Violation in Mixing

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14 �È Violation in the �� System ¡Ñ� � ÅÀ ÅÄ� ¡ � � À Ä� so that ¡Ñ� is positive by definition and so that, through those definitions, one can also define the difference between eigenvalues �Ä�À: ¡� � ¡Ñ� � ¡ �� (1.11) Going back to the Schrödinger equation (1.5), one can extract the eigenvalues �Ä�À and from these, one can also obtain constraints for the weights Ô and Õ as functions of the Hamiltonian matrix elements: � À À À À from this, for the two eigenvalues one gets: �Ä � À À Õ Ô Ô � À À Õ �À � À À Õ Ô � À À Ô Õ �� Ô Ô ��Ä�À � ¦Õ ¦Õ and thus the relation: Looking at the difference ¡�, the diagonal term À vanishes and one gets: Now squared: ¡� �¡Ñ� ¡� � ¡Ñ� � ¡ � �¡Ñ�¡ � ��À � � Õ À � Ô À � (1.12) � ¡ � � À Õ � (1.13) Ô � � Õ À ��À Ô À ��À À � where, to get to the last equalities, Eqs. 1.12 has been used. Now from the latter expression, one obtains: �� ¡Ñ� � Ê� Å �� ¡ � � � ÁÑ Å � � �� Å £ �� Å £ By substituting the elements of the matrix À with the elements of the two matrices Å and , one can split ¡� in its real and imaginary parts: MARCELLA BONA � £ � £ ��
1.2 Neutral � Mesons 15 � ¡� �� Å from which one obtains: � �� Å £ � £ � � �� Å � � � � � Å £ Å £ ¡Ñ� � ¡ � ��Å � � � � � (1.14) ¡Ñ�¡ � � � Å £ Å £ ℄��Ê�Å £ From the expressions 1.12, 1.11 and 1.13, one can rewrite the Õ�Ô ratio: Õ Ô � ¡� À � ¡Ñ� � ¡ � Å � Also the time-dependent Schrödinger equation becomes: and thus the solutions are � � where Ø is the proper time of the �Ä�À meson. 1.2.4 Phase Conventions �Ø � �Ä�À � ÑÄ�À � (1.15) � Å £ � £ � (1.16) � ¡Ñ� ¡ � � Ä�À �Ä�À � �� ÅÄ�À Ø � Ä�À Ø � �Ä�À The states � and � , as already shown in the previous section (Eq. 1.4), are related through �È transformation: �È �� È �� (1.17) The phase �� is arbitrary since flavour conservation is a symmetry of the strong interactions and a phase transformation [12] �� (1.18) has no physical effects. In the new basis, �È transformations take the form: �È �� È �� 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: 1.2 Neutral � Mesons 13 ��
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 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
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entries per mrad 2.2 The BABAR dete
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2.2 The BABAR detector. 67 entries
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2.2 The BABAR detector. 69 energies
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2.2 The BABAR detector. 71 The main
Page 79 and 80:
Efficiency 2.2 The BABAR detector.
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2.2 The BABAR detector. 75 Table 2-
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3 Ã Ë reconstruction and efficien
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3.3 Studies on data 79 19.5° BINS
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3.3 Studies on data 81 ¯ off-reson
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3.3 Studies on data 83 0.504 0.502
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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.
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3.3 Studies on data 95 0.508 0.506
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4 Strategy and Tools for Charmless
Page 105 and 106:
4.2 Event selection 99 channel sele
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4.2 Event selection 101 ÀÐ �
Page 109 and 110:
4.2 Event selection 103 Figure 4-2.
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4.3 Background fighting 105 Figure
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4.4 PID selection 107 In the case o
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4.4 PID selection 109 θ c Cut Effi
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4.5 Tracking Corrections 111 Ë�
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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
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5.2 Ã Ë reconstruction 123 1 0.8
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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
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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:
Page 189 and 190:
Page 191 and 192:
7.8 Validation studies 185 Figure 7
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7.8 Validation studies 187 100 75 5
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7.8 Validation studies 189 Figure 7
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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
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7.10 Systematic studies 197 Categor
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7.11 Summary 199 �Ã� Ë��
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7.11 Summary 201 �Ã� Ë��
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7.11 Summary 203 Variation �Ã�
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BIBLIOGRAFIA 205 Bibliografia Chapt
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BIBLIOGRAFIA 207 [38] J. Charles, P
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BIBLIOGRAFIA 209 Chapter 7 [67] D.
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