Violation in Mixing

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10 �È Violation in the �� System Moreover one can demonstrate that, given the invariance for �È symmetry, the off-diagonal terms of Å and should be real. As a matter of fact defining a Hermitian matrix À (which represents Å or ): � � �À� � � � � � � �È �È À �È �È � � � � ��È � � � �À� � � � ��À�� £ � (1.4) assuming that �È À �È � À and remembering that �È �È � (from which the eigenvalues satisfy the ��È � � ) and that �È � � � � ��È � � �. We can obtain one more constraint if we consider again the �ÈÌ invariance: the off-diagonal terms of Å and have to be one the complex conjugate of the other. This can be shown using the generic Hermitian matrix À: � � �À� � � � � � � �ÈÌ �ÈÌ À �ÈÌ �ÈÌ �� ÈÌ� ��À�� À�� £ where we have been using �ÈÌ À �ÈÌ � À and �ÈÌ�� ÈÌ��. The À eigenvalues (complex in the most general case) can be written as Ñ� � � where one can demonstrate that � � : the matrix is defined the decay matrix. As a matter of fact, if one defines the eigenstates �� of the non-Hermitian matrix À, the evolution in time of such a state is given by (as Eq. 1.3 shows): � Ø �� ÀØ �� Ñ�Ø �Ø �� from which one can extract the probability of the initial particle not to be decayed yet at a given time Ø: �� Ø � � � �Ø � Ý � �� This quantity depends only from � that can be considered the decay rate of the given À eigenstate ��. On the other hand, the matrix Å is called the Hermitian part of the mass matrix. 1.2.2 The � system: general formalism A generic neutral meson �È � together with its anti-particle �È � 2 can be considered as a set of eigenstates of the imperturbed Hamiltonian À with eigenvalues Ñ and Ñ , respectively: assuming that À conserves �ÈÌ, Ñ can be considered equal to Ñ and thus: À �È � � Ñ �È �� À �È � � Ñ �È �� These two states (particle and anti-particle) belong to Ñ that is the degenerate eigenvalue of À . Thus if an arbitrary linear combination of them is considered: 2 Here È and È label each neutral meson anti-meson pair. MARCELLA BONA
1.2 Neutral � Mesons 11 �È � � ��È � ��È � this gives another À eigenstate. In this particular case, this state must satisfy the Schrödinger equation where the matrix À is a ¢ matrix (together with the matrices Å and ): � � �Ø � � � � À � � � � � Å � Taking into account the � system and the relation �� Ô�� Õ�� that satisfies: � � �Ø � � Ô � À Õ - 0 B - 0 B b d - b d - � � �� Å Å � � Å £ Å � � � � (u,c) t (u,c) - - t - W - W W (u,c) t - + (u,c) - - - t W + � � � � (1.5) � £ d b - d b - �� Ô � Õ 96/10/24 11.13 Figure 1-1. Feynman’s box diagrams describing � � oscillations. The off-diagonal terms should be one the complex conjugate of the other, since the matrices are Hermitian. �È conservation would imply also the reality of those terms. The off-diagonal terms in these matrices, Å and , are particularly important in the discussion of �È violation: they are the dispersive and absorptive parts respectively of the transition amplitude from � to � . Å contributes to the transition amplitude from � to � through intermediate states described by box diagrams (see Fig. 1-1). The box diagrams have four vertices and so they are fourth order diagrams: in the Standard Model, they correspond to second B 0 B 0 �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: 1.2 Neutral � Mesons 9 Applying t
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
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4.2 Event selection 99 channel sele
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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
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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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5.5 Maximum likelihood analysis 133
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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
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6.3 The maximum likelihood analysis
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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
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6.7 Determination of the branching
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7 Analysis of the time-dependent
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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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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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