.186) KK × � S(¯qi, qj; ¯q ′ k , q′ l ) � ¯qLγ µ T q 3 (1 − δQ)qL + ¯qRγ µ T q 3 δqqR − s 2 w Qq ¯qγ µ q � q,q ′ + 4παL � � × + s 2 wc 2 w M 2 KK � q,q ′ � × ¯q ′ LγµT q′ 3 (1 − δQ ′)q′ L + ¯q ′ RγµT q′ 3 δq ′q′ R − s 2 w Qq ′ ¯q′ γµq ′� S(¯qi, qj; ¯q ′ k , q′ l ) � − ¯qLγµT q 3 (1 − δQ)qL + ¯qRγµT q 3 δqqR − s 2 � w Qq ¯qγµq ¯q ′ Lγ µ T q′ 3 (∆Q ′ − εQ ′)q′ L + ¯q ′ Rγ µ T q′ 3 εq ′q′ R − s 2 w Qq ′ � ¯qLγ µ T q 3 ( � ∆Q − �εQ)qL + ¯qRγ µ T q 3 �εqqR − s 2 w Qq � ′ ¯q Lγ µ ∆Q ′q′ L + ¯q ′ Rγ µ ∆q ′q′ � R � � ¯qLγ µ � ∆QqL + ¯qRγ µ � ∆qqR � ⊗ ¯q ′ LγµT q′ 3 ( � ∆Q ′ − �εQ ′)q′ L + ¯q ′ RγµT q′ 3 �εq ′q′ R − s 2 w Qq ′ � ¯q ′ Lγµ � ∆Q ′q′ L + ¯q ′ Rγµ � ∆q ′q′ � � R � . Here, the δ and ε matrices appear due to the non-vectorial coupling of the Z. Also, by writing mZ, we implicitly assume the RS corrections (2.111) to be absorbed. Here and in the following, mW,Z ≡ mRS . To a good approximation one can neglect these W,Z terms and ends up with a considerably simpler expression, which holds up to order v4 /M 4 KK corrections, H (Z) eqq = 4πα s2 wc2 w m2 � 1 + Z m2 Z in which − 8πα s 2 wc 2 w m 2 Z − 4παL s 2 wc 2 w M 2 KK � q 2M 2 KK � q + 4παL s2 wc2 w M 2 � KK q,q ′ S(¯q, q; ¯q ′ , q ′ ) J µ Z � 1 − 1 � m4 Z + O 2L M 4 �� S(¯q, q; ¯q KK ′ , q ′ )J µ Z JZµ S(¯qi, qj; ¯q ′ k , q′ l ) [¯qLγ µ T q 3 δQqL − ¯qRγ µ T q 3 δqqR] JZµ ≡ � q � � (2.187) S(¯q, q; ¯q ′ , q ′ � �T q ) 3 − s2 � w Qq ¯qLγ µ ∆QqL − s 2 w Qq ¯qRγ µ � ∆qqR JZµ � � �T q 3 − s2 w Qq � ¯qLγ µ � ∆QqL − s 2 w Qq ¯qRγ µ � ∆qqR � � ⊗ T q′ 3 − s2w Qq ′ � ¯q ′ Lγµ � ∆Q ′q′ L − s 2 w Qq ′ ¯q′ Rγµ � ∆q ′q′ � R � , �� q T3 − s2 � w Qq ¯qLγ µ qL − s 2 w Qq ¯qRγ µ � qR . (2.188) �
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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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138 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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