On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

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32 Chapter 1. Introduction: Problems beyond the Standard Model orders in λ) (Cd 11 + Cd 22 )λ2 Λ2 ( Fl ¯ Q2 d c 1)( ¯ Q2 d c 1) , (1.69) and there is no reason why, considering an abelian symmetry, the diagonal coefficients Cd ii should be related. In Nelson Strassler type models (or in warped extra dimensions) there is also no reason to assume this, but the coefficients are individually small, because they are proportional to the localizations of the involved light flavors or the corresponding mixing angles (compare the discussion at the end of 1.1). Further, models with Yukawa hierarchies induced from strong renormalization group running do not actually have an underlying flavor symmetry. At the flavor scale, the Yukawa coupling is an anarchic order one matrix, which does not increase the global symmetry group of the Lagrangian compared to the low energy theory. This is an advantage, because in the Froggatt Nielsen model the Yukawa couplings emerge as effective operators from a fundamental flavor symmetric theory. The breaking of this continuous symmetry by the vev of φ results in massless Goldstone bosons. As a consequence, it can not be realized exactly and even then, light Would-be Nambu Goldstone bosons are bound to appear. This is the reason why, after it became known that a rich flavor sector is realized in nature (basically after the discovery of the charm quark), most of the proposals for a underlying symmetry were based on discrete groups, most of them non-abelian ones (see [73, Ref.5]). Non Abelian Flavor Symmetry Another way to avoid the FCNC problem of ablian flavor theories is to assume nonabelian flavor symmetries, which can ensure the coefficient matrix of (1.68) to be universal. In these approaches, the maximal U(3) 3 = U(3)Q × U(3)u × U(3)d quark flavor symmetry, or a non-abelian subgroup, is considered to be broken by flavon fields which transform in a χ ∼ (¯3, 3) under the respective U(3)s (or a corresponding equivalent for smaller flavor groups) in order to make the Yukawa couplings 1 ΛFl χu ¯ Q ˜ Hu c + 1 ΛFl χd ¯ QHd c (1.70) invariant. The problems regarding continuous groups mentioned at the end of the last section have lost nothing of their validity and it is therefore preferable to consider a discrete flavor symmetry, in this case a discrete subgroup of U(3) 3 . In principle, another solution would be to gauge the flavor group in order to get rid of the massless Goldstone modes. However, the small breaking in the light flavor sector will generate very light flavor gauge bosons, as was already pointed out in the very first papers considering gauged flavor groups [73]. But also if a discrete non-abelian subgroup is realized, it should emerge from a gauged group via spontaneous breaking, because fundamental global symmetries are not respected by quantum gravity (e.g. a black hole may eat a proton and Hawking radiate a positron). Assuming such an explanation is found, renormalizable terms in the flavon potential might still be invariant under an accidental continuous symmetry, leading to Goldstone bosons. Higher dimensional
1.2. Solutions to the Flavor Problem 33 operators will give masses to these Goldstone bosons, but they are again expected to be light, because the breaking must be proportional to powers of λ. It is anyway a challenge even for a gauged flavor symmetry to realize a potential which is able to generate the Yukawa structures [75]. 23 This is already an impressive list of reasons, why non-abelian models with low flavor scales are problematic. Since FCNCs at a low scale are unavoidable if new physics (with no additional structure in the flavor sector) is assumed to solve the hierarchy problem, such a flavor structure is often simply imposed without further motivation. The practice to assume new physics to respect the flavor symmetries of the SM gauge sector, i.e. it is flavor blind, is called Minimal Flavor Violation (MFV). It was first implemented in [76], in the context of composite models and thoroughly defined as an effective field theory in [77]. In practice, that means that any flavored structure may be constructed only by insertions of SM Yukawa matrices, pretending they are fields transforming in the same representation as an actual U(3) 3 flavon. Afterwards these spurion fields are replaced by the constant Yukawa matrix again. For example the Wilson coefficient in (1.68) is in MFV given by C d = 1 Λ2 χdχd Fl = Y d Y d � � + 1 Λ2 χdχ Fl † 1 d + Λ2 χuχ Fl † u + 1 Λ4 χuχ Fl † uχdχ † d + . . . � � + Yd Y † d + Yu Y † u + YuY † u Yd Y † d + . . . � � , (1.71) which is to a good approximation diagonal in the mass eigenbasis. Deviations are due to insertions of flavor invariant terms formed from the Yukawas, which are however further suppressed by the flavor scale. The concept of MFV is very popular and an ingredient of many BSM models which do not address the flavor sector inherently. Very similar at first sight, however a qualitatively different approach can be realized in models with extra dimensions. Here, it is possible to separate the source of flavor breaking from the SM by confining it to a distant brane. In the simplest setup one has n flat extra dimensions, with the SM assumed to reside on one brane and the flavor breaking fields χ on another brane [69]. Flavor breaking is then only communicated to the SM by a bulk field φ which couples to both branes. Because there is no direct contact between the SM fields and the flavor breaking fields, one says the flavor violation is shined to the IR brane. The relevant terms in the Lagrangian read � L ∋ d 4 xdz n φkl Λ n/2−2 Fl χklδ n � (z − z0) + d 4 xdz n φkl Λ n/2+1 Fl ¯QkHd c l δn (z) . (1.72) After solving the equations of motion, the mediator will inherit the flavor structure from the source (its boundary condition), so that φkl = χkl f(z) and the effective Yukawa coupling from the Lagrangian above will therefore be (Yd)kl ∼ χkl(z = 0) Λ n/2+1 Fl , (1.73) This is exactly what one would expect from a spurion analysis. However, a plethora of phenomena can appear if one considers multiple sources sitting at different branes 2.2]. 23 For the full flavor group it is not even possible to find a renormalizable potential, see [74, Sec
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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Page 46 and 47: 36 Chapter 2. The Randall Sundrum M
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82 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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