Can FourCan Four--ZeroZero--Texture mass matrix model Texture mass matrix model reproduce the quark and lepton mixing reproduce the quark and lepton mixing

angles and CP violating phases angles and CP violating phases ??

K. Matsuda (K. Matsuda (TsinghuaTsinghua Univ.)Univ.)T. Fukuyama (T. Fukuyama (RitsumeikanRitsumeikan Univ.), Univ.), H. Nishiura (Osaka Inst. of Tech.)H. Nishiura (Osaka Inst. of Tech.)

International Workshop on Neutrino Masses and MixingsUniversity of Shizuoka, Dec. 18, 2006

Our Strategy for the FourOur Strategy for the Four--ZeroZero--Texture (FZT) Texture (FZT) mass matrix modelmass matrix model

Many people have discussed Many people have discussed the FZT mass matrix model;the FZT mass matrix model;H. Nishiura, K. M, and T. Fukuyama, PRD 60,013006 (1999).

And they applied the FZT mass And they applied the FZT mass matrix model to SO(10)GUT;matrix model to SO(10)GUT;K. M, T. Fukuyama, and H. Nishiura,PRD 61, 053001 (2000).

Recently, some people say “the FZT in quark sector is dying”;

We check this opinion.K. M and H. Nishiura,

PRD 74, 033014 (2006)

We will check the our model.In this talk

Recently, some people say “the some FZT in the SO(10) GUT is dying”;

Perhaps, if there are solutions, people will say ” it must be fine-tuning.”

We check this opinion.In this talk (if there is a time.)

the solutions is remained

modify

Apply the results

Is there solutions?

Is the bi-large mixing stable?

change the parameters

FourFour--zerozero--texture (FZT) mass matrix modeltexture (FZT) mass matrix model

We would like to discuss the FourWe would like to discuss the Four--ZeroZero--Texture mass Texture mass matrix model, quickly .matrix model, quickly .

If we assume the Hermite matrix, we getIf we assume the Hermite matrix, we get

The number of parameters The number of parameters in the quark sectorin the quark sector

The number of parameters in the quark sector.The number of parameters in the quark sector.

The number of constraints from experiments.The number of constraints from experiments.

The following two phase parameters can not be The following two phase parameters can not be determined.determined.

DiagonalizationDiagonalizationThe FZT matrix is diagonalized as followsThe FZT matrix is diagonalized as follows

The The CKMCKM matrixmatrixThe CKM quark mixing matrix UCKM ≡ Uu† Ud is given by

Note that Note that mmuu//mmtt and and mmcc//mmtt are not sensitive to CKM matrixare not sensitive to CKM matrix..

Numerical estimationNumerical estimationWe fix the quark masses by the observed masses.We fix the quark masses by the observed masses.Two component parameters Two component parameters xxuu and and xxdd and two phase and two phase parameters parameters αα22 and and αα33 are left as free parameters.are left as free parameters.In our former paper, we find that if In our former paper, we find that if αα22 takes a value takes a value as as αα22 ≅ ≅ ππ/2/2 , there are the allowed region in the dotted , there are the allowed region in the dotted regions.regions.

1 1.2 1.4 1.6 1.8 2a30.8

0.85

0.9

0.95

1

x d

1 1.2 1.4 1.6 1.8 2α3

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0.85

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0.95

1

x u

0.8 0.85 0.9 0.95 1xu0.8

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0.95

1

x d

The numerical resultsThe numerical resultsWe show the best fit values as a exampleWe show the best fit values as a example

Our results The values estimated from exp data in MSSM (tanβ=10)

General SO(10)General SO(10)Each SM family + a rightEach SM family + a right--handed neutrino in a single handed neutrino in a single 1616--dim rep.dim rep.

Here the matrices Here the matrices YY1010, , YY126126 are symmetric, and are symmetric, and YY120120 is is antianti--symmetric.symmetric.

Each terms include the following mass terms, Each terms include the following mass terms, respectively.respectively.

The relations from the SO (10) GUTThe relations from the SO (10) GUT

The resulting tree level mass matrices as followsThe resulting tree level mass matrices as follows

The relations of quark and charged leptons The relations of quark and charged leptons in the SO (10) GUTin the SO (10) GUT

First, we only discuss the mass matrices of up, down and First, we only discuss the mass matrices of up, down and charged lepton.charged lepton.

Here, we define r ≡ δ’/δ for the sake of later discussion.

The number of parameters in SO(10) GUTThe number of parameters in SO(10) GUT

The number of parameters in our model.The number of parameters in our model.

The number of constraints from equations.The number of constraints from equations.

The number of constraints from experiments.The number of constraints from experiments.

The number of free parameters in our model.The number of free parameters in our model.

After summarizing these After summarizing these eqseqs., two parameters ., two parameters ddee andand rrremain as free parameters in one equation.remain as free parameters in one equation.

The contour lines on which the following equation is satisfied.The contour lines on which the following equation is satisfied.

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de/ m τ

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-5

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r

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de/ m τ

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5

10

rThe additional conditions which come from the phases in the The additional conditions which come from the phases in the charged lepton mass matrix.charged lepton mass matrix.

Sol. (a)

Sol. (b)

0.3 0.302 0.304 0.306 0.308 0.31

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de/ m τr

0.92 0.925 0.93 0.935 0.94 0.945 0.95

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r

Zoomed view

Zoomed view

Sol. (a)

Sol. (b)

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de/ m τ

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rThe additional conditions which come from the phases in the The additional conditions which come from the phases in the charged lepton mass matrix.charged lepton mass matrix.

Sol. (a)

Sol. (b)

0.3 0.302 0.304 0.306 0.308 0.31

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de/ m τr

0.92 0.925 0.93 0.935 0.94 0.945 0.95

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de/ m τ

r

In this talk, I will concentrate at this point. Namely, the contribution of 120 is small.

The neutrino mass matrix predictedThe neutrino mass matrix predictedfrom FZT in the SO (10) GUTfrom FZT in the SO (10) GUT

As we have shown, the quark and charged lepton parts are OK. As we have shown, the quark and charged lepton parts are OK. Next, LetNext, Let’’s discuss the neutrino mass matrix.s discuss the neutrino mass matrix.

The neutrino mass matrix predictedThe neutrino mass matrix predictedfrom FZT in the SO (10) GUTfrom FZT in the SO (10) GUT

As we have shown, the quark and charged lepton parts are OK. As we have shown, the quark and charged lepton parts are OK. Next, LetNext, Let’’s discuss the neutrino mass matrix. s discuss the neutrino mass matrix. We use the following global analysis of neutrino experiments.We use the following global analysis of neutrino experiments.

The allowed regions of neutrino masses and mixing anglesThe allowed regions of neutrino masses and mixing anglesin the case of the normal hierarchy at Sol. (b)in the case of the normal hierarchy at Sol. (b)

− 5× 107− 1× 108

5× 105

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1× 106

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− 1× 106

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β γ

δ′′

1× 1070

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5× 104

1× 105

− 5× 104

− 1× 1052× 107 3× 107 4× 107

β γ

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1× 1070

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10− 1

1

0.380.35

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10− 3

10− 42× 107 3× 107 4× 107

β γ

sin2

θ ij

1× 1070

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105

1010

1015

2× 107 3× 107 4× 107

β γ

(mi

γ)2 [

MeV

2 ]

Sol. (b)Sol. (b)

The uncertainty from the RG effectsThe uncertainty from the RG effectsWe donWe don’’t know how much the true masses of fermions t know how much the true masses of fermions are in the GUT scale because nobody know what is are in the GUT scale because nobody know what is really happened above TeV scale.really happened above TeV scale.However, in the most case, the ratio However, in the most case, the ratio mmff11//mmff22 is stable.is stable.Therefore, by changing only Therefore, by changing only mmff33 in the very wide range in the very wide range even where it is unreasonable value theoretically, we even where it is unreasonable value theoretically, we will check the stability of the biwill check the stability of the bi--large mixing.large mixing.

I regret that the discussions about the uncertainty from I regret that the discussions about the uncertainty from the CKM matrix and so on are not finished.the CKM matrix and so on are not finished.

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37500 38000 38500 39000

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βγ βγ βγ

βγ βγ βγ

sin2

θ ij

sin2

θ ij

ikjjjjIndeterminate -10.7516 Indeterminate-10.7516 -5.03675 -3.10112Indeterminate -3.10112 -2.81708 y{zzzzikjjjjjIndeterminate 1.22501 Indeterminate1.22501 Indeterminate 1.20363Indeterminate 1.20363 Indeterminatey{zzzzzikjjjjIndeterminate 1.17063 Indeterminate1.17063 0.0894184 -0.0262295Indeterminate -0.0262295 -0.0407902y{zzzz

S122 = 0.33

S232 = 0.48

S132 = 0.0003

Δm122/Δdm232= 0.0003

S122 = 0.33

S232 = 0.50

S132 = 0.0002

Δm122/Δdm232= 0.007

S122 = 0.32

S232 = 0.57

S132 = 0.0003

Δm122/Δdm232= 0.0003

S122 = 0.33

S232 = 0.46

S132 = 0.0002

Δm122/Δdm232= 0.0003

S122 = 0.32

S232 = 0.57

S132 = 0.0003

Δm122/Δdm232= 0.0004

S122 = 0.33

S232 = 0.57

S132 = 0.00013

Δm122/Δdm232= 0.0004

There is always the peak of There is always the peak of sinsin22θθ1212 around around sinsin22θθ1212=1/2=1/2, even if , even if mmuu33 and and mmee33 are changed.are changed.

The rates of change of each elements with respect to The rates of change of each elements with respect to mmee33 are incoherent as follows.are incoherent as follows.

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Note that we can not take the biNote that we can not take the bi--large mixing for granted in large mixing for granted in the general FZT.the general FZT.If you change other masses, the biIf you change other masses, the bi--large mixing is sometimes large mixing is sometimes forbidden. For example, if muon was more heavy, the large forbidden. For example, if muon was more heavy, the large sinsin22θθ1212 can not be derived.can not be derived.

S122 = 0.004S232 = 0.50S132 = 0.0001Δm122/Δdm232= 0.053

Summary ofSummary ofour FZT mass matrix modelour FZT mass matrix model

Many people have discussed Many people have discussed the FZT mass matrix model;the FZT mass matrix model;H. Nishiura, K. M, and T. Fukuyama, PRD 60,013006 (1999).

And they applied the FZT mass And they applied the FZT mass matrix model to SO(10)GUT;matrix model to SO(10)GUT;K. M, T. Fukuyama, and H. Nishiura,PRD 61, 053001 (2000).

Recently, some people say “the FZT in quark sector is dying”;

We check this opinion.K. M and H. Nishiura,

PRD 74, 033014 (2006)

We will check the our model.In this talk

Recently, some people say “the some FZT in the SO(10) GUT is dying”;

Perhaps people will say ”if there are solutions, it must be fine-tuning.”

We’ll also check this opinion.In this talk (if there is time.)

the solutions is remained

modified

Apply the results

We find the solutions in the SO(10) GUT.

There is the hidden property in FZT which always makes the peak of sin2θ12 around sin2θ23=1/2.

change the parameters

The remaining problems The remaining problems in the FZT in SO(10) GUTin the FZT in SO(10) GUT

This is the model under consideration.This is the model under consideration.

ΔΔmm121222 is too small.is too small.

We donWe don’’t know the reasons or the mechanisms whicht know the reasons or the mechanisms whichalways makemake the peak of sin2θ12 around sin2θ23 =1/2under some conditions.

The stability of the biThe stability of the bi--large mixing is not sufficiently large mixing is not sufficiently discussed yet.discussed yet.

We donWe don’’t know where the FZT come from. t know where the FZT come from.

The allowed regions of neutrino masses and mixing anglesThe allowed regions of neutrino masses and mixing anglesin the case of the inverse hierarchy at Sol. (b)in the case of the inverse hierarchy at Sol. (b)

− 5× 107− 1× 108

5× 105

0

1× 106

− 5× 105

− 1× 106

0 5× 107 1× 108

β γ

δ′′

1× 1070

0

5× 104

1× 105

− 5× 104

− 1× 1052× 107 3× 107 4× 107

β γ

δ′′

1× 1070

10− 2

10− 1

1

0.380.35

0.65

0.25

0.03

10− 3

10− 42× 107 3× 107 4× 107

β γ

sin2

θ ij

1× 1070

1

105

1010

1015

2× 107 3× 107 4× 107

β γ

(mi

γ)2 [

MeV

2 ]

The allowed regions of neutrino masses and mixing anglesThe allowed regions of neutrino masses and mixing anglesin the case of the normal hierarchy at Sol. (a)in the case of the normal hierarchy at Sol. (a)

10− 2

10− 1

1

0.380.35

0.65

0.25

0.03

10− 3

10− 4

sin2

θ ij

0 0.5× 106 1.0× 106 1.5× 106

β γ0

1

105

10-5

1010

1015

5× 105 1.0× 106 1.5× 106

β γ

(mi

γ)2 [

MeV

2 ]

The allowed regions of neutrino masses and mixing anglesThe allowed regions of neutrino masses and mixing anglesin the case of the inverse hierarchy at Sol. (a)in the case of the inverse hierarchy at Sol. (a)

10− 2

10− 1

1

0.380.35

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sin2

θ ij

0 0.5× 106 1.0× 106 1.5× 106

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β γ

(mi

γ)2 [

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Sol. (a)

Sol. (b)

The numerical resultsThe numerical results

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The allowed regions of neutrino masses and mixing anglesThe allowed regions of neutrino masses and mixing anglesin the case of the normal hierarchy at Sol. (b)in the case of the normal hierarchy at Sol. (b)

1× 1070

10− 2

10− 1

1

0.380.35

0.65

0.25

0.03

10− 3

10− 42× 107 3× 107 4× 107

β γ

sin2

θ ij

Can Four-Zero-Texture mass matrix model �reproduce the quark and lepton mixing �angles and CP violating phases ?Our Strategy for the Four-Zero-Texture (FZT) �mass matrix modelFour-zero-texture (FZT) mass matrix modelThe number of parameters �in the quark sectorDiagonalizationThe CKM matrixNumerical estimationThe numerical resultsGeneral SO(10)The relations from the SO (10) GUTThe relations of quark and charged leptons �in the SO (10) GUTThe number of parameters in SO(10) GUTThe neutrino mass matrix predicted�from FZT in the SO (10) GUTThe neutrino mass matrix predicted�from FZT in the SO (10) GUTThe uncertainty from the RG effectsSummary of�our FZT mass matrix modelThe remaining problems �in the FZT in SO(10) GUTThe numerical results