The Origin of CP Violation in the Standard Model

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The Origin of CP Violationin the Standard Model

• Topical Lectures
• July 1-2, 2004
• Marcel Merk

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Contents

• Introduction symmetry and non-observables
• CPT Invariance
• CP Violation in the Standard Model Lagrangian
• Re-phasing independent CP Violation quantities
• The Fermion masses
• The matter anti-matter asymmetry

Theory Oriented!
3
Literature
References

• C.Jarlskog, Introduction to CP Violation,
• Advanced Series on Directions in High Energy
  Physics Vol 3
• CP Violation, 1998, p3.
• Y.Nir, CP Violation In and Beyond the Standard
  Model,
• Lectures given at the XXVII SLAC Summer
  Institute, hep-ph/9911321.
• Branco, Lavoura, Silva CP Violation,
• International series of monographs on physics,
• Oxford univ. press, 1999.
• Bigi and Sanda CP Violation,
• Cambridge monographs on particle physics,
  nuclear physics and cosmology,
• Cambridge univ. press, 2000.
• T.D. Lee, Particle Physics and Introduction to
  Field Theory,
• Contemporary Concepts in Physics Volume 1,
• Revised and Updated First Edition, Harwood
  Academic Publishers, 1990.

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Introduction Symmetry and non-Observables
T.D.Lee The root to all symmetry principles
lies in the assumption that it is impossible to
observe certain basic quantities the
non-observables

• There are four main types of symmetry
• Permutation symmetry
• Bose-Einstein and Fermi-Dirac Statistics
• Continuous space-time symmetries
• translation, rotation, acceleration,
• Discrete symmetries
• space inversion, time inversion, charge
  inversion
• Unitary symmetries gauge invariances
• U1(charge), SU2(isospin), SU3(color),..

• If a quantity is fundamentally non-observable it
  is related to an exact symmetry

• If a quantity could in principle be observed by
  an improved measurement
• the symmetry is said to be broken

Noether Theorem
symmetry
conservation law
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Symmetry and non-observables
Simple Example Potential energy V between two
particles
Absolute position is a non-observable The
interaction is independent on the choice of 0.
Symmetry V is invariant under arbitrary space
translations
0
0
Consequently
Total momentum is conserved
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Symmetry and non-observables
Non-observables Symmetry Transformations Conservation Laws or Selection Rules
Difference between identical particles Permutation B.-E. or F.D. statistics
Absolute spatial position Space translation momentum
Absolute time Time translation energy
Absolute spatial direction Rotation angular momentum
Absolute velocity Lorentz transformation generators of the Lorentz group
Absolute right (or left) parity
Absolute sign of electric charge charge conjugation
Relative phase between states of different charge Q charge
Relative phase between states of different baryon number B baryon number
Relative phase between states of different lepton number L lepton number
Difference between different co- herent mixture of p and n states isospin
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Parity Violation
Before 1956 physicists were convinced that the
laws of nature were left-right symmetric. Strange?
A gedanken experiment Consider two perfectly
mirror symmetric cars
Gas pedal
Gas pedal
driver
driver
L and R are fully symmetric, Each nut, bolt,
molecule etc. However the engine is a black box
R
L
Person L gets in, starts, .. 60 km/h
Person R gets in, starts, .. What happens?
What happens in case the ignition mechanism uses,
say, Co60 b decay?
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CPT Invariance

• Local Field theories always respect
• Lorentz Invariance
• Symmetry under CPT operation (an electron a
  positron travelling back in time)
• gt Consequence mass of particle mass of
  anti-particle

(Lüders, Pauli, Schwinger)
(anti-unitarity)
gt Similarly the total decay-rate of a particle
is equal to that of the anti-particle

• Question 1
• The mass difference between KL and KS Dm 3.5
  x 10-6 eV gt CPT violation?
• Question 2
• How come the lifetime of KS 0.089 ns while the
  lifetime of the KL 51.7 ns?
• Question 3
• BaBar measures decay rate B-gt J/y KS and Bbar-gt
  J/y KS. Clearly not the same how can it be?

Answer 1 2 A KL ? an anti-KS particle!
Answer 3 Partial decay rate ? total decay rate!
However, the sum over all partial rates (gt200 or
so) is the same for B and Bbar.
(Amazing! at least to
me)
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CP in the Standard Model Lagrangian(The origin
of the CKM-matrix)
LSM contains LKinetic fermion fields LHiggs
the Higgs potential LYukawa the Higgs
Fermion interactions

• Plan
• Look at symmetry aspects of the Lagrangian
• How is CP violation implemented?
• ? Several miracles happen in symmetry
• breaking

Standard Model gauge symmetry
Note Immediately The weak part is explicitly
parity violating

• Outline
• Lorentz structure of the Lagrangian
• Introduce the fermion fields in the SM
• LKinetic local gauge invariance fermions ?
  bosons
• LHiggs spontaneous symmetry breaking
• LYukawa the origin of fermion masses
• VCKM CP violation

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Lagrangian Density
Local field theories work with Lagrangian
densities
with the fields
taken at
The fundamental quantity, when discussing
symmetries is the Action
If the action is (is not) invariant under a
symmetry operation then the symmetry in question
is a good (broken) one
gt Unitarity of the interaction requires the
Lagrangian to be Hermitian
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Structure of a Lagrangian
Lorentz structure a Lagrangian in field theory
can be built using combinations of S Scalar
fields 1 P Pseudoscalar
fields g5 V Vector fields
gm A Axial vector fields
gmg5 T Tensor fields smn
Dirac field ?
Scalar field f
Example Consider a spin-1/2 (Dirac) particle
(nucleon) interacting with a spin-0 (Scalar)
object (meson)
Nucleon field
Meson potential
Nucleon meson interaction
Exercise What are the symmetries of this theory
under C, P, CP ? Can a and b be any complex
numbers? Note the interaction term contains
scalar and pseudoscalar parts
Violates P, conserves C, violates CP a and b must
be real from Hermeticity
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Transformation Properties
(Ignoring arbitrary phases)
Transformation properties of Dirac spinor
bilinears
c?c
c?c
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The Standard Model Lagrangian

• LKinetic Introduce the massless fermion
  fields
• Require local gauge
  invariance gt gives rise to existence of gauge
  bosons

gt CP Conserving

• LHiggs Introduce Higgs potential with ltfgt ? 0
• Spontaneous symmetry breaking

The W, W-,Z0 bosons acquire a mass
gt CP Conserving

• LYukawa Ad hoc interactions between Higgs
  field fermions

gt 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

gt CP-violating
gt CP-conserving!

• LKinetic in mass eigenstates CKM matrix

gt CP violating with a single phase
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Fields Notation
Q T3 Y
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

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Fields Notation
Q T3 Y
Explicitly

• The left handed quark doublet

• Similarly for the quark singlets

• The left handed leptons

• And similarly the (charged) singlets

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Intermezzo Local Gauge Invariance in a single
transparancy
Basic principle The Lagrangian must be invariant
under local gauge transformations
Example massless Dirac Spinors in QED
global U(1) gauge transformation
local U(1) gauge transformation
Is the Lagrangian invariant?
Not invariant!
Then
Then it turns out that
gt Introduce the covariant derivative
and demand that Am transforms as
is invariant!

• Introduce charged fermion field (electron)
• Demand invariance under local gauge
  transformations (U(1))
• The price to pay is that a gauge field Am must
  be introduced at the same time (the photon)

Conclusion
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The
Kinetic Part
Fermions gauge bosons interactions
Procedure Introduce the Fermion fields and
demand that the theory is local gauge invariant
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
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The
Kinetic Part
Exercise Show that this Lagrangian formally
violates both P and C Show that this Lagrangian
conserves CP
LKin CP conserving
For example the term with QLiI becomes
and similarly for all other terms
(uRiI,dRiI,LLiI,lRiI).
Writing out only the weak part for the quarks
W (1/v2) (W1 i W2) W- (1/v 2) (W1 i W2)
LJmWm
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The Higgs
Potential
?Note LHiggs CP conserving
V(f)
V(f)
Symmetry
Broken Symmetry
f
f
246 GeV
Spontaneous Symmetry Breaking The Higgs field
adopts a non-zero vacuum expectation value
Procedure
Substitute
And rewrite the Lagrangian (tedious)

2. The W,W-,Z0 bosons acquire mass
3. The Higgs boson H appears

(The other 3 Higgs fields are eaten by the W, Z
bosons)
The realization of the vacuum breaks the
symmetry
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The Yukawa
Part
Since we have a Higgs field we can add (ad-hoc)
interactions between f and the fermions in a
gauge invariant way.
doublets
L must be Her- mitian (unitary)
The result is
singlet
With
To be manifestly invariant under SU(2)
are arbitrary complex matrices which operate in
family space (3x3) gt Flavour physics!
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The Yukawa
Part
Writing the first term explicitly
Question In what aspect is this Lagrangian
similar to the example of the nucleon-meson
potential?
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The Yukawa
Part
In general LYukawa is CP violating
Formally, CP is violated if
Exercise (intuitive proof) Show that

• The hermiticity of the Lagrangian implies that
  there are terms in pairs of the form

• However a transformation under CP gives

CP is conserved in LYukawa only if Yij Yij
and leaves the coefficients Yij and Yij
unchanged
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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
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The Fermion
Masses
S.S.B
Start with the Yukawa Lagrangian
After which the following mass term emerges
with
LMass is CP violating in a similar way as LYuk
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The Fermion
Masses
S.S.B
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)
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The Fermion
Masses
S.S.B
In terms of the mass eigenstates
CP Conserving?
In flavour space one can choose Weak basis The
gauge currents are diagonal in flavour space, but
the flavour mass matrices are
non-diagonal Mass basis The fermion masses are
diagonal, but some gauge currents (charged weak
interactions) are not
diagonal in flavour space
In the weak basis LYukawa
CP violating In the mass basis LYukawa ?
LMass CP conserving
gtWhat happened to the charged current
interactions (in LKinetic) ?
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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 gt There
is no mixing in the lepton sector
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Flavour Changing Neutral Currents
To illustrate the SM neutral current take the W3m
and Bm term of the Kinetic Lagrangian
and
And consider the Z-boson field
Take further QLiIdLiI
Use
In terms of physical fields no non-diagonal
contributions occur for the neutral Currents. gt
GIM mechanism
Standard Model forbids flavour changing neutral
currents.
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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
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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
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Where were we?

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The Standard Model Lagrangian (recap)

• LKinetic Introduce the massless fermion
  fields
• Require local gauge
  invariance gt gives rise to existence of gauge
  bosons

gt CP Conserving

• LHiggs Introduce Higgs potential with ltfgt ? 0
• Spontaneous symmetry breaking

The W, W-,Z0 bosons acquire a mass
gt CP Conserving

• LYukawa Ad hoc interactions between Higgs
  field fermions

gt 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

gt CP-violating
gt CP-conserving!

• LKinetic in mass eigenstates CKM matrix

gt CP violating with a single phase
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Quark field re-phasing
Under a quark phase transformation
and a simultaneous rephasing of the CKM matrix
or
the charged current
is left invariant
2 generations
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 third
family of fermions!
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The LEP collider _at_ CERN
Aleph
L3
Opal
Delphi
Geneva Airport Cointrin
MZ
Maybe the most important result of LEP
There are 3 generations of neutrinos
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The lepton sector (Intermezzo)

• N. Cabibbo Phys. Rev.Lett. 10, 531 (1963)
• 2 family flavour mixing in quark sector (GIM
  mechanism)
• M.Kobayashi and T.Maskawa, Prog. Theor. Phys 49,
  652 (1973)
• 3 family flavour mixing in quark sector
• Z.Maki, M.Nakagawa and S.Sakata, Prog. Theor.
  Phys. 28, 870 (1962)
• 2 family flavour mixing in neutrino sector to
  explain neutrino oscillations!

• In case neutrino masses are of the Dirac type,
  the situation in the lepton sector is very
  similar as in the quark sector VMNS VCKM.
• The is one CP violating phase in the lepton MNS
  matrix
• In case neutrino masses are of the Majorana type
  (a neutrino is its own anti-particle ? no freedom
  to redefine neutrino phases)
• There are 3 CP violating phases in the lepton MNS
  matrix
• However, the two extra phases are unobservable in
  neutrino oscillations
• There is even a CP violating phase in case Ndim
  2

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Lepton mixing and neutrino oscillations
Question

• In the CKM we write by convention the mixing for
  the down type quarks in the lepton sector we
  write it for the (up-type) neutrinos. Is it
  relevant?
• If yes why?
• If not, why dont we measure charged lepton
  oscillations rather then neutrino oscillations?

However, observation of neutrino oscillations is
possible due to small neutrino mass differences.
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Rephasing Invariants
The standard representation of the CKM matrix is
However, many representations are possible. What
are the invariants under re-phasing?

• Simplest Uai Vai2 is independent of quark
  re-phasing
• Next simplest Quartets Qaibj Vai Vbj Vaj
  Vbi with a?b and i?j
• Each quark phase appears with and without
• VV1 Unitarity triangle Vud Vcd Vus Vcs
  Vub Vcb 0
• Multiply the equation by Vus Vcs and take the
  imaginary part
• Im (Vus Vcs Vud Vcd) - Im (Vus Vcs Vub Vcb)
• J Im Qudcs - Im Qubcs
• The imaginary part of each Quartet combination is
  the same (up to a sign)
• In fact it is equal to 2x the surface of the
  unitarity triangle
• ImVai Vbj Vaj Vbi J ?eabg eijk where J
  is the universal Jarlskog invariant
• Amount of CP Violation is proportional to J

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The Unitarity Triangle
unitarity
VCKM VCKM 1
The db triangle
Area ½ Im Qudcb ½ J
Under re-phasing
the unitary angles are invariant
(In fact, rephasing implies a rotation of the
whole triangle)
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Wolfenstein Parametrization
Wolfenstein realised that the non-diagonal CKM
elements are relatively small compared to the
diagonal elements, and parametrized as follows
Normalised CKM triangle
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CP Violation and quark masses
Note that the massless Lagrangian has a global
symmetry for unitary transformations in flavour
space.
Lets now assume two quarks with the same charge
are degenerate in mass, eg. ms mb
Redefine s Vus s Vub b
Now the u quark only couples to s and not to b
i.e. V13 0
Using unitarity we can show that the CKM matrix
can now be written as
CP conserving
Necessary criteria for CP violation
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The Amount of CP Violation
Using Standard Parametrization of CKM
(eg. JIm(Vus Vcb Vub Vcs) )
(The maximal value J might have 1/(6v3) 0.1)
However, also required is
All requirements for CP violation can be
summarized by
Is CP violation maximal? gt One has to
understand the origin of mass!
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Mass Patterns
Observe
Mass spectra (m Mz, MS-bar scheme) mu 1 - 3
MeV , mc 0.5 0.6 GeV , mt 180 GeV
md 2 - 5 MeV , ms 35 100 MeV , mb
2.9 GeV
me 0.51 MeV , mm 105 MeV , mt
1777 MeV
Why are neutrinos so light? Related to the fact
that they are the only neutral fermions? See-saw
mechanism?

• Do you want to be famous?
• Do you want to be a king?
• Do you want more then the nobel prize?
• - Then solve the mass Problem
• R.P. Feynman

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Matter - antimatter asymmetry

• In the visible universe matter dominates over
  anti-matter
• There are no antimatter particles present in
  cosmic rays
• There are no intense g-ray sources in the
  universe due to matter anti-matter collisions

Hubble deep field - optical
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Big Bang Cosmology
Equal amounts of matter antimatter
qq? gg
Matter Dominates ! CMB
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The matter anti-matter asymmetry
Cosmic Microwave Background
WMAP satellite
Almost all matter annihilated with antimatter,
producing photons
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The Sakharov conditions
Convert 1 in 109 anti-quarks into a quark in an
early stage of universe
Anti-

• A matter dominated universe can evolve in case
  three conditions occur simultaneous
• Baryon number violation L(DB)?0
• C and CP Violation G(N?f) ? G(N?f)
• Thermal non-equilibrium
• otherwise CPT invariance gt CP invariance

Sakharov (1964)
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Baryogenesis at the GUT Scale
Conceptually simple
GUT theories predict proton decay mediated by
heavy X gauge bosons
X boson has baryon number violating (1)
couplings X ?q q, X?q l Proton lifetime t gt
1032 s
Efficiency of Baryon asymmetry build-up
A simple Baryogenesis model
CP Violation (2) r ? r
Decay process Decay fraction DB
X ? q q r 2/3
X ? q l 1-r 1/3
X ? q q r -2/3
X ? q l 1-r -1/3
Assuming the back reaction does not occur (3)
Initial X number density
Initial light particle number density
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Baryogenesis at Electroweak Scale
Conceptually difficult
SM Electroweak Interactions 1) Baryon number
violation in weak anomaly Conserves B-L
but violates BL 2) CP Violation in the CKM 3)
Non-equilibrium electroweak phase transition
Electroweak phase transition wipes out GUT Baryon
asymmetry!
Can it generate a sufficiently large asymmetry?
Problems 1. Higgs mass is too heavy. In order to
have a first order phase transition
Requirement mH lt 70 GeV/c2 , from LEP mH gt
100 GeV/c2 2. CP Violation in CKM is not enough

• Leptogenesis
• Uses the large right handed majorana neutrino
  masses in the see-saw mechanism to generate a
  lepton asymmetry at high energies (using the MNS
  equivalent of CKM).
• Uses the electroweak sphaleron (B-L
  conserving) processes to communicate this to a
  baryon asymmetry, which survives further
  evolution of the universe.

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Conclusion

• Key questions in B physics
• Is the SM the only source of CP Violations?
• Does the SM fully explain flavour physics?

Biertje?