Seesaw Realization of Bi－Large Mixing and

フレーバーの離散対称性と
ニュートリノフレーバー混合
22 February 2008
仙台市作並温泉
谷本盛光
(新潟大学 )
1
Introduction
Neutrinos: Windows to New Physics
Neutrino Oscillations provided information
●
●
Tiny Neutrino Masses
Large Neutrino Flavor Mixings
Flavor Symmetry
Global fit for 3 flavors
Maltoni et al : hep-ph/0405172 ver.6 (Sep 2007)
Two Large Mixings
Tri-bi maximal
2
2
(Δmsol / |Δmatm| )1/2 = 0.16 - 0.20 ≒ λ
Tri-Bi-Maximal
Harrison, Perkins,
Scott (2002)
sin2θ12 =1/3 , sin2θ23 =1/2
Neutrino Mixing closes to
Tri-bi maximal mixing !
Tri-bi maximal mixing provides
good theoretical motivation
to search flavor symmetry.
A key to looking for “hidden” flavor
symmetry.
Mixing angles are independent of mass eigenvalues
Different from quark mixing angles
2 Discrete Flavor Symmetry
Non-Abelian Flavor Symmetry is appropriate
for lepton flavor physics.
Quark Sector
Discrete Symmetry
Non-Abelian discrete groups have non-singlet irreducible
representations which can be assigned to interrelate families.
order
6
SN : permutation groups
S3
DN : dihedral groups
D3
QN : quaternion groups
8
10
12
14
...
...
D4
D5
D6
Q4
T : tetrahedral groups
D7
...
Q6
...
T(A4)
...
Discrete symmetric models have long history . . .
Pakvasa and Sugawara (’78) : S3 Chang, Keung and Senjanovic, (’90)
Frampton and Kephart (’94), Frampton and Kong (’95)
Frampton and Rasin (’99) : D4, Q4 Grimus and Lavoura (’03) : D4
Kubo et al. (’03,’04,’05) : S3 Frigerio, S.K., Ma and Tanimoto (’04) : Q4
Babu and Kubo (’04) : Q6
...........
Need some ideas to realize Tri-bi maximal mixing by S3
flavor symmetry
3
1
1’
A4
1”
Model
3
by E. Ma
by E. Ma
3 × 3
→ (1, 1’,1”)
1’ × 1” → 1
Off Diagonal terms come from 3 × 3 ×3 → 1
Diagonal terms come from
hi
are yukawa couplings;
vi are VEV
Move to diagonal basis of
the charged lepton mass matrix
What is the origin of b=c and e=f=0 ?
Can one predict the deviation from
Tri-bi maximal mixing ?
In order to answer this question,
we should discuss the model:
Altarelli, Feruglio, Nucl.Phys.B720:64-88,2005
Tri-bimaximal neutrino mixing from discrete symmetry in extra dimensions
hd (1) , hu (1)
: gauge doublets
gauge singlets
b=c and e=f=0 is required for Tri-bi maximal.
4 Deviations from Tri-bi maximal mixing
M.Honda and M. Tanimoto, arXiv:0801.0181
Deviations in Charged Lepton Sector
CP violating phases
Deviations in Charged Lepton Sector
b=c=0
e=f=0
5 Discussions
Experiments indicate
Tri-bi maximal mixing for Leptons,
which is easily realized in A4 flavor symmetry.
Desired vacuum
Deviation from Tri-bi maximal mixing
is important to test A4 flavor symmetry.
does not deviate from 1 largely due to A4 phase.
can deviate from 0.5 largely.
can be as large as 0.2.
Can we predict CKM Quark Mixing angles
in A4 flavor symmetry ?
Quark mass matrices are given as
There is no Quark mixing
while tri-bi maximal mixing for Leptons.
Deviation is a clue to deeper
understanding of flavor symmetry !
What is the origin of the Discrete Symmetry ?
Stringy origin of non-Abelian discrete flavor symmetries:
Tatsuo Kobayashi, Hans Peter Nilles, Felix Ploger , Stuart Raby , Michael Ratz
Nucl.Phys.B768:135-156,2007.
arXiv:0802.2310
Hajime Iashimori, Tatsuo Kobayashi, Ohki Hiroshi
Yuji Omura, Ryo Takahashi, Morimitsu Tanimoto
SUSY化が容易にできる D4モデルが構成できる。
・FCNCの抑制の大きさが予言できる。
・Slepton の質量行列の構造が予言できる。
LHCでのテスト可能
再びクォークセクターは？
Hirsch, Ma, Moral, Valle: Phys. Rev. D72(2005)091301(R)
L lcΦi 3 ×3× (1,1’,1”) ← Diagonal matrix
LLηi 3 ×3 × (1,1’,1”) LLξ 3 ×3 × 3
< Φi >=v1, v2, v3
Bi - Maximal
θ12 = θ23 =π/4 , θ13 =0
Tri - Bi-maximal
θ12 ≒35°, θ23 =π/4 , θ13 =0
A4 flavor symmetry can easily realize
(approximate or exact) Tri-Bi-maximal Mixing
A4 symmetry
(Tetrahedral Symmetry)
Landau and Lifschitz
（理論物理学教程量子力学１２章対称性の理論
点群）
群T（正四面体群）：正４面体の対称軸系
立方体の向かい合った面の中心を通る３っの２回対称軸と
この立方体の空間対角線である４っの３回対称軸（二面的ではない）
二つの同じ角度の回転は、もしも群の元の中に、一方の回転軸を
他の回転軸に重ねるような変換があれば、同じ類に属する。
定義：ある物体がある軸のまわりを角度 2π/n回転するとき自分自身に
重なり合うとすれば、このような軸はｎ回対称軸と呼ばれる。
同じ軸の周りの、同じ角度の、反対方向の回転が共役ならば、
この軸を二面的と呼ぶ。
従って、群Tの１２の元（回転）は４っの類に分類される。
E（単位元） C２（４っの回転） C３（４っの回転） C４（３っの回転）
Tri - Bi-maximal
θ12 ≒35°, θ23 =π/4 , θ13 =0
A, B, C are independent complex parameters
S-Kam Atmospheric Neutrino Data
MINOS Experiment
SK atmospheric neutrinos
KamLand
Numerical Results: Deviations from Tri-bi maximal mixing.

Download Report