Violation in Mixing

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38 �È Violation in the �� System At the same way, for � Ô�Ý× Ø and � Ô�Ý× Ø states decaying into �: where � � �Ô�Ý× Ø � � � � Ø � ¢ Ò Ó Ê Ó× ¡Ñ�Ø � ×�Ò �Å � � ×�Ò ¡Ñ�Ø � � �Ô�Ý× Ø � � � � Ø Ò � ¢ Ó Ê Ó× ¡Ñ�Ø �×�Ò �Å �� ×�Ò ¡Ñ�Ø � �� Ê � �� (1.62) � (1.63) �È invariance requires that �È conjugated processes should have identical rates, this leading to these conditions: �� (1.64) �È violation arises if any of these equalities is not satisfied. ×�Ò �Å �� ×�Ò �Å � � � � � (1.65) Eqs. 1.63 are completely general. Assuming that one single amplitude contributes to the decay taken into account and that penguins can be neglected, we can express the phases of the amplitudes on the basis of their �È relationships as: �� Æ � � � �� Æ � � � �� Æ �� Æ � (1.66) where �� and �� represent the weak phases, while Æ and Æ are the strong phases. Substituting in Eq. 1.65, one gets: ×�Ò �Å �� ×�Ò �Å �� ¡Æ � ×�Ò �Å � � � � � ×�Ò �Å �� ¡Æ � (1.67) where ¡Æ � Æ Æ . The �È -violating weak phase is given by ¨ � �Å �� . From measurements of the timedependent decay distributions one can obtain the quantities: and from these one can extract ×�Ò ¨: Ë � ×�Ò ¨ ¡Æ Ë � ×�Ò ¨ ¡Æ ×�Ò ¨ � � Õ ËË¦ Ë Ë � � (1.68) The two solutions correspond to ×�Ò ¨ and to Ó× ¡Æ: this ambiguity can be removed analyzing other decays with final states which have the same weak phase ¨, but different strong phases. MARCELLA BONA
1.5 Determination of « 39 Assuming negligible the penguins contributions, the previous method can be used to extract the angle « through measurements of decays of � Ô�Ý× Ø and � Ô�Ý× Ø to final states like � � or � � . When penguins are not negligible, this method measures a quantity, denoted «�«, which differs from the true « by the unknown amount �� . This quantity is mode-dependent because it depends on the ratio of treedominated to penguin-only contributions. 1.5.2 Extraction of « in the Presence of Penguins In most of the decays modes, more than one amplitude is present and in the expression of the total amplitude, contributions from tree or penguin diagrams can be split: in the particular case of the channels of interest here, the weak phase difference between these terms is «. Going back to the case of a final state � being a �È eigenstate, we can factorize the decay amplitudes: � � � ��È ��È �Ì��Ì � �ÆÌ È� ��È � �ÆÈ � � � ��È ��È �Ì� ��Ì � �ÆÌ È� ��È � �ÆÈ (1.69) where Ì , �Ì and ÆÌ (È , �È and ÆÈ ) are the magnitude, the weak phase and the strong phase of the treedominated (penguin-only) amplitude. Thus, assuming the presence of penguin contributions, if �Ì �� È , ��È (Eq. 1.22) becomes a function of tree and penguin diagram parameters and as a consequence it does not corresponds to a clean measure of the CKM phase. The presence of non-negligible penguin contributions also leads to direct �È violation (see Sec. 1.5.3), that is ��È � �� . In the presence of direct �È violation, the time-dependent �È asymmetry contains a Ó× ¡Ñ�Ø term, the coefficient of which can also be measured. In case the strong phases are equal, ÆÌ � ÆÈ , then ��È is a pure phase (i.e., ��È � � ) and so no direct �È violation is present. However, like in the previous case, this phase depends on both tree and penguin parameters, so that there is still a shift in « due to penguin contributions, even though there is no direct �È violation. The method to separate the tree and penguin contributions is isospin analysis. Isospin amplitudes Á¡Á�Á� can be labeled by the ¡Á value of the �-quark decay and by the Á� of the final state, which includes the spectator quark. A gluon is pure Á � , so that the dominant gluonic � � � penguin diagrams are pure ¡Á � . On the other hand, the tree-level � � ÙÙ� decays have both ¡Á � and ¡Á � components. If the ¡Á � part can be isolated, then the tree contribution, which contains the weak phase to be measured, can be isolated. The inclusion of the spectator quark gives the final isospin value of or for the gluonic penguin contributions, but , or for the tree contributions. The same arguments apply to � � × penguins and � � ÙÙ× tree amplitudes. Table 1-2 lists the isospin amplitudes for all relevant channels for these states. Note that, in all cases, there is at least one isospin amplitude which can be reached only via tree diagrams: � � � for � � �� and � � � for � � �Ã. Isolation of such isospin amplitudes allows the removal of penguin pollution. �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 43: 1.5 Determination of « 37 1.5 Dete
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
Page 99 and 100:
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 �
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4.2 Event selection 103 Figure 4-2.
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4.3 Background fighting 105 Figure
Page 113 and 114:
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
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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
Page 135 and 136:
Page 137 and 138:
Page 139 and 140:
5.5 Maximum likelihood analysis 133
Page 141 and 142:
Page 143 and 144:
Page 145 and 146:
Page 147 and 148:
Page 149 and 150:
Page 151 and 152:
5.6 Counting analysis 145 Table 5-1
Page 153 and 154:
5.7 Determination of branching frac
Page 155 and 156:
6 Measurement of Branching Fraction
Page 157 and 158:
6.3 The maximum likelihood analysis
Page 159 and 160:
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
Page 167 and 168:
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
Page 175 and 176:
7.4 Data samples and event selectio
Page 177 and 178:
7.4 Data samples and event selectio
Page 179 and 180:
7.5 Background characterization 173
Page 181 and 182:
7.5 Background characterization 175
Page 183 and 184:
Page 185 and 186:
Page 187 and 188:
Page 189 and 190:
Page 191 and 192:
7.8 Validation studies 185 Figure 7
Page 193 and 194:
7.8 Validation studies 187 100 75 5
Page 195 and 196:
7.8 Validation studies 189 Figure 7
Page 197 and 198:
7.8 Validation studies 191 Paramete
Page 199 and 200:
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
Page 213 and 214:
BIBLIOGRAFIA 207 [38] J. Charles, P
Page 215 and 216:
BIBLIOGRAFIA 209 Chapter 7 [67] D.
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Violation in Mixing

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