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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 ). Introduction Neutrinos: Windows to New Physics. Neutrino Oscillations provided information. ● Tiny Neutrino Masses ● Large Neutrino Flavor Mixings. Flavor Symmetry. Global fit for 3 flavors - PowerPoint PPT Presentation
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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 )
11 IntroductionIntroduction
Neutrinos: Windows to New Neutrinos: Windows to New PhysicsPhysics
● Tiny Neutrino Masses● Large Neutrino Flavor Mixings
Flavor Symmetry
Neutrino Oscillations provided information
Global fit for 3 flavorsMaltoni et al : hep-ph/0405172 ver.6 (Sep 2007)
Two Large MixingsTri-bi maximal
(Δmsol / |Δmatm| )1/2 = 0.16 - 0.20 ≒ λ
22
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 motivationto search flavor symmetry.
A key to looking for “hidden” flavor symmetry.
Mixing angles are independent of mass eigenvalues
Different from quark mixing angles
Non-Abelian Flavor Symmetry is appropriatefor lepton flavor physics.
22 Discrete Flavor SymmetryDiscrete Flavor Symmetry
Quark SectorQuark Sector
order 6 8 10 12 14 ...
SN : permutation groups S3 ...
DN : dihedral groups D3 D4 D5 D6 D7 ...
QN : quaternion groups Q4 Q6 ...
T : tetrahedral groups T(A4) ...
Discrete Symmetry Discrete SymmetryNon-Abelian discrete groups have non-singlet irreducible representations which can be assigned to interrelate families. Non-Abelian discrete groups have non-singlet irreducible representations which can be assigned to interrelate families.
Pakvasa and Sugawara (’78) : S3Pakvasa and Sugawara (’78) : S3
Frampton and Rasin (’99) : D4, Q4 Frampton and Rasin (’99) : D4, Q4 Frigerio, S.K., Ma and Tanimoto (’04) : Q4 Frigerio, S.K., Ma and Tanimoto (’04) : Q4
Babu and Kubo (’04) : Q6 Babu and Kubo (’04) : Q6
Frampton and Kephart (’94), Frampton and Kong (’95)Frampton and Kephart (’94), Frampton and Kong (’95)Chang, Keung and Senjanovic, (’90) Chang, Keung and Senjanovic, (’90)
Kubo et al. (’03,’04,’05) : S3Kubo et al. (’03,’04,’05) : S3
. . . . . . . . . . .. . . . . . . . . . .
Grimus and Lavoura (’03) : D4 Grimus and Lavoura (’03) : D4
Discrete symmetric models have long history . . .Discrete symmetric models have long history . . .
Need some ideas to realize Tri-bi maximal mixing by S3 flavor symmetry
by E. Ma1 1’ 1” 3
33 A4A4 ModelModel
by E. Ma
Diagonal terms come from 3 × 3 → (1, 1’,1”) 1’ × 1” → 1 Off Diagonal terms come from 3 × 3 ×3 → 1
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 fromTri-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
55 DiscussionsDiscussions Experiments indicate Tri-bi maximal mixing for Leptons, which is easily realized in 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.
Deviation from Tri-bi maximal mixing is important to test A4 flavor symmetry.
Desired vacuum
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 RatzNucl.Phys.B768:135-156,2007.
arXiv:0802.2310Hajime Iashimori, Tatsuo Kobayashi, Ohki HiroshiYuji 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. Hirsch, Ma, Moral, Valle: Phys. Rev. D72(2005)091301(R)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 - MaximalBi - Maximal θθ1212 == θθ2323 =π/4 , θ =π/4 , θ1313 =0 =0
Tri - Bi-maximalθ12 ≒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 LifschitzLandau and Lifschitz(理論物理学教程 量子力学12章対称性の理論 (理論物理学教程 量子力学12章対称性の理論 点群点群)) 群群 TT (正四面体群):正4面体の対称軸系(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。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:定義: ある物体がある軸のまわりを角度 ある物体がある軸のまわりを角度 2π/n2π/n 回転するとき自分自身に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。 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、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。 この軸を二面的と呼ぶ。
従って、 群従って、 群 TT の12の元(回転)は4っの類に分類される。の12の元(回転)は4っの類に分類される。 EE (単位元) (単位元) CC 2(4っの回転) 2(4っの回転) CC 3(4っの回転) 3(4っの回転) CC 4(3っの回4(3っの回転)転)
Tri - Bi-maximalθ12 ≒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.
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