On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

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158 Chapter 4. The Asymmetry in Top Pair Production in which the operators are Q (V,8) q¯q,AB = (¯qγµT a PAq)(¯tγ µ T a PB t) , Q (V,8) tū,AB = (ūγµT a PAt)(¯tγ µ T a PB u) , Q (V,1) tū,AB = (ūγµPAt)(¯tγ µ PB u) , Q (S,1) tū,AB = (ūPAt)(¯tPB u) , (4.46) and the sum over q (u) involves all light (up-type) quark flavors. In addition, PL,R = (1 ∓ γ5)/2 project onto left- and right-handed chiral quark fields, and T a are the generators of SU(3)C. The superscripts V and S (8 and 1) label vector and scalar (color-octet and -singlet) contributions, respectively. Using the effective Lagrangian (4.45) it is straightforward to calculate the interference between the tree-level matrix element describing s channel SM gluon exchange and the s and t channel new-physics contributions arising from the Feynman graphs displayed in Figure 4.5. In terms of the following combinations of Wilson coefficients C (P,a) ij,� = Re K (0) βρ CF tū,RS = 32 Nc � C (P,a) ij,LL + C(P,a) ij,RR � , C (P,a) ij,⊥ = Re � C (P,a) ij,LR + C(P,a) ij,RL � , (4.47) the resulting hard-scattering kernels take the form K (0) � βρ CF t2 1 q¯q,RS = 32 Nc ˆs C(V,8) q¯q,⊥ + u21 ˆs C(V,8) q¯q,� + m2 � t C (V,8) � q¯q,� + C(V,8) q¯q,⊥ � , (4.48) �� u2 1 ˆs + m2 � � � � 1 t2 1 t + ˆs + m2 � t C (S,1) � tū,⊥ . C Nc (V,8) tū,� − 2C(V,1) tū,� (4.49) Note, that per definition a factor of αs or αe has been absorbed in the Wilson coefficients. As in (4.22), the coefficient K (0) by simply replacing cos θ with − cos θ. ¯qq,RS � (0) � (0) K ¯tu,RS is obtained from K q¯q,RS � (0) � K tū,RS After integrating over cos θ, one obtains the LO corrections to the symmetric and asymmetric parts of the cross section defined in (4.28). In the case of the symmetric part it follows S (0) βρ uū,RS = (2 + ρ) ˆs 216 S (0) d ¯ βρ = (2 + ρ) ˆs d,RS 216 � C (V,8) 1 uū,� + C(V,8) uū,⊥ + � C (V,8) d ¯ d,� 3 C(V,8) tū,� − 2C(V,1) tū,� while the asymmetric part in the partonic CM frame reads A (0) uū,RS = β2 ρ 144 ˆs A (0) d ¯ d,RS = β2 ρ 144 ˆs � C (V,8) 1 uū,� − C(V,8) uū,⊥ + � C (V,8) d ¯ d,� � + fS(z) ˜ C S tū , (4.50) + C(V,8) d ¯ � , (4.51) d,⊥ 3 C(V,8) tū,� − 2C(V,1) tū,� � + fA(z) ˜ C S tū , (4.52) − C(V,8) d ¯ � . (4.53) d,⊥
q 4.4. Cross Section and Asymmetry in the Minimal RS Model 159 G (k) q t q t Z, Z (k) γ k q t t q t G (k) q t q t h q t Figure 4.5: Upper row: Tree-level contributions to the q¯q → t¯t (left) and the uū → t¯t (right) transition arising from s and t channel exchange of KK gluons. Lower row: Tree-level contributions to the uū → t¯t transition arising from t channel exchange of the Z boson, of KK photons and Z bosons as well as of the Higgs boson. The s channel (t channel) amplitudes receive corrections from all light up- and down-type (up-type) quark flavors. Note, that coefficients involving down-type quarks do not receive corrections from flavor-changing t channel transitions, as the exchange of the W and its KK modes is not considered. The coefficients C (V,8) q¯q,� and C(V,8) q¯q,⊥ enter in (4.50) always in the combination CV q¯q ≡ � C (V,8) � q¯q,� + C(V,8) q¯q,⊥ , while in (4.52) they always appear in the form CA q¯q ≡ � C (V,8) � q¯q,� − C(V,8) q¯q,⊥ . This feature expresses the fact that the symmetric (asymmetric) LO cross section σs (σa) measures the product g q V gt � q V gAgt � A of the vector (axial-vector) parts of the couplings of the KK gluons to light quarks and top quarks. One can alternatively express (4.48) in the parameters given in (4.18) and compare it with the general expression (4.38) in the limit mA ≫ ˆs. In order to be able to incorporate a light Higgs boson with mh ≪ MKK into our analysis, we have kept the full Higgs-boson mass dependence arising from the t channel propagator. This dependence is described by the phase-space factors fS(z) = − βρ ⎡ � ⎣1 ρ (1 − z) ρ 4 + ρ (1 − z) + + 72 2 2� � � 2 (1 + β) − ρ (1 − z) ln 8β 2 (1 − β) − ρ (1 − z) ⎤ ⎦ , fA(z) = ρ ⎡ � ρ 4 + ρ (1 − z) ⎣1 − ρ + 144 2� ⎛ ln ⎝ 4 ρ � 4z + ρ (1 − z) 2� (2 − ρ (1 − z)) 2 ⎞⎤ ⎠⎦ , (4.54) with z ≡ m 2 h /m2 t , so that the new Wilson coefficient ˜ C S tū is the dimensionless counterpart of C (S,1) tū,⊥ .
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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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92 Chapter 3. Solving the Flavor Pr
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100 Chapter 3. Solving the Flavor P
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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
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Page 204 and 205: 194 Bibliography [13] S. Dimopoulos
Page 206 and 207: 196 Bibliography [43] G. Nordström
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210 Bibliography [239] V. Ahrens, A
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