Lectures%20on%20B-physics%2019-20%20April%202011%20Vrije%20Universiteit%20Brussel

1
Lectures on B-physics 19-20 April 2011Vrije
Universiteit Brussel

• N. Tuning

2
Menu
Time Topic
Lecture 1 1400-1500 C, P, CP and the Standard Model
1530-1630 CKM matrix
Lecture 2 1000-1045 Flavour mixing in B-decays
1100-1145 CP Violation in B-decays
1200 -1245 CP Violation in B/K-decays
Lecture 3 1400-1445 Unitarity Triangle
1500-1545 New Physics?
3
Grand picture.
4
Introduction its 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

5
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

6
Introduction its 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

7
Introduction its 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
• ? Symmetry related to particle anti-particle

8
Motivation 2 Understanding the universe

• Its 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

9
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 matter1, 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

10
Introduction its 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

11
Motivation 3 Sensitive to find new physics

• CP violation is about the weak interactions,
• In particular, the charged current interactions

• Are heavy particles running around in loops?

12
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

13
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 eif
• Different bases to express quark states, d0.97
  d 0.22 s 0.003 b
• Oscillations (mixing) of mesons K0gt
  ? ?K0gt
• Complicated calculations?
• Many decay modes? Beetopaipaigamma
• PDG reports 347 decay modes of the B0-meson
• G1 l ?l anything ( 10.33 0.28 ) 10-2
• G347 ? ? ? lt4.7 10-5 CL90
• And for one decay there are often more than one
  decay amplitudes

14
Start slowly P and C violation
15
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

16
Three Discrete Symmetries

• Parity, P
• Parity reflects a system through the origin.
  Convertsright-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

17
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.

18
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 (SZ5) the
  experiment is essentially the same (i.e. there is
  only one spin solution for the decay) 5,5gt ?
  4,4gt ½ ,½gt ½,½gt

S4
19
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

H1 (right-handed)
H-1 (left-handed)
20
The Wu experiment 1956

• Experimental challenge how do you obtain a
  sample of Co(60) where the spins are aligned in
  one direction
• Wus solution adiabatic demagnetization of
  Co(60) in magnetic fields at very low
  temperatures (1/100 K!). Extremely challenging
  in 1956.

21
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 ratew.r.t unpolarized rate
60Co polarization decreasesas function of time
Forward Counting ratew.r.t unpolarized rate
22
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!

23
So P is violated, whats next?

• Wus 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

p
m
nm
OK
OK
24
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
p
m
nm(LH)
C
nm(LH)
p-
m-
OK
25
The Weak force and C,P parity violation

• What about CP ? CP symmetry?
• CP symmetry is parity conjugation (x,y,z ?
  -x,-y,z)
• followed by charge conjugation (X ? X)

?
??
??
P
C
CP appears to be preservedin weakinteraction!
?
?
??
??
?
??
CP
26
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 p 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

27
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 hadronPygt pygt
• 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 P1 then Pygt 1ygt (wave function
  symmetric in space)if P-1 then Pygt -1 ygt
  (wave function anti-symmetric in space)

28
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-N1/2)
• 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 P1111, anti-baryons have
  P-1-1-1-1
• (Anti-)mesons have P1-1 -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 JLS formalism, it has an
  intrinsic component and an orbital component
• NB Photon is spin-1 particle has intrinsic P of
  -1

29
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
• Orbital ground state ? no extra term
• P(n) 1
• The K1(1270)
• Quark anti-quark composite intrinsic P
  (1)(-1) -1
• Orbital excitation with L1 ? extra term (-1)1
• P(K1) 1

• Experimental proof J.Steinberger (1954)
• pd?nn
• n are fermions, so (nn) anti-symmetric
• Sd1, Sp0 ? Lnn1
• Pnngt (-1)Lnngt -1 nngt
• Pdgt P pngt (1)2pngt 1 dgt
• ?To conserve parity Ppgt -1 pgt

Meaning Ppgt -1pgt
30
Figuring out C eigenvalues for hadrons

• Only particles that are their own anti-particles
  are C eigenstates because Cxgt ? xgt cxgt
• 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)LS
• 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 C1(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(p0????)lt3.1
10-5
31

• This was an introduction to P and C
• Lets change gear

32
CP violation in the SM Lagrangian

• Focus on charged current interaction (W) lets
  trace it

33
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 ltfgt ?
  0
• Spontaneous symmetry
  breaking

The W, W-,Z0 bosons acquire a mass

• LYukawa Ad hoc interactions between Higgs
  field fermions

Niels Tuning (33)
34
Fields Notation
Y Q - T3
Fermions
with y QL, uR, dR, LL, lR, nR
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)
35
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)
36
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
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)
37
The
Kinetic Part
For example, the term with QLiI becomes
Writing out only the weak part for the quarks
W (1/v2) (W1 i W2) W- (1/v 2) (W1 i W2)
LJmWm
Niels Tuning (37)
38
The Higgs Potential
And rewrite the Lagrangian (tedious)
(The other 3 Higgs fields are eaten by the W, Z
bosons)
Niels Tuning (38)
39
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.
The result is

i, j indices for the 3 generations!
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)
40
The Yukawa Part
Writing the first term explicitly
Niels Tuning (40)
41
The Yukawa Part
There are 3 Yukawa matrices (in the case of
massless neutrinos)

• 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)
42
The Fermion
Masses
Start with the Yukawa Lagrangian
After which the following mass term emerges
with
LMass is CP violating in a similar way as LYuk
Niels Tuning (42)
43
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
Niels Tuning (43)
44
The Fermion
Masses
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
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)
45
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 neutrinos VLn
arbitrary. Choose it such that VMNS 1 ? There
is no mixing in the lepton sector
Niels Tuning (45)
46
Charged Currents
The charged current term reads
(Together with (x,t) -gt (-x,t))
Under the CP operator this gives
A comparison shows that CP is conserved only if
Vij Vij
In general the charged current term is CP
violating
Niels Tuning (46)
47
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 ltfgt ? 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)
48
Recap

• Diagonalize Yukawa matrix Yij
• Mass terms
• Quarks rotate
• Off diagonal terms in charged current couplings

Niels Tuning (48)
49
Ok. Weve got the CKM matrix, now what?

• Its unitary
• probabilities add up to 1
• d0.97 d 0.22 s 0.003 b (0.9720.2220.0032
  1)
• How many free parameters?
• How many real/complex?
• How do we normally visualize these parameters?

Niels Tuning (49)
50
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 eif
• Different bases to express quark states, d0.97
  d 0.22 s 0.003 b
• Oscillations (mixing) of mesons K0gt
  ? ?K0gt
• Complicated calculations?
• Many decay modes? Beetopaipaigamma
• PDG reports 347 decay modes of the B0-meson
• G1 l ?l anything ( 10.33 0.28 ) 10-2
• G347 ? ? ? lt4.7 10-5 CL90
• And for one decay there are often more than one
  decay amplitudes

51
Break
Time Topic
Lecture 1 1400-1500 C, P, CP and the Standard Model
1530-1630 CKM matrix
Lecture 2 1000-1045 Flavour mixing in B-decays
1100-1145 CP Violation in B-decays
1200 -1245 CP Violation in B/K-decays
Lecture 3 1400-1445 Unitarity Triangle
1500-1545 New Physics?
52
Recap from last hour

• Diagonalize Yukawa matrix Yij
• Mass terms
• Quarks rotate
• Off diagonal terms in charged current couplings

Niels Tuning (52)
53
Ok. Weve got the CKM matrix, now what?

• Its unitary
• probabilities add up to 1
• d0.97 d 0.22 s 0.003 b (0.9720.2220.0032
  1)
• How many free parameters?
• How many real/complex?
• How do we normally visualize these parameters?

54
Quark field re-phasing
Under a quark phase transformation
and a simultaneous rephasing of the CKM matrix
or
In other words
55
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
56
Intermezzo Kobayashi Maskawa
57
Timeline

• Timeline
• Sep 1972 Kobayashi Maskawa predict 3
  generations
• Nov 1974 Richter, Ting discover J/? fill 2nd
  generation
• July 1977 Ledermann discovers ? discovery of
  3rd generation

58
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
59
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)
60
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 ratecorrelations (expected rate ?
  g8sin2qccos2qc)

d
?s
cosqc
sinqc
u
W
W
nm
m
m-
Niels Tuning (60)
61
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
dcos(qc)dsin(qc)s
s-sin(qc)dcos(qc)s
Niels Tuning (61)
62
The Cabibbo-GIM mechanism

• How does it solve the K0 ? mm- 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
?s
d
-sinqc
cosqc
cosqc
sinqc
c
u
nm
nm
m
m-
m
m-
Niels Tuning (62)
63
From 2 to 3 generations

• 2 generations d0.97 d 0.22 s (?c13o)
• 3 generations d0.97 d 0.22 s 0.003 b
• NB probabilities have to add up to 1
  0.9720.2220.00321
• ? Unitarity !

Niels Tuning (63)
64
From 2 to 3 generations

• 2 generations d0.97 d 0.22 s (?c13o)
• 3 generations d0.97 d 0.22 s 0.003 b
• Parameterization used by Particle Data Group (3
  Euler angles, 1 phase)

65
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 Maskawas original)
• Different but equivalent
• Physics is independent of choice of
  parameterization!
• But for any choice there will be complex-valued
  elements

66
Possible forms of 3 generation mixing matrix
? Different parametrizations! Its about phase
differences!
KM
Re-phasing V
PDG
3 parameters ?, t, s 1 phase f
Niels Tuning (66)
67
How do you measure those numbers?

• Magnitudes are typically determined from ratio of
  decay rates
• Example 1 Measurement of Vud
• Compare decay rates of neutrondecay and muon
  decay
• Ratio proportional to Vud2
• Vud 0.97418 0.00027
• Vud of order 1

68
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)

69
How do you measure those numbers?

• Example 3 Measurement of Vcb
• Compare decay rates of B0 ? D-ln and muon
  decay
• Ratio proportional to Vcb2
• Vcb 0.0412 0.0011
• Vcb is of order sin(qc)2 0.0484

70
How do you measure those numbers?

• Example 4 Measurement of Vub
• Compare decay rates of B0 ? D-ln and B0 ?
  p-ln
• Ratio proportional to (Vub/Vcb)2
• Vub/Vcb 0.090 0.025
• Vub is of order sin(qc)3 0.01

71
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

72
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

73
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

Ratio of frequencies for B0 and Bs
Vts ?2 Vtd ?3 ? ?ms (1/?2)?md 25 ?md
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
75
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)
76
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
Why the ranking?We dont know (yet)! If you
figure this out,you will win the nobelprize
l
l3
l
l2
l3
l2
lsin(qc)0.23
77
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!

78
Wolfenstein parameterization
3 real parameters A, ?, ? 1 imaginary
parameter ?
79
Wolfenstein parameterization
3 real parameters A, ?, ? 1 imaginary
parameter ?
80
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)

81
1995 What do we know about A, ?, ? and ??

• Fit all known Vij values to Wolfenstein
  parameterization and extract A, ?, ? and ?
• Results for A and l most precise (but dont 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

82
Deriving the triangle interpretation

• Starting point the 9 unitarity constraints on
  the CKM matrix
• Pick (arbitrarily) orthogonality condition with
  (i,j)(3,1)

83
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)
84
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)
85
Visualizing the unitarity constraint

• Sum of three complex vectors is zero ? Form
  triangle when put head to tail

(Wolfenstein params to order l4)
86
Visualizing the unitarity constraint

• Phase of base is zero ? Aligns with real
  axis,

87
Visualizing the unitarity constraint

• Divide all sides by length of base
• Constructed a triangle with apex (r,h)

(r,h)
(0,0)
(1,0)
88
Visualizing arg(Vub) and arg(Vtd) in the (r,h)
plane

• We can now put this triangle in the (r,h) plane

89
The Unitarity triangle

• We can visualize the CKM-constraints in (r,h)
  plane

90
ß

• We can correlate the angles ß and ? to CKM
  elements

91
Deriving the triangle interpretation

• Another 3 orthogonality relations
• Pick (arbitrarily) orthogonality condition with
  (i,j)(3,1)

Niels Tuning (91)
92
The other Unitarity triangle

• Two of the six unitarity triangles have equal
  sides in O(?)

• NB angle ßs introduced. But not phase invariant
  definition!?

93
The Bs-triangle ßs

• Replace d by s

94
The phases in the Wolfenstein parameterization
95
The CKM matrix

• Couplings of the charged current
• Wolfenstein parametrization

• Magnitude

• Complex phases

96
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
97
Consistency with other measurements in (r,h) plane
Precise measurement ofsin(2ß) agrees
perfectlywith other measurementsand CKM model
assumptionsThe CKM model of CP violation
experimentallyconfirmed with high precision!
98
Whats going on??

• ??? Edward Witten, 17 Feb 2009
• See From F-Theory GUTs to the LHC by Heckman
  and Vafa (arXiv0809.3452)

99
Menu
Time Topic
Lecture 1 1400-1500 C, P, CP and the Standard Model
1530-1630 CKM matrix
Lecture 2 1000-1045 Flavour mixing in B-decays
1100-1145 CP Violation in B-decays
1200 -1245 CP Violation in B/K-decays
Lecture 3 1400-1445 Unitarity Triangle
1500-1545 New Physics?