1

Unification of Quarks and LeptonsUnification of Quarks and Leptonsor Quark-Lepton Complementaritiesor Quark-Lepton Complementarities

Bo-Qiang Ma Peking University (PKU) Peking University (PKU)

in collaboration with Nan Li in collaboration with Nan Li

November 16, 2007, Talk at NCTS @ NTHUNovember 16, 2007, Talk at NCTS @ NTHU

?

2

Basic structure of matterBasic structure of matter Basic interactionsBasic interactions

3

Properties of Fermions

4

The unification of quarks and leptonsThe 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 ？

Parametrization of Lepton Mixing Matrix

Are there connections between the parametrizations of quark and lepton

mixing matrices ？

6

Cabibbo, Kobayashi, Maskawa (CKM) Matrix

b

s

d

b

s

d

b

s

d

VVV

VVV

VVV

b

s

d

w

w

w

tbtstd

cbcscd

ubusud

w

w

w

99.004.0005.0

04.097.022.0

005.022.0975.0

Unitarity (or lack thereof) of CKM matrix tests existence of further quark generations

and possible new physics (eg. Supersymmetry)

u,c,t quarks couple to superposition of other quarks

Weak eigenstates Mass eigenstates

w/ ParticleData Group ‘01Central Values

e

.

.

Neutrino Oscillations Neutrino Oscillations

This is only possible if neutrinos have mass new physics beyond the Standard Model This is only possible if neutrinos have mass new physics beyond the Standard Model

8

Two flavor case

Assuming are flavor eigenstates, are mass eigenstates.

Their mixing are

Assuming at , there is only electron neutrino, i.e.,

1

2

cos sin.

sin cos

e

,e 1 2 ，

0t

1 2(0) cos sin ,e

9

at t,

Express in terms of ， one obtains

Then the probablity of finding neutrino at t is

using and 2

2 2 11 1 2

mE p m p

E

1 21 2( ) cos sin ,iE t iE tt e e

,e

2 2

2 22 2 2 2 1 2( ) sin cos =sin 2 sin .2

iE t iE t E Et e e t

1 2,

22 2 2

2 2 ,2

mE p m p

E

1 2 1 22 2( ) (cos sin ) sin cos ( ) .iE t iE t iE t iE tet e e e e

10

one obtains

The oscillation period is

And the oscillation probability is

Generating to three generations ， 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.

2 222 2 1 2( ) sin 2 sin

4

m mt t

E

2 22 2 1 2sin 2 sin .

4

m m L

E c

2 21 2

4,

pc

m m

2 222 2 ( )

( ) sin 2 sin 1.27 ( ).( )

m eVt L km

E GeV

,lU U U

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Mixing matrices of quarks and leptons

2 2 3 3

1 1 3 3

1 1 2 2

2 3 2 3 2

1 3 1 2 3 1 3 1 2 3 1 2

1 3 1 2 3 1 3 1 2 3 1 2

1

1

1

i

i

i

i i

i i

c s e c s

c s s c

s c s e c

c c c s s e

c s s s c e c c s s s e s c

s s c s c e s c c s s e c c

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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

1

2

3

2.4 0.11 ,

0.2 0.04 ,

12.9 0.12 ,

59 13 .

CKM

CKM

CKM

CKM

1

2

3

45.0 6.5 ,

0 7.4 ,

32.6 1.6

uncertain.

PMNS

PMNS

PMNS

PMNS

，

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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

0.77 0.88 0.47 0.61 0.20

0.19 0.52 0.42 0.73 0.58 0.82

0.20 0.53 0.44 0.74 0.56 0.81

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Parametrization of quark mixing matrix夸克混合矩阵的参数化

L. Wolfenstein, Phys. Rev. Lett. 51 (1983) 1945.

0.82,A

2 3

2 2

3 2

11 ( )

21

12

(1 ) 1

A i

V A

A i A

0.2243 0.0016, 0.20, 0.33.

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Parametrization of lepton mixing matrix

轻子混合矩阵的参数化Base matrix ： （ 1 ） Bimaximal Matrix

（ 2 ） Tribimaximal Matrix2 20

2 2

1 1 2

2 2 2

1 1 2

2 2 2

6 30

3 3

6 3 2

6 3 2

6 3 2

6 3 2

1 2 345 0 45 . ， ， 1 2 345 0 35.3 . ， ，

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1.Parametrization Based on Bimaximal Matrix

Introduce parameters a, b, λ:

W. Rodejohann, Phys. Rev. D 69 (2004) 033005;

N. Li and B.-Q. Ma, Phys. Lett. B 600 (2004) 248.

2

2

2 2

2 2

1 1 2

2 2 2

1 1 2

2 2 2

b

a

17

We get trigonometric functions ：

22s b 2

2

11

2c b

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Expansion of Lepton Mixing （ PMNS ） Matrix：

U

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Ranges of

Data ： Ranges of Parameters ：

The expansion is reasonable ， and converges fast 。 To take phase angle into account ， set Ue3= bλe-iλ.

2 2 41

2 2,

2 4s a b

, ,a b

22 ,s b

2 3 2 43

2 12 12 2 2 .

2 4s b -2

1

2

3

0.58 0.81,

0 0.16,

0.48 0.61.

s

s

s

0.08 0.17,

0.35 1.56,

1.56 7.03.

a

b

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The Jarlskog parameter to describe CP violation

In this parametrization ，

The value of J is

C. Jarlskog, Phys. Rev. Lett. 55 (1985) 1039;C. Jarlskog, Z. Phys. C 29 (1985) 491.

4

2 22sin (1 4 ).

2J b

* * 22 3 3 2 1 2 3 1 2 3Im( ) sin .e eJ U U U U s s s c c c

0.00996 0.01096.J

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2. Parametrization Based on Tribimaximal Matrix

N. Li and B.-Q. Ma, Phys. Rev. D 71 (2005) 017302.

6 3

3 3

6 3 2

6 3 2

6 3 2

6 3 2

ib e

a

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Expansion of Lepton Mixing （ PMNS ） Matrix：

The Tribimaximal expansion converges more fast than the Bimaximal expansion, but is of less symmetry 。

0.3,a

The range of parameters , ,a b0.03 0.07, 1.5.b

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Quark-Lepton Complementarity

Lepton mixings Quark mixings

Present experimental data allows for relations like the bimaximal complementarity

relation:

Is such quark-lepton complementarity a hint of an underlying quark-lepton unification?

Two possibilities: 1. Bimaximal complementarity 2. Tri-bimaximal complementarity

12 45C

23CKM

13CKMC

Raidal ('04) Smirnov, Minakata ('04)

SFK (’05)

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3. Parametrization based on

QLC Quark mixing angles Lepton mixing angles

1

2

3

2.4 0.11 ,

0.2 0.04 ,

12.9 0.12 ,

59 13 .

CKM

CKM

CKM

CKM

1

2

3

45.0 6.5 ,

0 7.4 ,

32.6 1.6

.

PMNS

PMNS

PMNS

PMNS

，

不确定

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夸克轻子互补性 (QLC)Quark-Lepton Complementarity

A. Yu. Smirnov, hep-ph/0402264;M. Raidal, Phys. Rev. Lett. 93 (2004) 161801.

1 1

2 2

3 3

47.4 6.6 ,

0 ,

45.5 1.7 .

CKM PMNS

CKM PMNS

CKM PMNS

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For quark mixing ：

The trigonometric functions are ：

2 3

2 2

3 2

11 ( )

21

12

(1 ) 1

A i

V A

A i A

21sin ,CKM A 1cos 1,CKM

32sin ( ),CKM ie A i 2cos 1,CKM

3sin ,CKM 21

1cos 1 .

2CKM

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Assuming

As there are uncertainties in data ， we assume

(1)

(2)

therefore obtain unified parametrization of quark and lepton mixing matrix.

N. Li and B.-Q. Ma, Phys. Rev. D 71 (2005) 097301.

1 1

3 3

45 ,

45 .

CKM PMNS

CKM PMNS

32sin ( ),PMNS ie A i

22sin ( ).PMNS ie A i

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(1) From above assumptions, we obtain ：

21

21

32

2

23

23

2sin (1 ),

2

2cos (1 ),

2

sin ( ),

cos 1,

2 1sin (1 ),

2 2

2 1cos (1 ).

2 2

PMNS

PMNS

PMNS i

PMNS

PMNS

PMNS

A

A

e A i

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Expansion of Lepton Mixing （ PMNS ） Matrix ：

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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 。

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3. The Bimaximal expansion at first-order can be naturnal obtained:

Bimaximal expansion Expansion base on QLC

By re-scaling

the Bimaximal expansion can be naturally obtained 。

1 1 0

2 20

2 2

2 20

2 2

2 20

2 21 1

02 21 1

02 2

1

2

2eU 1

2(1 )

2eU

23

2

2U a

23

2(1 )

2U A

2

2 2a A

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4. The values of λ ： after re-scaling in Bimaximal expansion, the value is in the right range of the Wolfenstein parameter.

5. The value of , at best value i.e., Jarlskog parameter

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

33 ( )eU A i

0.2243 0.0016

0.08 0.17 0.1586

2

3 0.006eU 2 2 8.2

3 21(1 2 ) 0.0022

4J A

，

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（ 2 ） Under the assumption The expansion of lepton mixing （ PMNS ） matrix

22sin ( )PMNS ie A i

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Relations between quark and lepton

mixing matrices

Possible relations between quark and lepton mixing matrices ：

（ 1 ） （ 2 ）And their connections with QLC 。

N. Li and B.-Q. Ma, Euro.Phys.J.C 42, 17 (2005).

.bimalUV U,bimalVU U

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A remark

1. 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.

2. 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.

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Relations between masses of quarks and leptonsRelations between masses of quarks and leptons夸克轻子质量的关系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？

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Koide mass relationKoide mass relation

Y. Koide, Lett. Nuovo Cimento 34, 201 (1982); Phys. Lett. B 120, 161 (1983).

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QLC of the masses

The extended Koide’s relations

Y. Koide, Lett. Nuovo Cimento 34, 201 (1982); Phys. Lett. B 120, 161 (1983).

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Some conjectures

The neutrino masses

, 1,l dk k , 1 ,uk k 2l d uk k k k

51

32

23

1.0 10 ,

8.4 10 ,

5.0 10 .

m eV

m eV

m eV

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夸克轻子质量的关系夸克轻子质量的关系

N. Li and B.-Q. Ma, Phys. Lett. B 609, 309 (2005).

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Energy scale insensitivity of Koide's relationEnergy scale insensitivity of Koide's relation

N. Li and B.-Q. Ma, Phys. Rev. D 73, 013009 (2006).

• 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.

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1. QLC of the mixing angles

2. QLC of the mixing matrices

3. QLC of the masses

12 12

23 23

31 31

,4

,40 .

CKM PMNS

CKM PMNS

CKM PMNS

max .biUV U

, 1, , 1 ,

2.l d u

l d u

k k k k

k k k k

Unification or Complementarity of Quarks and LeptonsUnification or Complementarity of Quarks and Leptons

夸克轻子的统一性或互补性夸克轻子的统一性或互补性 (QLC)(QLC) ？？

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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.

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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).