Transcript Discrete Flavor Symmetry and Neutrino Mass Matrix

Masses and Mixings of Quark-Lepton
in
the non-Abelian Discrete Symmetry
VIth Rencontres du Vietnam
August 9 , 2006
Morimitsu Tanimoto
Niigata University
This talk is based on collaborated work with
E.Ma and H. Sawanaka
Plan of the talk
1 Introduction : Motivations
2 A4 Symmetry
3 A4 Model for Leptons
4 A4 Model for Quarks
5 Summary
1 Introduction : Motivations
Neutrino Oscillation Experiments already taught us
θsol ~ 33°, θatm ~ 45°, θCHOOZ < 12°
2
Δmatm
~ 2×10-3 eV2, Δmsol2 ~ 8×10-5 eV2, δ :unknown
Two Large Mixing Angles and One Small MixingAngle
2
2 1/2
) =
(Δmsol / Δmatm
Ideas
0.2
observed
values
structure of
mass matrix
Θij , mi
texture zeros,
flavor democracy,
μ-τ symmetry, ...
≒ λ
flavor
symmetry
？
Discrete Symmetry
S3, D4, Q4, A4...
Quark/Lepton mixing
Lepton : θ12 = 30〜35°, θ23 = 38〜52°, θ13 < 12°
Quark : θ12 ~ 13°, θ23 ~ 2.3°, θ13 ~ 0.2° (90% C.L.)
by M.Frigerio
Quark ⇔ Lepton : ● Comparable in 1-2 and 1-3 mixing.
● Large hierarchy in 2-3 mixing.
(Maximal 2-3 mixing in Lepton sector ?)
Tri-Bi maximal mixing ?
Bi-Maximal
Barger,Pakvasa,
Weiler,Whisnant
(1998)
Tri-Bi-Maximal
Harrison, Perkins,
Scott (2002)
θ12 ≒35°
Bi - Maximal
θ12 = θ23 =π/4 , θ13 =0
Tri - Bi-maximal
θ12 ≒35°, θ23 =π/4 , θ13 =0
What is Origin of the maximal 2-3 mixing ?
Discrete Symmetries are nice candidate.
Flavor Symmetry
S3, D4, Q4, A4 ...
Tri-Bi-Maximal mixing is easily
realized in A4 .
2 A4 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
QN : quaternion groups
D3
T : tetrahedral groups
8
10
12
14
...
...
D4
Q4
D5
D6
Q6
T(A4)
D7
...
...
...
1
1’
1”
3
by E. Ma
by E. Ma
3 A4 Model for Leptons
E.Ma
c
L=(νi , li )～3 li ～1, 1’, 1”
－
0
0
（Φi, Φi）～3 < Φi, >=v1, v2, v3
MνLL 3 ×3
L lcΦ 3 ×(1,1’,1”)× 3
Taking
b=c , e=f=0 , v1=v2=v3=v
Seesaw Realization
He, Keum, Volkas hep-ph/0601001
L=(νi , li )～3 li ～1, 1’, 1”
－
0
0
（Φi, Φi）～3 < Φi, >=v1, v2, v3
－
0
νRi ～3 （Φ, Φ ）～1
χi ～3
c
0
LνR Φ + νRiνRj χk + M0νRiνRj
Another assignment: Altarelli, Feruglio, hep-ph/0512103
Quark Sector ?
If the A4 assignments are
c
c
Q=(ui , di )～3 di , ui ～1, 1’, 1”
0
0
v1=v2=v3=v
（Φi, Φi）～3 <Φi> = v1, v2, v3
with
VCKM = UU† UD = I
CKM mixings come from higher operators!
4 A4 Model for Quarks
Ma, Sawanaka, Tanimoto, hep-ph/0606103
Quark-Lepton Unification in SU(5)
5*i (νi , li , dic ) ～3
c
c
c
10i ( li , uic , uic, dic)～1, 1’, 1”
－
0
0
－
0
0
－
0
－
0
（Φi, Φi）D～3 <Φi>D = v1D, v2D, v3D
（Φi, Φi）E～3 <Φi>E = v1E, v2E, v3E
（Φ1,Φ1）U～1’
<Φ1>U = v1U
0
（Φ2,Φ2）U～1” <Φ2>U = v2U
0
With
v1E=v2E=v3E
v1D << v2D << v3D in order to get quark mass hierarchy
v1E=v2E=v3E
in order to get Tri-Bi-maximal mixing
D
Parameters in Quarks: hi ,
1 1’1”
1 1’1”
1’1’1’
1”1”1”
viD , μ2 , μ3 , m2 , m3
Taking account in phase ω and Im(μ3 )
CP violation is predicted.
↑
O(λ)
comes from
A4 phase ω
How to test the quark mass matrices :
Since Vub depends on the phase of μ3 ,
We expect the correlation between Vub and sin2β.
5
Summary
A4 Flavor Symmetry gives us
Tri-Bi-maximal neutrino mixing
and CKM Quark Mixings
in the SU(5) unification of quarks/leptons.
0
-
0
（Φi, Φi）E～3 <Φi>E = v1E, v2E, v3E v1E=v2E=v3E
0
0
（Φi, Φi）D～3 <Φi>D = v1D, v2D, v3D v1D<< v2D<< v3D
（Φ1,Φ1）U ～1’
0
（Φ2,Φ2）U ～1”
0
★JCP comes from mainly A4 phase ω.
★Strong correlation between Vub and sin2β.