On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

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10 Chapter 1. Introduction: Problems beyond the Standard Model Here the subscript at the vertical bar reads “evaluated at the respective scale”. Therefore, the quark masses are given by mq ∼ b 〈QQ〉� � ETC Λ 2 ETC = b NTCΛ 3 TC Λ 2 ETC , (1.13) where we have assumed that 〈QQ〉 � � = 〈QQ〉 ETC � � = NTC Λ TC 3 TC . Since ΛTC is set by the electroweak breaking scale, for fixed NTC, the scale ΛETC can be extracted from (1.13) if the physical quark masses are used as an input. Allowing for a step-wise breaking of the extended symmetry at different scales Λ1 > Λ2 > Λ3 would make it even possible to dynamically generate the hierarchy between the three generations which will therefore obtain masses mq3 > mq2 > mq1 . However, ΛETC is severely constrained from the third type of contributions to (1.11), the c-terms, which generate FCNC processes suppressed by the same scale that enters the effective Yukawas. From Table 1.1 we know that even in more optimistic scenarios (assuming no additional CP violation), this bound amounts to at least ΛETC > 10 3 TeV, which translates in a maximal quark mass of (NTC < 10, ΛTC ∼ 1TeV) mq the large mass of the top quark poses therefore a problem for theories where the Higgs is described by a bound state. This problem can be attenuated by considering theories that are not like QCD [20]. In a QCD-like theory, the running of the coupling is fast because asymptotic freedom sets in quickly above ΛQCD, as illustrated on the upper left panel of Figure 1.2. The techniquark condensate will therefore stay roughly the same between the ETC scale and the TC scale and the assumption going into (1.13) is justified. More precisely, the running is given by 〈QQ〉 � �� ΛETC � = exp ETC ΛTC dµ µ γm(α(µ)) � 〈QQ〉 � � , (1.15) TC and QCD-like running corresponds to an anomalous dimension γm(α(µ)) ≈ γα(µ) ≈ γ/ ln(µ), which results in a power-logarithmic enhancement factor proportional to ln(ΛETC) γ / ln(ΛTC) γ , similar to QCD radiative corrections to semileptonic electroweak processes. For general strongly coupled theories, such a behaviour is not mandatory. It may well be, that the coupling evolves slowly for a large range of scales, before asymptotic freedom sets in, as depicted on the lower left panel of Figure 1.2. In such a case, the coupling stays close to a constant, so that γm(α(µ)) ≈ γα(µ) ≈ γα ∗ and the radiative corrections give a power-law enhancement factor (ΛETC/ΛTC) γ . In terms of the beta function, this behaviour corresponds to the convergence to a conformal fixed point, but not quite reaching it, as illustrated in the lower right panel of Figure 1.2. Yang- Mills theories with this behaviour are called walking and the corresponding walking technicolor (WTC) theories allow for a significant amplification of the a-and b-terms in (1.11), while the FCNC inducing c-terms, which do not couple to the technicolor sector, still feel the full ETC scale suppression. In the WTC scenario, we can thus
g(µ) g(µ) ΛTC ΛTC 1.1. Solutions to the Gauge Hierarchy Problem 11 µ µ β(g) β(g) Figure 1.2: Couplings and beta functions of strongly coupled theories. rewrite the operators in (1.11) at the TC scale symbolically as (symbolically in the sense that a Lagrangian itself should be independent of the energy scale) Ld=6 = aij QT i Q QT j Q Λ 2−2γ ETC + bij QT i Q qT j q Λ 2−γ ETC + cij g∗ qT i q qT j q Λ 2 ETC g g . (1.16) For walking technicolor theories, analyses based on the Schwinger-Dyson equation show that the anomalous dimension has an upper bound of γ ∼ 1, see e.g. [27, p.75]. Equivalently, one can say that the scaling dimension of the techniquark condensate ∆ ¯ QQ ≡ dim〈QQ〉= 3 − γ, is bound from below by ∆ ¯ QQ = 2. This generates an upper bound mq � bNTC Λ 3−γ TC Λ 2−γ ETC which allows for quark masses as large as (NTC < 10, ΛTC ∼ 1TeV) , (1.17) mq ine-tuned parameters in order to explain the large top mass. Equation (1.17) suggests, that this bound can be loosened further by choosing a larger number of colors NTC, or increasing the number of flavors NF of technifermions which take part in the condensation. However, the β-function of an SU(NC) gauge theory
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
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60 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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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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