CP violation lectures

Transcription

1 CP violation lectures MARCELLA BONA University of London Intercollegiate Postgraduate Course in Elementary Particle Physics Lecture 1

2 Outline essential symmetries: P, C, T, CP and CPT brief introduction to flavour physics and to B physics some basic concepts from the past: analysis strategies in electron colliders (B factory experiments) oscillations (the experimental view) 2

3 Essential Symmetries See the Symmetries and Conservation Laws course for more on this topic! More mathematically rigorous treatments of these symmetries are given in Vol I of Weinberg. 3

4 What do we mean by conservation/violation of a symmetry? Define a quantum mechanical operator O. If O describes a good symmetry: Physics looks the same before and after applying the symmetry i.e. the observed quantity associated with O is conserved (same before and after the operator is applied). e.g. conservation of energy momentum etc. If this condition is not met the symmetry is broken. That is, the symmetry is not respected by nature. So O is (at best) a mathematical tool used to help our understanding of nature. Slightly broken symmetries (like isospin in EW interactions) can be very useful)! e.g. Isospin symmetry assumes that mu=md. In doing so we can estimate branching fractions where the final state differs by a 0 vs a etc. The difference comes from a Clebsch Gordan coefficient. 4

5 Parity: P Reflection through a mirror, followed by a rotation of around an axis defined by the mirror plane. Space is isotropic, so we care if physics is invariant under a mirror reflection. P is violated in weak interactions: [P, HW] 0 Vectors change sign under a P transformation, pseudo vectors or axial vectors do not. P is a unitary operator: P 2=1. T. D. Lee & G. C. Wick Phys. Rev. 148 p1385 (1966) showed that there is no operator P that adequately represents the parity operator in QM. 5

6 Parity: P Equivalent to a reflection in an x, y mirror plus a rotation through 180 about the z axis 6

7 Parity: P In 1956 if was found that β decay violates parity conservation. It was subsequently found that all weak decays violate parity conservation In the decay of nuclei with spins aligned in a strong magnetic field and cooled to 0.01 K J=5 J=4 J=½ J=½ Spin direction It was found that electrons were emitted predominantly in direction (a) opposite the 60Co spin. If parity were conserved one would expect (a) and (b) equally 7

8 Parity: P Spin and momentum of electron are opposite directions Mirror Spin and momentum of electron are the same direction 8

9 Aside: helicity Signed projection of a particle s spin along the direction of its momentum: h ss. p p s p p P (h) = h C (h) = h T (h) = h h { 1, +1}

10 Parity: P This is understood as interference between the Vector (V) and Axial Vector (A) parts of the Weak Interaction p, r etc are Vectors p -p under mirror transformations Spin s is an Axial Vector s s under mirror transformations p Any law depending on (Vector) (Axial Vector) will not conserve parity The Fermi Matrix Element MF MV MA Interference term gives parity violation 10

11 Charge Conjugation: C Change a quantum field into, where has opposite U(1) charges: baryon number, electric charge, lepton number, flavour quantum numbers like strangeness & beauty etc. Change particle into antiparticle. the choice of particle and antiparticle is just a convention. C is violated in weak interactions, so matter and antimatter behave differently, and: [C, HW] 0 C is a unitary operator: C 2=1. 11

12 Parity and Charge Conjugation: CP Note that Charge Conjugation C is also violated but CP is (usually) conserved Charge Conjugation C changes particle into antiparticle (3) is again a favourite configuration from the point of view of weak interaction, just like (1) was. 12

13 Parity and Charge Conjugation: CP The fundamental point is that CP symmetry is broken in any theory that has complex coupling constants in the Lagrangian which cannot be removed by any choice of phase redefinition of the fields in the theory. Weak interactions are left right asymmetric. It is not sufficient to consider C and P violation separately in order to distinguish between matter and antimatter. i.e. if helicity is negative (left) or positive (right). CP is a unitary operator: CP 2=1 13

14 Time reversal: T Not to be confused with the classical consideration of the entropy of a macroscopic system. Flips the arrow of time Reverse all time dependent quantities of a particle (momentum/spin). Complex scalars (couplings) transform to their complex conjugate. It is believe that weak decays violate T, but EM interactions do not. T is an anti unitary operator: T 2= 1. 14

15 Summary on discrete symmetries Three discrete operations are potential symmetries of a field theory Lagrangian. Two of them, parity and time reversal are space time symmetries. Parity sends (t; x) (t; x), reversing the handedness of space. Time reversal sends (t; x) ( t; x), interchanging the forward and backward light cones. A third (non space time) discrete operation is: Charge conjugation: it interchanges particles and anti particles. The operators associated to these symmetries have different properties: P and C operators are: unitary and thus they satisfy the relation UT = U 1 linear and thus U ( a + b ) = U a + U b. T operator is anti unitary, that means that is satisfies the unitary relation: AT = A 1 but it is anti linear: A ( a + b ) = A a + A b. 15

16 CPT All locally invariant Quantum Field Theories conserve CPT CPT CPT.1 is anti unitary: CPT 2= 1. can be violated by non local theories like quantum gravity. These are hard to construct. see work by Mavromatos, Ellis, Kostelecky etc. for more detail. If CPT is conserved, a particle and its antiparticle will have The same mass and lifetime. Symmetric electric charges. Opposite magnetic dipole moments (or gyromagnetic ratio for point like leptons). See Weinberg volume I and references therein (Lueders 1954) for a proof of this. 1 16

17 Broken symmetries The symmetry CPT is conserved in the Standard Model. The other symmetries introduced here are broken by some amount. CP violation has been seen in kaon and B meson decays. These symmetries are broken for weak interactions only! They are conserved (as far as we know) in strong and electromagnetic interactions. 17

18 Summary on CP and CPT The combination CPT is an exact symmetry in any local Lagrangian field theory: The CPT theorem is based on general assumptions of field theory and relativity and states that every Hamiltonian that is Lorentz invariant is also invariant under combined application of CPT, even if it is not invariant under C, P and T separately. One of the consequences of this theorem is that particles and anti particles should have exactly the same mass and lifetime. From experiment, it is observed that electromagnetic and strong interactions are symmetric with respect to C, P and T. The weak interactions violate C and P separately, but preserve CP and T to a good approximation. Only certain rare processes have been observed to exhibit CP violation. All these observations are consistent with exact CPT symmetry. 18

19 Examples The u quark has JP = ½+, so the P operator acting on u has an eigenvalue of +1. The C operator changes particle to antiparticle. CP u = u CP 0 = CP = ± The 0 has JPC = 0 +, so the minus sign comes from the parity operator acting on the 0 meson. The C operator changes particle to antiparticle. A 0 is its own antiparticle. The has JP = 0, so the minus sign comes from the parity operator acting on the meson. The C operator changes the particle to antiparticle. 19

20 What is flavour physics? The term flavor was first used in particle physics in the context of the quark model of hadrons. It was coined in 1971 by Murray Gell Mann and his student at the time, Harald Fritzsch, at a Baskin Robbins ice cream store in Pasadena. Just as ice cream has both color and flavor so do quarks. RMP 81 (2009)

21 Parameters of the Standard Model 3 gauge couplings 2 Higgs parameters Cabibbo Kobayashi Maskawa 6 quark masses 3 quark mixing angles + 1 phase CKM matrix 3 (+3) lepton masses PMNS matrix (3 lepton mixing angles + 1 phase) flavour parameters Pontecorvo Maki Nakagawa Sakata ( ) = with Dirac neutrino masses 21

22 What is heavy flavour physics? mu 3 MeV md 5 MeV ms 100 MeV mc 1300 MeV mb 4200 MeV mt MeV m MeV m MeV m MeV me 0.5 MeV m 100 MeV m 1800 MeV The neutrinos have their own phenomenology Studies of the u and d quarks are the realm of nuclear physics Rare decays of kaons provide sensitive tests of the SM Studies of electric and magnetic dipole moments of the leptons test the Standard Model Searches for lepton flavour violation are another hot topic The top quark has its own phenomenology (since it does not hadronise) 22

23 Heavy quark flavour physics Focus in these lectures will be on: flavour changing interactions of beauty quarks charm is also very interesting and I will mention it at the end of the course.. But quarks feel the strong interaction and hence hadronise: various different charmed and beauty hadrons many, many possible decays to different final states hadronisation greatly increases the observability of CP violation effects 23

24 Why is heavy flavour physics interesting? Hope to learn something about the mysteries of the flavour structure of the Standard Model CP violation and its connection to the matter antimatter asymmetry of the Universe Discovery potential far beyond the energy frontier via searches for rare or SM forbidden processes 24

25 What breaks the flavour symmetries? In the Standard Model, the vacuum expectation value of the Higgs field breaks the electroweak symmetry Fermion masses arise from the Yukawa couplings of the quarks and charged leptons to the Higgs field (taking m =0) The CKM matrix arises from the relative misalignment of the Yukawa matrices for the up and down type quarks Consequently, the only flavour changing interactions are the charged current weak interactions no flavour changing neutral currents (GIM mechanism) not generically true in most extensions of the SM flavour changing processes provide sensitive tests 25

26 What causes the difference between matter and antimatter? The CKM matrix arises from the relative misalignment of the Yukawa matrices for the up and down type quarks: It is a 3x3 complex unitary matrix described by 9 (real) parameters: 5 can be absorbed as phase differences between the quark fields 3 can be expressed as (Euler) mixing angles the fourth makes the CKM matrix complex (i.e. gives it a phase) weak interaction couplings differ for quarks and antiquarks CP violation 26

27 CKM matrix in the Standard Model flavour structure encoded in flavour: flavour in the SM. 27

28 Unitary matrix independent parameters in general, an n n unitary matrix has n2 real and independent parameters: a n n matrix would have 2n2 parameters the unitary condition imposes n normalization constraints n(n - 1) conditions from the orthogonality between each pair of columns: thus 2n2 - n - n(n - 1) = n2. In the CKM matrix, not all of these parameters have a physical meaning: given n quark generations, 2n - 1 phases can be absorbed by the freedom to select the phases of the quark fields Each u, c or t phase allows for multiplying a row of the CKM matrix by a phase, while each d, s or b phase allows for multiplying a column by a phase. thus: n2 - (2n - 1) = (n 1)2. Among the n2 real independent parameters of a generic unitary matrix: ½ n(n - 1) of these parameters can be associated to real rotation angles, so the number of independent phases in the CKM matrix case is: n2 - ½ n(n- 1) - (2n 1) = ½ (n 1)(n - 2) 28

29 CKM matrix parameterisations 29

30 CKM matrix parameterisations we'll talk about the fit at the end 30

31 CKM matrix parameterisations Cabibbo's angle 31

32 A brief history of.... Mixing, CP violation, B factories and Nobel prizes 1963: Cabibbo introduced his angle for the quark mixing with 2 families 1964: Christensen, Cronin, Fitch and Turlay discover CP violation in the K 0 system. 1967: A. Sakharov: 3 conditions required to generate a baryon asymmetry: Period of departure from thermal equilibrium in the early universe. Baryon number violation. C and CP violation. 1973: Kobayashi and Maskawa propose 3 generations 1980: Nobel Prize to Cronin and Fitch 1981: I. Bigi and A. Sanda propose measuring CP violation in B J/ K0 decays. 1987: P. Oddone realizes how to measure CP violation: convert the PEP ring into an asymmetric energy e+e collider. 1999: BaBar and Belle start to take data. By 2001 CP violation has been established (and confirmed) by measuring sin2 0 in B J/ K0 decays. 2008: Nobel Prize to Kobayashi and Maskawa 32

33 A brief history of.... Mixing, CP violation, B factories and Nobel prizes 1963: Cabibbo introduced his angle for the quark mixing with 2 families 1964: Christensen, Cronin, Fitch and Turlay discover CP violation in the K0 system. 1967: A. Sakharov: 3 conditions required to generate a baryon asymmetry: Period of departure from thermal equilibrium in the early universe. Baryon number violation. C and CP violation. 1973: Kobayashi and Maskawa propose 3 generation 1980: Nobel Prize to Cronin and Fitch 1981: I. Bigi and A. Sanda propose measuring CP violation in B J/ K0 decays. 1987: P. Oddone realizes how to measure CP violation: convert the PEP ring into an asymmetric energy e+e collider. 1999: BaBar and Belle start to take data. By 2001 CP violation has been established (and confirmed) by measuring sin2 0 in B J/ K0 decays. 2008: Nobel Prize to Kobayashi and Maskawa 33

34 Dynamic generation of baryon asymmetry Suppose equal amounts of matter ( X ) and antimatter ( X ): X decays to A (baryon number NA) with probability p B (baryon number NB) with probability (1 p) X decays to A (baryon number NA) with probability p B (baryon number NB) with probability (1 p) Generated baryon asymmetry: NTOT = NA p + NB (1 p) NA p NB (1 p) = (p p) (NA NB) NTOT 0 requires p p & NA NB 34

35 CP violation and the baryon asymmetry We can estimate the magnitude of the baryon asymmetry of the Universe caused by KM CP violation The Jarlskog parameter J is a parametrization invariant measure of CP violation in the quark sector: J ~ O(10 5) The mass scale M can be taken to be the electroweak scale O(100 GeV) This gives an asymmetry O(10 17): much much below the observed value of O(10 10) 35

36 We need more CP violation To create a larger asymmetry, require: new sources of CP violation that occur at high energy scales Where might we find it? lepton sector: CP violation in neutrino oscillations quark sector: discrepancies with KM predictions gauge sector, extra dimensions, other new physics: precision measurements of flavour observables are generically sensitive to additions to the Standard Model 36

37 Flavour for new physics discoveries A lesson from history: FCNC suppressed S=2 suppressed wrt S=1 New physics shows up at precision frontier before energy frontier GIM mechanism before discovery of charm CP violation / CKM before discovery of bottom & top Neutral currents before discovery of Z Particularly sensitive loop processes Standard Model contributions suppressed / absent flavour changing neutral currents (rare decays) CP violation lepton flavour / number violation / lepton universality we are going to talk a little about what we can do with current flavour data.. 37

38 a quick example from our recent past: B physics at the B factories 38

39 PEP II and KEKB PEP II 9 GeV e on 3.1 GeV e+ (4S) boost: = 0.56 KEKB 8 GeV e on 3.5 GeV e+ (4S) boost: =

40 Experimental technique l ( ) (4S) = 0.56 eb mesons pair oscillating in a coherent state Y(4S) z t c z ~200 m K+ Btag e+ + p+ Breco K+ K0S z K p+ p 40

41 BABAR and Belle BABAR IFR EMC e+ SOLENOID e The differences between the two detectors are small. Both have: Asymmetric design. Central tracking system Particle Identification System Electromagnetic Calorimeter Solenoid Magnet Muon/K0L Detection System High operation efficiency SVT DCH DIRC 41

42 What does an event look like? 0 0 B0 0 0 event 42

43 Producing B mesons Collide electrons and positrons at center of mass energy s = GeV/c2 (4S) off-peak many types of interaction occur. We re interested in e+e (4S) BB (for B physics). where we have 43

44 How do BB pairs behave? The B0 and B0 mesons from the (4S) are in a coherent L = 1 state: The (4S) is a bb state with JPC = 1. B mesons are scalars (JP = 0 ) total angular momentum conservation the BB pair has to be produced in a L = 1 state. The (4S) decays strongly so B mesons are produced in the two flavour eigenstates B0 and B0: After production, each B evolves in time, but in phase so that at any time there is always exactly one B0 and one B0 present, at least until one particle decays: If at a given time t one B could oscillate independently from the other, they could become a state made up of two identical mesons: but the L = 1 state is anti symmetric, while a system of two identical mesons (bosons!) must be completely symmetric for the two particle exchange. Once one B decays the other continues to evolve, and so it is possible to have events with two B or two B decays. 44

45 Neutral meson oscillation We have flavour eigenstates M0 and M0: M0 can be K0 (sd), D0 (cu), Bd0 (bd) or Bs0 (bs) flavour states (defined flavour) Heff eigenstates: (defined m1,2 and 1,2) if we consider only strong or electromagnetic interactions only, these flavour eigenstates would correspond to the physical ones However due to the weak interaction, the physical eigenstates are different from the flavour ones. This means that they can mix into each other: via short distance or long distance processes and then the flavour superposition decays M = p M0 ± q M0 45

46 Neutral meson oscillation (II) We have flavour eigenstates M0 and M0: M0 can be K0 (sd), D0 (cu), Bd0 (bd) or Bs0 (bs) flavour states (defined flavour) Heff eigenstates: (defined m1,2 and 1,2) Time dependent Schrödinger eqn. describes the evolution of the system: H is the hamiltonian; M and are 2x2 hermitian matrices ( aij = aji ) M = ½ (H+H ) and = i(h H ) CPT theorem: M11 = M22 and 11 = 22 particle and antiparticle have equal masses and lifetimes 46

47 Solving the Schrödinger equation Physical states: eigenstates of effective Hamiltonian: MS,L (or ML,H) = p M0 ± q M0 label can be either S,L (short, long lived) or L,H (light, heavy) depending on values of m & Γ (labels 1,2 usually reserved for CP eigenstates) p & q complex coefficients that satisfy p 2 + q 2 = 1 CP conserved if physical states = CP eigenstates ( q/p =1) Eigenvalues ( ) and mass ( m) and lifetime ( ) differences can be derived with this formalism: L,H = ml,h i/2 L,H = (M11 i/2 11) (q/p) (M12 i/2 12) m = mh ml and = H L other useful ( m)2 ¼ ( )2 = 4 ( M ¼ 12 2 ) definitions: m = 4 Re (M12 12*) x m/ 2 (q/p) = (M12* i/2 12*)/(M12 i/2 12) y /2 47

48 Simplistic picture of mixing parameters Δm: value depends on rate of mixing diagram together with various other constants... remaining factors can be obtained from lattice QCD calculations that can be made to cancel in ratios ΔΓ: value depends on widths of decays into common final states (CP eigenstates) large for K0, small for D0 & Bd0 q/p 1 if arg(γ12/m12) 0 q/p 1 if M12 << Γ12 or M12 >> Γ12 CP violation in mixing when q/p 1 in B system: << m negligible CP violation in mixing / = (usually set = 0) 48

49 Bd oscillations E signal region Method reconstructed flavour specific decays Method t distribution Af=0, <<1 measure t distribution 1 dγ P 0 f Δt =[ 1 1 2w cos xδt ] e R Δt 2 dδt A f 0 f dγ P 1 Δt =[ 1 1 2w cos xδt ] e R Δt 2 dδt A f : wrong tag probability (reduces ampl. of oscillations) R( t): resolution function intrinsic detector resolution on position of both B vertices smearing due to non primary tracks smearing due to B meson CMS momentum aver( t)=1.43 ps 49

50 Bd oscillations HFAG, d ( B 0 f ) / d t d ( B f ) / d t d ( B 0 f ) / d t d ( B 0 f ) / d t (1 2 w) cos( x t ) R( t ) md = (0.507 ± 0.005) ps 1 x = md Bd = ±

51 B oscillations B0 B0 transition box diagram at quark level so same phenomenology also in the Bs, except that the oscillation frequency is very different ~ B s0 probab. to observe an initially produced X0 as X0 after time t probab. to observe an initially produced X0 as X0 after time t 51

52 Bs oscillations At the Tevatron on the Bs: amplitude method, instead of extracting directly ms (à la LEP) 1 Af 2 d ( P 0 ( P 0 ) f ) [1 A(1 2 w) cos( x t )]e t d t fit A at different values of ms; if A=1 oscillations at this ms value Very precise determination from the Tevatron: ms=(17.77±0.10±0.07) ps 1 x= ms Bs= 25.5±0.6 52

53 B oscillations S = Inami Lim function for the t t contribution (from perturbative calculations) BBd and fbd from lattice QCD We can use these results to extract possible values for fundamental SM parameters: 2 of the parameters of the CKM matrix: and md ms/ md 53

54 backup 54

55 PEP II and KEKB performances X LHC: in

56 the luminosity emittance : it is a measurement of the area occupied by the beam in the phase space function: describes the oscillation of a particle around the nominal trajectory in the phase space, a bunch occupies a volume shaped like a tilted ellipse. the emittance measures the ellipse area, the beta function measures the amplitude in the spatial axis X (or Y) the relation with the beam dimension in X or Y: x=( x*emittanceh) 56

57 measuring the quantity of data instantaneous luminosity usually measured in cm 2s 1 (flux per unit area per unit time) integrating over time you are left with cm 2 usually we use the unity 1 barn = cm2 integrated luminosity is a measure of the quantity of accumulated data (inverse barn) multiplied with a given cross section gives the number of events of the given process N = L 57

58 Data 58

59 Time evolution and CP violation at the (4S): At (4S), the BB pairs are produced in coherent P wave Three interference effects can be observed: both neutral CP violation in the mixing ( q/p 1 ) (direct) CP violation in the decays ( A/A 1 ) CP violation in interference between mixing and decay ( Im 0 ) neutral B and charged B Time evolution of the BB system (assuming =0) direct CP violation CP violation in interference C 0 S 0 59

60 Direct CP violation: both neutral and charged B's tagging not always necessary (charged and "self tagging" modes) higher efficiency interference between (at least) two amplitudes contributing to the same final state the measured asymmetry becomes: i: strong phase CP even i: weak phase CP odd ACP interesting interesting modes: modes: + K K+ :: penguin+tree penguin+tree 0 K K0 ++:: pure pure penguin penguin 60

61 CP violation in interference between mixing and decay: B0 decays in final state f accessible to both a B or a B ( f is not necessarily a CP eigenstate) mixing t=0 Af CP ay c e d 0 B if Im 0 then CP violation t f Af t is is the the mixing mixing phase phase examples examples r r parameter 61

62 Measuring t + B0 e- e+ Fully reconstruct decay to state or admixture under study (BRECO) B0 Asymmetric energies produce boosted (4S), decaying into coherent BB pair 62

63 Measuring t + B0 e- e+ Fully reconstruct decay to state or admixture under study (BRECO) B0 Asymmetric energies produce boosted (4S), decaying into coherent BB pair = 0.56 (BaBar) = (Belle) t = t1 corresponds to the time that BTAG decays. Δz=(βγc)Δt Determine time between decays from vertices t=t2 t=t1 t2 t1= t 63

64 Measuring t + B0 e- e+ Fully reconstruct decay to state or admixture under study (BRECO) B0 Asymmetric energies produce boosted (4S), decaying into coherent BB pair = 0.56 (BaBar) = (Belle) t = t1 corresponds to the time that BTAG decays. Δz=(βγc)Δt Determine time between decays from vertices t=t2 KlDetermine flavour and vertex position of other B decay (BTAG) t=t1 t2 t1= t Then fit the t distribution to obtain the amplitude of sine and cosine terms. 64

65 Analysis strategy isolation of signal events: B candidate selection through kinematic variables ( E, mes) background fighting: contamination especially from continuum light quark production (signal to background ratios at the level of 10 3 or worse) through topological variables particle identification K/ separation is crucial also to isolate the signal flavour tagging identification of the flavour of the second (not completely reconstructed) B, called BTAG maximum likelihood fits rather than cut based analysis to take full advantage of the characteristic variables taking into account also their different shapes 65

66 Isolating signal events Beam energy is known very well at an e+e collider use an energy difference and effective mass to select events: ( E) ~ MeV (mode dependent) signal (MES) ~ 3 MeV/c2 B background 66

67 More background suppression Use the shape of an event to distinguish between (4S) BB and e+e qq. s=10.58 GeV: compare mbb = GeV/c2 with muu, mdd, mss ~ few to 100 MeV/c2 mcc ~ 1.25 GeV/c2 B pair events decay isotropically continuum (ee qq) events are jetty Analyses combine several event shape variables in a single discriminating variable: either Fisher or artificial Neural Network. This allows for some discrimination between B and continuum events Arbitrary Units Legendre Moments (L0, L2), Sphericity, cos B Signal u,d,s,c background Fisher Discriminant: F i xi i 67

68 On the control samples: DK reconstruction with D contamination. Continuum and BB backgrounds also included 68

69 Particle identification DIRC measurement of the Cherenkov angle K/ separation: 2 GeV/c and 4 GeV/c de/dx from SVT and DCH up to 0.7 GeV/c Kaon selection: efficiency ~90%, mis id <10% 69

70 Flavour tagging Decay products of BTAG are used to determine its flavour. At t=0, the flavour of BRECO is opposite to that of other BTAG. BRECO continues to mix until it decays. B0 B0 Different BTAG final states have different purities and different mis tag probabilities. Lepton Can (bottom) split information by physical category or (top) use a continuous variable to distinguish particle and anti particle. BaBar s flavour tagging algorithm splits events into mutually exclusive categories ranked by signal purity and mis tag probability. Belle opts to use a continuous variable output. Kaon 1 Kaon 2 Kaon-Pion 70

71 Flavour tagging Don t always identify BTAG flavour correctly: the mistakes dilute the asymmetry by a factor: = probability for assigning the wrong flavour (mistag probability). Effect is slightly different for B0 and B0 tags: 71

72 Flavour tagging Don t always identify BTAG flavour correctly: the mistakes dilute the asymmetry by a factor: = probability for assigning the wrong flavour (mistag probability). Effect is slightly different for B0 and B0 tags: Define an effective tagging efficiency: Use a modified f ( t): Example: The BaBar tagging algorithm: 72

73 Fitting for CP asymmetries and mixing parameters Perform an extended un binned ML fit in several dimensions (2 to 8). Pji is the probability density functions for the ith event and jth component (type) of event. nj is the event yield of the jth component. N is the total number of events. Usually replicate the likelihood for each tagging category (BaBar) or include a variable in the fit that incorporates flavour tagging information (Belle). In practice we minimize lnl in order to obtain the most probable value of our experimental observables with a 68.3% confidence level (1 error). md and/or S and/or C are observables that are allowed to vary when we fit the data. 73