A Mathematica based Version of the CKMfitter Package

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42 Chapter 5. Probing the Standard Model The quality of the agreement between the SM predictions and the experimental data , which is a can be obtained from the interpretation of the test statistics χ2 min,ymod probe of the goodness-of-fit for the SM hypothesis. The p-value P (χ2 min,ymod |SM) for the validity of the Standard Model needs to be calculated using Toy Monte Carlo simulations [4]. Since this is beyond the scope of this thesis, an approximative method is chosen instead. Assuming naively a gaussian behavior of the global likelihood function L(ymod) with Ndof = Nexp − Nmod = 6, the p-value can be approximately obtained from the Prob function, which is given in Equation (3.6). The global χ2-minimum of the Standard Global CKM Fit is: = 4.25 . (5.28) This leads to a p-value of χ 2 min,ymod P (χ 2 min,ymod |SM) ≤ P rob(χ2min,ymod ) = 64.2 % (5.29) for the validity of the Standard Model. The main contribution to the global χ 2 -minimum stems from the slight disagreement of the measurements of |Vub| and sin 2β. The plots in Figure 5.5 show the SM prediction compared to the direct measurement for both observables, |Vub|incl. and sin 2β. In addition, the result of the Standard Global CKM Fit is shown. 1 - CL 1 0.8 0.6 0.4 0.2 0 CKM f i t t e r ICHEP 2006 CKM fit prediction measurement 0.0035 0.004 0.0045 0.005 0.0055 | Vub | incl. 1 - CL 1 0.8 0.6 0.4 0.2 CKM f i t t e r ICHEP 2006 CKM fit prediction measurement 0 0.5 0.6 0.7 0.8 0.9 1 sin(2 β ) Figure 5.5: Confidence level on |Vub|incl (left) and sin 2β (right) obtained from the Standard Global CKM Fit (blue curve), the SM predictions (red curve) and the direct measurements (green shaded). Another possibility for the determination of the CKM matrix element |Vub| is provided by purely leptonic decays of charged B-mesons. Since the SM expectations for the corresponding branching fractions are rather small, only the decay B + → τ + ντ has been seen yet. Figure 5.6 shows the SM predictions from the Standard Global CKM Fit for B(B + → τ + ντ ) and B(B + → µ + νµ). In addition, the combined B + → τ + ντ measurement from BABAR and Belle and the corresponding fit result is shown.
5.2. Standard Model Fit Results 43 1 - CL 1 0.8 0.6 0.4 0.2 CKM f i t t e r ICHEP 2006 CKM fit prediction measurement 0 0 0.5 1 1.5 2 2.5 3 + + 4 BR(B → τ ν τ ) × 10 1 - CL 1 0.8 0.6 0.4 0.2 CKM f i t t e r ICHEP 2006 0 0.2 0.3 0.4 0.5 0.6 0.7 + + 6 BR(B → μ ν μ ) × 10 Figure 5.6: Confidence level on B(B + → τ + ντ ) and B(B + → µ + νµ) obtained from the SM prediction (red curve). Furthermore, the results of the direct measurement (green shaded) and the fit (blue curve) are given for the decay B + → τ + ντ . Since the SM prediction of the branching fraction of purely leptonic B-meson decays depends also on the decay constant fBd , a precise measurement can help to reduce theoretical uncertainties in the Standard Global CKM Fit. Figure 5.7 shows the constraint on (¯ρ,¯η) coming from the individual constraints B(B + → τ + ντ ), ∆md and a combination of both inputs. Since B(B + → τ + ντ ) depends on the matrix element |Vub|, its 95 % CL belt is to a very good approximation a ring around (¯ρ, ¯η) = (0, 0). In the combination with ∆md, the theoretical uncertainties are reduced due to the cancelation of the decay constant contribution. The remaining theoretical uncertainty stems from the bag factor Bd in the SM prediction of ∆md. η 1.5 1 0.5 0 -0.5 -1 excluded area has CL > 0.95 + + B → τ ντ CKM fit γ α CKM + + f i t t e r Constraint from B → τ ντ and Δmd ICHEP 06 -1.5 -1 -0.5 0 0.5 1 1.5 2 Figure 5.7: Confidence level in the (¯ρ,¯η) plane obtained from B(B + → τ + ντ ) (blue ring), ∆md (yellow ring) and a combination of both (green area). ρ β Δmd
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 18 and 19: 8 Chapter 2. Theory 2.4 Neutral Mes
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
Page 26 and 27: 16 Chapter 3. CKMfitter
Page 28 and 29: 18 Chapter 4. A Mathematica based V
Page 40 and 41: 30 Chapter 5. Probing the Standard
Page 60 and 61: 50 Chapter 6. New Physics Beyond th
Page 68 and 69: 58 Chapter 7. Conclusions and Persp
Page 70 and 71: 60 Appendix A. The Inami-Lim Functi
Page 72 and 73: 62 Appendix B. Additional Figures o
Page 74 and 75: 64 Appendix B. Additional Figures o
Page 76 and 77: 66 Appendix C. Testjob } globalMinS
Page 78 and 79: 68 Appendix D. Source Code else k =
Page 80 and 81: 70 Appendix D. Source Code do i = 1
Page 82 and 83: 72 Appendix E. User Guide E.2 Datac
Page 84 and 85: 74 Appendix E. User Guide scanMin T
Page 86 and 87: 76 Appendix E. User Guide E.3 Input
Page 88 and 89: 78 Appendix E. User Guide
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92 Appendix E. User Guide
Page 104 and 105:
Page 106 and 107:
Page 108 and 109:
98 BIBLIOGRAPHY [12] L. Wolfenstein
Page 110 and 111:
100 BIBLIOGRAPHY [43] A. G. Akeroyd
Page 112 and 113:
102 BIBLIOGRAPHY [74] T. Inami and
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Erklärung Hiermit versichere ich,
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A Mathematica based Version of the CKMfitter Package

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