Violation in Mixing

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12 �È Violation in the �� System order terms with respect to «Û expansion, the weak interaction coupling constant. These contributions arise from the box diagrams with two Ï exchanges: the quark contribution can come from Ù, or Ø exchanges. Theoretically the matrix element Å can be related to the squared mass of the exchanged quark Õ, to the squared mass of the � mesons and to the product ÎÕ�Î £ Õ� where the Î�� terms are the �ÃÅ matrix elements (1.46). Thus, since the elements ÎÙ� and Î � are strongly suppressed with respect to ÎØ� (more than how ÎØ� is suppressed with respect to ÎÙ� and Î �), and the Ø mass dominates on the and Ù masses, the dominant contribution is due to Ø exchange. Therefore we can write [11]: Å �� Ñ� � � �ÎØ�ÎØ�℄ � ÑØ Ñ � Ñ � ÐÒ Ñ Ø Ñ � � Ç Ñ � Ñ � � Ñ Ø where �� is the Fermi constant for the weak interaction when considered point-like and �� is a constant called � decay constant which rules the purely leptonic decays and represents the probability of the two constituent quarks to annihilate. Thanks to the fact that the QCD effective coupling constant becomes smaller in the processes where high values of momentum are transferred (if we call � the transferred momentum, this happens for � � £É�� where £É�� is a typical scale for QCD interaction), these quark diagrams are, to a good approximation, the main contribution to Å , according to the Standard Model. £É�� Î corresponds to the order of magnitude according to which one can distinguish the small coupling constant region from the strong coupling constant region, the latter being non-perturbative: if the quark É mass satisfy the ÑÉ � £É��, É is then called a heavy quark. According to this criterion, quarks Ù, � and × are light quarks, while , � and Ø are heavy. For heavy quarks, the effective coupling constant «× ÑÉ is small, so strong interactions can be considered perturbative and can be treated in a similar way as we do with the electromagnetic interactions. In the � system, long-distance contributions are expected to be negligible (unlike in the Ã system), so that a good approximation is taking into account only the leading orders of the expansion with respect to the strong coupling. The matrix can be related to the decay amplitude of the À eigenstates: it describes the processes that rule the meson decay. The off-diagonal element represents the absorptive part of processes like � � � � � and � � � � � , where � is an on-shell intermediate state. In the case of decays through on-shell Figure 1-2. � � mixing through virtual intermediate states (Å ) or through on-shell intermediate states ( ). intermediate states, the top quark cannot contribute (because of the energy conservation) and so the leading contribution becomes the term containing the mass Ñ� of the � mesons. Theoretically, one obtains the expression [11]: MARCELLA BONA � (1.6)
1.2 Neutral � Mesons 13 �� Ñ� � � �� Ñ ¡ Ñ� � Ñ� � � �Î �Î �℄ Ñ� � Ñ Ñ � Ñ � � � � Ñ � Ñ� � Ñ Ñ� � � �Î �Î �℄�ÎÙ�ÎÙ�℄ ¡ �ÎÙ�ÎÙ�℄ Ñ � � (1.7) that contains the contributions of and Ù. Using one of the unitary relation of the �ÃÅ matrix (see Sec. 1.4.2), it can be simplified into [11]: �� Ñ� � � �ÎØ�ÎØ�℄ Ñ� �Î�Î �℄�ÎØ�ÎØ�℄Ñ Ç �� Ñ� Ñ� So now the leading term is the one containing mass Ñ�, that is of the same order of the mass Ñ� of quark �: talking about orders of magnitude, we can write: This allows to write also: � �� Å �� Ñ � �ÎØ�ÎØ�℄ Ñ Ø �ÎØ�ÎØ�℄ � Ñ � Ñ Ø This relation is crucial for the � system and it will be used in Sec. 1.2.5. 1.2.3 The � system: mass eigenstates � �� (1.8) � �� Å �� (1.9) The states with definite mass and lifetime are eigenstates of the whole Hamiltonian À and they can be written like �Ä (the lighter) and �À (the heavier), linear combination of the � flavour eigenstates: ��Ä� � Ô �� Õ �� À� � Ô �� Õ�� with the normalization condition: �Õ� �Ô� � � (1.10) where Ô and Õ are complex coefficients. Being these eigenstates of À, they correspond to two eigenvalues that can be written as: �Ä�À � ÅÄ�À � Ä�À� eigenvalues of ��Ä� and ��À�, respectively. The mass difference ¡Ñ� and the width difference ¡ � between the neutral � mesons are defined as follows: �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: 1.2 Neutral � Mesons 11 �È �
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
Page 67 and 68: 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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