Transcript ppt
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 Large
Neutrino Masses 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 (Δm 2 2 sol / |Δmatm| ) 1/2 = 0.16 - 0.20 ≒ λ
Tri-Bi-Maximal
Harrison, Perkins, Scott (2002)
sin
2
θ
12
=1/3
,
sin
2
θ
23 =
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
”
symmetry.
flavor
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
SN
: permutation groups
DN
: dihedral groups 6
S3 D3
8
D4
10
D5
12
D6
14
D7 QN
: quaternion groups
T
: tetrahedral groups
Q4 Q6 T(A4)
...
...
...
...
...
Discrete symmetric models have long history . . .
Pakvasa and Sugawara (’78) : S 3 Chang, Keung and Senjanovic, (’90) Frampton and Kephart (’94), Frampton and Kong (’95) Frampton and Rasin (’99) : D 4 , Q 4 Grimus and Lavoura (’03) : D 4 Kubo et al. (’03,’04,’05) : S 3 Frigerio, S.K., Ma and Tanimoto (’04) : Q 4 Babu and Kubo (’04) : Q 6 . . . . . . . . . . .
Need some ideas to realize Tri-bi maximal mixing by S3 flavor symmetry
3 A4 Model
1 1
’
1
”
3
by E. Ma
by E. Ma
Diagonal terms come from Off Diagonal terms come from 3 × 3 → (1, 1’,1”) 3 1’ × 3 × × 1” 3 → 1 → 1
h i
are yukawa couplings;
v i
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
h d (1) , h u (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 Nucl.Phys.B768:135-156,2007. , Michael Ratz
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 l c Φi 3 × 3 × LL η i 3 × 3 × (1,1’,1”) (1,1’,1”) ← LL ξ Diagonal matrix 3 × 3 × 3 < Φi >= v 1 , v 2 , v 3
Bi - Maximal
θ 12 = θ 23 =π/4 , θ 13 =0 θ 12
Tri - Bi-maximal
≒35°, θ 23 =π /4 , θ 13 = 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）
θ 12
Tri - Bi-maximal
≒35°, θ 23 =π /4 , θ 13 = 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.