Transcript The Origin of CP Violation in the Standard Model Topical Lectures

The Origin of CP Violation
in the Standard Model
Topical Lectures
July 1-2, 2004
Marcel Merk
July 1-2, 2004
1
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!
July 1-2, 2004
2
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.
•
C. Quigg, “Gauge Theories of the Strong, Weak and Electromagnetic Interactions”,
Frontiers in Physics, Benjamin-Cummings, 1983.
•
H. Fritsch and Z. Xing, “Mass and Flavour Mixing Schemes of Quarks and Leptons”, hep-ph/9912358.
•
Mark Trodden, “Electroweak Baryogenesis”, hep-ph/9803479.
July 1-2, 2004
3
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:
July 1-2, 2004
symmetry
conservation law
4
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.
r1
r2
Symmetry:
V is invariant under arbitrary
space translations:
r2  r2  d
r1  r1  d
0’
Consequently:

V  V r1  r2
July 1-2, 2004
d
0
Total momentum is conserved:


d
p1  p2
dt

 F1  F2


  1   2 V
 0
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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
Absolute time
Time translation
t  t 
energy
Absolute spatial direction
Rotation
r  r
angular momentum
Absolute velocity
Lorentz transformation
Absolute right (or left)
Absolute sign of electric charge
r r
r  r
e  e
momentum
generators of the Lorentz group
parity
charge conjugation
Relative phase between states of
different charge Q
 e 
Relative phase between states of
different baryon number B
  eiN
Relative phase between states of
different lepton number L
  eiL
lepton number
 p
 p

U
 
 
n
n
isospin
Difference between different coherent mixture of p and n states
July 1-2, 2004
iQ
charge
baryon number
6
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
“L” and “R” are fully symmetric,
Each nut, bolt, molecule etc.
However the engine is a black box
“L”
driver
“R”
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?
July 1-2, 2004
7
CPT Invariance
Local Field theories always respect:
• Lorentz Invariance
(Lüders, Pauli, Schwinger)
• Symmetry under CPT operation (an electron = a positron travelling back in time)
=> Consequence: mass of particle = mass of anti-particle:
M  p   p H p  p  CPT   CPT  H  CPT 
†
 p  CPT  H  CPT 
1

p  p H
 CPT 

p  M  p
1
p

(anti-unitarity)
=> Similarly the total decay-rate of a particle is equal to that of the anti-particle
• Question 1:
Answer 1 + 2: A KL ≠ an anti-KS particle!
The mass difference between KL and KS: m = 3.5 x 10-6 eV => 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-> J/ KS and Bbar-> J/ KS. Clearly not the same: how can it be?
Answer 3:
Partial decay rate ≠ total decay rate! However, the sum over all partial rates (>200 or so)
is the same for B and Bbar.
(Amazing! – at least to me)
July 1-2, 2004
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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:
GSM  SU (3)C  SU (2) L U (1)Y
 SU (3)C U (1) EM
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
•
July 1-2, 2004
VCKM
: CP violation
9
Lagrangian Density
Local field theories work with Lagrangian densities:
L  x , t   L  j  x , t  ,    j  x , t  
with  j  x, t  , j  1, 2,..., N the fields taken at
x, t
The fundamental quantity, when discussing symmetries is the Action:
A   d 4 x L  x, t 
If the action is (is not) invariant under a symmetry operation then
the symmetry in question is a good (broken) one
=> Unitarity of the interaction requires the Lagrangian to be Hermitian
July 1-2, 2004
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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
: 5
V: Vector fields
: 
A: Axial vector fields
: 5
T: Tensor fields
: sn
Example:
Consider a spin-1/2 (Dirac) particle (“nucleon”)
interacting with a spin-0 (Scalar) object (“meson”)
Dirac field  :
(i     m)  0
Scalar field :
(i     m2 )  0
L  x , t   i  x , t       x , t   m  x , t   x , t 
2
1 

   x , t      x , t   V   x , t  
2
   x , t   a  ib 5    x , t    x , t 
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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?
Violates P, conserves C, violates CP
Note: the interaction term contains scalar and pseudoscalar parts
a and b must be real from Hermeticity
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Transformation Properties
P
C
( x, t )
 ( x , t )
( x, t )
 † ( x, t )
Scalar Field :
( x, t )
 ( x, t )


Dirac Field :
 ( x, t )
  0 (  x , t )
V† ( x , t )
Axial Field :
A ( x , t )   A ( x , t )
A† ( x , t )

Q 0  Q0
, Q k  Qk
i 2 0 T ( x , t )
V  ( x, t )
Vector Field : V ( x , t )
Feynman Metric:
(Ignoring arbitrary phases)
Transformation properties of Dirac spinor bilinears:
P
C
CP
T
CPT
S:
 1 2

 1 2
 2 1
 2 1
 1 2
 2 1
P:
 1 5 2

 1 5 2
 2 5 1
 2 5 1
 1 5 2
 2 5 1
V:
 1  2

 1  2
 2  1
 2  1
 1  2
 2  1
 2   5 1
 2   5 1
 1   5 2
 2  5 1
 2s n 1
 1s n 2
 2s n 1
 2s n 1
c→c*
c→c*
A :  1   5 2
  1   5 2
T :  1s n 2

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 1s n 2
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The Standard Model Lagrangian
LSM  LKinetic  LHiggs  LYukawa
• LKinetic : •Introduce the massless fermion fields
•Require local gauge invariance => gives rise to existence of gauge bosons
=> CP Conserving
• LHiggs : •Introduce Higgs potential with <> ≠ 0
GSM  SU (3)C  SU (2) L  U (1)Y  SU (3)C  U (1)Q
•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
• LKinetic
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in mass eigenstates: CKM – matrix
=> CP-violating
=> CP-conserving!
=> CP violating with a single phase
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Fields: Notation
Fermions:
 1  5 
L  

 2 
 u I (3, 2,1 6) 
 I

d
(3,
2,1
6)

L i
•
I
uRi
(3,1, 2 3)
•
Leptons:
n I (1, 2,  1 2) 


I
l
(1,1,

1)

L i
•
I
l
• Ri (1,1, 1)
Scalar field:
July 1-2, 2004
with  = QL, uR, dR, LL, lR, nR
Interaction rep.
Quarks:
Under SU2:
Left handed doublets
Right hander singlets
 1  5 
; R  

 2 
Q = T3 + Y
•
  
 (1, 2, 1 2)   0 
 

I
QLi
(3, 2,1 6)
SU(3)C SU(2)L
Leftgeneration
handed index
•

•
Y
d RiI (3,1, 1 3)
LILi (1, 2,  1 2)
n RiI
Note:
Interaction representation: standard model
interaction is independent of generation
number
14
Fields: Notation
Q = T3 + Y
Explicitly:
• The left handed quark doublet :
I
I
I
I
I
I
I
I
I






u
,
u
,
u
c
,
c
,
c
t
,
t
,
t
r
g
b
r
g
b
r
g b
I
QLi (3, 2,1 6)   I I I  ,  I I I  ,  I I I 
 d , d , d   s , s , s  b ,b ,b 
 r g b L  r g b L  r g b L
T3   1
2
T3   1
2
(Y  1 6)
• Similarly for the quark singlets:
uRiI (3,1, 2 3) 
d RI i (3,1, 1 3) 
u , u , u  , c , c , c  , t , t , t 
 d , d , d  ,  s , s , s  , b , b , b 
• The left handed leptons:
I
r
I
r
I
r
I
r
I
r
I
r
R
I
r
I
r
I
r
I
r
I
r
R
I
r
R
I
r
I
r
I
r
I
r
 y  2 3
R
I
r
I
I
I


n



n
n

e
 
I
LLi (1, 2, 1 2)   I  ,   ,  I 
 e    I   
 L  L  L
• And similarly the (charged) singlets:
July 1-2, 2004
R
I
r
R
 y   1 3
T3   1 2
T3   1 2
lRiI (1,1, 1)  eRI ,  RI , RI
15
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:
L  i      
  x     x   ei  x 
  x     x   ei  x   x 
  x   ei  x   x  ;   x   ei  x   x 
Is the Lagrangian invariant?
   x   ei  x    x   iei  x   x     x 
Then:
i     i   
=> Introduce the covariant derivative:
and demand that A transforms as:
Conclusion:
July 1-2, 2004
     x 
D     ieA
1
A  A   A     x 
e
Not
invariant!
Then it turns out that:
L  L  L
is invariant!
• Introduce charged fermion field (electron)
• Demand invariance under local gauge transformations (U(1))
• The price to pay is that a gauge field A must be introduced at the same time (the photon)
16
LSM  LKinetic  LHiggs  LYukawa
LKinetic :
: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:
   D      ig s Ga La  igWb Tb  ig B  Y
Ga : 8 gluons
Wb : weak bosons: W1, W2, W3
B : hypercharge boson
Fields:
Generators:
July 1-2, 2004
L  i (    )
La : Gell-Mann matrices:
Tb : Pauli Matrices:
Y : Hypercharge:
½ la
½ b
(3x3)
(2x2)
SU(3)C
SU(2)L
U(1)Y
For the remainder we only consider Electroweak: SU(2)L x U(1)Y
17
LSM  LKinetic  LHiggs  LYukawa : The Kinetic Part


L kinetic : i (   )  i ( D   )
with   QLiI , uRiI , d RiI , LILi , lRiI
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:
L kinetic (QLiI )  iQLiI   D  QLiI
 iQLiI   (  
i
i
i
g s Ga la  gWb b  g B  ) QLI i
2
2
6
and similarly for all other terms (uRiI,dRiI,LLiI,lRiI).
Writing out only the weak part for the quarks:
I
Weak
Lkinetic
(u, d ) LI
i
I

 u 
 i u, d  L      g W1 1  W2 2  W3 3    
2

  d L
g I
 iuLI     uLI  id LI     d LI 
uL W   d LI 
2
uLI
g
July 1-2, 2004
dL
I
W+
L=JW
g I
d L  W   uLI
2
W+ = (1/√2) (W1+ i W2)
W- = (1/√ 2) (W1 – i W2)
 ...
18
LSM  LKinetic  LHiggs  LYukawa :
The Higgs Potential
LHiggs  D † D   VHiggs
→Note LHiggs = CP conserving
2
1 2 †
†
VHiggs       l   
2
V()
Symmetry
V
Broken
Symmetry
2  0 :
   0
2  0 :

 0 

    v


2


v   2 l
~ 246 GeV
Spontaneous Symmetry Breaking: The Higgs field adopts a non-zero vacuum expectation value
Procedure:
    e   im  

0
0
0 


e


i

m

  

 


And rewrite the Lagrangian (tedious):
(The other 3 Higgs fields are “eaten” by the W, Z bosons)
July 1-2, 2004

Substitute:
v H0
e  
2
0
1. GSM :  SU (3)C  SU (2)L U (1)Y    SU (3)C U (1)EM 
2. The W+,W-,Z0 bosons acquire mass
3. The Higgs boson H appears
“The realization of the vacuum breaks the symmetry”
19
LSM  LKinetic  LHiggs  LYukawa
: The Yukawa Part
Since we have a Higgs field we can add (ad-hoc) interactions
between  and the fermions in a gauge invariant way.
doublets
The result is:

Yij  Li 
 LYukawa 
L must be Hermitian (unitary)
singlet

 
Y L  l
Rj

h.c.


 Yijd QLiI  d RjI
 Yiju QLiI  uRjI

 h.c.
l
ij
I
Li
I
Rj
 0 1 *

1 0 

To be manifestly invariant under SU(2)
With:
Yijd
July 1-2, 2004
, Yiju
, Yijl
  is 2  *  
are arbitrary complex matrices which
operate in family space (3x3)
=> Flavour physics!
20
LSM  LKinetic  LHiggs  LYukawa
: The Yukawa Part
Writing the first term explicitly:


 I

d
I
I
Yij (uL , d L )i  0  d Rj
 
 d I I
 Y11 uL , d L


 Y d cI , d I
 21 L L


 Y31d t LI , d LI





  






d
I
I
d
I
I
 0  Y12 uL , sL  0  Y13 u L , bL  0  
 
 
  
 d RI 



  



  I 


d
I
I
d
I
I
 0  Y22 cL , sL  0  Y13 cL , bL  0     sR 
 
 
     I 
bR 


 
  






d
I
I
d
I
I
 0  Y32 t L , sL  0  Y33 t L , bL  0  
 
 
  


















Question:
In what aspect is this Lagrangian similar to the example of the
nucleon-meson potential?
July 1-2, 2004
21
LSM  LKinetic  LHiggs  LYukawa


 LYukawa  Yij  Li   Rj
 h.c.
Formally, CP is violated if:

: The Yukawa Part

m det Y d Y d † , Y uY u †   0
In general LYukawa is CP violating
Exercise (intuitive proof)
Show that:
• The hermiticity of the Lagrangian implies that there
are terms in pairs of the form:
Yij Li Rj
 Yij* Rj † Li
• However a transformation under CP gives:
 Li Rj
  Rj † Li
and leaves the coefficients Yij and Yij* unchanged
July 1-2, 2004
CP is conserved in LYukawa
only if Yij = Yij*
22
LSM  LKinetic  LHiggs  LYukawa
: The Yukawa Part
There are 3 Yukawa matrices (in the case of massless neutrino’s):
Yijd
, Yiju
, Yijl
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
July 1-2, 2004
23

S.S.B
LYukawa
LMass
: The Fermion Masses
Start with the Yukawa Lagrangian
 LYuk


 I

d
I
I
 Yij (uL , d L )i  0  d Rj
 
S .S .B. : e  0  
 Yiju ...  Yijl ...
vH
2
After which the following mass term emerges:
 LYuk 
 L Mass
 d LiI M ijd d RjI
 lLiI M ijl lRjI
with
v d
M 
Yij
2
d
ij
 uLiI M iju uRjI
 h.c.
v u
, M 
Yij
2
u
ij
v l
, M 
Yij
2
l
ij
LMass is CP violating in a similar way as LYuk
July 1-2, 2004
24
 LMass
S.S.B
LYukawa
: The Fermion Masses
Writing in an explicit form:





LMass   d , s , b  M d
I
I
I
L
 I
 d 
  sI  
 I
 b 
 R

u , c , t   M
I
I
I
L
uI 
u   c I  
 
  tI 
 R
 e ,  ,  M
 I
e 
l  I
 I
  
 R
f
f
I
I
I
L
V




V
 h.c.
The matrices M can always be diagonalised by unitary matrices
L and R such that:

 d I 
 I I I f† f
f
f
f†
f  I 
d
,
s
,
b
V
V
M
V
V
VLf M f VRf †  M diagona
L
L
R
R  s 

l

 bI 
 



Then the real fermion mass eigenstates are given by:
d Li  VLd   d LjI
d Ri  VRd   d RjI
uLi  VLu   uLjI
uRi  VRu   uRjI
lLi  VLl   lLjI
lRi  VRl   lRjI
ij
ij
ij
dLI , uLI , lLI
dL , uL , lL
July 1-2, 2004
ij
ij
ij
are the weak interaction eigenstates
are the mass eigenstates (“physical particles”)
25
LYukawa
 LMass
S.S.B
In terms of the mass eigenstates:
 L Mass 
 d , s, b 
L

 e,  , 
L
L Mass 
 md

ms



 me

m



: The Fermion Masses



mb 
d 
 
 s   u , c, t
b
 R
 e
 
     h.c.
m     R


L
 mu




mc



mt 
u 
 
c
t
 R
mu uu  mc cc  mt tt
 md dd
 ms ss  mbbb
 me ee  m 
= CP Conserving?
 m
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
July 1-2, 2004
=>What happened to the charged current interactions (in LKinetic) ?
26
LW
 LCKM
: The Charged Current
The charged current interaction for quarks in the interaction basis is:
 LW 
g
2


uLiI
d LI i W
The charged current interaction for quarks in the mass basis is:
 LW 

g
2
uLi VLu
  VLd † d Li W
VCKM  VLu VLd † 
The unitary matrix:
With:
†
VCKM  VCKM
1
is the Cabibbo Kobayashi Maskawa mixing matrix:
LW 

g
2
d 
 u , c , t L VCKM   s 
b
 L
Lepton sector: similarly
  W
VMNS  VLn  VLl † 
However, for massless neutrino’s: VLn = arbitrary. Choose it such that VMNS = 1
=> There is no mixing in the lepton sector
July 1-2, 2004
27
Flavour Changing Neutral Currents
To illustrate the SM neutral current take the W3 and B term of the Kinetic Lagrangian:
g 
g  I
- L NC (Q )   iQ  ( W3  3  B ) QLi
2
6
And consider the Z-boson field:
Z   cos W W3  sin W B 
I
Li
I
Li 
- LZ (QLiI )  
g
cos W
and
tan W  g  g
 1 1 2  I
 I
   sin W  QLi  Z QLi
 2 3

Take further QLiI=dLiI
 1 1 2  I
 I
   sin W  d Li  Z d Li
 2 3

g  1 1 2 
d
d†

 
   sin W  d Li  VL VL d Li Z
cos W  2 3

- L Z (d LiI )  
g
cos W
- LZ (QLi )  
g
cos W
 1 1 2 

   sin W  d Li  d Li Z
 2 3

In terms of physical fields no non-diagonal
contributions occur for the neutral
Currents. => GIM mechanism
July 1-2, 2004
Use:
(VRu †VRu  VRd †VRd  1)
 ...   uLi  uLi Z 
Standard Model forbids flavour changing
neutral currents.
28
Charged Currents
The charged current term reads:
g I   I
g I   I


uLi W d Lj 
d Lj  W u Li  J CC
W  J CC
W
2
2
 1  5    *  1  5 
g  1  5     1  5 
g

ui 
dj 
  W Vij 
dj 
  W Vij 
 ui
2  2 
2  2 
 2 
 2 
g
g

ui WVij 1   5  d j 
d j  WV ji* 1   5  ui
2
2
LCC 
(Together with (x,t) -> (-x,t))
Under the CP operator this gives:
CP
LCC 

g
d j WVij 1   5  ui
2

g
ui WiVij* 1   5  d k
2
A comparison shows that CP is conserved only if
Vij = Vij*
In general the charged current term is CP violating
July 1-2, 2004
29
Charged Currents
The charged current term reads:
g I   I
g I   I


uLi W d Lj 
d Lj  W u Li  J CC
W  J CC
W
2
2
 1  5    *  1  5 
g  1  5     1  5 
g

ui 
dj 
  W Vij 
dj 
  W Vij 
 ui
2  2 
2  2 
 2 
 2 
g
g

ui WVij 1   5  d j 
d j  WV ji* 1   5  ui
2
2
LCC 
(Together with (x,t) -> (-x,t))
Under the CP operator this gives:
CP
LCC 

g
d j WVij 1   5  ui
2

g
ui WiVij* 1   5  d k
2
A comparison shows that CP is conserved only if
Vij = Vij*
In general the charged current term is CP violating
July 1-2, 2004
30
Where were we?
July 1-2, 2004
31
The Standard Model Lagrangian (recap)
LSM  LKinetic  LHiggs  LYukawa
• LKinetic : •Introduce the massless fermion fields
•Require local gauge invariance => gives rise to existence of gauge bosons
=> CP Conserving
• LHiggs : •Introduce Higgs potential with <> ≠ 0
GSM  SU (3)C  SU (2) L  U (1)Y  SU (3)C  U (1)Q
•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
• LKinetic
July 1-2, 2004
in mass eigenstates: CKM – matrix
=> CP-violating
=> CP-conserving!
=> CP violating with a single phase
32
Quark field re-phasing
Under a quark phase transformation:
d Li  eidi d Li
u Li  eiui u Li
and a simultaneous rephasing of the CKM matrix:
 e u

V 


e c
e t
  Vud

  Vcd
  Vtd

Vus Vub   e d

Vcs Vcb  
Vts Vtb  
the charged current

J CC
 uLi Vij d Lj
Degrees of freedom in VCKM in
Number of real parameters:
Number of imaginary parameters:
Number of constraints (VV† = 1):
Number of relative quark phases:
Total degrees of freedom:
Number of Euler angles:
Number of CP phases:
July 1-2, 2004
e  s



b 
e 
or


V j  exp i  j    V j
is left invariant
3
N generations
9
+ N2
9
+ N2
9
- N2
5
- (2N-1)
----------------------4
(N-1)2
3
N (N-1) / 2
1
(N-1) (N-2) / 2
2 generations:
VCKM
 cos 

  sin 
sin  

cos  
No CP violation in SM.
This is the reason Kobayashi
and Maskawa first suggested
a third family of fermions!
33
The LEP collider @ CERN
Aleph
L3
Delphi
Opal
MZ
Maybe the most important result of LEP:
“There are 3 generations of neutrino’s”
Light, left-handed, “active”
July 1-2, 2004
34
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 antiparticle → 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
July 1-2, 2004
35
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 don’t we measure charged lepton oscillations rather then neutrino
oscillations?
W+
lL I

J CC

νL I
ne

n
 
g
g
li  Vijn j  
liVij  n j
2
2
However, observation of neutrino oscillations is
possible due to small neutrino mass differences.
W
July 1-2, 2004
36
Rephasing Invariants
The standard representation of the CKM matrix is:
 Vud

V   Vcd
V
 td
Vus Vub  
c12 c13
 
Vcs Vcb     s12c23  c12 s23 s13ei
Vts Vtb   s12 s23  c12 c23 s13ei
s12 c13
c12c23  s12c23 s13e i
c12 s23  s12 c23 s13ei
s13e i
s23c13
c23c13





cij  cos ij
sij  sin ij
However, many representations are possible. What are the invariants under re-phasing?
•
•
Simplest: Ui = |Vi|2 is independent of quark re-phasing
Next simplest: Quartets: Qibj = Vi Vbj Vj* Vbi* with ≠b and i≠j
– “Each quark phase appears with and without *”
•
V†V=1: 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
Im[Vi Vbj Vj* Vbi*] = J ∑eb eijk where J is the universal Jarlskog invariant
Amount of CP Violation is proportional to J
July 1-2, 2004
37
The Unitarity Triangle
The “db” triangle:
VudVub*  VcdVcb*  VtdVtb*  0
 VtdVtb* 
  arg  
 arg  Qubtd 
* 
 VudVub 

Vud Vub
unitarity: VCKM† VCKM = 1
 VcdVcb*
b  arg  
*
 VtdVtb
Vtd Vtb*
*

  arg  Qtbcd 

 VudVub* 
  arg  
 arg  Qcbud 
* 
V
V
 cd cb 
Area = ½ |Im Qudcb| = ½ |J|

b
Vcd Vcb*


Under re-phasing: V j  exp i  j    V j
the unitary angles are invariant
(In fact, rephasing implies a rotation of the whole triangle)
July 1-2, 2004
38
Wolfenstein Parametrization
Wolfenstein realised that the non-diagonal CKM elements are relatively
small compared to the diagonal elements, and parametrized as follows:
 1 l2 / 2
l

V 
l
1 l 2 / 2
 Al 3 1    i   Al 2

Al 3    i  

4
Al 2
  O l 

1

 Vud

  Vcd
 Vtd eib

Normalised CKM triangle:
,
Vus
Vcs
Vts
Vub ei
Vcb
Vtb







(0,0)
July 1-2, 2004
b
(1,0)
39
CP Violation and quark masses
Note that the massless Lagrangian has a global symmetry for unitary
transformations in flavour space.
Let’s 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:
 cos 

VCKM     sin  cos 
 sin  sin 

sin 
0 

cos  cos  sin  
 cos  sin  cos  
Necessary criteria for CP violation:
July 1-2, 2004
CP conserving
mu  mc
, mc  mt
, mt  mu
md  ms
, ms  mb
, mb  md
,
40
The Amount of CP Violation
Using Standard Parametrization of CKM:

c12 c13

V    s12 c23  c12 s23 s13ei
 s12 s23  c12c23 s13ei

s12c13
c12c23  s12c23 s13ei
c12 s23  s12c23 s13ei
s13e  i
s23c13
c23c13





J  c12c23c132 s12 s23s13 sin    3.0  0.3 105
cij  cos ij
sij  sin ij
(eg.: J=Im(Vus Vcb Vub* Vcs*) )
(The maximal value J might have = 1/(6√3) ~ 0.1)
However, also required is:
m
2
t
 mc2   mc2  mu2   mt2  mu2   mb2  ms2   ms2  md2   mb2  md2   0
All requirements for CP violation can be summarized by:


m det  M d M d† , M u M u† 
  2 J  mt2  mc2   mc2  mu2   mu2  mt2 
  mb2  ms2   ms2  md2   md2  mb2 
 6 105  4 1010 (GeV12 )  0  CP Violation
July 1-2, 2004
Is CP violation maximal? => One has to understand the origin of mass!
41
Mass Patterns
Mass spectra ( = Mz, MS-bar scheme)
Observe:
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
mu
mc
mc
mt
l4 ,
me = 0.51 MeV , m = 105 MeV
md
ms
ms
mb
l2
, m = 1777 MeV
Why are neutrino’s 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
July 1-2, 2004
42
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 -ray sources in the universe due to matter anti-matter collisions
July
1-2, 2004
Hubble
deep
field - optical
43
Big Bang Cosmology
Equal amounts
of matter &
antimatter
q+q⇄ +
July 1-2, 2004
Matter Dominates !
+ CMB
44
The matter anti-matter asymmetry
Cosmic Microwave Background
WMAP satellite
Angular Power Spectrum
 T 1 , 1 
 T 2 , 2 

July 1-2, 2004
l
N baryons
N photons
2.7248K
2.7252K
 T  ,    almYlm  , 
10
 (6.50.4
)

10
0.3
Almost all matter annihilated with
antimatter, producing photons…
45
The Sakharov conditions
NB  NB

NB  NB
Convert 1 in 109 anti-quarks into a quark in an early stage of universe:
NB  NB
 109
N
Anti-
A matter dominated universe can evolve in
case three conditions occur simultaneous:
1) Baryon number violation: L(B)≠0
2) C and CP Violation: G(N→f) ≠ G(N→f)
3) Thermal non-equilibrium:
otherwise: CPT invariance => CP invariance
July 1-2, 2004
Sakharov (1964)
46
Baryogenesis at the GUT Scale
Conceptually simple
GUT theories predict proton decay mediated by heavy X gauge bosons:
d
u
u
e+
X
u- 0
p
u
proton
X boson has baryon number violating (1)
couplings: X →q q, X→q l
Proton lifetime:   1032 s
Efficiency of Baryon asymmetry build-up:
A simple Baryogenesis model:
CP Violation (2) : r ≠ r
BX
 r   2 / 3 
BX
 r   2 / 3 
Decay
process
Decay
fraction
B
X→qq
r
2/3
X→ql
1-r
1/3
X→qq
r
-2/3
nq
X→ql
1-r
-1/3
n
BX BX

1  r   1/ 3
1  r    1/ 3
1 / 3   r  r 
Assuming the back reaction does not occur (3):

n
1
r  r  X
3
n0
Initial X number density
Initial light particle
number density
July 1-2, 2004
47
Baryogenesis at Electroweak Scale
Conceptually difficult
SM Electroweak Interactions:
1) Baryon number violation in weak anomaly:
Conserves “B-L” but violates “B+L”
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 < ~ 70 GeV/c2 , from LEP mH > ~ 100 GeV/c2
2. CP Violation in CKM is not enough:
Requirement:
N  B
 109
N  
from CKM:
N B  N B J CKM
 12  1020
NB  NB
Tc
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.
July 1-2, 2004
48
Conclusion
Key questions in B physics:
• Is the SM the only source of CP Violations?
• Does the SM fully explain flavour physics?
Biertje?
July 1-2, 2004
49