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Discrete Symmetries Discrete Symmetries by Prof.dr Ing. J. F. J. van den Brand Vrije Universiteit Amsterdam, The Netherlands www.nikhef.nl/ jo 1 2 Contents 1 INTRODUCTION 5 2 Symmetries 7 2.1SymmetriesinQuantumMechanics...................... 7 2.2ContinuousSymmetryTransformations.................... 9 2.2.1 ConservationofMomentum...................... 9 2.2.2 ChargeConservation.......................... 10 2.2.3 LocalGaugeSymmetries........................ 11 2.2.4 ConservationofBaryonNumber.................... 14 2.2.5 ConservationofLeptonNumber.................... 15 2.3DiscreteSymmetryTransformations...................... 19 2.3.1 Parity or Symmetry under Spatial Reflections . ......... 19 2.3.2 Parity Violation in β-decay...................... 20 2.3.3 HelicityofLeptons........................... 22 2.3.4 ConservationofParityintheStrongInteraction........... 24 2.4ChargeSymmetry................................ 26 2.4.1 Particles and Antiparticles, -Theorem.............. 26 2.4.2 ChargeSymmetryoftheStrongInteraction.............CPT 30 2.5 Invariance under Time Reversal ........................ 31 2.5.1 PrincipleofDetailedBalance..................... 34 2.5.2 ElectricDipoleMomentoftheNeutron................ 34 2.5.3 Triple Correlations in β-Decay..................... 36 3SU(2) U(1) Symmetry or Standard Model 37 3.1NotesonFieldTheory.............................× 37 3.1.1 DiracEquation............................. 38 3.1.2 QuantumFields............................. 38 3.2ElectroweakInteractionintheStandardModel............... 42 3.2.1 ElectroweakBosons........................... 42 3.2.2 FermionsintheStandardModel.................... 47 4 Quark Mixing 53 4.1Introduction................................... 53 4.2CabibboFormalism............................... 54 4.3GIMScheme................................... 54 4.3.1 SummaryofQuarkMixinginTwoGenerations........... 55 4.4TheCabibbo-Kobayashi-MaskawaMixingMatrix.............. 57 5 The neutral Kaon system 63 5.1ParticleMixingfortheNeutralK-Mesons.................. 63 5.2MassandDecayMatrices........................... 67 5.3Regeneration.................................. 70 5.4 CP Violation with Neutral Kaons . .................... 72 5.5IsospinAnalysis................................. 76 5.6CPViolationinSemileptonicDecay...................... 80 5.7 CPT Theorem and the Neutral Kaon System ................. 81 3 5.7.1 GeneralizedFormalism......................... 81 5.7.2 CPSymmetry.............................. 82 5.7.3 TimeReversalInvarianceandCPTInvariance............ 83 5.7.4 IsospinAmplitudes........................... 83 5.8ParticleMixingandtheStandardModel................... 84 6 The B-meson system 89 6.1CPViolationintheStandardModel..................... 89 6.1.1 CPViolationinneutralBdecays................... 91 6.1.2 MeasurementofRelevantParameters................. 95 6.1.3 MeasurementoftheAnglesoftheUnitarityTriangle........100 7 Status of Experiments 105 7.1TheLHCbExperimentatCERN.......................105 8 PROBLEMS 107 8.1TimeReversal..................................107 8.2ChargeConjugation...............................107 8.3 -Theorem..................................107 8.4CabibboAngle.................................107CPT 8.5UnitaryMatrix.................................107 8.6QuarkPhases..................................108 8.7ExploitingUnitarity..............................108 8.8D-mesonDecay.................................108 8.9MassandDecayMatrix............................108 8.10 Kaon Intensities . ...............................108 8.11OpticalTheorem................................109 8.12 Neutral Kaon States in Matter .........................109 8.13RegenerationParameter............................109 8.14IsospinAnalysis.................................109 9 APPENDIX B: SOLUTIONS 111 9.1Cabibboangle..................................111 9.2UnitaryMatrix.................................111 9.3QuarkPhases..................................111 9.4ExploitingUnitarity..............................112 9.5D-mesonDecay.................................112 9.6MassandDecayMatrix............................113 9.7 Kaon Intensities . ...............................114 9.8OpticalTheorem................................115 9.9 Neutral Kaon States in Matter .........................116 9.10RegenerationParameter............................117 9.11IsospinAnalysis.................................117 4 1 INTRODUCTION Symmetries and conservation laws play a fundamental role in physics. The invariance of a system under a continuous symmetry transformation leads to a conservation law by Noethers’ theorem. For example, the invariance under space and time translations results in momentum and energy conservation. Besides these continuous symmetries one has discrete symmetries that play an important role. Particularly, three such discrete symmetries are a topic of interest in modern particle physics. The parity transformation performs a reflection of the space coordinates at the origin (~r ~r). Position and momentumP change sign, while spin is unaffected. The charge conjugation→− operator transforms a particle into its antiparticle and vice versa. All intrinsic ‘charges’ change sign,C but motion and spin are left unchanged. Time reversal operates on the time coordinate. Now also spin changes sign, like momentum and velocity.T Composed symmetries, such as and , can also be considered. It was long thought that was an exact symmetry inCP nature.CPT In 1964 -violation was discovered in the neutral kaonCP system by Christensen, Cronin, Fitch andCP Turlay. A few years later Kobayashi and Maskawa demonstrated that a third quark generation could accommodate violation in the Standard Model by a complex phase in the CKM matrix. Since then,CP however, (or ) violation has not been observed in any other system and we do not understandCP its mechanismsT and rare processes. The discovery of the b-quark in 1977 opened a new possibility to test the Standard Model in B-mesons studies. This might answer unresolved questions in the Standard Model or lead to new physics. With experiments that study B-meson decay one mainly addresses the following questions: Is the phase of the CKM-matrix the only source of -violation? • CP What are the exact values of the components of the CKM-matrix? • Is there new physics in the quark region? • This introductory course is structured as follows. In chapter 2, an introduction to quark mixing is given. In our discussions we will follow a historic route. The present status of violation in the kaon system is discussed in chapter 3. Chapter 4 gives a brief overviewCP of violation in the neutral B-meson system. The merit of various experiments for theCP study of violation in the B sector is discussed in chapter 5. The LHCb experiment is describedCP in somewhat more detail. Appendix A provides a set of exercises. Solutions are presented in appendix B. 5 6 2 Symmetries 2.1 Symmetries in Quantum Mechanics The concept of symmetries and their associated conservation laws has proven extraordi- nary useful in particle physics. From classical physics we know, for example, that the demand that laws need to be invariant under translation in time, leads to conservation of energy. In addition, one has that invariance with respect to spatial rotations leads to conservation of angular momentum. While the conservation laws for energy, momentum and angular momenta are always strictly valid, we know that other symmetries are bro- ken in certain interactions. It was for example quite a surprise for physicists when it was demonstrated that mirror symmetry is violated in the weak interaction (and only in this interaction!); even maximally violated. Furthermore, we presently do not understand why this is the case, or why certain other symmetries ( , ) are only ‘slightly violated’. CP T Here, we first want to summarize the quantum mechanical basis, which we will need for our discussion of the various phenomena. A system is described by a wave function, ψ. A physical observable is represented by a quantum mechanical operator, O,whose expectation values are given by the eigenvalues of this operator. The eigenvalues corre- spond to the results of measurements, and the expectation value of O in the state ψa is defined as1 <O>= ψa∗OψadV. (3) Z Since the expectation values can be determined experimentally, they need to be real quantities, and consequently O needs to be hermitian. When O is an operator, then its hermitian conjugate operator O† is defined as (Oψ)∗φdV = ψ∗O†φdV, (4) Z Z and the operator O is called hermitian when one has O† = O. The time dependence of the wave function, ψ, is given by the Schr¨odinger equation, ∂ψ ih¯ = Hψ. (5) ∂t In case the hamiltonian H is real, one also has ∂ψ∗ ih¯ =(Hψ)∗ = ψ∗H. (6) − ∂t 1When we consider two states, one can write in analog fashion Oba = b∗O adV; (1) Z and Oba is called the transition matrix element between the states a en b. The expectation value of O in state a is the diagonal element of Oba for b = a, <O>= Oaa: (2) The non-diagonal elements do not directly correspond to classical observables. However, the transitions between states a and b are related to Oba. 7 For the time dependence of an observable, O, we then find ∂ ∂ ∂t <O> = ∂t ψ∗OψdV R ∂ψ∗ ∂ψ = ∂t Oψ + ψ∗O ∂t dV (7) R i = ψ∗(HO OH)ψdV. ¯h − R Thus, we conclude that <O>does not chance, and corresponds to a constant of motion, in case the commutator [H, O]vanishes, [H, O] HO OH =0. (8) ≡ − Consequently, a wave function can be found, that is simultaenously an eigenfunction of O and of H, Hψ = Eψ and Oψ = oψ, (9)
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