unification of quarks and leptons or quark-lepton complementarities
DESCRIPTION
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. ?. Basic structure of matter. Basic interactions. Properties of Fermions. - PowerPoint PPT PresentationTRANSCRIPT
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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
?
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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
12
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
18
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
22
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
32
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
,
33
( 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).
39
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
40
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).