Generation of Quark and Lepton
Masses in the Standard Model
International WE Heraeus Summer School on
Flavour Physics and CP Violation
Dresden, 29 Aug – 7 Sep 2005
Aug 29-31, 2005
M. Jezabek
1
Preliminaries
Metric: g00 = -g11 = -g22 = -g33 = 1
Dirac field Ψ (s = ½ )
{ γμ , γν } = 2gμν , { γμ , γ5 } = 0
(μ,ν = 0, 1, 2, 3)
γ5† = γ5 , Tr γ5 = 0, γ5 2 = I4
Left- (right-) handed fields:
γ5 ΨL = - ΨL, γ5 ΨR = ΨR
ΨL = ½ (1 - γ5) Ψ, ΨR = ½ (1 + γ5) Ψ
Aug 29-31, 2005
M. Jezabek
2
Weak charged current (one generation)
J  2[ L  e  uL  d L ]

L
and weak interaction (four fermion) Hamiltonian
HW
GF
† 

J J
2
are invariant under chiral transformations
e
Mass terms
i  

m  m(R L  L R )
flip chirality and break the chiral invariance of
the weak interaction theory
Aug 29-31, 2005
M. Jezabek
3
Weyl spinors
In the chiral (Weyl) basis
with

 0

  



 1 0

I  
0 1
 L 
  ,
 R 
Aug 29-31, 2005
 5 i   
0

0 

 I, 
i

0 1

  
1 0
1
1
2
3
 I

 0

0

I

   I ,    
0  i

  
i
0


2
1 0 

  
 0  1
3
L , R
M. Jezabek
4
Dirac equation (m = 0):

p   0
 0

p 
 
with:

p  L 
   0
0  R 

p  E,  p
 
 ER  p   R

 
 EL   p   L
for m = 0 chirality ↔ helicity
Aug 29-31, 2005
M. Jezabek
5
Lorentz transformations
 x   S x'   e
with
    
i   
 x' 
i
[    ]
2
The generators of rotations Ji and of boosts Κi satisfy the
commutation relations:
[ J i , J j ]  i ijk J k
[ i ,  j ]  i ijk J k
In a more convenient basis
1
N i  ( J i  i i )
2
[ Ni , N j ]  i ijk Nk
[ Ni , N ]  0
'
i
Aug 29-31, 2005
[ J i ,  j ]  i ijk  k
1
N i  ( J i  i i )
2
'
[ Ni' , N 'j ]  i ijk Nk'
 SU2  SU2
M. Jezabek
6
For Weyl fields ( [  ,   ]  0
L  S L L
R  S R R
For
LR :
)
SL  e
with
SR  e
1
Ji  
2
M. Jezabek







   
i  
i
 i   
2
right – handed fermions → ( ½, 0 )
left– handed fermions → ( 0, ½ )
Aug 29-31, 2005

   
i  
of
SU2  SU2
7
Parity
For the generators of rotations and boosts
 i   i
(pseudo - vectors)
(vectors)
Ni  Ni ' ,
Ni '  Ni
Ji  Ji
and
Under parity transformation:
 L 
 R   0 I 

  


 P 
 R 
 L   I 0 
P  P  0 
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M. Jezabek
8
Charge conjugation
For the Pauli matrices
  *  
2
 2 S L * 2  2e

and
i
2
i


i 
  *   i 
2
 2  SR
 S R *  S L
2
2
Under a Lorentz transformation
R   L   SL L   SL   L  SRR
2
*
L   R  S LL
2
Aug 29-31, 2005
*
2
*
*
M. Jezabek
2
*
2
2
*
9
Charge conjugated bi-spinor:
 C
c
with
C  i 
2

0
*




L 
L
C
2 0 0T
  i    *  
 
 

 R
 R 
C
 0
 


i


Aug 29-31, 2005
i    L *   i  R * 

 *  
*

0   R    i L 
M. Jezabek
10
Charge – conjugation matrix C
For a Dirac particle in em field

[ (i   eA )  m]
Complex conjugation flips the relative sign
between   and A

[ * (i   eA )  m] 
*
  († )T  (  )T    C 1C
T
with
Aug 29-31, 2005
 C
C
M. Jezabek
11
If
*
T
T

C [ (i   eA )  m] C  (i   eA )  m
one obtains

1

[ (i   eA )  m] 
C

0T
0T
C
 
*
 
*
0T
0T
 
 ( 
0

†
 ) 
0 T
T
The charge conjugation matrix fulfills the relations:
C  C
T
1
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  ,

C C ,
†
M. Jezabek
-1
C  C
T
12
Fermion masses
1. Dirac mass term
For two Weyl spinors L and R
†
†
LD  m(R L L R )
is invariant under Lorentz and parity transformations:
L   L e
1   
( i  )
2
  
1
(

i

 )
† 2
†
, R   R e
In bi-spinor notation
 0 I   L 
   
R L L R  (L , R ) 
 I 0   R 
†
†
†
†
  †    
Aug 29-31, 2005
M. Jezabek
13
2. Majorana mass term
For a left – handed Weyl spinor L
R  i L
2
*
is a right –handed object and
T
†
*
LM  M L (L i L  L i L )
is Lorentz invariant
In bi-spinor notation:
1
L  (1   5 )
2
1
T
T
C
T
( L )  CL  C (1   5 )  * 
2
1
1
1
T
T
*
T
 C (1   5 )   (1   5 )C  (1   5 )  C
2
2
2
is a right – handed field
Aug 29-31, 2005
M. Jezabek
14
The Majorana mass term reads


LM L  M L L L  h.c.
C
and for a right – handed field R


LM R  M R R R  h.c.
Aug 29-31, 2005
C
M. Jezabek
15
GWS Theory of Electroweak
Interactions
SU2  U1
local gauge symmetry
 L 
L    2,
 eL 
doublet SU2
R  eR  1
singlet SU2
The most general unitary transformation
L  UL ,
R  VR
includes the lepton – number phase transformation
L  ei L ,
R  ei R
which is not a local gauge symmetry.
Aug 29-31, 2005
M. Jezabek
16
Generators:
1
Ti   i
2
(i  1, 2, 3);  i- Pauli matrices
Y  Q  T3
1
Y ( L )   ,
2
Q - em charge operator
Y ( R )  1
The gauge fields interact with matter fields L
and R through the covariant derivative
 
D     igW  T  ig ' BY
ˆ  i D
ˆ
L  iL D
L
R ' R
where
  g '
'
D     igW   i B , D     ig ' B
2
2 
Aug 29-31, 2005
M. Jezabek
17
Spontaneous Symmetry Breaking
Scalar field:
  
  0 ,
 
1
Y
2
1 
1
D  (   ig   W  i g ' B )
2
2
with non – zero vacuum expectation value

0 



   exp i  x   / 2v 
v / 2 
In the unitary gauge


  ,

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M. Jezabek
v  '
 
2
0
18
Higgs mechanism:
1
  '   '
2

( D ) ( D  ) 
†
1 2 1 1
2
2
3
'
3
[ g (W W  W W )  ( gW   g B )(gW  g ' B  )]v 2  ...
8
Mass eigenstates:
A  cos w B  sin  w W 3 
Z    sin  w B  cos w W
W
with:
Aug 29-31, 2005

mA  0
3
cos w 
g2  g
'2
,
sin  w 
M. Jezabek
mW
2

1

(W 1  iW 2 )
2
g
mZ
2
'2
g g 2

v
4
g2 2

v
4
2
g'
g2  g'
2
19
The electromagnetic field
1 3
1
A  W   ' B
g
g
couples to
1
1 '
( gT3 )  ' ( g Y )  T3  Y  Q
g
g
and U(1)Q gauge symmetry remains unbroken
Aug 29-31, 2005
M. Jezabek
20
Quarks
u
L    2,
 d L
u R  1,
d R  1,
Aug 29-31, 2005
1
Y
6
2
Y
3
1
Y 
3
M. Jezabek
21
Fermion masses
Yukawa couplings:
   
LY   f e  e , e L   eR
 
  
 f d u, d L   d R
 
~
 f u u, d L u R  h.c

where:


0* 


  i 2 *    
  
~
Note: in some extended models
different Higgs doublets
Aug 29-31, 2005
M. Jezabek

and
~
 may be
22
LY
is
 1 1
 2  2 1

 1 1 1
Y    
 6 2 3
 1  1  2
 6 2 3
Aug 29-31, 2005
SU2  U1 invariant. For example:



for










,
e

e
R
L
 e

 u , d L  d R

~
 u , d L  u R

M. Jezabek



respectively




23
Under weak isospin transformations
 U ,
where
U e
~
L  UL ,
i k k
  i 2
*
R  R
is a unitary matrix and detU = 1

~
i 2U    2U  2
* *
*
 2 i  2   i

 2U * 2  U
~
~
  U transforms as a SU2 doublet.
*
For the mass terms:
L  R

~
L  R
Aug 29-31, 2005

LU †U R  L  R
~
L  R
M. Jezabek
24
The vacuum expectation value of
symmetry. For example:
f e  e ,

breaks chiral
 0  
v


e L 
v / 2
eR  f e 2 eL eR



Dirac masses for charged leptons and quarks
me  f e
v
,
2
md  f d
v
,
2
If a right – handed neutrino exists:

 f  , e

Problem: why
Aug 29-31, 2005

L
~
 νR

f  f charged
leptons
M. Jezabek
mu  f u
 R  1,
v
2
Y 0
v
m  f
2
 f quarks
?
25
Generations
Three generations of quarks and leptons (   ):
Q L :
u
  ,
 d L
u R :
s
  ,
 c L
b
 
 t L
uR ,
cR ,
tR
d R :
dR ,
sR ,
bR
L L :
 e 
  ,
 e L
l R :
 R :
Aug 29-31, 2005
  
  ,
  L
R ,
eR ,
  
 
  L
R
?
M. Jezabek
26
Yukawa couplings + SSB
f  R   L  mD  R  L
†
with:
mD  f   

The mass matrices mD are complex and can be
diagonalised by bi – unitary transformations
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M. Jezabek
27
Any (n x n) complex matrix A can be diagonalised by
a bi – unitary transformation
U † AV  D
with D diagonal, and U and V unitary.
Proof
†
AA† and A A hermitian
•
•
The eigenvalues of
•

AA† and A† A are the same
( A† A) X  X

( AA† ) AX   ( AX )
real and non – negative

X † A† AX  X † X  ( AX )† ( AX )  0
unitary U and V:
U 1 AA†U  D 2
V 1 A† AV  D 2
with D diagonal and real. The columns of AV and U are
proportional
AV  U  D
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M. Jezabek
28
Diagonalisation of the mass matrices
~
V 1mDW  m
D
~
with mD diagonal leads to relations between the mass eigenstates
'
L, R
and the weak interactions eigenstates  L, R :
 ' L  WL ,
 ' R  VR
Weak charged current for quarks
J   L' (Q  2 / 3)  L' (Q  1/ 3)  L (Q  2 / 3) Uq L (Q  1/ 3)
Quark mixing matrix
Uq  W † (Q  2 / 3)W (Q  1/ 3)
(Cabibbo; Kobayashi, Maskawa)
Note: in GWS theory only U q is observable
Aug 29-31, 2005
M. Jezabek
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For (n x n) unitary matrix U q:
n real parameters
-(2n - 1) phases of u L and d L

p = ( n – 1 )2 observable real parameters
A common convention:
where
prs
1
n(n  1)
2
1
s  (n  1)(n  2)
2
rotation angles
r
complex phases
For n = 3 (e.g. PDG):
0   c13
0 s13e i   c12 s12 0 
1 0




VCKM   0 c23 s23    0
1
0     s12 c12 0 
 0  s c    s e i 0 c   0
0 1 
23
23  
13
13  

with
cij  cosij ,
sij  sin ij
Parameters: 12 ,  23 , 13
Aug 29-31, 2005
and
M. Jezabek
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30
Masses of neutrinos
A.
Dirac neutrinos
Three right – handed sterile neutrinos   R ,
  
LD    R mD  L  h.c.
 
B.
Dirac neutrinos in Majorana form
  
nL   L c 
 ( R ) 
with
 c  C 
1
LD   (nL ) C Mn L  h.c.
2
with
 e L 


 L    L ,
 
 tL 
 o
M  
 mD
 1 R 


 R   2 R 
 
 3R 
T
mD 

o 
1
1 C
T
C
LD    R mD L  ( L )mD  R   R mD L
2
2
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M. Jezabek
31
C  C 1    ,
C †  C 1 ,
T
C T  C
( L )  (C L )†     L C †    L   C 1    L C 1 (  )    L C 1
C
T
*
( L ) mD  R   L mD C 1C R
C
T
C
T
T
T
T
T
( R mD L )T   R mD L

T
T
↑
anticommutation of fermion fields
The matrices of Majorana masses are symmetric:
 (  ) c M      C 1M      C 1M     (  ) c M  
T
 
T
 
1
↑
 
 
and C are antisymmetric,
anticommutation of fermion fields
C
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M. Jezabek
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Neutrino masses – general case
 L 

nL  
c
 ( R ) 
DM
L
and
M
with
 1 
 
R    
 
 nR  R
1
C
DM
  (nL ) M
nL  h.c.
2
D M
ML
 
 mD
ML  ML,
T
M D M
 e 
 
 L     and
 
 t L
T
mD
MR




MR  MR
T
is a symmetric complex matrix of dimension
(3 + nR) x (3 + nR)
Note: for quarks and charged leptons M L  M R  0
due to electric charge conservation
Aug 29-31, 2005
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Any symmetric complex n x n matrix can be diagonalised
by a transformation:

M
U MU  
T
where U is unitary,
and i  0
  diag (1  n ) with i
real
M  A  iB
Let
where
()
A
and
B
are real and symmetric, and
A B 

M  
 B  A
is a 2n x 2n real and symmetric matrix
Aug 29-31, 2005
M. Jezabek
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Let
and
 ν i ,1 


v i    ,
ν 
 i ,n 
 w i ,1 


w i    ,
w 
 i ,n 
i  1,2,, n
 vi 
 vi 
M    i  
 wi 
 wi 
i. e.
Av i  Bw i  i v i
Bv i  Aw i  i w i
which implies that

 wi 
 wi 
  i 

M 
  vi 
  vi 
Eigenvalues of M are real and equal to  i ,
(i  1,, n)
and at least n of them are non - negative
Aug 29-31, 2005
M. Jezabek
35
()

MU  U *
(' )
Let U  V  iW where V and W are real matrices which fulfill
the following system of equations:
Solution:

 AV  BW  V

BV  AW  W







( )
V  ( v1 , v 2 ,, v n ),
W  (w1 , w 2 ,, w n ),
For k-th columns in ( ) one obtains
 vk 
 A B  v k 

   k  
 B  A  w k 
 wk 
It follows that () is fulfilled and M is diagonalised by
U  ( v1  iw1, v 2  iw 2 ,, v n  iw n )  (u1, u2 ,, un )
Unitarity of U:
†
T
T
T v 
u m u n  ( v m  iw m )T ( v n  iw n )  v m v n  w m w n  v m , w m   n    mn
 wn 
Aug 29-31, 2005
M. Jezabek
36
Massive neutrinos in the Standard Model
Before 1998 (SuperK):
m  0
A simple extention of SM:
 1 
 
R    
 
 nR  R
and
 
nL   CL 
 R  L
The right – handed neutrinos  R are sterile. For singlets of gauge
group SU3 x SU2 x U1 explicit Majorana masses are allowed
 a new mass scale|MR|
Two mass scales:  mD   v  102 GeV
| mR |

?

The Majorana masses of the active neutrinos
are
L
forbidden by the electroweak SU2 x U1 gauge symmetry
ML = 0
Aug 29-31, 2005
M. Jezabek
37
Seesaw Mechanism
DM
L
with:
1
  (nL )C M D  M nL  h.c.
2
M
DM
 0
 
 mD
T
mD
MR




  mD  / | M R | 1
M D  M  U T M D  M U , nL  nL '  U †nL
with a unitary matrix
and
 I  A† 
  O ( A2 ,)
U  
I 
A
| A |  1
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M. Jezabek
38
 I
M   *
 A
T
A
I
'
with



 0 mD   I  A   

 
  
 mD M R   A I   

T
 

'
MR 

  mD  M R A  A*mD A
T
  AT mD  mDT A  AT M R A
M R  M R  mD A†  A*mD  M R  
'
T
 M is in a block – diagonal form
/
   
if
1
A   M R mD    
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M. Jezabek
39

with

M  
0
'
0 
  
MR 
1
 m M R mD
T
D
†

  L 
I
A
'
  C 
nL  
 

A
I

  R

 '    A† C    m † ( M 1 )† C  
L
R
L
D
R
R
 L
 C'
1
C




M
 R
R
R mD L  
Aug 29-31, 2005
M. Jezabek




40
Low mass sector
|  |  | mD | 
 L   L  O ( )
'
 R   R  O ( )
'
For
Aug 29-31, 2005
| mD |  1MeV
   
| mD |  10 2 GeV
   12
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41
Aug 29-31, 2005
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42