Violation in Mixing

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18 �È Violation in the �� System From Eqs. 1.12 and 1.9, one can then deduce that is a good approximation. Õ Ô Å £ � � � ÁÑ �Å � Å ¬ ¬ ¬ Õ Ô¬ � 1.2.6 Time Formalism for Coherent �� States �� (1.26) In an � � collider operating at the § �Ë resonance (called a � factory, see the following chapter 2), the � and � mesons produced from the decay of the § are in a coherent Ä � state. The § �Ë is a resonant � � state with quantum numbers Â È� � and it can decay into � � or � � pairs: � mesons are scalars (Â È � ) and so, because of the total angular momentum conservation, the � � pair has to be produced in a Ä � state. Since the § �Ë decays strongly, � mesons are produced in the two flavour eigenstates � and � . After the production, one can imagine that each of the two particles evolve in time as described above for a single �. However they evolve in phase, so that at any time there is always exactly one � and one � present, at least until one particle decays. As a matter of fact, if at a given time Ø one � could oscillate independently from the other, they could become a state made up of two identical mesons: but this cannot happen since the Ä � state is anti-symmetric, while a system of two identical mesons (that are bosons) must be completely symmetric for the two particle exchange, However once one of the particles decays the other continues to evolve, and thus there are possible events with two � or two � decays, whose probability is governed by the time between the two decays. Identifying the two particles from the § �Ë decay by the angle � that they form with the � beam direction in the § �Ë rest frame (and thus calling them a backward and a forward �), the two-� state can be written as [12]: Ë Ø� �Ø� � Ô � �Å Ø� Ø� Ò � Ø� �� Ø�� Ø� �� Ø�� Ó ×�Ò � (1.27) where Ø� is the proper time of the ��, the � particle in the forward half-space at angle �� and Ø� is the proper time for the backward-moving ��, at � �� ). In Eq. 1.27, it has been taken into account that the § �Ë is a spatially asymmetric state (it is a parity eigenstate with eigenvalue ) and it is a � eigenstate with eigenvalue . Therefore also the system to which it decays, must be spatially anti-symmetric as well as charge-conjugation anti-symmetric at a MARCELLA BONA
1.2 Neutral � Mesons 19 given time (Ø� � Ø�): in this case (a particle-anti-particle state), È and � transformations correspond to the same one. As a matter of fact, through parity � �� goes to � � �� and therefore the state � �� results in a spatially anti-symmetric one. As a consequence, the spatial contribution coming from the Ä � condition in the spherical functions � Ñ Ð� must result symmetric: ×�Ò � has been included. On the other hand, by applying �, � �� goes into � �� so that the state in the Eq. 1.27 is asymmetric for particle-antiparticle exchange as requested by the negative � eigenvalue of the § �Ë . Since the coherent time evolution of the two particles can be treated like a single particle evolution, in equations 1.24 and 1.25 one can substitute � Ø�� Ô�Ý× Ø�� Ø�� Ô�Ý× Ø�� After this substitution, Eq. 1.27 can be written extracting the time dependence (and using addiction and subtraction trigonometric rules): Ë Ø� �Ø� � Ô � �Å Ø� Ø� � ¡Ñ� Ø� Ø� �×�Ò �� Ô Õ�� Ò � ¡Ñ� Ø� Ø� Ó× �� Õ Ô�� Ó � ×�Ò �� (1.28) Since the �’s have equal (though back-to-back) momenta in the center-of-mass frame, before the decay of the first of the two �’s, Ø� is equal to Ø� and Eq. 1.28 contains one � and one � . However decay stops the clock for the decayed particle so the terms that depend on ×�Ò�¡Ñ� Ø� Ø� � ℄ begin to play a role. From Eq. 1.28 one can derive the amplitude for decays where one of the two �’s decays to any state � at time Ø and the other decays to � at time Ø : � Ø �Ø � Ô � �Å Ø Ø � Ø �Ø �×�Ò � ¡Ñ� Ø Ø �� Ô � � Õ Ò � ¡Ñ� Ø Ø Ó× �� Õ � � Ô � �Ó ×�Ò � � (1.29) where �� is the amplitude for a � to decay to the state ��, �� is the amplitude for a � to decay to the same state �� (see Eqs. 1.20). To keep signs consistent with Eq. 1.28, the symbol � Ø �Ø � � Ø � Ø� � Ø � Ø�, Ø � Ø�� Ø � Ø� is introduced, but this overall sign factor will disappear in the rate. Any state that identifies the flavor of the parent � (tagging) has either �� or �� . In Eq. 1.29, the sum Ø� Ø� remains only in the factorized exponential and is vanished from ×�Ò� or Ó×�Ò� arguments. �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: 1.2 Neutral � Mesons 17 and � a
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
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-
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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 �
Page 109 and 110:
4.2 Event selection 103 Figure 4-2.
Page 111 and 112:
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�
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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
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
Page 133 and 134:
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
Page 201 and 202:
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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