Lectures on B-physics 19-20 April 2011 Vrije Universiteit Brussel N. Tuning Niels Tuning (1)

Menu Time Topic Lecture 1 14:00-15:00 C, P, CP and the Standard Model 15:30-16:30 CKM matrix Lecture 2 10:00-10:45 Flavour mixing in B-decays 11:00-11:45 CP Violation in B-decays 12:00 -12:45 CP Violation in B/K-decays Lecture 3 14:00-14:45 Unitarity Triangle 15:00-15:45 New Physics? Niels Tuning (2)

Grand picture…. Niels Tuning (3)

Introduction: it’s all about the charged current “CP violation” is about the weak interactions, In particular, the charged current interactions: The interesting stuff happens in the interaction with quarks Therefore, people also refer to this field as “flavour physics” Niels Tuning (4)

Motivation 1: Understanding the Standard Model “CP violation” is about the weak interactions, In particular, the charged current interactions: Quarks can only change flavour through charged current interactions Niels Tuning (5)

Introduction: it’s all about the charged current “CP violation” is about the weak interactions, In particular, the charged current interactions: In 1st hour: P-parity, C-parity, CP-parity  the neutrino shows that P-parity is maximally violated Niels Tuning (6)

Introduction: it’s all about the charged current “CP violation” is about the weak interactions, In particular, the charged current interactions: b W- u gVub W+ gV*ub In 1st hour: P-parity, C-parity, CP-parity  Symmetry related to particle – anti-particle Niels Tuning (7)

Motivation 2: Understanding the universe It’s about differences in matter and anti-matter Why would they be different in the first place? We see they are different: our universe is matter dominated Equal amounts of matter & anti matter (?) Matter Dominates ! Niels Tuning (8)

Where and how do we generate the Baryon asymmetry? No definitive answer to this question yet! In 1967 A. Sacharov formulated a set of general conditions that any such mechanism has to meet You need a process that violates the baryon number B: (Baryon number of matter=1, of anti-matter = -1) Both C and CP symmetries should be violated Conditions 1) and 2) should occur during a phase in which there is no thermal equilibrium In these lectures we will focus on 2): CP violation Apart from cosmological considerations, I will convince you that there are more interesting aspects in CP violation Niels Tuning (9)

Introduction: it’s all about the charged current “CP violation” is about the weak interactions, In particular, the charged current interactions: Same initial and final state Look at interference between B0  fCP and B0  B0  fCP Niels Tuning (10)

Motivation 3: Sensitive to find new physics “CP violation” is about the weak interactions, In particular, the charged current interactions: b s “Box” diagram: ΔB=2 b s μ “Penguin” diagram: ΔB=1 g̃ Bs Bs b s s b x b̃ s̃ Bs b s μ x s̃ b̃ g̃ K* b s μ x s̃ b̃ g̃ B0 d Are heavy particles running around in loops? Niels Tuning (11)

Recap: CP-violation (or flavour physics) is about charged current interactions Interesting because: Standard Model: in the heart of quark interactions Cosmology: related to matter – anti-matter asymetry Beyond Standard Model: measurements are sensitive to new particles Matter Dominates ! b s Niels Tuning (12)

Personal impression: People think it is a complicated part of the Standard Model (me too:-). Why? Non-intuitive concepts? Imaginary phase in transition amplitude, T ~ eiφ Different bases to express quark states, d’=0.97 d + 0.22 s + 0.003 b Oscillations (mixing) of mesons: |K0> ↔ |K0> Complicated calculations? Many decay modes? “Beetopaipaigamma…” PDG reports 347 decay modes of the B0-meson: Γ1 l+ νl anything ( 10.33 ± 0.28 ) × 10−2 Γ347 ν ν γ <4.7 × 10−5 CL=90% And for one decay there are often more than one decay amplitudes… Niels Tuning (13)

Start slowly: P and C violation Niels Tuning (14)

Continuous vs discrete symmetries Space, time translation & orientation symmetries are all continuous symmetries Each symmetry operation associated with one ore more continuous parameter There are also discrete symmetries Charge sign flip (Q  -Q) : C parity Spatial sign flip ( x,y,z  -x,-y,-z) : P parity Time sign flip (t  -t) : T parity Are these discrete symmetries exact symmetries that are observed by all physics in nature? Key issue of this course Niels Tuning (15)

Three Discrete Symmetries Parity, P 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 Charge conjugation turns a particle into its anti-particle e +  e- , K -  K + Time Reversal, T Changes, for example, the direction of motion of particles t  -t +  Niels Tuning (16)

Example: People believe in symmetry… Instruction for Abel Tasman, explorer of Australia (1642): “Since many rich mines and other treasures have been found in countries north of the equator between 15o and 40o latitude, there is no doubt that countries alike exist south of the equator. The provinces in Peru and Chili rich of gold and silver, all positioned south of the equator, are revealing proofs hereof.” Niels Tuning (17)

A realistic experiment: the Wu experiment (1956) Observe radioactive decay of Cobalt-60 nuclei The process involved: 6027Co  6028Ni + e- + ne 6027Co is spin-5 and 6028Ni is spin-4, both e- and ne are spin-½ If you start with fully polarized Co (SZ=5) the experiment is essentially the same (i.e. there is only one spin solution for the decay) |5,+5>  |4,+4> + |½ ,+½> + |½,+½> S=1/2 S=4 Niels Tuning (18)

Intermezzo: Spin and Parity and Helicity We introduce a new quantity: Helicity = the projection of the spin on the direction of flight of a particle H=+1 (“right-handed”) H=-1 (“left-handed”) Niels Tuning (19)

The Wu experiment – 1956 Experimental challenge: how do you obtain a sample of Co(60) where the spins are aligned in one direction Wu’s solution: adiabatic demagnetization of Co(60) in magnetic fields at very low temperatures (~1/100 K!). Extremely challenging in 1956. Niels Tuning (20)

The Wu experiment – 1956 The surprising result: the counting rate is different Electrons are preferentially emitted in direction opposite of 60Co spin! Careful analysis of results shows that experimental data is consistent with emission of left-handed (H=-1) electrons only at any angle!! ‘Backward’ Counting rate w.r.t unpolarized rate 60Co polarization decreases as function of time ‘Forward’ Counting rate w.r.t unpolarized rate Niels Tuning (21)

The Wu experiment – 1956 Physics conclusion: Angular distribution of electrons shows that only pairs of left-handed electrons / right-handed anti-neutrinos are emitted regardless of the emission angle Since right-handed electrons are known to exist (for electrons H is not Lorentz-invariant anyway), this means no left-handed anti-neutrinos are produced in weak decay Parity is violated in weak processes Not just a little bit but 100% How can you see that 60Co violates parity symmetry? If there is parity symmetry there should exist no measurement that can distinguish our universe from a parity-flipped universe, but we can! Niels Tuning (22)

So P is violated, what’s next? Wu’s experiment was shortly followed by another clever experiment by L. Lederman: Look at decay p+  m+ nm Pion has spin 0, m,nm both have spin ½  spin of decay products must be oppositely aligned  Helicity of muon is same as that of neutrino. Nice feature: can also measure polarization of both neutrino (p+ decay) and anti-neutrino (p- decay) Ledermans result: All neutrinos are left-handed and all anti-neutrinos are right-handed m+ p+ nm OK OK Niels Tuning (23)

Charge conjugation symmetry Introducing C-symmetry The C(harge) conjugation is the operation which exchanges particles and anti-particles (not just electric charge) It is a discrete symmetry, just like P, i.e. C2 = 1 C symmetry is broken by the weak interaction, just like P OK m+ p+ nm(LH) C m- p- nm(LH) OK Niels Tuning (24)

The Weak force and C,P parity violation What about C+P  CP symmetry? CP symmetry is parity conjugation (x,y,z  -x,-y,z) followed by charge conjugation (X  X) +   Intrinsic spin P C CP appears to be preserved in weak interaction! + +  +   CP Niels Tuning (25)

What do we know now? C.S. Wu discovered from 60Co decays that the weak interaction is 100% asymmetric in P-conjugation We can distinguish our universe from a parity flipped universe by examining 60Co decays L. Lederman et al. discovered from π+ decays that the weak interaction is 100% asymmetric in C-conjugation as well, but that CP-symmetry appears to be preserved First important ingredient towards understanding matter/anti-matter asymmetry of the universe: weak force violates matter/anti-matter(=C) symmetry! C violation is a required ingredient, but not enough as we will learn later Niels Tuning (26)

Conserved properties associated with C and P C and P are still good symmetries in any reaction not involving the weak interaction Can associate a conserved value with them (Noether Theorem) Each hadron has a conserved P and C quantum number What are the values of the quantum numbers Evaluate the eigenvalue of the P and C operators on each hadron P|y> = p|y> What values of C and P are possible for hadrons? Symmetry operation squared gives unity so eigenvalue squared must be 1 Possible C and P values are +1 and -1. Meaning of P quantum number If P=1 then P|y> = +1|y> (wave function symmetric in space) if P=-1 then P|y> = -1 |y> (wave function anti-symmetric in space) Niels Tuning (27)

Figuring out P eigenvalues for hadrons QFT rules for particle vs. anti-particles Parity of particle and anti-particle must be opposite for fermions (spin-N+1/2) Parity of bosons (spin N) is same for particle and anti-particle Definition of convention (i.e. arbitrary choice in def. of q vs q) Quarks have positive parity  Anti-quarks have negative parity e- has positive parity as well. (Can define other way around: Notation different, physics same) Parity is a multiplicative quantum number for composites For composite AB the parity is P(A)*P(B), Thus: Baryons have P=1*1*1=1, anti-baryons have P=-1*-1*-1=-1 (Anti-)mesons have P=1*-1 = -1 Excited states (with orbital angular momentum) Get an extra factor (-1) l where l is the orbital L quantum number Note that parity formalism is parallel to total angular momentum J=L+S formalism, it has an intrinsic component and an orbital component NB: Photon is spin-1 particle has intrinsic P of -1 Niels Tuning (28)

Parity eigenvalues for selected hadrons The p+ meson Quark and anti-quark composite: intrinsic P = (1)*(-1) = -1 Orbital ground state  no extra term P(p+)=-1 The neutron Three quark composite: intrinsic P = (1)*(1)*(1) = 1 P(n) = +1 The K1(1270) Quark anti-quark composite: intrinsic P = (1)*(-1) = -1 Orbital excitation with L=1  extra term (-1)1 P(K1) = +1 Experimental proof: J.Steinberger (1954) πd→nn n are fermions, so (nn) anti-symmetric Sd=1, Sπ=0 → Lnn=1 P|nn> = (-1)L|nn> = -1 |nn> P|d> = P |pn> = (+1)2|pn> = +1 |d> To conserve parity: P|π> = -1 |π> Meaning: P|p+> = -1|p+> Niels Tuning (29)

Figuring out C eigenvalues for hadrons Only particles that are their own anti-particles are C eigenstates because C|x>  |x> = c|x> E.g. p0,h,h’,r0,f,w,y and photon C eigenvalues of quark-anti-quark pairs is determined by L and S angular momenta: C = (-1)L+S Rule applies to all above mesons C eigenvalue of photon is -1 Since photon is carrier of EM force, which obviously changes sign under C conjugation Example of C conservation: Process p0  g g C=+1(p0 has spin 0)  (-1)*(-1) Process p0  g g g does not occur (and would violate C conservation) Experimental proof of C-invariance: BR(π0→γγγ)<3.1 10-5 Niels Tuning (30)

This was an introduction to P and C Let’s change gear… Niels Tuning (31)

CP violation in the SM Lagrangian Focus on charged current interaction (W±): let’s trace it dLI g W+m uLI Niels Tuning (32)

The Standard Model Lagrangian LKinetic : • Introduce the massless fermion fields • Require local gauge invariance  gives rise to existence of gauge bosons LHiggs : • Introduce Higgs potential with <f> ≠ 0 • Spontaneous symmetry breaking The W+, W-,Z0 bosons acquire a mass LYukawa : • Ad hoc interactions between Higgs field & fermions Niels Tuning (33) 33

Fields: Notation Y = Q - T3 Fermions: with y = QL, uR, dR, LL, lR, nR SU(3)C SU(2)L Hypercharge Y (=avg el.charge in multiplet) Left- handed generation index Interaction rep. Quarks: Under SU2: Left handed doublets Right hander singlets Leptons: Scalar field: Note: Interaction representation: standard model interaction is independent of generation number Niels Tuning (34) 34

Fields: Notation Q = T3 + Y Y = Q - T3 Explicitly: The left handed quark doublet : Similarly for the quark singlets: The left handed leptons: And similarly the (charged) singlets: Niels Tuning (35) 35

:The Kinetic Part : Fermions + gauge bosons + interactions Procedure: Introduce the Fermion fields and demand that the theory is local gauge invariant under SU(3)CxSU(2)LxU(1)Y transformations. Start with the Dirac Lagrangian: Replace: Gam : 8 gluons Wbm : weak bosons: W1, W2, W3 Bm : hypercharge boson Fields: Replace the derivative with the covariant derivative introducing the force carriers: gluons, weak vector bosons and photon. The constants g_s, g and g’ are the corresponding coupling constants of the gauge particles. Generators: La : Gell-Mann matrices: ½ la (3x3) SU(3)C Tb : Pauli Matrices: ½ tb (2x2) SU(2)L Y : Hypercharge: U(1)Y For the remainder we only consider Electroweak: SU(2)L x U(1)Y Niels Tuning (36) 36

: The Kinetic Part uLI W+m L=JmWm dLI For example, the term with QLiI becomes: Writing out only the weak part for the quarks: Writing out the weak part leads to terms that can be illustrated by Feynman diagrams. The interactions are usually written as a current J times a gauge field. Exercise: To show this use the defitions of the parity and charge operators. Start by setting q_l = ½ (1+g_5) q, etc. Use also (1-g_5)g_0 = g_0(1+g_5). Show for a few terms that Parity leads to q_l -> q_R. For C it goes similarly using transformation properties under C. In fact, one can see immediately that, since the lagrangian contains vector terms and axial vector terms, which transform with opposite sign, that it cannot be Parity or charge conserving. Under CP, both the vector and axial currents are odd. dLI g W+m uLI W+ = (1/√2) (W1+ i W2) W- = (1/√ 2) (W1 – i W2) L=JmWm Niels Tuning (37) 37

: The Higgs Potential V(f) V(f) f f Broken Symmetry Symmetry ~ 246 GeV Spontaneous Symmetry Breaking: The Higgs field adopts a non-zero vacuum expectation value Procedure: Substitute: The Higgs potential has an asymmetric minimum in one of the 4 degrees of freedom: Re(phi_0). Only this component remains in the final Lagrangian: the Higgs field. The other three components are absorbed in the mass terms of the W+, W-, Z bosons. The photon remains massless. And rewrite the Lagrangian (tedious): . The W+,W-,Z0 bosons acquire mass The Higgs boson H appears (The other 3 Higgs fields are “eaten” by the W, Z bosons) Niels Tuning (38) 38

: The Yukawa Part Since we have a Higgs field we can (should?) add (ad-hoc) interactions between f and the fermions in a gauge invariant way. doublets singlet The result is: ~ i, j : indices for the 3 generations! Remember that the Lagrangian is a scalar object (it must respect Lorentz invariance). How can we make scalars with left handed doublets, higgs doublets and right handed singlets? Y_ij are Yukawa coupling constants, the lefthanded fermion doublet and the higgs doublet together make a scalar to be multiplied by the righthanded fermion scalar. With: (The CP conjugate of f To be manifestly invariant under SU(2) ) are arbitrary complex matrices which operate in family space (3x3)  Flavour physics! Niels Tuning (39) 39

: The Yukawa Part Writing the first term explicitly: 40 Answer: The lagrangian involves scalar and pseudoscalar interactions (since d_r = ½ (1-g_5) d etc.) in a similar way as the nuclean-meson potential. Niels Tuning (40) 40

: 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 Niels Tuning (41) 41

: The Fermion Masses S.S.B Start with the Yukawa Lagrangian After which the following mass term emerges: In the vacuum, the only non-zero component of the Higgs field is phi^0. The L_mass is CP conserving if the Mass matrices are real. with LMass is CP violating in a similar way as LYuk Niels Tuning (42) 42

S.S.B : The Fermion Masses Writing in an explicit form: The matrices M can always be diagonalised by unitary matrices VLf and VRf such that: Then the real fermion mass eigenstates are given by: dLI , uLI , lLI are the weak interaction eigenstates dL , uL , lL are the mass eigenstates (“physical particles”) Niels Tuning (43)

: The Fermion Masses S.S.B In the weak basis: LYukawa = CP violating In terms of the mass eigenstates: 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 Use the fact that: d ½ (1+g_5) m_d ½ (1+g_5) d = m d ½ (1+g5) d. And similarly: d ½ (1-g_5) m_d ½ (1-g_5) d = m d ½ (1-g5) d. Sum them up to get: m d d In the weak basis: LYukawa = CP violating In the mass basis: LYukawa → LMass = CP conserving  What happened to the charged current interactions (in LKinetic) ? Niels Tuning (44) 44

: 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: With: is the Cabibbo Kobayashi Maskawa mixing matrix: Lepton sector: similarly However, for massless neutrino’s: VLn = arbitrary. Choose it such that VMNS = 1  There is no mixing in the lepton sector Niels Tuning (45) 45

Charged Currents In general the charged current term is CP violating The charged current term reads: Under the CP operator this gives: (Together with (x,t) -> (-x,t)) A comparison shows that CP is conserved only if Vij = Vij* In general the charged current term is CP violating Niels Tuning (46) 46

The Standard Model Lagrangian (recap) LKinetic : •Introduce the massless fermion fields •Require local gauge invariance  gives rise to existence of gauge bosons  CP Conserving LHiggs : •Introduce Higgs potential with <f> ≠ 0 •Spontaneous symmetry breaking The W+, W-,Z0 bosons acquire a mass  CP Conserving LYukawa : •Ad hoc interactions between Higgs field & fermions  CP violating with a single phase LYukawa → Lmass : • fermion weak eigenstates: - mass matrix is (3x3) non-diagonal • fermion mass eigenstates: - mass matrix is (3x3) diagonal  CP-violating  CP-conserving! LKinetic in mass eigenstates: CKM – matrix  CP violating with a single phase Niels Tuning (47) 47

Recap Diagonalize Yukawa matrix Yij Mass terms Quarks rotate Off diagonal terms in charged current couplings Replace the derivative with the covariant derivative introducing the force carriers: gluons, weak vector bosons and photon. The constants g_s, g and g’ are the corresponding coupling constants of the gauge particles. Niels Tuning (48) 48

Ok…. We’ve got the CKM matrix, now what? It’s unitary “probabilities add up to 1”: d’=0.97 d + 0.22 s + 0.003 b (0.972+0.222+0.0032=1) How many free parameters? How many real/complex? How do we normally visualize these parameters? Niels Tuning (49) 49

Personal impression: People think it is a complicated part of the Standard Model (me too:-). Why? Non-intuitive concepts? Imaginary phase in transition amplitude, T ~ eiφ Different bases to express quark states, d’=0.97 d + 0.22 s + 0.003 b Oscillations (mixing) of mesons: |K0> ↔ |K0> Complicated calculations? Many decay modes? “Beetopaipaigamma…” PDG reports 347 decay modes of the B0-meson: Γ1 l+ νl anything ( 10.33 ± 0.28 ) × 10−2 Γ347 ν ν γ <4.7 × 10−5 CL=90% And for one decay there are often more than one decay amplitudes… Niels Tuning (50)

Break Time Topic Lecture 1 14:00-15:00 C, P, CP and the Standard Model 15:30-16:30 CKM matrix Lecture 2 10:00-10:45 Flavour mixing in B-decays 11:00-11:45 CP Violation in B-decays 12:00 -12:45 CP Violation in B/K-decays Lecture 3 14:00-14:45 Unitarity Triangle 15:00-15:45 New Physics? Niels Tuning (51)

uI W dI u W d,s,b Diagonalize Yukawa matrix Yij Recap from last hour Mass terms Quarks rotate Off diagonal terms in charged current couplings u d,s,b W Replace the derivative with the covariant derivative introducing the force carriers: gluons, weak vector bosons and photon. The constants g_s, g and g’ are the corresponding coupling constants of the gauge particles. Niels Tuning (52) 52

Ok…. We’ve got the CKM matrix, now what? It’s unitary “probabilities add up to 1”: d’=0.97 d + 0.22 s + 0.003 b (0.972+0.222+0.0032=1) How many free parameters? How many real/complex? How do we normally visualize these parameters? Niels Tuning (53)

Quark field re-phasing Under a quark phase transformation: and a simultaneous rephasing of the CKM matrix: or In other words: Niels Tuning (54)

Quark field re-phasing Under a quark phase transformation: and a simultaneous rephasing of the CKM matrix: or the charged current is left invariant. Degrees of freedom in VCKM in 3 N generations Number of real parameters: 9 + N2 Number of imaginary parameters: 9 + N2 Number of constraints (VV† = 1): -9 - N2 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 3rd family of fermions! 2 generations: Niels Tuning (55)

Intermezzo: Kobayashi & Maskawa Niels Tuning (56)

Timeline: Timeline: Sep 1972: Kobayashi & Maskawa predict 3 generations Nov 1974: Richter, Ting discover J/ψ: fill 2nd generation July 1977: Ledermann discovers Υ: discovery of 3rd generation Niels Tuning (57)

Quark field re-phasing Under a quark phase transformation: and a simultaneous rephasing of the CKM matrix: or the charged current is left invariant. Degrees of freedom in VCKM in 3 N generations Number of real parameters: 9 + N2 Number of imaginary parameters: 9 + N2 Number of constraints (VV† = 1): -9 - N2 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 3rd family of fermions! 2 generations: Niels Tuning (58)

Cabibbos theory successfully correlated many decay rates Cabibbos theory successfully correlated many decay rates by counting the number of cosqc and sinqc terms in their decay diagram Niels Tuning (59) 59

Cabibbos theory successfully correlated many decay rates There was however one major exception which Cabibbo could not describe: K0  m+ m- Observed rate much lower than expected from Cabibbos rate correlations (expected rate  g8sin2qccos2qc) d s cosqc sinqc u W W nm m+ m- Niels Tuning (60) 60

The Cabibbo-GIM mechanism Solution to K0 decay problem in 1970 by Glashow, Iliopoulos and Maiani  postulate existence of 4th quark Two ‘up-type’ quarks decay into rotated ‘down-type’ states Appealing symmetry between generations u c W+ W+ d’=cos(qc)d+sin(qc)s s’=-sin(qc)d+cos(qc)s Niels Tuning (61) 61

The Cabibbo-GIM mechanism How does it solve the K0  m+m- problem? Second decay amplitude added that is almost identical to original one, but has relative minus sign  Almost fully destructive interference Cancellation not perfect because u, c mass different d s d s cosqc +sinqc -sinqc cosqc u c nm nm m+ m- m+ m- Niels Tuning (62) 62

From 2 to 3 generations 2 generations: d’=0.97 d + 0.22 s (θc=13o) 3 generations: d’=0.97 d + 0.22 s + 0.003 b NB: probabilities have to add up to 1: 0.972+0.222+0.0032=1  “Unitarity” ! Niels Tuning (63) 63

From 2 to 3 generations 2 generations: d’=0.97 d + 0.22 s (θc=13o) 3 generations: d’=0.97 d + 0.22 s + 0.003 b Parameterization used by Particle Data Group (3 Euler angles, 1 phase):

Possible forms of 3 generation mixing matrix ‘General’ 4-parameter form (Particle Data Group) with three rotations q12,q13,q23 and one complex phase d13 c12 = cos(q12), s12 = sin(q12) etc… Another form (Kobayashi & Maskawa’s original) Different but equivalent Physics is independent of choice of parameterization! But for any choice there will be complex-valued elements Niels Tuning (65)

Possible forms of 3 generation mixing matrix  Different parametrizations! It’s about phase differences! Re-phasing V: KM PDG 3 parameters: θ, τ, σ 1 phase: φ Niels Tuning (66) 66

How do you measure those numbers? Magnitudes are typically determined from ratio of decay rates Example 1 – Measurement of Vud Compare decay rates of neutron decay and muon decay Ratio proportional to Vud2 |Vud| = 0.97418 ± 0.00027 Vud of order 1 Niels Tuning (67)

How do you measure those numbers? Example 2 – Measurement of Vus Compare decay rates of semileptonic K- decay and muon decay Ratio proportional to Vus2 |Vus| = 0.2255 ± 0.0019 Vus  sin(qc)

How do you measure those numbers? Example 3 – Measurement of Vcb Compare decay rates of B0  D*-l+n and muon decay Ratio proportional to Vcb2 |Vcb| = 0.0412 ± 0.0011 Vcb is of order sin(qc)2 [= 0.0484]

How do you measure those numbers? Example 4 – Measurement of Vub Compare decay rates of B0  D*-l+n and B0  p-l+n Ratio proportional to (Vub/Vcb)2 |Vub/Vcb| = 0.090 ± 0.025 Vub is of order sin(qc)3 [= 0.01]

How do you measure those numbers? Example 5 – Measurement of Vcd Measure charm in DIS with neutrinos Rate proportional to Vcd2 |Vcd| = 0.230 ± 0.011 Vcb is of order sin(qc) [= 0.23]

How do you measure those numbers? Example 6 – Measurement of Vtb Very recent measurement: March ’09! Single top production at Tevatron CDF: |Vtb| = 0.91 ± 0.13 D0: |Vtb| = 1.07 ± 0.12

How do you measure those numbers? Example 7 – Measurement of Vtd, Vts Cannot be measured from top-decay… Indirect from loop diagram Vts: recent measurement: March ’06 |Vtd| = 0.0081 ± 0.0006 |Vts| = 0.0387 ± 0.0023 Vts Ratio of frequencies for B0 and Bs Vts ~ 2 Vtd ~3  Δms ~ (1/λ2)Δmd ~ 25 Δmd  = 1.210 +0.047 from lattice QCD -0.035

What do we know about the CKM matrix? Magnitudes of elements have been measured over time Result of a large number of measurements and calculations Magnitude of elements shown only, no information of phase Niels Tuning (74)

What do we know about the CKM matrix? Magnitudes of elements have been measured over time Result of a large number of measurements and calculations Magnitude of elements shown only, no information of phase Niels Tuning (75) 75

Approximately diagonal form Values are strongly ranked: Transition within generation favored Transition from 1st to 2nd generation suppressed by cos(qc) Transition from 2nd to 3rd generation suppressed bu cos2(qc) Transition from 1st to 3rd generation suppressed by cos3(qc) CKM magnitudes u d t c b s Why the ranking? We don’t know (yet)! If you figure this out, you will win the nobel prize l l3 l l2 l3 l2 l=sin(qc)=0.23 Niels Tuning (76)

Intermezzo: How about the leptons? We now know that neutrinos also have flavour oscillations thus there is the equivalent of a CKM matrix for them: Pontecorvo-Maki-Nakagawa-Sakata matrix a completely different hierarchy! Niels Tuning (77)

Wolfenstein parameterization 3 real parameters: A, λ, ρ 1 imaginary parameter: η

Wolfenstein parameterization 3 real parameters: A, λ, ρ 1 imaginary parameter: η

Exploit apparent ranking for a convenient parameterization Given current experimental precision on CKM element values, we usually drop l4 and l5 terms as well Effect of order 0.2%... Deviation of ranking of 1st and 2nd generation (l vs l2) parameterized in A parameter Deviation of ranking between 1st and 3rd generation, parameterized through |r-ih| Complex phase parameterized in arg(r-ih) Niels Tuning (80)

~1995 What do we know about A, λ, ρ and η? Fit all known Vij values to Wolfenstein parameterization and extract A, λ, ρ and η Results for A and l most precise (but don’t tell us much about CPV) A = 0.83, l = 0.227 Results for r,h are usually shown in complex plane of r-ih for easier interpretation Niels Tuning (81)

Deriving the triangle interpretation Starting point: the 9 unitarity constraints on the CKM matrix Pick (arbitrarily) orthogonality condition with (i,j)=(3,1) Niels Tuning (82)

Deriving the triangle interpretation Starting point: the 9 unitarity constraints on the CKM matrix 3 orthogonality relations Pick (arbitrarily) orthogonality condition with (i,j)=(3,1) Niels Tuning (83) 83

Deriving the triangle interpretation Starting point: the 9 unitarity constraints on the CKM matrix Pick (arbitrarily) orthogonality condition with (i,j)=(3,1) Niels Tuning (84) 84

Visualizing the unitarity constraint Sum of three complex vectors is zero  Form triangle when put head to tail (Wolfenstein params to order l4) Niels Tuning (85)

Visualizing the unitarity constraint Phase of ‘base’ is zero  Aligns with ‘real’ axis, Niels Tuning (86)

Visualizing the unitarity constraint Divide all sides by length of base Constructed a triangle with apex (r,h) (r,h) (0,0) (1,0) Niels Tuning (87)

Visualizing arg(Vub) and arg(Vtd) in the (r,h) plane We can now put this triangle in the (r,h) plane Niels Tuning (88)

“The” Unitarity triangle We can visualize the CKM-constraints in (r,h) plane

β We can correlate the angles β and γ to CKM elements:

Deriving the triangle interpretation Another 3 orthogonality relations Pick (arbitrarily) orthogonality condition with (i,j)=(3,1) Niels Tuning (91) 91

The “other” Unitarity triangle Two of the six unitarity triangles have equal sides in O(λ) NB: angle βs introduced. But… not phase invariant definition!? Niels Tuning (92)

The “Bs-triangle”: βs Replace d by s: Niels Tuning (93)

The phases in the Wolfenstein parameterization Niels Tuning (94)

The CKM matrix W- b gVub u Couplings of the charged current: Wolfenstein parametrization: b W- u gVub Magnitude: Complex phases: Niels Tuning (95)

Back to finding new measurements Next order of business: Devise an experiment that measures arg(Vtd)b and arg(Vub)g. What will such a measurement look like in the (r,h) plane? Fictitious measurement of b consistent with CKM model CKM phases b g Niels Tuning (96)

Consistency with other measurements in (r,h) plane Precise measurement of sin(2β) agrees perfectly with other measurements and CKM model assumptions The CKM model of CP violation experimentally confirmed with high precision! Niels Tuning (97)

What’s going on?? ??? Edward Witten, 17 Feb 2009… See “From F-Theory GUT’s to the LHC” by Heckman and Vafa (arXiv:0809.3452)

Menu Time Topic Lecture 1 14:00-15:00 C, P, CP and the Standard Model 15:30-16:30 CKM matrix Lecture 2 10:00-10:45 Flavour mixing in B-decays 11:00-11:45 CP Violation in B-decays 12:00 -12:45 CP Violation in B/K-decays Lecture 3 14:00-14:45 Unitarity Triangle 15:00-15:45 New Physics? Niels Tuning (99)