On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

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160 Chapter 4. The Asymmetry in Top Pair Production The expressions (4.50) and (4.52) encode in a model-independent way possible newphysics contributions to σs,a that arise from tree-level exchange of color-octet vectors in the s and t channels, as well as from t channel corrections due to both new color- singlet vector and scalar states. Based on the general expressions compiled in Section take the form 2.6, the Wilson coefficients appearing in S (0) ij,RS C (V,8) q¯q,� C (V,8) q¯q,⊥ C (V,8) tū,� C (V,1) tū,� = − αsπ 2M 2 KK = − αsπ 2M 2 KK � 1 αsπ = − M 2 L KK � and A(0) ij,RS � (∆ ′ a)11 + (∆ ′ a)33 − 2L ( � ∆a)11 ⊗ ( � � ∆a)33 � , (4.55) � − L a=Q,q � 1 � � − (∆ L a=Q,q ′ a)11 + (∆ ′ � a)33 � + 2L ( � ∆Q)11 ⊗ ( � ∆q)33 + ( � ∆q)11 ⊗ ( � � ∆Q)33 � , a=U,u αeπ = − M 2 L KK s2 wc2 w − αeπ M 2 L Q KK 2 u � ( � ∆a)13 ⊗ ( � � ∆a)31 , ��T u 3 − s 2 �2 wQu ( ∆U)13 � ⊗ ( � ∆U)31 + � s 2 �2 wQu ( ∆u)13 � ⊗ ( � ∆u)31 � a=U,u � ( � ∆a)13 ⊗ ( � � ∆a)31 , for q = u, d and Q = U, D. Since the coefficient C (S,1) tū,⊥ is formally of O(v4 /M 4 KK ), its explicit form is not presented here. Further, while the expressions for C (V,8) q¯q,� and C (V,8) tū,� v 2 /M 2 KK are exact, in the coefficient C(V,1) tū,� � , C(V,8) q¯q,⊥ , we have only kept terms leading in . In the numerical analysis however, we implement the Higgs exchange and the subleading terms as well. Note, that these coefficients are subject to a symmetry factor Suutt = 1/4 compared to the 1/2 in the case of neutral meson mixing. This is rooted in the fact, that t and s channel exchanges are matched to different operators here, while in Section 2.6 these diagrams were indistinguishable in the full theory. For the custodially protected RS model based on an SU(2)R ×SU(2)L ×U(1)X ×PLR bulk gauge group, the expressions for the Wilson coefficients can simply be obtained from (4.55) by multiplying the left-handed part of the Z-boson contribution to C (V,1) tū,� by a factor of around 3, while the right-handed contribution is protected by custodial symmetry and thus smaller by a factor of roughly 1/L ≈ 1/37. Note, that this couplings are enhanced for the opposite chirality compared to the down sector, which is a consequence of the embedding of the fermions, which in return was fixed by the requirement to protect the Zbb vertex from huge corrections, see [114] for details. The custodial protection does not really play a role here, as the electroweak corrections turn out to be subleading for all realistic choices of bulk mass parameters and the KK , remain unchanged. gluon contributions, encoded in C (V,8) q¯q,� , C(V,8) q¯q,⊥ , and C(V,8) tū,� The Wilson coefficients (4.55) are understood to be evaluated at the KK scale. The renormalization group (RG) running down to the top-quark mass scale at leadinglogarithmic accuracy, i.e. at one-loop order, neglecting tiny effects that arise from the
4.4. Cross Section and Asymmetry in the Minimal RS Model 161 mixing with QCD penguin operators gives for P = V, A, � � ˜C P q¯q(mt) = 2 η2/7 + 3η4/7 3 ˜C P q¯q(MKK) , (4.56) where η ≡ αs(MKK)/αs(mt) is the ratio of strong coupling constants evaluated at the relevant scales MKK and mt. The impact of RG effects is however limited, as can be seen from evaluating (4.56) using αs(MZ) = 0.139, MKK = 1 TeV, and mt = 173.1 GeV, so that η = 0.803 at one-loop order. We obtain ˜C P q¯q(mt) = 1.07 ˜ C P q¯q(MKK) , (4.57) which makes for an effect of the order of a few percent. We anticipate, that the t channel contributions are suppressed, because they always feel the localization of the up quarks, which yields an exponential suppression due to the zero mode profiles and will therefore neglect renormalization group effects for the corresponding Wilson coefficients. Restricting ourselves to the corrections proportional to αs and suppressing relative O(1) factors as well as numerically subleading terms, one finds from the ZMA expansion of the Wilson coefficients of the results given in (4.55) that the coefficient functions S (0) quark like S (0) uū,RS A (0) uū,RS ij,RS ∼ αsπ M 2 KK and A(0) ij,RS introduced in (4.50) and (4.52) scale in the case of the up � A=L,R αsπ ∼ − M 2 L KK F 2 (ctA ) , (4.58) � � q=t,u � F 2 (cqR ) − F 2 (cqL ) � + 1 3 where ctL ≡ cQ3 , ctR ≡ cu3 , cuL ≡ cQ1 , and cuR ≡ cu1 . � A=L,R F 2 (ctA )F 2 (cuA ) � , (4.59) As a consequence of the composite character of the top, its bulk mass parameters are typically ctA > −1/2, while the up quark, being mostly elementary is located close to the UV brane. The relevant F 2 (cqA ) factors can therefore be approximated by F 2 (ctA ) ≈ 1 + 2ctA , F 2 (cuA ) ≈ (−1 − 2cuA ) eL(2cu A +1) , (4.60) with A = L, R, as elaborated in Section 2.4. The difference of bulk mass parameters for light quarks (cuL − cuR ) is typically small and positive, whereas (ctL − ctR ) can be of O(1) and is usually negative, because the localization parameter of the SU(2)L doublet also enters the mass formula for the bottom quark (2.158), which typically results in ctR > ctL . Using the above approximations and expanding in powers of
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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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92 Chapter 3. Solving the Flavor Pr
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100 Chapter 3. Solving the Flavor P
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Page 156 and 157: 146 Chapter 4. The Asymmetry in Top
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Page 188 and 189: 178 Appendix A. Generating Paramete
Page 190 and 191: 180 Appendix A. Generating Paramete
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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,
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Page 212 and 213: 202 Bibliography [133] M. Baak, M.
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Page 216 and 217: 206 Bibliography [185] C. Csaki, G.
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210 Bibliography [239] V. Ahrens, A
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