B Physics Lecture 1 Steven Robertson

Transcription

1 B Physics Lecture 1 Canadian Institute of Particle Physics McGill University TRIUMF Summer Institute Vancouver B.C., July

2 Outline Lecture 1 Why study B physics? CKM formalism Parameterizations Unitarity triangles Types of CP violation Time dependent CP asymmetries Angles measurements β α γ Unitarity triangle constraints from angles 2

3 Why study B physics? Quark masses, mixing and CP violation all arise from a common source within the Standard Model Many new physics scenarios have implications for CP violation and/or the flavour sector in general E.g. SUSY may be found at the LHC (or elsewhere), but only sensitive to diagonal element of SUSY mass matrices -need off-diagonal elements to help understand the SUSY-breaking scenario Third generation couplings potentially sensitive to New Physics e.g. Non-SM Higgs sector Unitarity of CKM matrix implies relations between disparate experimental measurements Heavy b quark permits use of powerful computational methods such as HQET to give (sometimes!) robust theoretical predictions Historically, many important discoveries resulted from measurements in the CP/flavour sector 3

4 Standard Model Describes the interactions of particles via three of the four fundamental forces Matter Higgs Boson (spin ½) (spin 0) u c t +2/3 d s b -1/3 νe e νµ ντ µ τ 0-1 H0 Forces (spin 1) EM: photon ( γ ) Weak: Z0 W+ W- Strong: 8 gluons Particles interact through forces mediated by the exchange of (virtual) gauge bosons Weak interaction couples to weak eigenstates rather than mass eigenstates permitting quark flavour changing interactions between generations 4

5 In the SM, fermion masses arise from Yukawa interactions, which relate left-handed (quark) doublets (QIL) with right-handed singlets (dir, uir) via the Higgs field (Φ): Y are complex 3x3 matrices Electroweak symmetry breaking results in Higgs field aquiring a vacuum expectation value Results in Dirac mass matrices in flavour eigenstate basis 5

6 CKM matrix To move to mass eigenstate basis, need to diagonalize the mass matrices via a transformation: UL are unitary 3x3 matrices. Note that the u and d type matrices are diagonalized via different transformations. Define CKM matrix V as product of transformation matrices: 6

7 Charged current weak interaction has form: V parameterizes weak interaction couplings between the three quark generations Vud s Vus d u u K+ Weak eigenstates essentially rotated in flavour space relative to mass eigenstates π0 u u How many independent parameters does V contain? NxN complex matrix contains 2N2 degrees of freedom Requirement of unitarity reduces this to N2 parameters 2N-1 arbitrary phases associated with quark fields can be removed by a global phase rotation Thus V contains (N-1)2 free parameters π+ N(N-1)/2 angles and (N-1)(N-2)/2 phases Three generations 3 angles, 1 phase 7

8 Standard CKM parameterization Standard CKM parameterization gives V as a product of three complex rotation matrices with Euler angles θ12, θ13 and θ23 and a single overall phase δ Angles represent mixing between the three generations Matrix satisfies unitarity exactly by construction: Apparent hierarchy in mixing between the three generations motivates an alternative representation. Define: 8

9 Wolfenstein parameterization Substituting in standard representation gives an exact expression for V, but can alternatively expand in powers of λ: (Note truncation at order λ4 - this is sufficient for current experimental precision, but not necessary) Since the choice of parameterization is somewhat arbitrary, only phase-convention invariant quantities can have physical meaning: Combinations of Vij or various quadri-products Vij Vkl V*il V*kj Note that both λ and A in Wolfenstein parameterization are phase invariant, but ρ and η are not 9

10 Jarlskog Invariant The Jarlskog invariant is given by (various possible) quadriproducts of CKM matrix elements: In standard parameterization: J must be non-zero if the CKM phase is non-vanishing, and hence for CP violation to occur More precisely, a measure of CP violation is given by: Both J 0 and non-degenerate u type and d type quark masses required! 10

11 How big is the CP violation in the Standard Model? Mathematically, J has a maximum of ~0.1, hence CP violation really is small in a quantitative sense the observed hierarchy of CKM elements strongly suppresses CP violation 11

12 Unitarity Triangles Unitarity of V implies several relations between matrix elements: Three independent off-diagonal (j k) equations produce socalled unitarity triangles relating to K, Bd and Bs systems e.g.: Shape of triangle depends on relative sizes of matrix elements, but all triangles have equal areas, given by J/2 Im B mesons: Bs mesons: Re Kaons: 12

13 The Unitarity Triangle Bd triangle has all sides of comparable length By convention, triangle is usually rescaled so that one side is real and as unit length: Lengths of the sides are related to the magnitudes of CKM matrix elements 13

14 Angles Non-zero Jarlskog invariant implies that angles are not 0 or π Note that angles are unchanged by global phase rotation or scaling of sides Apex of the rescaled triangle is related to the Wolfenstein parameters (to all orders in λ): 14

15 Kaon and Bs triangles Can define analogous quantities for the triangles relating to the Kaon and Bs systems Kaons: Bs mesons: 15

16 C and P Parity (P) transforms a vector r = (x,y,z) into -r = (-x,-y,-z) equivalent to a mirror reflection followed by a rotation by π Define categories of mathematical objects based on their transformation properties: scalar, pseudoscalar, vector... E.g. helicity (projection of the spin of a particle along the direction of its momentum), is pseudoscalar since p is a vector but s is a pseudovector Charge conjugation take a field Φ to Φ with the opposite U(1) charges (no classical analogue) Electric charge, baryon and lepton number, flavour quantum numbers (e.g. strangeness ) etc. C and P both separately conserved by strong and EM interactions, and hence combined CP operation also conserved Parity maximally violated by weak interaction, but combined CP is almost conserved in weak interactions 16

17 CP violation CP violation arises due to the presence of complex phases in transition amplitudes Conceptually, CP violation T violation since CPT is conserved, but T is an antiunitary operator i.e. It changes quantities into their complex conjugates. Hence CP violation can occur when physical observables depend on a complex phase Overall, quantum mechanical phases of initial and final states are arbitrary and states can be rephased at will, hence the only physically meaningful phases are relative phases between coherent contributions to a particular transition amplitude CP violation only observable if two or more coherent phases are present weak phases (CP-odd) strong phases (CP-even) 17

18 Why study CP violation? Gauge field theories require CPT to be a good symmetry: CP violation CP violation is required in order to explain the observed Baryon Asymmetry of the Universe (BAU): (A. Sakharov, 1967) Time reversal (T) violation Baryon number violation Departure from thermal equilibrium C, CP violation (GUTS) (expanding universe)??? Weak interaction and CKM matrix is only known source of CP violation (non-zero neutrino masses implies similar matrix in lepton sector) CP studies are sensitive probe of possible new CP-violating phases associated with New Physics 18

19 Types of CP violation (1) CP violation in decay (aka direct CP violation): Compare the rate of a process with the rate of its CP conjugate process B0 Acp = N( B0 f ) N( B0 f ) N( B f ) + N( B f ) 0 0 A dec f CP ay f CP CP Requires at least two contributing amplitudes with non-zero relative weak and strong phases Predicted to be potentially large in the B system and observed in both kaon and B decays, but interpretation is model dependent due to dependence on strong phases 19

20 Direct CP violation in 0 B + Kπ Rate asymmetry between a decay and it CP conjugate mode Requires at least two contributing decay amplitudes carrying different weak and strong phases B0 K+π- occurs via both tree and penguin processes hence direct CP violation possible Total of candidates identified in 227M BB events It took 40 years to observe direct CP violation in kaon decays, but only 4 years in B decays! AKπ = (stat) (syst) 4.2σ hep-ex/

21 Types of CP violation (2) CP violation in mixing (aka indirect CP violation): Look for final states which cannot be directly produced by the initial flavour of the B t=0 Acp(t) = B0 N( B0(t) f ) N( B0(t) f ) CP N( B0(t) f ) + N( B0(t) f ) t ng decay f CP A f CP B0 Heavy (H) and light (L) mass eigenstates are linear combinations of * flavour eigenstates Vtd Vtb B mi xi 0 b t d d t b * Vtd B0 Vtb CP violation occurs if mass eigenstates are not CP eigenstates Well established in Kaon system but expected to be extremely small in Bd system (and even smaller in Bs system) 21

22 Types of CP violation (3) CP violation in interference between decays with and without mixing Can be observed in decays to final states which are accessible directly from both B and B initial states B mixing ~e-2iβ Can occur even in the absence of CP violation in mixing or decay Measure time-dependent asymmetry: 0 B 0 t=0 A f CP CP ay c e d t f CP A f CP 22

23 If final state is a CP eigenstate: If process is dominated by an amplitude with a single weak phase: Hence, amplitude of time dependent asymmetry directly related to CP violation! 23

24 Tree level decays In Bd system, gold-plated modes are tree-level decays to CP J/Ψ eigenstates containing charmonium c e.g. 0 0 B J/Ψ K S,L c s d b B0 d K0 (quark subprocess: b ccs) q/p from B mixing A/A for b ccs pk/qk from K0 mixing Hence, measurement of the time-dependent CP-asymmetry yields a direct measurement of the UT angle β 24

25 Time dependent CP measurement µ+ ϒ(4S) e- e+ B 0rec J/Ψ π+ - µ Exclusive B Meson Reconstruction π- K 0S Exploit correlation between B0 flavour and lepton charge, K charge etc K- 0 Btag Reconstruct decay vertex positions Δz l- B-Flavour Tagging 25

26 Time dependent CP Asymmetries Measure rate of flavour-tagged B0 and B0bar events as a function of the time Δt between the decays of the tag and signal B sin 2β = 0.7 Observed asymmetry will depend on time resolution and quality of the flavour tag Parameterized by dilution factor ω 26

27 27

28 BELLE B0 J/ ( ) K S0 Candidate with kaon tag K K r z 28

29 sin(2β) from charmonium (ηcp=-1) B J/Ψ Ks0, Ψ(2S)Ks0, χc1ks0, ηcks0 (ηcp=+1) B J/Ψ KL0 (ηcp mixed) B J/Ψ K*0( Ks0π0) 227M BB CP odd modes CP even modes Δt [ps] 29

30 Unitarity triangle constraints 30

31 Trees and Loops B decays can also occur via loop diagrams, referred to as gluonic penguins b ccs is clean tree process in SM sin2β with ~1% uncertainties b sss is purely penguin decay: sin2β in SM with ~0.1 hadronic uncertainties b qqs (q=u,d) mixture of tree and penguin contributions W+ u, c, t b B0 b ccs g d b dds b sss W+ B0 b g d u,c, t s s s d K0 b New Physics d ~ g ~ b ~ s g s d K0 d d π0 s s /d s/ d d 31

32 sin(2β) from penguins Enhance sensitivity to new physics by looking for processes which are CKM-suppressed or forbidden at tree-level Provided there is only a single dominant amplitude, can still extract meaningful time dependent CP asymmetries New physics amplitudes/phases can lead to results incompatible with SM expectations: 32

33 α from charmless B decays Extraction of α relies on modes which possess non-negligible (and unknown) contributions from penguin processes B0 π+π-, B0 ρ+π- and B0 ρ+ρ- (π is pseudoscalar and ρ is a vector) Unitarity implies: where T and P represent tree and penguin contributions with distinct weak phases In absence of penguins, B0 h+h- yields S=sin2α, but in reality αeff= α + δα and C is potentially non-zero 33

34 B ππ Use SU(2) isospin to relate various charged and neutral B hh final states and hence estimate penguin contributions Relative phase between A+- and A+- gives 2δα Neglect EWP ( ~2 ) and other SU(2) symmetry breaking effects Simplest case is where final state mesons are pseudoscalar: B0 π+π-, B+ π+π0, B0 π0π0 Measure branching fractions, plus time dependent CP asymmetry in B0 π+π- and (time-integrated) CP asymmetry in other modes 34

35 B ρρ B0 ρ+ρ- believed to have relatively small penguin pollution due to small experimental bound on BF( B0 ρ0ρ0 ) CP analysis is complicated by existence of three helicity states, corresponding to longitudinal polarization (H=0; CP even) and two transverse polarization states which are not CP eigenstates Need to extract the fraction, f L of longitudinal polarization Longitudinal polarization state has been found to dominate Full angular analysis not needed! 35

36 Ultimate precision of γ extraction from B0 ρ+ρ- analysis will depend on the measured branching fraction and CP content for B0 ρ0ρ0 mode Has advantage over B0 π+ π-, due to fact that B0 ρ0ρ0 has a final state which is all charged tracks, hence can measure S as well as C B0 ρ+/-π-/+ analysis further complicated by fact that final state is not a CP eigenstate full Dalitz plot analysis of π+ππ0 final state Current experimental results do not significantly constrain angle γ 36

37 Summary of α measurements 37

38 Measurements of γ Measure direct CP violation resulting from interference between b c and b u transition amplitudes, where γ is the relative weak phase between the two B decay amplitudes Does not rely on mixing, hence can use both charged and neutral B decays Use final states that have contributions from both D0 and D0, resulting in interference between leading b c and CKM- and color-suppressed b u B+ DK+ where D = D(*)0 or D(*)0 38

39 Physical observables are rates R and charge asymmetries: δb is the relative strong phase and rb is the relative size of the contributing amplitudes: Theory uncertainties quite small (no penguins!), but also consequently expect no sensitivity to non-sm contributions Sensitivity depends critically on size of rb ~0.1 experimental measurements rely on CKM suppressed amplitudes, hence results currently statistically limited D0K- with D0 K+π B- D0K- with D0 K+π- Small branching ratio, but expect large asymmetry since amplitudes are of similar sizes 39

40 GLW method: neutral D in CP eigenstates D0 K+K-, D0 π+ π-, D0 K0sπ0, D0 K0sω, D0 K0sΦ Clean but statistically limited; currently no sensitivity to γ ADS method: combine dominant b c transition with a suppressed D0 decay to enhance sensitivity to rb Neither method currently has statistical sensitivity to rb i.e. has observed suppressed amplitude 40

41 GGSZ Dalitz method GGSZ method: Dalitz plot analysis of 3-body phase space in (self-conjugate) D0 Ks0π+π- mode Improved sensitivity to the suppressed amplitude, but some additional uncertainties due to Dalitz plot model γ=75,δ=180,rb=0.125 DCS K*(1430) Interference between B- D0K-, D0 K0Sρ0 B- D0K-, D0 K0Sρ0 GLW like ρ(770) Interference between B- D0K-, D0 K*+πB- D0K, D0 K*+πADS like DCS K*(892) 41

42 Status of γ 42

43 UT constraints from angles Using information only from angle measurements: 43

44 Assignment 1 1) Draw the three unitarity triangles to scale 2) Use unitarity of V to derive nine conditions, including six trangle relationships 3) Demonstrate that V is unitary using standard representation 4) Use the Wolfenstein parameterization of V to show that the three sides of the Bd triangle are all of the same order in λ. Show that this is not the case for the K and Bs triangles. 5) Show that the area of the Bd triangle is J/2 44