Quark and Lepton Mixing Quark and Lepton Mixing in S in S44 Flavor Model Flavor Model

September 28, 2010

Max-Planck-Institut für Kernphysik

Heidelberg, Germany

Morimitsu Tanimoto (Niigata University)with H. Ishimori and Y. Shimizu

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NiigataNiigata

↓↓

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Niigata UniversityNiigata University

Niigata CityPopulation: 811,996Niigata City is an urban center developed by its port. Even though it is located on a substantial expansion of agricultural landscapes, it has also easy accesses to major cities by airplanes, express omnibuses, and bullet trains. Also from its international airport, there are regular flights to Harbin, Shanghai, Seoul, Vladivostok , Khabarovsk, Guam. Niigata aspires to be a gateway to the East Asia.

1 Tri-bi maximal mixing and Flavor Symmetry

2 S4 Flavor Model in Quarks and Leptons

3 S4 Flavor Model in Sleptons

4 Summary

Plan of my talk

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Three Flavor analysis strongly suggests Tri-bimaximal Mixing of Neutrinos

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Harrison, Perkins, Scott (2002)

1 Tri-bimaximal mixing and Flavor symmetry

Recent experiments of the neutrino oscillations go into a new phaseof precise determination of mixing angles and mass squared differences.

indicates Non-Abelian Flavor Symmetry ?

Mixing angles are independent of mass eigenvalues

Different from quark mixing angles

Consider the structure of Neutrino Mass Matrix,which gives Tri-bi maximal mixing

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Quark Sector

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Let us consider Flavor Symmetry.

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Need some ideas to realize Tri-bi maximal mixing by S3 flavor symmetry

×

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○

T’ , ST’ , S4 4 , Δ(54), Δ(54) flavor models also give Tri-bi maximal mixing !flavor models also give Tri-bi maximal mixing !

A4 Symmetry may be hidden.

　　　　 triplet (νe ,νμ,ντ)L

A4 should be broken !

3L×3L3L×3L×3H

Δ （ 27 ） , Δ （ 54 ） , Σ （ 81 ） 11

Stringy origin of non-Abelian discrete flavor symmetriesStringy origin of non-Abelian discrete flavor symmetries T. Kobayashi, H. Niles, F. Ploeger, S. Raby, M. Ratz, NPB768,135(2007) hep-ph/0611020T. Kobayashi, H. Niles, F. Ploeger, S. Raby, M. Ratz, NPB768,135(2007) hep-ph/0611020

Non-Abelian Discrete Flavor Symmetry from TNon-Abelian Discrete Flavor Symmetry from T22/Z/ZNN Orbifolds Orbifolds A.Adulpravitchai, A. Blum, M. Lindner, JHEP0907, 053 (2009), 0906.0468A.Adulpravitchai, A. Blum, M. Lindner, JHEP0907, 053 (2009), 0906.0468

Non-Abelian Discrete Flavor Symmetries from Magnetized/Intersecting Non-Abelian Discrete Flavor Symmetries from Magnetized/Intersecting Brane ModelsBrane ModelsH. Abe, K-S. Choi, T. Kobayashi, H. Ohki, H. Abe, K-S. Choi, T. Kobayashi, H. Ohki, NPB820, 317 (2009)NPB820, 317 (2009) , 0904.26310904.2631

Origin of the non-Abelian Flavor Origin of the non-Abelian Flavor symmetry ?symmetry ?

Tri-bimaximal neutrino mixing from orbifolding,G.Altarelli, F.Feruglio, Y.Lin, NPB775, 31 (2007) hep-ph/0610165

Non-Abelian Discrete Groups from the Breaking of Continuous Non-Abelian Discrete Groups from the Breaking of Continuous

Flavor SymmetriesFlavor Symmetries A.Adulpravitchai, A. Blum, M. Lindner, JHEP0909, 018 (2009), 0907.2332A.Adulpravitchai, A. Blum, M. Lindner, JHEP0909, 018 (2009), 0907.2332

Non-Abelian Discrete Symmetries in Particle Physics

Hajime Ishimori, Tatsuo Kobayashi, Hiroshi Ohki, Hiroshi Okada, Yusuke Shimizu, Morimitsu Tanimoto,

e-Print: arXiv:1003.3552 [hep-th]

Prog.Theor.Phys.Suppl.183:1-163,2010 We review pedagogically non-Abelian discrete groups and show some applications for physical aspects.This article includes a brief view on general aspects of group theory, i.e. something basic and useful theorems.

Reference

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●●CKM mixing in Quarks ? CKM mixing in Quarks ? Cabibbo angle?Cabibbo angle? We need Quark-lepton unification in a GUT.We need Quark-lepton unification in a GUT.

●Ue3=0 in Tri-bimaximal mixing!　　 There are hints Non-zero Ue3 in experiments.　　 How can one predict Ue3 ?

Flavor Symmetry of Neutrinos is related with Physical Phenomena.

●●SUSY Flavor Sector,SUSY Flavor Sector, SUSY FCNC , EDM SUSY FCNC , EDM

We discuss the case of SWe discuss the case of S44 symmetry.symmetry.

1 1’ 1” 3

Four irreducible representations in A4

symmetry

Before discussing S4 model, let us understand how to get the tri-bimaximal mixing in the example of A4 flavor model. E. Ma and G. Rajasekaran, PRD64(2001)113012

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1’ × 1” → 1

3L × 3L → 1

3L × 3flavon → 1

3L × 3flavon → 1”

3L × 3flavon → 1’

3L × 3L × 3flavon → 1

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Can one get Desired Vacuum in Spontaneous Symmetry Breaking ? Scalar Potential Analysis

These mass matrices do not yet predict tri-bimaximal mixing !

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----------- -------------

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As seen in this A4 model,in order to reproduce the tri-bi maximal mixing, we need

Non-Abelian Discrete Symmetry(A4, T’, S4 … )

and

Symmetry BreakingVacuum Alignment of flavons.

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Spontaneous Breaking ? ( Scalar potential )Explicit Breaking ? (Boundary condition in extra-dim.)

H. Ishimori, K. Saga, Y. Shimizu, M. Tanimoto, arXiv:1004.5004 

2 S4 Flavor Model in Quarks and Leptons

S4×Z4 with SUSY SU(5) GUT⇒ Tri-bimaximal, Cabibbo angle

C.Hagedorn, M.Lindner, R.N.Mohapatra, JHEP 0606, 042 (2006) SO(10)B.Dutta, Y. Mimura, R.N. Mohapatra, arXiv:0911.2242 SO(10)C.Hagedorn, S. F. King, C. Luhn, arXiv:1003.4249 SU(5)R.d.A. Toorop, F. Bazzocchi, L. Merlo, arXiv: 1003.4502 Pati-Salam

Upquarks

MR Dirac Neutrinos

Charged leptonsDown quarks

We take l=m=1, n=2.

S4×Z4×U(1)FN with SUSY SU(5) GUT

S4 invariant superpotential for leptons

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3L×2R×3flavon

3L×1R×3flavon

2R×2R×2flavon

3L×1R×3flavon

2R×2R1R×1R

3L×2R×3flavon

We take VEV’s

We get Lepton Mass Matrices

Due to m-n<0○ ○○

○

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No mixing in the left-hand !Θ12=60°in the right-hand !

Vacuum alignment

After seesaw, we get the tri-bimaximal mixing

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Deviation from the Tri-bimaximal mixing due to Higher dimensional mass operators Superpotential of next-to-leading order

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The charged lepton mass matrix including the next-to-leading terms 　　

Since the lepton mixing is given as

we have non-zero Ue3

0.003

Next-to-leading in Neutrino sector

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Determination of magnitudes

Putting observed masses and M=1012 GeV, we get

FN charges l=m=1, n=2Desired Vacuum Alignments

We can predict mixing angles.

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Quark Sector is predicted. Down Quarks

Left-handed mixing is given as

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Including next-to-leading order, we get

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Up Quark Sector

We add the next-to-leading mass matrix

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Direct Yukawa coupling

We take alignment , we get

After rotating it by the orthogonal matrix,

We obtain

Up Quarks

We obtain CKM matrix elements

In the leading order, we predict

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Including next-to-leading order corrections, we get

The parameter set

reproduces observed values very well.

Values of parameters are consistent with our mass matrices.

37CP violation can be discussed !

Flavor symmetry constrains not only quark/lepton mass matrices,but also mass matrices of their superpartner, i.e. squark/sleptonSpecific patterns of squark/slepton mass matrices could be tested in future experiments.

In this talk, we concentrate on lepton FCNC.

3. S4 Flavor Symmetry in Sleptons

Consider Soft SUSY Breaking Term in Supergravity.

Second order

Slepton mass matrices are derived from

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For the left-handed sector, higher dimensional terms are given as

Left-handed Slepton mass matri ｘ is

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Right-handed Slepton mass matrix is

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Move to Super-CKM basis (Diagonal Basis of Charged Lepton)

in order to estimate magnitudes of FCNC.

Mass Insertion Parameters

F. Gabbiani, E. Gabrielli, A. Masiero and L. Silvestrini, Nucl. Phys. B477(1996) 321

Experimental Constraint from μ→eγ

Numerical analyses are required.

where

○Dominant term

A terms are obtained as

Experimental Constraint

We need numerical analyses of μ→eγ .

Dangerous !

μ→eγ Decay

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○

○ ○

EDM of Electron

J.Hisano, M. Nagai, P. Paradisi, Phys.Rev.D80:095014,2009.

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Preliminary

Assume the maximal phase

Assume the maximal phase

Assume the maximal phase

44 Summary Summary

☆ S4 Flavor Symmetry in SU(5) can give realistic quark and lepton mixing matrices.　　 Tri-bimaximal mixing, Cabibbo angle

☆ S4 discrete symmetries work to suppress FCNC in the framework of gravity mediation in SUSY breaking.

★ Squark sectors in S4 Symmetry ?★ Origin of S4 Symmetry ?

50★ Mass Spectrum ?

One of Future ProblemsOne of Future Problems

Can we predict Neutrino Can we predict Neutrino Masses?Masses?Symmetry cannot predict mass spectrum. Symmetry cannot predict mass spectrum. Symmetry breaking gives mass spectrum.Symmetry breaking gives mass spectrum.

Normal mass hierarchy Normal mass hierarchy

Inverted mass hierarchyInverted mass hierarchy

T2KT2K and NOνand NOν Ａ Ａ !! 51

We know We know KoideKoide formulaformula ! !

Can we predict neutrino spectrum Can we predict neutrino spectrum consistent with Tri-bimaximal Mixingconsistent with Tri-bimaximal Mixing ？？

We need more studies of Symmetry We need more studies of Symmetry Breakings !Breakings !

Accuracy Accuracy 1010-5-5

for neutrinosfor neutrinos

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Thank you !Thank you !

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★Tiny Neutrino Masses ? Seesaw, Extra-Dimensions　　　　　　　　　　　　　　 Mass Spectrum?★Large Flavor Mixings ? However Θ13 ? CP?

★Majorana Neutrinos ？　 L number violation?

★Right-handed Neutrinos ？ LHC, Leptogenesis

★Sterile Neutrinos ？

★New Interaction of Neutrinos ？

★Neutrino Soft Mass ？ ★ Cosmic Neutrinos?

What is interesting in Neutrino Physics ?

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A.D. Dolgov, arXiv: A.D. Dolgov, arXiv: 0803.38870803.3887

mmνν< 0.63 eV < 0.63 eV 95% c.l95% c.l

K.Ichikawa, M. Fukugita, M. kawasaki, PRD71(2005)043001K.Ichikawa, M. Fukugita, M. kawasaki, PRD71(2005)043001M. Fukugita, K. Ichikawa, M. Kawasaki, O. Lahav, M. Fukugita, K. Ichikawa, M. Kawasaki, O. Lahav, PRD74(2006)027302PRD74(2006)027302

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Our Multiplication Rule of S4

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S4 invariant superpotential

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Realization of Vacuum Alignment

Introduce driving fields with R charge 2

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We obtain Desired Vacuum Alignment

Scalar potential

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The first neutrino was detected on 24thFeb 2010.

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　　　 A4 symmetry (Tetrahedral Symmetry) 　

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EDM of Electron

J.Hisano, M. Nagai, P. Paradisi, Phys.Rev.D80:095014,2009.

Anomalous Magnetic Moment of Muon

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Neutrino Parameters Global fit for 3 flavors

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