Unification of Quarks and Leptons
or Quark-Lepton Complementarities
?
Bo-Qiang Ma
Peking University (PKU)
in collaboration with Nan Li
November 16, 2007, Talk at NCTS @ NTHU
1
Basic structure of matter
Basic interactions
2
Properties of Fermions
3
The unification of quarks and leptons
• Why there are three generations？
• The origin of masses and their
relations？
• The mixing between different
generations：why mixing？
• Is it possible for a unification of
quarks and leptons？
4
Parametrization of Lepton Mixing Matrix
Are there connections between the
parametrizations of quark and lepton
mixing matrices？
Cabibbo, Kobayashi,
Maskawa (CKM) Matrix
u,c,t quarks couple to
superposition of other quarks
Weak eigenstates
 d w   Vud
 ÷
 s w ÷=  Vcd
b ÷  V
 w   td
Vus
Vcs
Vts
Vub  d 
 ÷
÷
Vcb ÷
s÷
b ÷
Vtb ÷
 
Mass eigenstates
 d w   0.975 0.22 0.005  d 
 ÷
 ÷
÷
 s w ÷=  0.22 0.97 0.04 ÷
s÷
 b ÷  0.005 0.04 0.99 ÷
 ÷
 w 
 b 
Unitarity (or lack thereof) of CKM matrix
tests existence of further quark generations
and possible new physics (eg. Supersymmetry)
w/ Particle
Data Group ‘01
Central Values
6
Neutrino Oscillations
e
.

.

This is only possible if neutrinos have mass
 new physics beyond the Standard Model
Two flavor case
Assuming  e ,   are flavor eigenstates,  1 ， 2
are mass eigenstates.
Their mixing are
 e

 

  cos
=
   sin 

sin     1 
   .
cos    2 
Assuming at t = 0 , there is only electron neutrino,
i.e.,
 (0) =  e = cos 1  sin   2 ,
8
at t,
 (t ) = cos 1 eiE t  sin   2 eiE t ,
1
1 ,  2
Express
in terms of
2
e , 
，one obtains
 (t ) = (cos 2  eiE t  sin 2  eiE t )  e  sin  cos (eiE t  eiE t )   .
1
2
1
2
Then the probablity of finding neutrino  at t is
   (t )
using
2
= sin  cos  e
E1 =
2
2
 iE2t
e
 iE2t 2
m12
2
2
p  m1  p 
and E2 =
2E
E1  E2
= sin 2 sin
t.
2
2
2
m22
p m  p
,
2E
2
2
2
9
one obtains
2
2
2
2
m

m
m

m
2
2
2
2
1
2
1
2 L
   (t ) = sin 2 sin
t = sin 2 sin
.
4E
4E c
4 pc
,
The oscillation period is 2
2
m1  m2
2
And the oscillation probability is
   (t )
2
2
2

m
(
eV
)
= sin 2 2 sin 2 1.27
L(km).
E (GeV )
Generating to three generations，U = U lU ,i.e., the
lepton mixing matrix，is called PMNS matrix.
B. Pontecorvo, JETP 7 (1958) 172;
Z. Maki, M. Nakagawa, and S. Sakata, Prog. Theor. Phys. 28 (1962)
870.
10
Mixing matrices of quarks and leptons
1

c1


 s1

  c2

s1  
c1    s2 ei

c2 c3

=  c1s3  s1s2 c3ei
 s1s3  c1s2 c3ei

s2 e  i   c3

1
   s3
c2  
c2 s3
c1c3  s1s2 s3ei
 s1c3  c1s2 s3ei
s3
c3



1 
s2 e  i 

s1c2 
c1c2 
11
Data for quark and lepton mixing angles
夸克与轻子混合角的实验数据
quark mixing angles

lepton mixing angles
= 2.4  0.11 ,
1PMNS = 45.0  6.5 ,

CKM
2
= 0.2  0.04 ,

CKM
3

CKM
CKM
1

PMNS
2
= 0  7.4 ,
= 12.9  0.12 ,

PMNS
3
= 32.6  1.6 ，
= 59  13 .

PMNS
uncertain.
12
Data for quark mixing matrix
夸克混合矩阵的实验数据
 0.9739  0.9751 0.221  0.227 0.0029  0.0045 


0.221

0.227
0.9730

0.9744
0.039

0.044


 0.0048  0.014

0.037

0.043
0.9990

0.9992


Data for lepton mixing matrix
轻子混合矩阵的实验数据
 0.20 
 0.77  0.88 0.47  0.61


0.19

0.52
0.42

0.73
0.58

0.82


 0.20  0.53 0.44  0.74 0.56  0.81


13
Parametrization of quark mixing matrix
夸克混合矩阵的参数化
1 2


 1 2 

1 2

V=

1 

2
 3
2
A

(1



i

)

A




A 3 (   i ) 


A 2


1


 = 0.2243  0.0016,
 = 0.33.
 = 0.20,
A = 0.82,
L. Wolfenstein, Phys. Rev. Lett. 51 (1983) 1945.
14
Parametrization of lepton mixing matrix
轻子混合矩阵的参数化
Base matrix：
（1）Bimaximal Matrix （2）Tribimaximal Matrix
 2

 2
 1

 2
 1

 2
2
2
1
2
1

2

0 

2

2 
2

2 
1 = 45 ， 2 = 0 ， 3 = 45 .
 6

 3

6

 6
 6

 6
3
3
3
3
3

3

0 

2

2 
2

2 
1 = 45 ， 2 = 0 ， 3 = 35.3 .
15
1.Parametrization Based on Bimaximal Matrix
Introduce parameters a, b, λ:
 2


 2

1
 
2

 1

 2
2
2
1
2
1

2

b



2
2
 a 
2

2 

2

2
W. Rodejohann, Phys. Rev. D 69 (2004) 033005;
N. Li and B.-Q. Ma, Phys. Lett. B 600 (2004) 248.
16
We get trigonometric functions：
s2 = b
2
1 2
c2 = 1  b
2
17
Expansion of Lepton Mixing（PMNS）Matrix：
U=
18
Ranges of  , a, b
2
2 2 4
2
s1 =
 a 
b ,
s2 = b 2 ,
2
4
2
1
2
3
s3 =
   2 - 2  12 2  2b 2  4 .
2
4

Data：

Ranges of Parameters：
0.58  s1  0.81,
0  s2  0.16,
0.48  s3  0.61.
0.08    0.17,
0.35  a  1.56,
1.56  b  7.03.
The expansion is reasonable，and converges fast。
To take phase angle into account，set Ue3= bλe-iλ.
19
The Jarlskog parameter to describe CP violation
J = Im(Ue 2U  3U U  2 ) = s s s c c c sin  .
*
e3
*
2
1 2 3 1 2 3
In this parametrization，
4
 2
2
2
J =
b

sin

(1

4

).

 2 
The value of J is 0.00996  J  0.01096.
C. Jarlskog, Phys. Rev. Lett. 55 (1985) 1039;
C. Jarlskog, Z. Phys. C 29 (1985) 491.
20
2.
Parametrization Based on Tribimaximal
Matrix
 6

 3

6

 6
 6

 6
3

3
3
3
3

3

b e



2
 a 
2

2 

2

 i
N. Li and B.-Q. Ma, Phys. Rev. D 71 (2005) 017302.
21
Expansion of Lepton Mixing（PMNS）Matrix：
The range of parameters  , a, b
0.03    0.07,
a
0.3,
b 1.5.
The Tribimaximal expansion converges more fast than
the Bimaximal expansion, but is of less symmetry。
22
Quark-Lepton Complementarity
Lepton mixings
Quark mixings
C
13CKM
CKM
 23
Present experimental data allows for relations
like the bimaximal complementarity relation:
12  C  45
Is such quark-lepton complementarity a hint of an underlying
quark-lepton unification?
Two possibilities:
Raidal ('04) Smirnov, Minakata ('04)
1.
Bimaximal complementarity
SFK (’05)
2.
Tri-bimaximal complementarity
23
3. Parametrization based on QLC
Quark mixing angles

Lepton mixing angles
= 2.4  0.11 ,


CKM
2
= 0.2  0.04 ,

CKM
3
= 12.9  0.12 ,
CKM
1
 CKM = 59  13 .
PMNS
1
= 45.0  6.5 ,

PMNS
2
= 0  7.4 ,

PMNS
3
= 32.6  1.6 ，

PMNS
不确定.
24
夸克轻子互补性 (QLC)
Quark-Lepton Complementarity

CKM
1

PMNS
1

CKM
2


CKM
3

PMNS
2
PMNS
3
= 47.4  6.6 ,
0,
= 45.5  1.7 .
A. Yu. Smirnov, hep-ph/0402264;
M. Raidal, Phys. Rev. Lett. 93 (2004) 161801.
25
For quark mixing：
1 2


 1 2 

1 2

V=

1 

2
 3
2
A

(1



i

)

A




A 3 (   i ) 


A 2


1


The trigonometric functions are：
sin 1CKM = A 2 ,
sin  2CKM e i = A 3 (   i ),
sin  3CKM =  ,
cos 1CKM = 1,
cos  2CKM = 1,
1 2
CKM
cos 1
= 1  .
2
26
Assuming
1CKM  1PMNS = 45 ,


CKM
3
PMNS
3
= 45 .
As there are uncertainties in data，we assume
(1)
sin 
(2)
sin 
PMNS
2
e
 i
PMNS  i
2
e
= A (  i ),
3
= A (  i ).
2
therefore obtain unified parametrization of quark and
lepton mixing matrix.
N. Li and B.-Q. Ma, Phys. Rev. D 71 (2005) 097301.
27
(1) From above assumptions, we obtain：
2
sin 
=
(1  A 2 ),
2
2
PMNS
cos 1
=
(1  A 2 ),
2
sin  2PMNS e  i = A 3 (  i ),
PMNS
1
cos  2PMNS = 1,
2
1 2
sin 
=
(1     ),
2
2
2
1 2
PMNS
cos  3
=
(1     ).
2
2
PMNS
3
28
Expansion of Lepton Mixing（PMNS）Matrix：
29
Significance of the expansion
1. The Wolfenstein parameter λcan measure both
the deviation of quark mixing matrix from the
unit matrix, and the deviation of lepton mixing
matrix from Bimaximal mixing pattern。
2. The Bimaximal mixing pattern is derived as the
leading-order term naturally 。
30
3. The Bimaximal expansion at first-order can be
naturnal obtained:
Bimaximal expansion

 1

2


2

2



 2
1
2
2
2

2
Expansion base on QLC

 2
0
2
2


 U e1 =

2
 2
2

0
1
 1

 U = 2  a 2  2
2
 3
1 1
2
0
 2

2

By re-scaling

2

2
2

(1   )
0  U e1 =
2


0
2
2
U
=
(1

A

)
3

2

0

a   2A
the Bimaximal expansion can be naturally obtained。
31
4. The values of λ：
 = 0.2243  0.0016
after re-scaling in
 = 0.1586
Bimaximal expansion,
0.08    0.17
the value is in the right range of the Wolfenstein parameter.
5. The value of U e 3 = A 3 (  i ), at best value
2
U e 3 = 0.006 i.e.,
 2   2 = 8.2
Jarlskog parameter
1 3
J = A  (1  2 2 ) = 0.0022
4
can be fixed by the CP violation of the lepton sector，thus
one obtain the range of ， . Therefore all of the 4
parameters can be fixed
32
PMNS  i
2
sin

e
=
A

(  i )
（2） Under the assumption
2
The expansion of lepton mixing（PMNS）matrix
33
Relations between quark and lepton
mixing matrices
Possible relations between quark and lepton
mixing matrices：
（1）VU = U bimal , （2） UV = U bimal .
And their connections with QLC。
N. Li and B.-Q. Ma, Euro.Phys.J.C 42, 17 (2005).
34
A remark
1.
2.
The quark and lepton mixing matrices can be
parametrized with the same set of Wolfenstein
parameters A and λ, therefore the two parameters can
describe both the derivation of the quark mixing matrix
from the unit matrix and the lepton mixing matrix from
the Bimaximal mixing pattern.
If λand A are different for quarks and leptons,
we
can consider the expansion of the lepton mixing matrix
as a general parametrization form in similar to the
Wolfenstein parametrization of quark mixing matrix.
35
Relations between masses of quarks and leptons
夸克轻子质量的关系:
质量的起源和关系？
Quark and lepton masses are 12 free
parameters in the standard model.
• Are there relations between masses
of different generations?
• Are there relations between masses
of quarks and leptons？
36
Koide mass relation
Y. Koide, Lett. Nuovo Cimento 34, 201 (1982); Phys. Lett. B 120, 161 (1983).
37
QLC of the masses
The extended Koide’s relations
Y. Koide, Lett. Nuovo Cimento 34, 201 (1982); Phys. Lett. B 120,
161 (1983).
38
Some conjectures
kl , kd  1,
ku , k  1   ,
kl  kd  ku  k  2
The neutrino masses
5
m1  1.0 10 eV ,
3
m2  8.4 10 eV ,
2
m3  5.0 10 eV .
39
夸克轻子质量的关系
N. Li and B.-Q. Ma, Phys. Lett. B 609, 309 (2005).
40
Energy scale insensitivity of Koide's relation
•
•
The relation is insensitive of energy scale in a
huge energy range from 1 GeV to 2x1016GeV.
the quark-lepton complementarity of masses is
also insensitive of energy scale.
N. Li and B.-Q. Ma, Phys. Rev. D 73, 013009 (2006).
41
Unification or Complementarity of Quarks and Leptons
夸克轻子的统一性或互补性(QLC)？
1. QLC of the mixing angles

CKM
12

CKM
23

PMNS
12

PMNS
23
 31CKM   31PMNS
=
=

4

,
,
4
0.
2. QLC of the mixing matrices
3. QLC of the masses
UV = U bi max .
kl , kd  1, ku , k  1   ,
kl  kd  ku  k  2.
42
Conclusions
1. Provide possible relations connecting quark and
lepton mixing and masses.
2. Provide useful hints for model construction
toward a unification of quarks and leptons.
2. Provide a general form of parametrization of
lepton mixing matrix, in similar to the
Wolfenstien parametrization of quark mixing
matrix.
43
Publications
1. N. Li and B.-Q. Ma, Phys. Lett. B 600, 248 (2004).
2. N. Li and B.-Q. Ma, Phys. Rev. D 71, 017302 (2005).
3. N. Li and B.-Q. Ma, Phys. Rev. D 71, 097301 (2005).
4. N. Li and B.-Q. Ma, Euro.Phys.J.C 42, 17 (2005).
5. N. Li and B.-Q. Ma, Phys. Lett. B 609, 309 (2005).
6. N. Li and B.-Q. Ma, Phys. Rev. D 73, 013009 (2006).
44