Violation in Mixing

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8 �È Violation in the �� System where À is the strong and electromagnetic Hamiltonian À � À× À�Ñ, while À is a small perturbation that represents the weak-interaction Hamiltonian À � ÀÛ. The system governed by this Hamiltonian must be solution of the time dependent Schrödinger equation: � � �Ø � � Ø �× � À� � Ø �× We can define a set of discreet eigenstates � � � and a set of eigenstates in the continuum � � � for which we can write: À � � � � � � � � and À � � � � �� At this point, using the Schrödinger picture, the generic system can be written like: � � Ø �× � � � «� Ø � � � � � ¬� Ø � � � where � � � represent the states to which the mesons �È � and �È � (or in a more general way the states � � �) can decay. For Ø � , the generic state is: Using the interaction picture, we have: � � Ø � �×� � « �� Ø � �Á �� Ø� �×� �� Ø �Á �� À Ø � � Ø �× � � the dynamic equation being written: � � �Ø � � Ø �Û � À Û� � Ø �Û � � �� Ø � � � � � �� Ø � � � that depends only on the weak Hamiltonian redefined like À Û � � �À Ø À � �À Ø . In terms of amplitude we have: � �� Ø �Ø MARCELLA BONA � �� Ø �Ø � � � � �ÀÛ� � �� Ø � � �� Ø ��ÀÛ��Ø � � � � � � �� Ø � � �ÀÛ� � �� Ø � � �� Ø ��ÀÛ��Ø (1.1) � �
1.2 Neutral � Mesons 9 Applying the Wigner-Weisskopf method [10], we introduce an approximation by leaving out in Eq. 1.1, the last term: this corresponds to neglecting the weak interaction for those particles to which the initial mesons can decay. Therefore the decay products are considered to be stable. With this method and with this approximation, we can write these two equations as functions of �� Ø that are a finite number Ò of functions. Defining the vector: � Ø � � � � � � � Ø � � � �Ò Ø � � � � � and where Ï is defined as: Ï � Ï�� that satisfies � Ø �� ÏØ « � where « � � ��À�� È � ��À�� À � � � � �� È and going back to the Schrödinger picture, one obtains: � � � � � « � � � « Ò � � � � � � � « Ø � �� È � Æ � �� À � � �� À � � � � (1.2) « Ø �� Ø � Ø �� ÏØ � �� ÀØ � (1.3) where À � � Ï. This matrix À is called the non-Hermitian mass (or energy) matrix. At this point, One can define two Hermitian matrices Å � Å Ý and � Ý : Å � À À Ý ��À À Ý from which one obtains À � Å The elements of these matrices can be extracted from Eq. 1.2: Å �� Æ �� À � � � � �� À � � �� À � � � � � �� È �Æ � �� À � � �� À � � � The �ÈÌ invariance guarantees the equality À�� À�� with the state � � � that represents the charge conjugate of � � � (they both belong to the same eigenvalue): in fact, from � � �À� � � � � � � �ÈÌ �ÈÌ À �ÈÌ �ÈÌ �� ÈÌ� ��À� � � � � � �À� � �� supposing �ÈÌ À �ÈÌ � Àand knowing that �ÈÌ �ÈÌ � (therefore the eigenvalues can be ��ÈÌ� � exclusively) and that �ÈÌ�� ÈÌ��. � �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: 1.2 Neutral � Mesons 7 Note, howe
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 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
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Efficiency 2.2 The BABAR detector.
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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
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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
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3.3 Studies on data 87 0.504 0.502
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3.3 Studies on data 89 N(π + π -
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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
Page 107 and 108:
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
Page 113 and 114:
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:
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5.5 Maximum likelihood analysis 133
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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:
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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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