non-zero |u e3 | and quark-lepton in discrete symmetry y.h.ahn based on phys.rev.d83:076012,2011....

1

Non-zero |Ue3| and Quark-Lepton in Discrete Symmetry

Y.H.Ahnbased on Phys.Rev.D83:076012,2011. working with Hai-Yang Cheng and S.C.Oh

2011 년 8 월 17 일 수요일

2

Outline

Present Knowledges and Motivations

Tri-Bimaximal Mixing and Non-zero |Ue3|

A4 symmetry+TBM and its Deviations in Seesaw

Charged fermion mixing angles

Low energy phenomenology and leptogenesis

Conclusion

2011 년 8 월 17 일 수요일

3

Present Knowledges and MotivationsNeutrino oscillation (arXiv: 1106, 6028 G.L.Fogli, E.Lisi, A.Marrone,A.Palazzo, A.M.Rotunno) Analysis by Fogli etal. Including the latest T2K and MINOS results

Bi-Large mixing angles These results should be compared with Theta13 which is very small , and with the quark mixing angles in the VCKM.

Some new flavor symmetries A clue to the nature among quark-lepton physics beyond SM

2011 년 8 월 17 일 수요일

4




2011 년 8 월 17 일 수요일

Solar=34o Atm.=40.4o

5




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The disparity that nature indicates between quark and lepton mixing angles has been viewed in terms of a "quark–lepton complementarity" which can be expressed in the relations “Raidal 2004” “Smirnov and Minakata”


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2011 년 8 월 17 일 수요일

The disparity that nature indicates between quark and lepton mixing angles has been viewed in terms of a "quark–lepton complementarity" which can be expressed in the relations “Raidal 2004” “Smirnov and Minakata”


Accidental or not ?

7

Present Knowledges and Motivations Nothing is known about all three CP-violating phases If δCP and θ13≠0, CP is violated in ν oscillations.

δCP : Not directly related to leptogenesis , but would be likely in most leptogenesis models.

Dirac phase : CP violation in ν oscillation Leptogenesis Majorana phases : Neutrinoless Double beta decay Leptogenesis

A relatively large Theta13>0 (T2K and MINOS) opens up : CP-violation in neutrino oscillations Exps. (T2K, NOνA…) Matter effects can experimentally determine the type of ν

mass spectrum : normal or inverted mass ordering (goal of future LBL ν

oscillation Exps. program)

1 2, , CP

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A relatively large Theta13>0 (T2K and MINOS) opens up : CP-violation in neutrino oscillations Exps. (T2K, NOνA…) Matter effects can experimentally determine the type of ν mass

spectrum : normal or inverted mass ordering (goal of future LBL ν

oscillation Exps. program) CP violations in the lepton sector are imperative, if the baryon asymmetry of the Universe (BAU) originated from leptogenesis scenario in the seesaw models. So any observation of the leptonic CP violation, or demonstrating that CP is not a good symmetry of the leptons, can strengthen our belief in leptogenesis.

1 2, , CP

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Reactor=9.1o

9







1 2, , CP

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Reactor=9.1o

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1 2, , CP

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Reactor=9.1o

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Present Knowledges Cosmological limit (including WMAP 3-years result) upper bound on neutrino masses (astro-ph/0604335 : Uros Seljak, Anze Slosar,

Patrick McDonald) Starting to disfavor the degenerate spectrum of neutrinos

BAU Astrophys. J. Suppl. 192 (2011) 18

Why is there only Matter in Universe but no antimatter ?

• No evidence of antimatter in our domain of Universe (~20Mpc≈108light-years)• How can we generate ηB ≈10-10 from an initial condition for Big-Bang ?

2011 년 8 월 17 일 수요일

B 10#of baryons(N )6.2

#of photons1

)0

(NB

http://www.slac.stanford.edu/spires/find/wwwhepau/wwwscan?rawcmd=fin+%22Seljak,%20Uros%22

http://www.slac.stanford.edu/spires/find/wwwhepau/wwwscan?rawcmd=fin+%22Slosar,%20Anze%22

http://www.slac.stanford.edu/spires/find/wwwhepau/wwwscan?rawcmd=fin+%22McDonald,%20Patrick%22

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Present Knowledges Cosmological limit (including WMAP 3-years result) upper bound on neutrino masses (astro-ph/0604335 : Uros Seljak, Anze Slosar,

Patrick McDonald) Starting to disfavor the degenerate spectrum of neutrinos

BAU Astrophys. J. Suppl. 192 (2011) 18

Why is there only Matter in Universe but no antimatter ?

• No evidence of antimatter in our domain of Universe (~20Mpc≈108light-years)• How can we generate ηB ≈10-10 from an initial condition for Big-Bang ?

2011 년 8 월 17 일 수요일

B 10#of baryons(N )6.2

#of photons1

)0

(NB

Sakharov 1967

B-violationC- and CP-violation

Departure from thermal equilibrium

http://www.slac.stanford.edu/spires/find/wwwhepau/wwwscan?rawcmd=fin+%22Seljak,%20Uros%22

http://www.slac.stanford.edu/spires/find/wwwhepau/wwwscan?rawcmd=fin+%22Slosar,%20Anze%22

http://www.slac.stanford.edu/spires/find/wwwhepau/wwwscan?rawcmd=fin+%22McDonald,%20Patrick%22

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All data can be explained in terms of oscillation between just 3 known species

: Two possible orderings of neutrino masses

Earth matter effects(LBL)

Quasi-Degenerate case

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νμ ντνe

14

Pontecorvo-Maki-NaKagawa-Sakata (PMNS) Matrix

Atmospheric and SBL reactor Solar and LBL accelerator LBL reactor

Majorana phases Neutrinoless Double beta decay

23 23

13 13

13 1323

12 12

12 12

23

1 0 0 0 0

0 0 1 0 0

0 0 0 0 1

CP

CP

PMN

i

iS

c s e

U P

s

c s

s c e c

c s

s c

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Goal: sin22q13 ~ 0.02 @ 90% CL in 3 years

15

Pontecorvo-Maki-NaKagawa-Sakata (PMNS) Matrix

Atmospheric and SBL reactor Solar and LBL accelerator LBL reactor

Majorana phases Neutrinoless Double beta decay

23 23

13 13

13 1323

12 12

12 12

23

1 0 0 0 0

0 0 1 0 0

0 0 0 0 1

CP

CP

PMN

i

iS

c s e

U P

s

c s

s c e c

c s

s c

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Goal: sin22q13 ~ 0.02 @ 90% CL in 3 years

Not ob-servable in oscillations

16

So far it is still unclear how to theoretically understand the observed neutrino mixing.

A tentative way is to start with the data of ν oscillation and conjecture a simple constant mixing pattern, for example, “Tri-Bimaximal mixing” matrix

(Harrison, Perkins and Scott; see also

Wolfenstein(1970) and He and Zee)

It suggests that flavor structure for mixing should be divorced from trying to understand

the mass eigenvalues.

Correlations between mass matrix elements It is suggestive of a flavor symmetry.

2011 년 8 월 17 일 수요일

TBM * diag †TBM T

12

1

12

2

BM ( )

(.

).

.

(

)

B

A B D

A B

A

m A B Dm

B

D

U U

17








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TBM * diag †TBM T

12

1

12

2

BM ( )

(.

).

.

(

)

B

A B D

A B

A

m A B Dm

B

D

U U

Tri-maximalΘ12=35.3O

Θ13=0

Θ23=45O

Bi-maximal

18








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TBM * diag †TBM T

12

1

12

2

BM ( )

(.

).

.

(

)

B

A B D

A B

A

m A B Dm

B

D

U U

19








If there exists such a flavor symmetry in Nature, the TBM pattern for the neutrino mixing will be a good zeroth order approximation to reality.2011 년 8 월 17 일 수요일

TBM * diag †TBM T

12

1

12

2

BM ( )

(.

).

.

(

)

B

A B D

A B

A

m A B Dm

B

D

U U

20

Deviations from Tri-Bimaximal It is clear by now that the TBM is not consistent with the recent experimental data on the reactor mixing angle θ13 because of the vanishing matrix element Ue3 in UTBM.

(arXiv: 1106.6028 ; G.L.Fogli, E. Lisi, A. Marrone, A. Palazzo and A.M. Rotunno)

Recent data of the T2K Collaboration and the analysis based on global fits of neutrino oscillations enter into a new phase of precise measurements of the neutrino mixing angles and mass-squared differences, indicating that the TBM mixing for three flavors of leptons should be modified.

2011 년 8 월 17 일 수요일

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Seesaw A simple and attractive explanation of the smallness of ν mass : Origin of

operator

SEESAW MECHANISM: (i) SM+RH ν (EW singlet) (ii) SM+SU(2) Triplet Higgs while Ł of the EW int. keeps invariant

SU(2)×U(1) (iii) SM+SU(2) Triplet fermions (R.Foot, H.Lew, X.-G. He, G.C. Joshi 1989)

3×3 Seesaw model has 18 parameters: 12 real+6 phases (cf. Casas Ibarra )

Integrating out the heavy fermions leaves us with observable mass matrix (9 observables: 6 real+3 phases) Half of the parameters of the model get lost

at low-E The 3 low-E CP-violating depend, in general, on all 6 seesaw phases.

The effects of high-E CP-violating phases control the generation of the BAU in the leptogenesis scenario, in |<m>| and in the leptonic CP-violating rephasing invariant Jcp.

Complicated & many unknown parameters !!

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B L

* d dD PMNS RY U m R M

22

A4 In approaches to reconstruct the high-energy physics from low-energy data, one can assume a flavor symmetry, which may reduce the unknown parameters.

Unless flavor symmetries are assumed, particle masses and mixings are generally undetermined in gauge theory: To understand the present data we consider A4 flavor symmetry. (E.Ma and G.Rajarasekaran; G.Altarelly and F.Feruglio; X.G.He,

Y.Y.Keum and R.Volkas)

Tri-Bimaximal in neutrino mixing & No Quark mixing Non-zero |Ue3| and correct CKM matrix : Higher order corrections

In seesaw + A4 : Leptogenesis scale ~ 1013-1015 GeV, due to the equal size

of moduli of neutrino Yukawa couplings (strong wash-out) Without the aid of higher order corrections or soft-

breaking term , non-zero |Ue3| as well as Leptogenesis are not possible

TeV-scale Resonant-Leptogenesis2011 년 8 월 17 일 수요일

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A4 A4 is the symmetry group of the tetrahedron and the finite groups of all twelve the even permutation of four objects: its irreducible representations contain one triplet 3 and three singlets 1,1’,1” with the multiplication rules 3×3=3+3+1+1’+1” and 1’×1’=1”

The 12 representation matrices for 3 Identity matrix : I Reflection matrices :

r1=diag.(1,-1,-1), r2=diag.(-1,1,-1), r3=diag.(-1,-1,1)

Cyclic, Anticyclic matrices :1 1

0 0 1 0 1 0

1 0 0 , 0 0 1 , ,

0 1 0 1 0 0i i i ic a a c rcr rar

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0 0 1 0 1 0

1 0 0 , 0 0 1 , ,


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Z2

Z3

Subgroup

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0 0 1 0 1 0

1 0 0 , 0 0 1 , ,


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Z2

Z3

SubgroupBAU ?UCKM≠1 ?

|Ue3|≈9.1°

26


Let’s denote two A4 triplets and

where

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Construction of Lagrangian

27

Under SU(2)×U(1)×A4×Z2

Hence its Yukawa interaction

Z2: forbidden

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Z2: forbidden

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Fermion Scalar


29



Z2: forbidden

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30



Z2: forbidden

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Same structure


31



Z2: forbidden

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Same structure

QUARK-LEPTON SYMMETRY

32

In the charged fermion sector: Assumption: the VEVs of A4 triplets can be equally aligned, i.e,

Charged fermion mass matrix comes from and has the form U(w)×Diag.(arbitrary eigenvalues)

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33



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Z3

34


“Tri-maximal” Lepton-sector


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Z3

35




Quark sector In a weak eigenstate basis When diagonalizing the charged fermion mass matrices by rotating

No Mixing !! it should be corrected !!

2011 년 8 월 17 일 수요일

Z3

36




Quark sector In a weak eigenstate basis When diagonalizing the charged fermion mass matrices by rotating

No Mixing !! it should be corrected !!

2011 년 8 월 17 일 수요일

Z3

37

In the neutrino sector: The Yukawa interaction , after EW symmetry breaking

: Z2 symmetry breaking

No Leptogenesis and No low-E CP-violation

Taking the scale of A4 symmetry breaking to be above EW scale,

Heavy Majorana mass matrix :

And assuming the vacuum alignment of heavy singlet scalar,

where

2011 년 8 월 17 일 수요일

Bi-maximal

38







where

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Bi-maximal

No (Tree+loop) interference No CP-Asymmetry

39







where

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Bi-maximal


40







where

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41







where

2011 년 8 월 17 일 수요일

Bi-maximal


42

In the neutrino sector:

If we consider Leptogenesis and low-energy CP-violation in our

scenario,

2011 년 8 월 17 일 수요일

Residual Z2

43

In a weak eigenstate basis, the Yukawa interaction and charged gauge

interaction Lagrangian

Neutrino mass matrix

as , which leads to an effective neutrino mass matrix at low energies

Diagonalizing matrix of light neutrino mass matrix is

2011 년 8 월 17 일 수요일

44


interaction lagrangian

Neutrino mass matrix

as , which leads to an effective neutrino mass matrix at low energies

Diagonalizing matrix of light neutrino mass matrix is

2011 년 8 월 17 일 수요일

Bi-m

axim

al

45



By rotating

From the charged lepton current

2011 년 8 월 17 일 수요일

46



By rotating


Tri-Bimaximal (Leptonic Sector)+No mixing (Quark Sector) Non-zero |Ue3|

QLC Leptogenesis

2011 년 8 월 17 일 수요일

Summary

47



By rotating


Tri-Bimaximal (Leptonic Sector)+No mixing (Quark Sector) Non-zero |Ue3|

QLC Leptogenesis Consider Naïve and very simple corrections to the Quarks and lepton sectors

2011 년 8 월 17 일 수요일

Summary

48

Deviations TBM from higher dimensional operators In the Lagrangian level, assume that above the cutoff scaleΛ there is

no CP violation term in the Dirac Yukawa neutrino and charged fermion Yukawa interactions,

which for scale below Λ is expressed in terms of 5-dimentional operators. In the presence of 5-D operators driven byχ -VEV alignment, the Yukawa

interactions in the fermion sector, which is invariant under SU(2)×U(1)×A4×Z2

2011 년 8 월 17 일 수요일

49





2011 년 8 월 17 일 수요일

50





2011 년 8 월 17 일 수요일

Neutrino sector

51





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Neutrino sector

High-E CP-Violation

52





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Neutrino sector

53





2011 년 8 월 17 일 수요일

Neutrino sector

CHARG

ED FERM

ION

SECTO

RS

54

In the neutrino sector: Taking the scale of A4 symmetry breaking to be above EW scale, and assuming the vacuum alignment of heavy singlet scalar,

where

2011 년 8 월 17 일 수요일

55


where

The Yukawa interaction , after EW symmetry breaking : Z2 symmetry breaking

The corrected neutrino Yukawa matrix :

2011 년 8 월 17 일 수요일

56


where

The Yukawa interaction , after EW symmetry breaking : Z2 symmetry breaking

The corrected neutrino Yukawa matrix :

Leptogenesis low-E CP violation 2011 년 8 월 17 일 수요일

Z2

57

In the charged fermion sector: VEV alignment

The corrected charged fermion mass matrix

where

2011 년 8 월 17 일 수요일

58



where

2011 년 8 월 17 일 수요일

Z35-Dim. Operatorsthrough <χ>

59



where

For the most natural case of hierarchical charged fermion Yukawa couplings

the corrected off-diagonal terms are not larger than the diagonal ones in size

2011 년 8 월 17 일 수요일

60

Only the mixing matrix takes part in PMNS and CKM mixing

matrices

Low-energy CP-violation

2011 년 8 월 17 일 수요일

61


matrices


There are empirical fermion mass ratios in the charged-lepton, up- and down-type quark sectors

calculated from the measured values

2011 년 8 월 17 일 수요일

62


matrices


There are empirical fermion mass ratios in the charged-lepton, up- and down-type quark sectors

calculated from the measured values

2011 년 8 월 17 일 수요일

What is the interpretation of the empirical relation ?

63

Quark-Lepton symmetry is broken by masses of quarks and

lepton, therefore one does not expect that the quark mixing is

transmitted to the lepton sector exactly.

2011 년 8 월 17 일 수요일

64




However, there is some interesting empirical relation

2011 년 8 월 17 일 수요일

65





The mass spectrum of the charged-leptons exhibit a similar hierarchy

to the down-type one’s, on the other hand, the mass spectrum of up-type quark shows a

strong hierarchy compared to the down-type one’s .

2011 년 8 월 17 일 수요일

66

Quark-Lepton symmetry is broken by masses of quarks and lepton, therefore one does not expect that the quark mixing is



The mass spectrum of the charged-leptons exhibit a similar hierarchy to the

down-type one’s, on the other hand, the mass spectrum of up-type quark shows a strong

hierarchy compared to the down-type one’s .

In terms of the Cabbibo angle the fermion masses can be scaled as

which may represent that the CKM matrix mainly generated from the down-type-quark sector,

as well as charged-lepton mixing matrix is similar to the CKM matrix.

2011 년 8 월 17 일 수요일

67

CKM Mixing MatrixUp-type quark and its mixing matrix Due to the measured up-quark mass hierarchy, it is impossible to

generate the Cabbibo angle :

if let then,

which is in discrepancy with the measured

does not affect the leading order predictions in λ

Cabbibo angle should arise from the first and second generation mixing in the down-type quark sector2011 년 8 월 17 일 수요일

68

CKM Mixing Matrix Down-type quark and its mixing matrix From the measured down-quark mass hierarchies

For letting in turn which means

should be roughly

Additional assumption to get a correct CKM matrix :

2011 년 8 월 17 일 수요일

69

CKM Mixing Matrix Down-type quark and its mixing matrix From the measured down-quark mass hierarchies


should be roughly

Additional assumption to get a correct CKM matrix :

2011 년 8 월 17 일 수요일

70

Down-type quark and its mixing matrix From the measured down-quark mass hierarchies


should be roughly

Assumption to get a correct CKM matrix :

2011 년 8 월 17 일 수요일

71

CKM Mixing Matrix CKM mixing matrix When diagonalizing the charged-fermion mass matrices by rotating

if we let

Effects caused by CP violation are always proportional to Jalskog-invariant

which is of the order

2011 년 8 월 17 일 수요일

72

CKM Mixing Matrix CKM mixing matrix When diagonalizing the charged-fermion mass matrices by rotating

if we let

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73

Charged-Lepton sectorFrom the measured charged-lepton mass hierarchies

Similar to down-type quark sector, if let , which means

Scenario-I

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Charged-Lepton sectorFrom the measured charged-lepton mass hierarchies

Similar to down-type quark sector, if let , which means

Scenario-I

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75

Charged-Lepton sector Scenario-II

Similar to Down-type quark mixing matrix

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76



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77



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Phased !!

78

Charged-Lepton sector Scenario-III

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79

Charged-Lepton sector Scenario-III

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80

In a weak eigenstate basis

When diagonalizing the charged-fermion mass matrices

we can rotate

In a mass eigenstate basis of the charged gauge interaction term

2011 년 8 월 17 일 수요일

81



we can rotate


2011 년 8 월 17 일 수요일

CKM-like + Tri-Bimaximal (deviated)

82



we can rotate


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Consider how the observed deviation from Tri-Bimaximal and QLC are realized

Neutrino sector

83

The effective low-energy neutrino mass matrix can not be diagonalized by Bi-Maxiaml mixing

An additional mixing matrix to diagonalize

with the mixing angle and phases

2011 년 8 월 17 일 수요일

θ=±π/4 +δ for δ«1

Neutrino sector

84

The effective low-energy neutrino mass matrix can not be diagonalized by Bi-Maxiaml mixing

An additional mixing matrix to diagonalize

with the mixing angle and phases

2011 년 8 월 17 일 수요일

θ=±π/4 +δ for δ«1

High-E CP violation

85

The neutrino mass eigenvalues are given as

From the above the solar and atmospheric mass squared differences are written as

2011 년 8 월 17 일 수요일

86

The neutrino mass eigenvalues are given as

From the above the solar and atmospheric mass squared differences are written as

2011 년 8 월 17 일 수요일

87

PMNS Mixing matrix that is, for the charged lepton correction is negligible

In the limit of exact TBM is recovered.

3σ(1σ) Exp. Bounds →

this is disfavored in 1σ Exp. Results due to the upper bounds

2011 년 8 월 17 일 수요일

88

PMNS Mixing matrix that is, for the charged lepton correction is negligible

In the limit of exact TBM is recovered.

3σ(1σ) Exp. Bounds →

this is disfavored in 1σ Exp. Results due to the upper bounds Therefore, sizable contributions from the charged lepton sector are

required to reach accordance with the 1σ Exp. results2011 년 8 월 17 일 수요일

89

PMNS Mixing (charged lepton correction)

Leptonic CP-violation at low-energies can be detected through neutrino

oscillations which are sensitive to the Dirac-phase, but insensitive to the Majorana phases.

2011 년 8 월 17 일 수요일

90

PMNS Mixing (charged lepton correction)

Leptonic CP-violation at low-energies can be detected through neutrino

oscillations which are sensitive to the Dirac-phase, but insensitive to the Majorana phases.

2011 년 8 월 17 일 수요일

Controlled by Leptogenesis

91

PMNS Mixing (charged lepton correction) Solar mixing angle

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92


For example , in the second scenario

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93


For example , in the second scenario

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QLC satisfied

Phased !!

94

PMNS Mixing (charged lepton correction) Reactor mixing angle

2011 년 8 월 17 일 수요일

95

PMNS Mixing (charged lepton correction) Reactor mixing angle

The parameter can be determined by the BAU if Leptogenesis is true.

Subsequently, the parameter Φ(φ21,θ,ψ3)

can be decided by the first QLC relation.

And then, we can predict the value of reactor angle.

2011 년 8 월 17 일 수요일

96

PMNS Mixing (charged lepton correction) Atmospheric mixing angle

where

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97

PMNS Mixing (charged lepton correction) Atmospheric mixing angle

where

If we consider Leptogenesis in this scenario, then we can pin down the value of ψ1.

2011 년 8 월 17 일 수요일

QLC satisfied

98

PMNS Mixing (charged lepton correction) Solar Mixing Angle versus ψ3 Atm. Mixing Angle

versus ψ1

θ12+θq12 =45ο θ23+θq23 =45ο

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99

PMNS Mixing (charged lepton correction) Non-zero |Ue3| Leptonic CP-

violation JCP

2011 년 8 월 17 일 수요일

Lower bound of |Ue3|≥5O

JCP~10-2

100

Another very attractive feature of Seesaw ? In addition to the explanation of neutrino masses, seesaw has another

appearing feature so-called “Leptogenesis”

Lepton asymmetry:

It is independent of mixing angles and CP phases of neutrino oscillation.

The loop function includes the 1-loop vertex and self energy corrections to the heavy

neutrino decay amplitude.

i

i

NB L BB

N L

nn n ns s

s n s n n n

*( ) ( )

MATTER antimatter

i iBr N Br N

0

i

i iN

n n

n

( ) 1 (1 ) ln1 1

x xf x x x

x x

2011 년 8 월 17 일 수요일

† 2

† 2

Im{( ) ( ) ( ) }

8 ( )ij i j j

ij i ii i

Y Y Y Y Mf

Y Y M

101

Another very attractive feature of Seesaw ? wash-out effects: The generated asymmetry survives if decays takes place out-of-

equilbrium at otherwise, inverse decay and scattering processes cancel the

asymmetry

We are in the energy scale where A4 symmetry is broken but the SM gauge group remains

unbroken. Choose leptogenesis scale

Flavor effects: (PRD49,6394

James M.

Cline, Kimmo Kainulainen, Keith A. Olive)

Interactions involving the charged τ are out-of-equilibrium at Interactions involving the charged μ are out-of-equilibrium at

Interactions involving the charged e are out-of-equilibrium at

23 2

EQPl

5 10 for T

y T H T TM

iT M

2011 년 8 월 17 일 수요일

1210 GeV ( )T H 910 GeV ( )T H

510 GeV ( )eT H

iN e ei

B Bi

Lii i

nn n

s s n

iN eB Bi i

ei i ii

L

nn n

s s n

102

A link between low-energy observables and Leptogenesis Go into the physical basis of the RH neutrino

Diagonalizing matrix

2011 년 8 월 17 일 수요일

JCP , ββ0ν Leptogenesis at TeV or

>1012GeV ?

103

Conclusions In the presence of 5-Dimensional operators driven by χ-field, A relatively large |Ue3| and CKM mixing matrix can be explained.

QLC can be naturally explained in A4 flavor symmetry through phased-effects.

The deviations from TBM can be fitted to Experimental data , especially a relatively large Theta13, through the phased effects from higher dimensional operators.

If we consider the BAU through Leptogenesis in this scenario, we can have more informations and give predictions about neutrino data.

2011 년 8 월 17 일 수요일

non-zero |u e3 | and quark-lepton in discrete symmetry y.h.ahn based on phys.rev.d83:076012,2011....

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