Transcript Neutrino Mass Matrix from Non

Prediction of Ue3 and cosθ２３
from
Discrete symmetry
Morimitsu TANIMOTO
Niigata University, Japan
XXXXth RENCONTRES DE MORIOND
March ６, 2005 @ La Thuile, Aosta Valley, Italy
Based on the work by W.Grimus, A.Joshipura, S.Kaneko,
L.Lavoura, H.Sawanaka and M.Tanimoto,
Nucl. Phys. B, 2005 (hep-ph/0408123)
I. Introduction
sin 22θatm > 0.92 (90% C.L.)
tan θ2sol = 0.33 - 0.49 (90% C.L.)
2
sin θ
chooz < 0.057 (3σ)
θ23 = 45° ±8°
θ12 = 33° ± 4°
θ13 < 12°
Neutrino Mixings are near Bi-Maximal
Questions:
★ Why are θ23 and θ12 so large ?
★ Why does sinθ12 deviate from maximal
although sinθ23 is almost maximal ?
★ Why is θ13 small ?
How small isθ13 ?
Expectation:
★ Flavor Symmetry prevents non-zero θ13
Plan of the talk
1. Introduction
2. Vanishing Ue3 and Discrete Symmetry
3. Symmetry Breaking and Neutrino Mixing Angles
Grimus, Joshipura, Kaneko, Lavoura, Sawanaka and Tanimoto
Nucl. Phys. B(2005) hep-ph/0408123
4. Summary and Discussions
2. Vanishing Ue3 and Discrete Symmetry
•
•
Basic Idea : Naturalness of Theory
Suppose a dimensionless small parameter a.
If a 0, the Symmetry is enhanced.
Neutrino Mass Matrix is constructed
in terms of neutrino masses and mixings
m1, m2, m3
• which has Z
2
Symmetry
2
2
2
１,
2
atm
2
atm
Normal : m1 < m2 < m3 (Δmsol = m2ーm Δm
Inverted : m3 < m1 < m2 (Δm2 sol = m22ーm2１, Δm
Quasi-Degenerate : m1 ~ m2 ~ m3
2
３
2
１
2
１
2
３
= m ーm )
= m ーm )
In the limit
γ=2θ23 =π/2, θ13 =0
μ-τ(Z ) Interchange Symmetry
2
Remark: θ12 is arbitrary !
Neutrino mass matrix
W.Grimus and L.Lavoura(2003)
Assumption : θ23 = 45°, θ13 = 0°
Framework : SM + 3νR (seesaw model)
θ12 = arbitrary, no Dirac phase, two Majorana phases
D4 × Z2 model
SM + 3νR + 3φ + 2χ
φ : gauge doublet Higgs
χ : gauge singlet Higgs
Charge assignment of D4 :
W.Grimus and L.Lavoura (2003)
D4 × Z2 model
origin of lepton mixings
3. Symmetry breaking and neutrino mixing angles
Two Independent Symmetry Breakings Terms should be considered.
|ε|,|ε’| are constrained
by experiments.
Small perturbations ε,ε’ give non-zero Ue3 and cos 2θ23
W.Grimus, A.S.Joshipura, S.Kaneko., L.Lavoura H.Sawanaka and M.T (’0５ N.P.B)
In the hierarchical case (m3 >> m2 > m1)
with Majorana phases ρ=σ=0
Normal hierarcy of ν mass
CHOOZ
|ε|,|ε’| < 0.3
ρ＝0, σ=0
|Ue3| < 0.2
|cos2θ23| < 0.28
atm.
symmetric
ρ＝π/4, σ=0
ρ＝π/2, σ=0
Inverted hierarcy of ν mass
CHOOZ
ρ＝0, σ=0
atm.
ρ＝π/4, σ=0
ρ＝π/2, σ=0
Quasi-degenerate of ν mass
|ε| < 0.3, |ε’| < 0.03, m = 0.3 eV
CHOOZ
ρ＝π/4, σ=0
ρ＝0, σ=0
atm.
ρ＝π/2, σ=0
ρ＝0, σ=π/2
1. Normal ν mass hierarchy :
small deviation of |Ue3|
large deviation of |cos2θ23|
2. Inverted ν mass hierarchy :
small / large deviation of |Ue3|
(depend on Majorana phases)
large deviation of |cos2θ23|
3. Quasi-Degenerate ν mass hierarchy :
small / large deviation of |Ue3|
(depend on Majorana phases)
small deviation of |cos2θ23|
Model of Symmetry Breaking: Radiatively generated Ue3 and cos2θatm
Assumption :
ε,ε’ are generated due to radiative corrections.
MEW
SM
MX
MSSM
Ue3 = non-zero
cos2θ23 =non- zero
GUT
Ue3 = ０
cos2θ23 = 0
tanβ is constrained :
tanβ< 23
~
Effect of Radiative correction is
significant in Qusi-Degenerate case
m = 0.3 eV,
ρ＝0, σ=π/2
4. Summary and Discussions
We easily find the Neutrino Models based on Discrete Symmetry
which predicts θ13
= 0°, θ23 = 45° in the symmetric limit.
|Ue3| and |cos2θ23| are deviated from zero
by small symmetry breaking (ex. Ｒadiative correction)
These deviations depend on Majorana phases ρ, σ.
Discrete Symmetry: S3 , D4, Q4, Q6
Q4(8) Model (Talk of M. Frigerio)
Models based on non-Abelian discrete groups
S3
S.Pakvasa and H.Sugawara, PLB 73(1978)61.
J.Kubo, A.Modragon, M.Mondragon and
E.Rodrigues-Jauregui, Prog.Theor.Phys.109(2003)795.
D4
W.Grimus and L.Lavoura, PLB 572 (2003) 189.
Grimus, Joshipura, Kaneko, Lavoura, Sawanaka and M.T
hep-ph/0408123
Q8(Q4)
M. Frigerio, S. Kaneko., E. Ma and M. T, hep-ph/0409187.
A4
Q12(Q6)
E.Ma and G.Rajasekaran, PRD 64 (2001) 113012.
K.S.Babu, E.Ma and J.W.F.Valle, PLB 552 (2003) 207.
K.S.Babu and J.Kubo, hep-ph/0411226.
Future
Unify the lepton and quark sectors
S-quark and S-lepton sector in SUSY
Higgs potential
2×２ Decompositions of D4
Mass matrix in D4 doublet basis
lepton : (lL1, lL2), (lR1,
Higgs : H1, H2
lR2)
Non-Abelian discrete groups
order : number of elements
order
6
SN
S3
DN
D3(=S3)
QN
8
10
12
14
...
...
D4
Q8(Q4)
D5
D6
D7
...
Q12(Q6)
...
T(A4)
...
T
Geometrical object :
D3(=S3) : rotations and reflections of △
D4 : rotations and reflections of □
A4 : rotations and reflections of tetrahedron
Group D4
2
1
3
4
C3
1
4
2
3
Σ(dim. of reps.)^2
= # of elements
# of reps.
= # of classes
n : # of elements
h : order of any elements
in that class(gh=1)