Violation in Mixing

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30 �È Violation in the �� System So going back to the Ò real independent parameters of a generic unitary matrix, Ò Ò of these parameters can be associated to real rotation angles: among the remaining Ò Ò Ò phases, Ò can be removed with quark field re-definitions, as already said. So the number of independent phase factors is: Ò Ò Ò Ò � Ò Ò � In table 1-1, the cases with , or � quark families are shown: note that at least three quark generations are necessary in order to have, in the �ÃÅ matrix, a �È -violating phase factor. A two-generation theory would not be able to accommodate �È violation without the addition of extra fields. It was this observation that led Kobayashi and Maskawa to suggest a third quark generation long before there was any experimental evidence for it [2, 3]. The unitarity of the �ÃÅ matrix can be made more explicit using a particular parameterization. There are various useful ways to parameterize it, but the standard choice is based on the definition of four angles � , � , � e Æ [14, 20]: Î � × × � �Æ × × × � �Æ × × × � �Æ × × × × � �Æ × × × � �Æ � � (1.47) where �� Ó× �� and ×�� ×�Ò ��, �� is the mixing angle between the � and � quark families and the phase Æ is responsible for �È violation. Considering that Î � when �È violation effect goes to zero, the representation 1.47 can be simplified: for example one can use the fact that �ÎÙ�� × � is very small and so is extremely close to unity. As a consequence, one can neglect terms proportional to × relative to terms of order unity and obtain: Î � × × � �Æ × × × × × � �Æ × � � (1.48) In this approximation, only ÎÙ� and ÎØ� carry phases. Another useful expansion of the �ÃÅ matrix has been first given by Wolfenstein [21] in the small parameter � � ×�Ò �� where �� is the Cabibbo angle: Î � � � �� Ç � � � (1.49) In this form, only four independent parameters remain: �, �, � and �. ToÇ � � � , the ¢ upper-left portion of the �ÃÅ matrix is the matrix associated with Cabibbo rotations of the � and × quarks and it is constructed to be nearly unitary: MARCELLA BONA
1.4 �È Violation in the Standard Model 31 � � � Ç � � accordingly with measurements which give Ô �ÎÙ�� ÎÙ×� � ��. 1.4.2 Unitarity of the �ÃÅ Matrix The unitarity of the �ÃÅ matrix implies various relations among its elements. Three of them are related to the study of �È violation within the Standard Model: ÎÙ�Î £ Ù× Î �Î £ × ÎØ�Î £ Ø× � � ÎÙ×Î £ Ù� Î ×Î £ � ÎØ×Î £ Ø� � � ÎÙ�Î £ Ù� Î �Î £ � ÎØ�Î £ Ø� � � (1.50) Each of these three relations corresponds to an orthogonality condition between columns and requires the sum of three complex quantities to vanish: as a consequence, it can be geometrically represented in the complex plane as a triangle. These are the unitarity triangles: the term Unitarity Triangle is traditionally reserved for the relation 1.50 only. The latter is the one involving the two smaller elements of the �ÃÅ matrix and every single element of the sum is of the order of � , as in the parameterization in 1.49. In the parameterization in 1.47, Î �, Î � and ÎØ� are real and, using the approximations ÎÙ� ÎØ� and the fact that Î � � , the relation 1.50 can be re-written as: Î £ Ù� �Î �Î �� ÎØ� � � �Î �Î �� In terms of the Wolfenstein parameterization, the coordinates of this triangle are � , � and �� (as a matter of fact, two sides are � �� and � �� ). All the three triangles can be drawn knowing the experimental values (within errors) for the various �Î��: this has been done in Fig. 1-5 in a common scale. This figure can be understood by looking at the order of magnitude: ÎÙ�Î £ Ù× Î �Î £ × ÎØ�Î £ Ø× Ç � Ç � Ç � � � � ÎÙ×Î £ Ù� Î ×Î £ � ÎØ×Î £ Ø� Ç � � Ç � Ç � � � ÎÙ�Î £ Ù� Î �Î £ � ÎØ�Î £ Ø� Ç � Ç � Ç � � � �È VIOLATION IN THE �� SYSTEM
Page 1 and 2: ÍÒ�Ú�Ö×�Ø��
Page 3 and 4: Contents Introduction . ...........
Page 5 and 6: 4 Strategy and Tools for Charmless
Page 7 and 8: 7.2 Branching fraction � � �
Page 9 and 10: Introduction The main goal of the B
Page 11 and 12: 1 �È Violation in the �� Sys
Page 13 and 14: 1.2 Neutral � Mesons 7 Note, howe
Page 15 and 16: 1.2 Neutral � Mesons 9 Applying t
Page 17 and 18: 1.2 Neutral � Mesons 11 �È �
Page 19 and 20: 1.2 Neutral � Mesons 13 ��
Page 21 and 22: 1.2 Neutral � Mesons 15 � ¡�
Page 23 and 24: 1.2 Neutral � Mesons 17 and � a
Page 25 and 26: 1.2 Neutral � Mesons 19 given tim
Page 27 and 28: 1.3 The Three Types of �È Violat
Page 33 and 34: 1.4 �È Violation in the Standard
Page 35: 1.4 �È Violation in the Standard
Page 43 and 44: 1.5 Determination of « 37 1.5 Dete
Page 45 and 46: 1.5 Determination of « 39 Assuming
Page 47 and 48: 1.5 Determination of « 41 Figure 1
Page 49 and 50: 1.5 Determination of « 43 b q (a)
Page 51 and 52: 1.5 Determination of « 45 � �
Page 53 and 54: 2 The BABAR Experiment 2.1 Physics
Page 55 and 56: 2.1 Physics at � � B Factory op
Page 57 and 58: 2.2 The BABAR detector. 51 Integrat
Page 59 and 60: 2.2 The BABAR detector. 53 The dete
Page 61 and 62: 2.2 The BABAR detector. 55 two oute
Page 63 and 64: 2.2 The BABAR detector. 57 SVT Effi
Page 65 and 66: 2.2 The BABAR detector. 59 324 1015
Page 67 and 68: Efficiency 2.2 The BABAR detector.
Page 69 and 70: 2.2 The BABAR detector. 63 2.2.4 Th
Page 71 and 72: entries per mrad 2.2 The BABAR dete
Page 73 and 74: 2.2 The BABAR detector. 67 entries
Page 75 and 76: 2.2 The BABAR detector. 69 energies
Page 77 and 78: 2.2 The BABAR detector. 71 The main
Page 79 and 80: Efficiency 2.2 The BABAR detector.
Page 81 and 82: 2.2 The BABAR detector. 75 Table 2-
Page 83 and 84: 3 Ã Ë reconstruction and efficien
Page 85 and 86: 3.3 Studies on data 79 19.5° BINS
Page 87 and 88:
3.3 Studies on data 81 ¯ off-reson
Page 89 and 90:
3.3 Studies on data 83 0.504 0.502
Page 91 and 92:
3.3 Studies on data 85 0.03 0.025 0
Page 93 and 94:
3.3 Studies on data 87 0.504 0.502
Page 95 and 96:
3.3 Studies on data 89 N(π + π -
Page 97 and 98:
3.3 Studies on data 91 0.508 0.506
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3.3 Studies on data 93 Figure 3-13.
Page 101 and 102:
3.3 Studies on data 95 0.508 0.506
Page 103 and 104:
4 Strategy and Tools for Charmless
Page 105 and 106:
4.2 Event selection 99 channel sele
Page 107 and 108:
4.2 Event selection 101 ÀÐ �
Page 109 and 110:
4.2 Event selection 103 Figure 4-2.
Page 111 and 112:
4.3 Background fighting 105 Figure
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4.4 PID selection 107 In the case o
Page 115 and 116:
4.4 PID selection 109 θ c Cut Effi
Page 117 and 118:
4.5 Tracking Corrections 111 Ë�
Page 119 and 120:
4.6 Analysis methods 113 Ñ�Ë si
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4.6 Analysis methods 115 Events/2.5
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4.6 Analysis methods 117 Figure 4-1
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4.6 Analysis methods 119 θ c radia
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5 Measurement of Branching Fraction
Page 129 and 130:
5.2 Ã Ë reconstruction 123 1 0.8
Page 131 and 132:
5.3 Analysis Strategy 125 0.35 0.3
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5.4 Background Suppression and PDF
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Page 137 and 138:
Page 139 and 140:
5.5 Maximum likelihood analysis 133
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Page 143 and 144:
Page 145 and 146:
Page 147 and 148:
Page 149 and 150:
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5.6 Counting analysis 145 Table 5-1
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5.7 Determination of branching frac
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6 Measurement of Branching Fraction
Page 157 and 158:
6.3 The maximum likelihood analysis
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Page 161 and 162:
Page 163 and 164:
6.4 Counting analysis 157 120 100 8
Page 165 and 166:
6.5 Analysis choice 159 6.5 Analysi
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6.6 Results on the Run1 data-set 16
Page 169 and 170:
6.7 Determination of the branching
Page 171 and 172:
7 Analysis of the time-dependent
Page 173 and 174:
7.3 Analysis strategy 167 Parameter
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7.4 Data samples and event selectio
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7.4 Data samples and event selectio
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7.5 Background characterization 173
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7.5 Background characterization 175
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Page 185 and 186:
Page 187 and 188:
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7.8 Validation studies 185 Figure 7
Page 193 and 194:
7.8 Validation studies 187 100 75 5
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7.8 Validation studies 189 Figure 7
Page 197 and 198:
7.8 Validation studies 191 Paramete
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7.9 Results 193 Parameter Fit Resul
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7.9 Results 195 Events/2 MeV/c 2 Ev
Page 203 and 204:
7.10 Systematic studies 197 Categor
Page 205 and 206:
7.11 Summary 199 �Ã� Ë��
Page 207 and 208:
7.11 Summary 201 �Ã� Ë��
Page 209 and 210:
7.11 Summary 203 Variation �Ã�
Page 211 and 212:
BIBLIOGRAFIA 205 Bibliografia Chapt
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BIBLIOGRAFIA 207 [38] J. Charles, P
Page 215 and 216:
BIBLIOGRAFIA 209 Chapter 7 [67] D.
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