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8 Chapter 2. Theory 2.4 Neutral Meson Oscillation The phenomenon of quark flavor oscillation is predicted in the four neutral meson systems: K 0 (¯sd) ↔ ¯ K 0 (s ¯ d) , D 0 (cū) ↔ ¯ D 0 (¯cu) , B 0 d (¯ bd) ↔ ¯ B 0 d (b ¯ d) , B 0 s ( ¯ bs) ↔ ¯ B 0 s (b¯s) . Since the mean life time of the neutral D mesons is very small compared to their oscillation frequency, neutral meson mixing has been only observed in the Kaon and B-meson systems, yet. The time evolution of these transitions is given by the Schrödinger Equation for two-state systems of instable particles: i ˙ ψ(t) = ˆ H ψ(t) , (2.20) where ψ(t) is the two-state system of the neutral mesons, e. g. for the Bd system: � |B0 (t)〉 ψ(t) = | ¯ B0 � . (2.21) (t)〉 The Hamilton Operator ˆ H contains the two hermitian 2 × 2 mass (Mij) and decay width (Γij) matrices: ⎛ ˆH = ˆ M − i 2 ˆ ⎜ Γ = ⎝ M11 − i 2 Γ11 M12 − i 2 Γ12 M21 − i 2 Γ21 M22 − i 2 Γ22 ⎞ ⎟ ⎠ . (2.22) Since CPT symmetry requires equal masses and decay rates for a particle and its anti-particle, the diagonal elements must be equal: Γ11 = Γ22 = Γ (2.23) M11 = M22 = m (2.24) and the eight real parameters can be reduced to six independent parameters. The hermiticity of ˆ Γ and ˆ M leads to: M21 = M ∗ 12 and Γ21 = Γ ∗ 12 . (2.25) Because of an arbitrary global phase, only five observables can be defined � � � � Γ12 Γ12 m , Γ , |M12| , Re and Im . (2.26) M12 The Schrödinger Equation (2.20) has well defined solutions ψ(t) for any ψ(0), but only two mass eigenstates with time-independent flavor composition for each neutral meson system. M12
2.4. Neutral Meson Oscillation 9 2.4.1 The K 0 - ¯K 0 System The flavor eigenstates of the neutral Kaon system mix via weak interaction through the box diagrams shown in Figure 2.2. These are effective flavor changing neutral current (FCNC) processes with |∆S| = 2 7 , where the main contributions in the loop to the observable |ɛK| as described below come from top- and also from charm-quark exchange. s u,c,t d 0 K W W u,c,t 0 K d s s d W u,c,t u,c,t Figure 2.2: Box diagram contribution to K 0 - ¯ K 0 mixing The two mass eigenstates of the Kaon system are defined according to their lifetimes: 0 K |K 0 S〉 = pK|K 0 〉 + qK| ¯ K 0 〉 (2.27) |K 0 L〉 = pK|K 0 〉 − qK| ¯ K 0 〉 (2.28) where K0 S is the short living, and K0 L is the long living normalized mass eigenstate, with |qK| + |pK| = 1. An interesting observable of the neutral Kaon system related to the CKM matrix analysis is the CP-violating parameter |ɛK|. εK is defined by: εK = 2 3 η+− + 1 3 η00 where η+− and η00 are the ratios of the K 0 L and K0 S and neutral pair of pions respectively: η+− = A � K 0 L → π+ π −� A � K 0 S → π+ π −� 2.4.2 The B 0 - ¯B 0 System W 0 K d s (2.29) decay amplitudes to a charged η00 = A � K0 L → π0π0� A � K0 S → π0π0� . (2.30) Neutral B-meson oscillations occur in the SM through a second order FCNC process. They are mediated by the |∆B| = 2 box diagrams shown in Figure 2.3, where the loop is dominated by W boson and up-type quark contributions. According to their mass, the eigenstates of the B-meson system are defined by: |B 0 L〉 ∼ pB|B 0 〉 + qB| ¯ B 0 〉 (2.31) |B 0 H〉 ∼ pB|B 0 〉 − qB| ¯ B 0 〉 , (2.32) 7 |∆F | expresses the flavor quantum number difference between the initial- and the final-state of a transition, e. g. |∆S| for Strangeness and |∆B| for Bottomness.
Page 1 and 2: A Mathematica based Version of the
Page 3: Kurzfassung Das CKMfitter Software-
Page 6 and 7: vi CONTENTS 6 New Physics Beyond th
Page 8 and 9: viii LIST OF FIGURES B.6 Constraint
Page 10 and 11: x LIST OF TABLES
Page 12 and 13: 2 Chapter 1. Introduction Unfortuna
Page 14 and 15: 4 Chapter 2. Theory the bosons of t
Page 16 and 17: 6 Chapter 2. Theory VCKM is a compl
Page 20 and 21: 10 Chapter 2. Theory b u,c,t d 0 B
Page 22 and 23: 12 Chapter 2. Theory CP violation i
Page 24 and 25: 14 Chapter 3. CKMfitter The quantit
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Page 40 and 41: 30 Chapter 5. Probing the Standard
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58 Chapter 7. Conclusions and Persp
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60 Appendix A. The Inami-Lim Functi
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62 Appendix B. Additional Figures o
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64 Appendix B. Additional Figures o
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66 Appendix C. Testjob } globalMinS
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68 Appendix D. Source Code else k =
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70 Appendix D. Source Code do i = 1
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72 Appendix E. User Guide E.2 Datac
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74 Appendix E. User Guide scanMin T
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76 Appendix E. User Guide E.3 Input
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78 Appendix E. User Guide
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98 BIBLIOGRAPHY [12] L. Wolfenstein
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100 BIBLIOGRAPHY [43] A. G. Akeroyd
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102 BIBLIOGRAPHY [74] T. Inami and
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Erklärung Hiermit versichere ich,
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