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Program Topical Lectures December 2007 118-12-2007 TimeTopicSpeaker Wednesday 9.45 – 10.30Introduction Discrete SymmetriesMarcel Merk 10.45 – 11.30CPV in the Standard ModelMarcel Merk 11.45 – 12.30Flavour mixing in B-decaysMarcel Merk 14.00 – 14.45Observing CPV in B-decaysMarcel Merk 15.00 – 15.45Measuring CP angle beta (phi1)Gerhard Raven Thursday 9.45 – 10.30Measuring CP angle alpha (phi2)Gerhard Raven 10.45 – 11.30Measuring CP angle gamma (phi3)Gerhard Raven 11.45 – 12.30CP Violation and BaryogenesisWerner Bernreuther 14.00 – 14.45CP Violation and BaryogenesisWerner Bernreuther Friday 9.45 – 10.30LHCb: OverviewHans Dijkstra 10.45 – 11.30LHCb: ExperimentHans Dijkstra 11.45 – 12.30LHCb: Sensitivity and UpgradeHans Dijkstra 14.00 – 16.00Discussion and DrinksAll

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Sept 28-29, 20052 CP Violation in the Standard Model Topical Lectures Nikhef Dec 12, 2007 Marcel Merk Part 1: Introduction: Discrete Symmetries Part 2: The origin of CP Violation in the Standard Model Part 3: Flavour mixing with B decays Part 4: Observing CP violation in B decays

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Apologies… 18-12-20073 Sometimes I can’t resist showing some silly pic’s…

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Sept 28-29, 20054 Preliminary remarks The lectures are aimed at graduate students who are non- experts in flavour physics and CP violation. Apologies for those who are active in the field. I will focus on concepts rather than on state of the art measurements and exact theoretical derivations. “Undemocratic” presentation: – Almost none of the beautiful kaon physics will be discussed. Very much looking forward to these sessions…

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Questions are Welcome (1) 18-12-20075

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Sept 28-29, 20056 Questions are Welcome (2) Answers might depend on the context…

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Sept 28-29, 20057 Literature C.Jarlskog, “Introduction to CP Violation”, Advanced Series on Directions in High Energy Physics – Vol 3: “CP Violation”, 1998, p3. Y.Nir, “CP Violation In and Beyond the Standard Model”, Lectures given at the XXVII SLAC Summer Institute, hep-ph/9911321. Branco, Lavoura, Silva: “CP Violation”, International series of monographs on physics, Oxford univ. press, 1999. Bigi and Sanda: “CP Violation”, Cambridge monographs on particle physics, nuclear physics and cosmology, Cambridge univ. press, 2000. T.D. Lee, “Particle Physics and Introduction to Field Theory”, Contemporary Concepts in Physics Volume 1, Revised and Updated First Edition, Harwood Academic Publishers, 1990. C. Quigg, “Gauge Theories of the Strong, Weak and Electromagnetic Interactions”, Frontiers in Physics, Benjamin-Cummings, 1983. References:

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Sept 28-29, 20058 Introduction: Symmetry and non-Observables T.D.Lee: “The root to all symmetry principles lies in the assumption that it is impossible to observe certain basic quantities; the non-observables” There are four main types of symmetry: Permutation symmetry: Bose-Einstein and Fermi-Dirac Statistics Continuous space-time symmetries: translation, rotation, acceleration,… Discrete symmetries: space inversion, time inversion, charge inversion Unitary symmetries: gauge invariances: U 1 (charge), SU 2 (isospin), SU 3 (color),.. If a quantity is fundamentally non-observable it is related to an exact symmetry If a quantity could in principle be observed by an improved measurement; the symmetry is said to be broken Noether Theorem:symmetry conservation law

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Sept 28-29, 20059 Symmetry and non-observables Simple Example: Potential energy V between two charged particles: Absolute position is a non-observable: The interaction is independent on the choice of the origin 0. Symmetry: V is invariant under arbitrary space translations: Consequently:Total momentum is conserved: 00’

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Sept 28-29, 200510 Symmetry and non-observables Non-observablesSymmetry TransformationsConservation Laws or Selection Rules Difference between identical particles PermutationB.-E. or F.-D. statistics Absolute spatial positionSpace translationmomentum Absolute timeTime translationenergy Absolute spatial directionRotationangular momentum Absolute velocityLorentz transformationgenerators of the Lorentz group Absolute right (or left)parity Absolute sign of electric chargecharge conjugation Relative phase between states of different charge Q charge Relative phase between states of different baryon number B baryon number Relative phase between states of different lepton number L lepton number Difference between different co- herent mixture of p and n states isospin

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Sept 28-29, 200511 Puzzling thought… (to me, at least) Can we use the “dipole asymmetry” in cosmic microwave background to define an absolute Lorentz frame in the universe? If so, what does it imply for Lorentz invariance? COBE: WMAP:

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Sept 28-29, 200512 Three Discrete Symmetries Parity, P unobs.: (absolute handedness) –Parity reflects a system through the origin. Converts right-handed coordinate systems to left-handed ones. –Vectors change sign but axial vectors remain unchanged x x, p -p, but L = x p L Charge Conjugation, C unobs.: (absolute charge) –Charge conjugation turns a particle into its anti-particle e e K K Time Reversal, T unobs.: (direction of time) –Changes, for example, the direction of motion of particles t t

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Sept 28-29, 200513 Parity The parity operation performs a reflection of the space coordinates at the origin: If we apply the parity operation to a wave function , we get another wave function ’ with: which means that P is a unitary operation. If P =a , then is an eigenstate of parity, with eigenvalue a. For example: The combination = cosx+sinx is not an eigenstate of P Spin-statistics theorem: bosons (1,2) + (2,1) symmetric fermions (1,2) - (2,1) antisymmetric

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Sept 28-29, 200514 Parity One can apply the parity operation to physical quantities: – Mass mP m = mscalar – Force FP F(x) = F(-x) = -F(x) vector – Acceleration aP a(x) = a(-x) = -a(x)vector It follows that Newton’s law is invariant under the parity operation There are also vectors which do not change sign under parity. They are usually derived from the cross product of two other vectors, e.g. the magnetic field: These are called axial vectors. Finally, there are also scalar quantities which do change sign under the parity operation. They are usually an inner product of a vector and a axial vector, e.g. the electric dipole moment ( is the spin): These are the pseudoscalars.

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Sept 28-29, 200515 Charge conjugation The charge conjugation C is an operation which changes the charge (and all other internal quantum numbers). Applied to the Lorentz force Generally, when applied to a particle, the charge conjugation inverts the charge and the magnetic moment of a particle leaving other quantities (mass, spin, etc.) unchanged. it gives: which shows that this law is invariant under the C operation. Only neutral states can be eigenstates, e.g. Evidently, and so C is unitary, too.

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C and P operators 18-12-200716 In Dirac theory particles are represented by Dirac spinors: Implementation of the P and C conjugation operators in Dirac Theory is However: In general C and P are only defined up to phase, e.g.: (See H&M section 5.4 and 5.6) Note: quantum numbers associated with discrete operations C and P are multiplicative in contrast to quantum numbers associated by continuous symmetries +1/2, -1/2 helicity solutions for the particle +1/2, -1/2 helicity solutions for the antiparticle Antimatter!

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Sept 28-29, 200517 Time reversal Time reversal is analogous to the parity operation, except that the time coordinate is affected, not the space coordinate Again the macroscopic laws of physics are unchanged under the operation of time reversal (although some people find it hard to imagine the time inverse of a broken mirror…), the law remains invariant since t appears quadratically. Other vectors, like momentum and velocity, which are linear motions, change sign under time reversal. So do the magnetic field and spin, which are due to the motion of charge.

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Sept 28-29, 200518 Time reversal: antiunitary Wigner found that T operator is antiunitary: As a consequence, T is antilinear: This leaves the physical content of a system unchanged, since: Antiunitary operators may be interpreted as the product of an unitary operator by an operator which complex-conjugates. Consider time reversal of the free Schrodinger equation: Complex conjugation is required to stay invariant under time reversal

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Sept 28-29, 200519 C-,P-,T-, Symmetry The basic question of Charge, Parity and Time symmetry can be addressed as follows: Suppose we are watching some physical event. Can we determine unambiguously whether: – we are watching this event in a mirror or not? Macroscopic asymmetries are considered to be accidents on life’s evolution rather then a fundamental asymmetry of the laws of physics. – we are watching the event in a film running backwards in time or not? The arrow of time is due to thermodynamics: i.e. the realization of a macroscopic final state is statistically more probably than the initial state. It is not assigned to a time-reversal asymmetry in the laws of physics. – we are watching the event where all charges have been reversed or not? Classical Theory (Newton mechanics, Maxwell Electrodynamics) are invariant under C,P,T operations, i.e. they conserve C,P,T symmetry

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18-12-200720 At each crossing: 50% - 50% choice to go left or right After many decisions: invert the velocity of the final state and return Do we end up with the initial state? Macroscopic time reversal (T.D. Lee)

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18-12-200721 At each crossing: 50% - 50% choice to go left or right After many decisions: invert the velocity of the final state and return Do we end up with the initial state? Macroscopic time reversal (T.D. Lee) Very unlikely!

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Sept 28-29, 200522 Parity Violation Before 1956 physicists were convinced that the laws of nature were left-right symmetric. Strange? A “gedanken” experiment: Consider two perfectly mirror symmetric cars: “L” and “R” are fully symmetric, Each nut, bolt, molecule etc. However the engine is a black box Person “L” gets in, starts, ….. 60 km/h Person “R” gets in, starts, ….. What happens? What happens in case the ignition mechanism uses, say, Co 60 decay? “L” “R” Gas pedal driver Gas pedal driver

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Sept 28-29, 200523 Parity Violation! e-e- Magnetic field Parity transformation e-e- 60 Co J J More electrons emitted opposite the J direction. Not random -> Parity violation! Sketch and photograph of apparatus used to study beta decay in polarized cobalt-60 nuclei. The specimen, a cerium magnesium nitrate crystal containing a thin surface layer of radioactive cobalt- 60, was supported in a cerium magnesium nitrate housing within an evacuated glass vessel (lower half of photograph). An anthracene crystal about 2 cm above the cobalt-60 source served as a scintillation counter for beta-ray detection. Lucite rod (upper half of photograph) transmitted flashes from the counter to a photomultiplier (not shown). Magnet on either side of the specimen was used to cool it to approximately 0.003 K by adiabatic demagnetization. Inductance coil is part of a magnetic thermometer for determining specimen temperature. B

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Sept 28-29, 200524 Weak Force breaks C and P, is CP really OK ? Weak Interaction breaks both C and P symmetry maximally! Despite the maximal violation of C and P symmetry, the combined operation, CP, seemed exactly conserved… But, in 1964, Christensen, Cronin, Fitch and Turlay observed CP violation in decays of Neutral Kaons! W+W+ e+Re+R L W+W+ e+Le+L R WW eReR L WW eLeL R P C

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Sept 28-29, 200525 Testing CP conservation Create a pure K L (CP=-1) beam: (Cronin & Fitch in 1964) Easy: just “wait” until the K s component has decayed… If CP conserved, should not see the decay K L → 2 pions Main background: K L -> + - 0 K2+-K2+- Effect is tiny: about 2/1000 … and for this experiment they got the Nobel price in 1980…

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Sept 28-29, 200526 Contact with Aliens ! Are they made of matter or anti-matter?

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Sept 28-29, 200527 Contact with Aliens ! CPLEAR, Phys.Rep. 374(2003) 165-270 Compare the charge of the most abundantly produced electron with that of the electrons in your body: If equal: anti-matter If opposite: matter

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Sept 28-29, 200528 CPT Invariance Local Field theories always respect: Lorentz Invariance Symmetry under CPT operation (an electron = a positron travelling back in time) => Consequence: mass of particle = mass of anti-particle: Question 1: The mass difference between K L and K S : m = 3.5 x 10 -6 eV => CPT violation? Question 2: How come the lifetime of K S = 0.089 ns while the lifetime of the K L = 51.7 ns? Question 3: BaBar measures decay rate B→J/ K S and B →J/ K S. Clearly not the same: how can it be? => Similarly the total decay-rate of a particle is equal to that of the anti-particle Answer 3: Partial decay rate ≠ total decay rate! However, the sum over all partial rates (>200 or so) is the same for B and B. (Amazing! – at least to me) Answer 1 + 2: A K L ≠ an anti- K S particle! (Lüders, Pauli, Schwinger) (anti-unitarity)

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CPT Violation… 18-12-200729

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Sept 28-29, 200530 CP Violation in the Standard Model Topical Lectures Nikhef Dec 12, 2007 Marcel Merk Part 1: Introduction: Discrete Symmetries Part 2: The origin of CP Violation in the Standard Model Part 3: Flavour mixing with B decays Part 4: Observing CP violation in B decays

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Sept 28-29, 200531 CP in the Standard Model Lagrangian (The origin of the CKM-matrix)

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Sept 28-29, 200532 L SM contains: L Kinetic : fermion fields L Higgs : the Higgs potential L Yukawa : the Higgs – fermion interactions Plan: Look at symmetry aspects of the Lagrangian How is CP violation implemented? → Several “miracles” happen in symmetry breaking CP in the Standard Model Lagrangian (The origin of the CKM-matrix) Standard Model gauge symmetry: Note Immediately: The weak part is explicitly parity violating Outline: Lorentz structure of the Lagrangian Introduce the fermion fields in the SM L Kinetic : local gauge invariance : fermions ↔ bosons L Higgs : spontaneous symmetry breaking L Yukawa : the origin of fermion masses V CKM : CP violation

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Sept 28-29, 200533 Lagrangian Density Local field theories work with Lagrangian densities: The fundamental quantity, when discussing symmetries is the Action: If the action is (is not) invariant under a symmetry operation then the symmetry in question is a good (broken) one => Unitarity of the interaction requires the Lagrangian to be Hermitian with the fields taken at

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Sept 28-29, 200534 Structure of a Lagrangian Example: Consider a spin-1/2 (Dirac) particle (“nucleon”) interacting with a spin-0 (Scalar) object (“meson”) Meson potential Nucleon field Nucleon – meson interaction Lorentz structure: interactions can be implemented using combinations of: S: Scalar currents : 1 P: Pseudoscalar currents : 5 V: Vector currents : A: Axial vector currents : 5 T: Tensor currents : Dirac field Exercise: What are the symmetries of this theory under C, P, CP ? Can a and b be any complex numbers? Note: the interaction term contains scalar and pseudoscalar parts Scalar field Violates P, conserves C, violates CP a and b must be real from Hermeticity

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Sept 28-29, 200535 Transformation Properties Transformation properties of Dirac spinor bilinears (interaction terms): (Ignoring arbitrary phases) c→c *

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Sept 28-29, 200536 The Standard Model Lagrangian L Kinetic : Introduce the massless fermion fields Require local gauge invariance => gives rise to existence of gauge bosons L Higgs : Introduce Higgs potential with ≠ 0 Spontaneous symmetry breaking L Yukawa : Ad hoc interactions between Higgs field & fermions L Yukawa → L mass : fermion weak eigenstates: -- mass matrix is (3x3) non-diagonal fermion mass eigenstates: -- mass matrix is (3x3) diagonal L Kinetic in mass eigenstates: CKM – matrix The W +, W -,Z 0 bosons acquire a mass => CP Conserving => CP violating with a single phase => CP-violating => CP-conserving! => CP violating with a single phase

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Sept 28-29, 200537 Fields: Notation SU(3) C SU(2) L Y Left- handed generation index Interaction rep. Quarks: Leptons: Scalar field: Q = T 3 + Y Under SU2: Left handed doublets Right hander singlets Note: Interaction representation: standard model interaction is independent of generation number Fermions: with = Q L, u R, d R, L L, l R, R

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Sept 28-29, 200538 Fields: Notation Explicitly: Similarly for the quark singlets: And similarly the (charged) singlets: The left handed leptons: The left handed quark doublet : Q = T 3 + Y

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Sept 28-29, 200539 Intermezzo: Local Gauge Invariance in a single transparancy Basic principle: The Lagrangian must be invariant under local gauge transformations Example: massless Dirac Spinors in QED: “global” U(1) gauge transformation: “local” U(1) gauge transformation: Is the Lagrangian invariant? Then: Not invariant! => Introduce the covariant derivative: and demand that transforms as: Then it turns out that: Introduce charged fermion field (electron) Demand invariance under local gauge transformations (U(1)) The price to pay is that a gauge field A must be introduced at the same time (the photon) is invariant! Conclusion :

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Sept 28-29, 200540 Fermions + gauge bosons + interactions Procedure: Introduce the Fermion fields and demand that the theory is local gauge invariant under transformations. Start with the Dirac Lagrangian: Replace: Fields: Generators: a : G a : 8 gluons W b : weak bosons: W 1, W 2, W 3 B : B : hypercharge boson L a : Gell-Mann matrices: ½ a (3x3) SU(3) C T b : Pauli Matrices: ½ b (2x2) SU(2) L Y : Hypercharge: U(1) Y :The Kinetic Part For the remainder we only consider Electroweak: SU(2) L x U(1) Y

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Sept 28-29, 200541 and similarly for all other terms ( u Ri I,d Ri I,L Li I,l Ri I ). : The Kinetic Part Exercise: Show that this Lagrangian formally violates both P and C Show that this Lagrangian conserves CP For example the term with Q Li I becomes: Writing out only the weak part for the quarks: uLIuLI dLIdLI g W+W+ L Kin = CP conserving W + = (1/√2) (W 1 + i W 2 ) W - = (1/√ 2) (W 1 – i W 2 ) L= J W

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Sept 28-29, 200543 : The Higgs Potential →Note L Higgs = CP conserving V V( ) SymmetryBroken Symmetry ~ 246 GeV Spontaneous Symmetry Breaking: The Higgs field adopts a non-zero vacuum expectation value Procedure:Substitute: And rewrite the Lagrangian (tedious): “The realization of the vacuum breaks the symmetry” (The other 3 Higgs fields are “eaten” by the W, Z bosons) 1.. 2.The W +,W -,Z 0 bosons acquire mass 3.The Higgs boson H appears

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Sept 28-29, 200544 : The Yukawa Part Since we have a Higgs field we can add (ad-hoc) interactions between and the fermions in a gauge invariant way. L must be Her- mitian (unitary) The result is: are arbitrary complex matrices which operate in family space (3x3) => Flavour physics! doublets singlet With : (The C-conjugate of To be manifestly invariant under SU(2) )

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Sept 28-29, 200545 : The Yukawa Part Writing the first term explicitly: Question: In what aspect is this Lagrangian similar to the example of the nucleon-meson potential? For For the nucleon potential we had an interaction term:

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Sept 28-29, 200546 : The Yukawa Part Exercise (intuitive proof) Show that: The hermiticity of the Lagrangian implies that there are terms in pairs of the form: However a transformation under CP gives: and leaves the coefficients Y ij and Y ij * unchanged CP is conserved in L Yukawa only if Y ij = Y ij * In general L Yukawa is CP violating Formally, CP is violated if:

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Sept 28-29, 200547 : The Yukawa Part There are 3 Yukawa matrices (in the case of massless neutrino’s): Each matrix is 3x3 complex: 27 real parameters 27 imaginary parameters (“phases”) many of the parameters are equivalent, since the physics described by one set of couplings is the same as another It can be shown (see ref. [Nir]) that the independent parameters are: 12 real parameters 1 imaginary phase This single phase is the source of all CP violation in the Standard Model ……Revisit later

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Sept 28-29, 200548 So far, so good…?

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Hope not… 18-12-200749

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Sept 28-29, 200550 : The Fermion Masses S.S.B Start with the Yukawa Lagrangian After which the following mass term emerges: with L Mass is CP violating in a similar way as L Yuk is vacuum expectation value of the Higgs potential

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Sept 28-29, 200551 : The Fermion Masses S.S.B Writing in an explicit form: The matrices M can always be diagonalised by unitary matrices V L f and V R f such that: Then the real fermion mass eigenstates are given by: d L I, u L I, l L I are the weak interaction eigenstates d L, u L, l L are the mass eigenstates (“physical particles”)

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Sept 28-29, 200552 : The Fermion Masses S.S.B In terms of the mass eigenstates: = CP Conserving? In flavour space one can choose: Weak basis: The gauge currents are diagonal in flavour space, but the flavour mass matrices are non-diagonal Mass basis: The fermion masses are diagonal, but some gauge currents (charged weak interactions) are not diagonal in flavour space In the weak basis : L Yukawa = CP violating In the mass basis: L Yukawa → L Mass = CP conserving =>What happened to the charged current interactions (in L Kinetic ) ?

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Sept 28-29, 200553 : The Charged Current The charged current interaction for quarks in the interaction basis is: The charged current interaction for quarks in the mass basis is: The unitary matrix: is the Cabibbo Kobayashi Maskawa mixing matrix: With: Lepton sector: similarly However, for massless neutrino’s: V L = arbitrary. Choose it such that V MNS = 1 => There is no mixing in the lepton sector

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Sept 28-29, 200554 Use to find in general : Flavour Changing Neutral Currents To illustrate the SM neutral current take W 3 and B term of the Kinetic Lagrangian: In terms of physical fields no non- diagonal contributions occur for the neutral Currents. => GIM mechanism And consider the Z-boson field: Take further Q Li I =d Li I : Standard Model forbids flavour changing neutral currents. and

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Sept 28-29, 200555 Charged Currents A comparison shows that CP is conserved only if V ij = V ij * (Together with (x,t) -> (-x,t)) The charged current term reads: Under the CP operator this gives: In general the charged current term is CP violating

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Sept 28-29, 200556 Where were we?

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Sept 28-29, 200557 The Standard Model Lagrangian (recap) L Kinetic : Introduce the massless fermion fields Require local gauge invariance => gives rise to existence of gauge bosons L Higgs : Introduce Higgs potential with ≠ 0 Spontaneous symmetry breaking L Yukawa : Ad hoc interactions between Higgs field & fermions L Yukawa → L mass : fermion weak eigenstates: -- mass matrix is (3x3) non-diagonal fermion mass eigenstates: -- mass matrix is (3x3) diagonal L Kinetic in mass eigenstates: CKM – matrix The W +, W -,Z 0 bosons acquire a mass => CP Conserving => CP violating with a single phase => CP-violating => CP-conserving! => CP violating with a single phase

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Sept 28-29, 200558 Quark field re-phasing Under a quark phase transformation: and a simultaneous rephasing of the CKM matrix: or the charged currentis left invariant. Degrees of freedom in V CKM in 3 N generations Number of real parameters: 9 + N 2 Number of imaginary parameters: 9 + N 2 Number of constraints ( VV † = 1): -9 - N 2 Number of relative quark phases: -5 - (2N-1) ----------------------- Total degrees of freedom: 4 (N-1) 2 Number of Euler angles: 3 N (N-1) / 2 Number of CP phases: 1 (N-1) (N-2) / 2 No CP violation in SM! This is the reason Kobayashi and Maskawa first suggested a third family of fermions! 2 generations: Exercise: Convince yourself that there are indeed 5 relative quark phases

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Sept 28-29, 200559 The LEP collider @ CERN Maybe the most important result of LEP: “There are 3 generations of neutrino’s” L3L3 Aleph Opal Delphi Geneva Airport “Cointrin” MZMZ Light, left-handed, “active”

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Sept 28-29, 200560 The lepton sector (Intermezzo) N. Cabibbo: Phys.Rev.Lett. 10, 531 (1963) – 2 family flavour mixing in quark sector (GIM mechanism) M.Kobayashi and T.Maskawa, Prog. Theor. Phys 49, 652 (1973) – 3 family flavour mixing in quark sector Z.Maki, M.Nakagawa and S.Sakata, Prog. Theor. Phys. 28, 870 (1962) – 2 family flavour mixing in neutrino sector to explain neutrino oscillations! In case neutrino masses are of the Dirac type, the situation in the lepton sector is very similar as in the quark sector: V MNS ~ V CKM. –There is one CP violating phase in the lepton MNS matrix In case neutrino masses are of the Majorana type (a neutrino is its own anti-particle → no freedom to redefine neutrino phases) –There are 3 CP violating phases in the lepton MNS matrix However, the two extra phases are unobservable in neutrino oscillations –There is even a CP violating phase in case N dim = 2

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Sept 28-29, 200561 Lepton mixing and neutrino oscillations In the CKM we write by convention the mixing for the down type quarks; in the lepton sector we write it for the (up-type) neutrinos. Is it relevant? –If yes: why? –If not, why don’t we measure charged lepton oscillations rather then neutrino oscillations? However, observation of neutrino oscillations is possible due to small neutrino mass differences. Question: e e, e

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Sept 28-29, 200562 Rephasing Invariants Simplest: U i = |V i | 2 is independent of quark re-phasing Next simplest: Quartets: Q i j = V i V j V j * V i * with ≠ and i≠j – “Each quark phase appears with and without *” V † V=1 : Unitarity triangle: V ud V cd * + V us V cs * + V ub V cb * = 0 – Multiply the equation by V us * V cs and take the imaginary part: – Im (V us * V cs V ud V cd * ) = - Im (V us * V cs V ub V cb * ) – J = Im Q udcs = - Im Q ubcs – The imaginary part of each Quartet combination is the same (up to a sign) – In fact it is equal to 2x the surface of the unitarity triangle Im[V i V j V j * V i *] = J ∑ ijk where J is universal Jarlskog invariant The standard representation of the CKM matrix is: However, many representations are possible. What are invariants under re-phasing? Amount of CP violation is proportional to J

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Sept 28-29, 200564 unitarity: The Unitarity Triangle V cd V cb * V td V tb * V ud V ub * Under re-phasing:the unitary angles are invariant (In fact, rephasing implies a rotation of the whole triangle) Area = ½ | Im Q udcb | = ½ | J | The “ db ” triangle: V CKM † V CKM = 1

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Sept 28-29, 200565 Wolfenstein Parametrization Wolfenstein realised that the non-diagonal CKM elements are relatively small compared to the diagonal elements, and parametrized as follows: Normalised CKM triangle: (0,0)(1,0)

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Sept 28-29, 200566 CP Violation and quark masses Note that the massless Lagrangian has a global symmetry for unitary transformations in flavour space. Let’s now assume two quarks with the same charge are degenerate in mass, eg.: m s = m b Redefine: s’ = V us s + V ub b Now the u quark only couples to s’ and not to b’ : i.e. V 13 ’ = 0 Using unitarity we can show that the CKM matrix can now be written as: CP conserving Necessary criteria for CP violation:

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Sept 28-29, 200567 The Amount of CP Violation However, also required is: All requirements for CP violation can be summarized by: (The maximal value J might have = 1/(6√3) ~ 0.1) Using Standard Parametrization of CKM: Is CP violation maximal? => One has to understand the origin of mass! (eg.: J=Im(V us V cb V ub * V cs * ) )

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Sept 28-29, 200568 Mass Patterns Mass spectra ( = M z, MS-bar scheme) m u ~ 1 - 3 MeV, m c ~ 0.5 – 0.6 GeV, m t ~ 180 GeV m d ~ 2 - 5 MeV, m s ~ 35 – 100 MeV, m b ~ 2.9 GeV Why are neutrino’s so light? Is it related to the fact that they are the only neutral fermions? See-saw mechanism? m e = 0.51 MeV, m = 105 MeV, m = 1777 MeV Do you want to be famous? Do you want to be a king? Do you want more then the nobel prize? - Then solve the mass Problem – R.P. Feynman Observe:

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Indian Yoga… 18-12-200769

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Russian Yoga… 18-12-200770

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Sept 28-29, 200571 CP Violation in the Standard Model Topical Lectures Nikhef Dec 12, 2007 Marcel Merk Part 1: Introduction: Discrete Symmetries Part 2: The origin of CP Violation in the Standard Model Part 3: Flavour mixing with B decays Part 4: Observing CP violation in B decays

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Sept 28-29, 200572 Before we saw how CP violation can be embedded in the CKM sector of the SM. Next we investigate how we can observe CP violation and measure the relevant parameters in the CKM description. Although CP violation was first observed with Kaons I will focus on B mesons. Again, questions are welcome.

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Sept 28-29, 200573 Back to serious business… Mixing in the neutral B decay system

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Sept 28-29, 200574 Dynamics of Neutral B (or K) mesons… Time evolution of B 0 and B 0 can be described by an effective Hamiltonian: No mixing, no decay… No mixing, but with decays… (i.e.: H is not Hermitian!) With decays included, probability of observing either B 0 or B 0 must go down as time goes by:

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Sept 28-29, 200575 As can be easily seen…. Since the particles decay:

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Sept 28-29, 200576 Describing Mixing… Time evolution of B 0 and B 0 can be described by an effective Hamiltonian: Where to put the mixing term? Now with mixing – but what is the difference between M 12 and 12 ? M 12 describes B 0 B 0 via off-shell states, e.g. the weak box diagram 12 describes B 0 f B 0 via on- shell states, eg. f= For details, look up “Wigner-Weisskopf” approximation…

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Sept 28-29, 200577 Solving the Schrödinger Equation From the eigenvalue calculation: Eigenvectors: m and follow from the Hamiltonian: Solution: ( and are initial conditions):

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Sept 28-29, 200578 B Oscillation Amplitudes For B 0, expect: ~ 0, |q/p|=1 For an initially produced B 0 or a B 0 it then follows: using: with

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Sept 28-29, 200579 Measuring B Oscillations Decay probability B0B0B0B0 B0B0B0B0 Proper Time For B 0, expect: ~ 0, |q/p|=1 Examples:

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Sept 28-29, 200580 Time Integrated B Mixing Decay probability B0B0B0B0 B0B0B0B0 Proper Time

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Sept 28-29, 200581 Computing m d : Mixing Diagrams Dominated by top quark mass: GIM(i.e. V CKM unitarity): if u,c,t same mass, everything cancels!

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Sept 28-29, 200582 B 0 B 0 Mixing: ARGUS, 1987 First hint of a really large m top ! Produce an b b bound state, (4S), in e + e - collisions: e + e - (4S) B 0 B 0 and then observe: ~17% of B 0 and B 0 mesons oscillate before they decay m ~ 0.5/ps, B ~ 1.5 ps Integrated luminosity 1983-87: 103 pb -1

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Sept 28-29, 200583 Exercise Given the reconstructed tracks and their particle ID, how many ways can you determine the flavour of B 1 ? And of B 2 ?

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Sept 28-29, 200584 Discovery of the Top Quark: CDF & D0, 1994/5 m t ~174 GeV

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Sept 28-29, 200585 Time Dependent Mixing Asymmetry imperfect flavour tagging dilutes the asymmetry! 1. Tag flavour at decay by reconstructing self-tagging mode, eg. B 0 D* - + or B 0 D* + - 2. Tag initial flavour with leptons, kaons, … from decay of recoiling B with a mistag probability ‘w’ B 0 oscillates fully in ~4 lifetimes m d = 0.502 +- 0.006 ps

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Sept 28-29, 200586 m d Measurements in Comparison Many measurements Dominated by B factories Theoretical hadronic uncertainties limit extraction of |V td | (PDG 2000) Either: Need lattice QCD computations to improve Or: get more from data eg. measure Br(B + + )

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Sept 28-29, 200587 D 0 meson K 0 meson B 0 mesonB s meson Q: Why does the B s oscillate so much faster than the B 0 ? Q: why has D 0 meson Oscillations not been observed? (or, why does it oscillate so extremely slow?) Q: do you expect any other (neutral) mesons to mix? Blue line: given a P 0, at t=0, the probability of finding a P 0 at t. Red Line: given a P 0, at t=0, the probability of finding a P 0 bar at t. Summary of Neutral Meson Mixing (Vts/Vtd) (Box diagram)

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Sept 28-29, 200588 B d mixing vs. B s mixing A more precise calculation leads to the SM expectation of ~18/ps Determined with “Amplitude scan”: For each value of m fit the value of parameter A:

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Sept 28-29, 200589 CP violation in Mixing? gV cb * gV cb t=0 t ?

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Sept 28-29, 200590 CP Violation in B 0 Mixing As expected, no asymmetry is observed… Look for a like-sign lepton Asymmetry with inclusive Dilepton events: Acp = (0.5±1.2(stat) ±1.4(syst) )% |q/p| = 0.998 ±0.006(stat)±0.007(syst)

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Sept 28-29, 200591 This was different in the case of Kaons… CPLEAR, Phys.Rep. 374(2003) 165-270 Kaons: CP Violation in mixing B-mesons: no CP Violation in mixing

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Sept 28-29, 200592 CP Violation in the Standard Model Topical Lectures Nikhef Dec 12, 2007 Marcel Merk Part 1: Introduction: Discrete Symmetries Part 2: The origin of CP Violation in the Standard Model Part 3: Flavour mixing with B decays Part 4: Observing CP violation in B decays

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It’s all about imaginary numbers 18-12-200793

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Sept 28-29, 200594 CP violation in neutral B decays

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Sept 28-29, 200595 B Decay Rate Look at a particular decay of a B into a final state f and calculate the decay rate: We had already from solving Schrodinger’s equation With: Introduce notation:

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Sept 28-29, 200596 Master Formula for B decay Writing it out gives the master equation for B decays valid for all neutral B decays: With:

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Sept 28-29, 200597 Master Formula for B decay Often they are also written in an alternative form: with (Only for final state f )

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Sept 28-29, 200598 CP Observables CP A single phase can not lead to an observable effect… Need a bit more! Reflecting a quark process involving W bosons in the CP mirror induces a CP-violating phase shift in the transition amplitude

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Sept 28-29, 200599 Observing CP violation CP-violating asymmetries can be observed from interference of two amplitudes with relative CP-violating phases But additional requirements exist to observe a CP asymmetry! Example: process B f via two amplitudes a 1 + a 2 = A. weak phase diff. 0, no CP-invariant phase diff. BfBfBfBf |A|=|A| No observable CP asymmetry

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Sept 28-29, 2005100 Observing CP violation BfBf A A BfBf a1a1 a1a1 a2a2 a2a2 A=a 1 +a 2 ++ -- |A|=|A| No observable CP asymmetry CP-violating asymmetries can be observed from interference of two amplitudes with relative CP-violating phases But additional requirements exist to observe a CP asymmetry! Example: process B f via two amplitudes a 1 + a 2 = A. weak phase diff. 0, no CP-invariant phase diff.

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Sept 28-29, 2005101 Example: process B f via two amplitudes a 1 + a 2 = A. weak phase diff. 0, CP-invariant phase diff. 0 Observing CP violation BfBfBfBf

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Sept 28-29, 2005102 Observing CP violation Example: process B f via two amplitudes a 1 + a 2 = A. weak phase diff. 0, CP-invariant phase diff. 0 Interference between two decay amplitudes gives decay time independent observable CP violated if BF(B f) ≠ BF(B f) CP-invariant phases provided by strong interaction part. Strong phases usually unknown this can complicate things… |A||A| Need also CP-invariant phase for observable CP violation BfBfBfBf A=a 1 +a 2 ++ -- a1a1 a1a1 a2a2 a2a2 A A

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Sept 28-29, 2005103 Three types of CP violation (always two amplitudes!): 1.CP violation in mixing (“indirect” CP violation): 2.CP violation in decay (“direct” CP violation): 3.CP violation in the interference: Classification of CP violation CP violating phase is imbedded in the parameter: Note that in the SM all these effects are caused by a single complex parameter in the CKM matrix!

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Sept 28-29, 2005104

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Sept 28-29, 2005105 Assume no CP in mixing and no CP in decay B decay into CP eigenstates Consider final states that are CP eigenstates: An example is the decay: In this case we have: To observe CP just fill in the Master equation and require:

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Sept 28-29, 2005106 Master Formula for B decay (recap) with

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Sept 28-29, 2005107 Combining Mixing and Decay t=0 t Amplitude CP For neutral B mesons, g - has a 90 o phase difference wrt. g +

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Sept 28-29, 2005108 Interfering Amplitudes t=0 t Amplitude

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Sept 28-29, 2005109 Interfering Amplitudes t=0 t Amplitude re im

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Sept 28-29, 2005110 Interfering Amplitudes t=0 t Amplitude mt/2=0 mt/2= mt/2= mt/2=3 Time Dependent CP Asymmetry!!!

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Sept 28-29, 2005111 EXERCISE The decay rate is obtained by taking the absolute square of the amplitude. To get the result, show that:

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Let’s expand a bit further… 18-12-2007112

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Sept 28-29, 2005113 Decays to CP Eigenstates t=0 t Amplitude

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Sept 28-29, 2005114 Time dependent CP asymmetries for B f CP Probe of “direct CP violation” since it requires Sensitive to the phase of even without direct CP Violation Finally we get:

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Sept 28-29, 2005115 As expected from Master Equation… with Putting =0 immediately yields the just derived result

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Sept 28-29, 2005116 Babar and Belle BaBar: 113 fb -1 Belle: 140 fb -1

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Sept 28-29, 2005117 Golden Decay Mode: B 0 → J/ K 0 S c J/ K0K0K0K0 B0B0B0B0 b c s d d K0K0K0K0 B0B0B0B0 c d s c b d b d d b Time-dependent CP asymmetry => See next lectures

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Concluding Remarks and Outlook 18-12-2007118 Remember the classical double slit experiment of Young and Feynman’s quantum equivalent…

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6-sept-2007Nikhef-evaluation119 “slit A”: A Quantum Interference B-experiment pp at LHCb: 100 kHz bb Decay time BsBs Ds-Ds- “slit B”: Measure decay time

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6-sept-2007Nikhef-evaluation120 CP Violation: matter – antimatter asymmetry BsBs DsDs An interference pattern: Decay time Decay time

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6-sept-2007Nikhef-evaluation121 CP Violation: matter – antimatter asymmetry BsBs Ds+Ds+ BsBs DsDs Matter Antimatter CP-mirror: Observation of CP Violation is a consequence of quantum interference!! Decay time Decay time An interference pattern:

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6-sept-2007Nikhef-evaluation122 Searching for new virtual particles (example: B s → J ) Standard Model BsBs J/ Standard Model Decay time

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6-sept-2007Nikhef-evaluation123 Searching for new virtual particles (example: B s → J ) Standard Model New Physics BsBs J/ Decay time

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6-sept-2007Nikhef-evaluation124 Searching for new virtual particles (example: B s → J ) Standard Model New Physics Mission: To search for new particles and interactions that affect the observed matter-antimatter asymmetry in Nature, by making precision measurements of B-meson decays. B->J/ BsBs J/ Search for a CP asymmetry: Decay time

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Outlook to the rest of the lectures How can the CKM angles be determined and how are they sensitive to physics beyond the Standard Model? What role does the CKM CP Violation mechnism play in Baryogenesis? An outlook to the LHCb experiment: B-physics at the LHC collider 18-12-2007130

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Sept 28-29, 2005131 The Quest for CP Violation in the B System ATLASATLASBTEV B A B AR BELLE 2009 1999 2007 2002 Mission Statement Obtain precision measurements in the domain of the charged weak interactions for testing the CKM sector of the Standard Model, and probing the origin of the CP violation phenomenon

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Exercise 18-12-2007132

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