第1,2世代ニュートリノの質量縮退から探る世代構造の謎
荒木威 (益川塾)
with 石田裕之 (益川塾)
Based on arXiv:1211.4452 [hep-ph].

Partially-(quasi-)Degenerate
Mass-Spectrum of Neutrinos
and Flavor Puzzles

Takeshi Araki (Maskawa Inst.)
with H. Ishida (Maskawa Inst.)

Based on arXiv:1211.4452 [hep-ph].

Neutrino mass hierarchy

Fixed:

Neutrino mass hierarchy

Hierarchical, like the quarks and charged leptons.

Neutrino mass hierarchy

All three neutrinos are quasi-degenerate in mass.

Neutrino mass hierarchy

Focus ofthis talk

Only the first two neutrinos are degenerate.

Contents●Introduction
A few more words about
the neutrino masses and mixings●Motivation●Model
DN family symmetric model●Summary and Future works

Neutrino oscillation

Two large mixing angles and one small but non-zero angle:
only two squared-mass differences are known:no information on the leptonic CP violation.

Partially-degenerate region

Motivation

On the experimental side
Would be observed? by
0nbb experiments and
CMB observations.
EXO(90% C.L.)
KamLAND-Zen(90% C.L.)
WMAP9(95% C.L.)

WMAPMoroi@KEK-FF 2013

On the theoretical side
In the limit of ¢m212 = 0 (mº2 = mº1 ); a Majorana neutrinomass matrix comes to enjoy the O(2) symmetry:
0mº1

RT @ 00

where0mº1010mº100 AR = PT @ 0mº300cos µ
R = @ ¡ sin µ0

In other words,sin µcos µ0100 A;
110mº10mº1000 AP = @ 0mº30011P =@ 00O(2)0¡1000 A:
10mº10100 Amº3fij
Li Lj HH¤ºof the O(2).

On the theoretical sideThis partial degenerate limit seems to conflict with thehierarchical mass spectrum of the charged leptons.
0me(¹)@ 000me(¹)
0100 Am¿me= 0:005m¹?

We can assign different reps. to the right-handed leptons.

+00@ 00000100 A
m¿

or00
@ 0m1

00m2

1µL1
L2

¶z 2n0000 A ! MMy = @ 0
m30O(2)L3 z 1000100 Am2¿

On the theoretical side
On one hand, small corrections generate small mixing in( Note V_12 can be large. )the charged lepton sector:

On the other hand, the small corrections can generatelarge mixing in the neutrino sector:
0mº1@ 000mº101000 A + ±º @mº3because mº1 ' mº3 :1AThus, large PMNS mixing.
Note: small CKM mixing may be possible.

On the theoretical side
To summarize, the partial degenerate limit may be a goodstarting point for understanding
1: ¢m212 ¿ ¢m223
2:

ffm1;2 ¿ m3

and suggests an O(2) family symmetry. (or its subgroup)
PurposeConstructing a comprehensive framework for the flavor
structure in view of the partial mass-degeneracy.

ModelO(2) symmetry
The O(2) consists of two singlet and an infinite number ofdoublet irreducible representations:(n ∈ N)We will use the D_N group because we are free fromgauge amomalies, massless NGBs and so on.

Model with DN symmetry
DN

LI
22L3
1

The charged lepton sector:[rank = 1]

`i1H1

SI
21

I = 1; 2i = 1; 2; 3gauge singlet

Model with DN symmetry
DN

LI
22L31

`i
1

The charged lepton sector:[rank = 1]

H1

SI
21I = 1; 2i = 1; 2; 3gauge singlet[rank = 1]

[rank = 1]yi
LI H`i (S 2 )I2¤Fyi0
LI H`i (S 4 )I4¤F

Thanks to a FN-like mechanism, a hierarchical mass
texture is obtained.
Especially, each matrix is responsible for each mass.

Model with DN symmetry
DN

LI
22L31

`i
1

The charged lepton sector:

H1

SI
21I = 1; 2i = 1; 2; 3gauge singlet[rank = 1]

However,yi2LH`(S)IIi¤2Fyi04LH`(S)IIi¤4F
Actually, [rank = 1] .

The electron remains massless,, even if onetakes further higher-order terms into account.

Model with DN symmetry
DN

LI
22L31

`i
1

The charged lepton sector:

H
1

SI
21

I = 1; 2i = 1; 2; 3gauge singlet

Suppose complex VEVs:
20y ±1 4@ 1 1y1 ±2¤2F0y2 ±1
y2 ±2010y3 ±1y3 ±2 A + @0y10 ±1¤
y10 ±2¤y20 ±1¤y20 ±2¤00[rank = 2]
1 0 000 ¤y3 ±1
y1 ±a00
y30 ±2¤ A + @ y1 ±b0000

y2 ±a00
y2 ±b000

13y3 ±a00
y3 ±b A50corresponds to the massless limit of the electron.

Model with DN symmetry
DN

LI
22L3
1

The charged lepton sector:

`i1

H
1

I = 1; 2i = 1; 2; 3SI
21gauge singlet

Suppose complex VEVs:[rank = 2]

corresponds to the massless limit of the electron.

Model with DN symmetry
DN0LI
22L31

`i1H
1

The neutrino sector:f1@ 000f10010010 A+ 2 @000
¤Ff3®±1 + ®0 ±1¤ + ® ±afLLHH¤º0000000000¯²1 + ¯ 0 ²¤1 + ¯ ²a + ¯ ²x + ¯ ²¤x1000000000+ 4 @ ¯²2 + ¯ 0 ²¤2 + ¯ ²b + ¯ ²y + ¯ ²¤y¤F0SI
21

I = 1; 2i = 1; 2; 3hSI i = (s1 eiÁ1 ; s2 eiÁ2 )0000®±2 + ®0 ±2¤ + ® ±b00®±1 + ®0 ±1¤ + ® ±a00®±2 + ®0 ±2¤ + ® ±b0®
LI L3 HHS 22¤º ¤F000000000¯²2 + ¯ 0 ²¤2 + ¯ ²b + ¯ ²y + ¯ ²¤y000000000¡(¯²1 + ¯ 0 ²¤1 + ¯ ²a + ¯ ²x + ¯ ²¤x )0¯LI LJ HHS 44¤º ¤F1A

100 A
0Model with DN symmetry
DN

LI
22

L3
1

`i
1H1SI
21

I = 1; 2i = 1; 2; 3hSI i = (s1 eiÁ1 ; s2 eiÁ2 )±1 = (s21 e2iÁ1 ¡ s22 e2iÁ2 ); ±2 = 2s1 s2 ei(Á1 +Á2 )±a = s21 ¡ s22 ; ±b = 2s1 s2 cos(Á1 ¡ Á2 )²1 = ±12 ¡ ±22 ; ²2 = 2±1±2²a = ±a2 ¡ ±b2 ; ²b = 2±a ±b²x = ±1 ±a ¡ ±2 ±b ; ²y = ±1 ±b + ±2 ±a

Model with DN symmetry
LI22DNL31`i1H1

I = 1; 2i = 1; 2; 3hSI i = (s1 eiÁ1 ; s2 eiÁ2 )SI
21

The charged lepton sector.

01 @+ 2¤F

y10 ±1¤y10 ±2¤y20 ±1¤y20 ±2¤y30 ±1¤y30 ±2¤0001000

y1 ±a1A+@ y 00 ±b1¤2F000y2 ±a00
y2 ±b000

1y3 ±a00y3 ±b A0

Model with DN symmetry
DN0LI
22L31`i1The neutrino sector.

f1@ 0001+ 4 @¤F0f101000
10 A+ 2 @000
¤Ff3®±1 + ®0 ±1¤ + ® ±a000000000¯²1 + ¯ 0 ²¤1 + ¯ ²a + ¯ ²x + ¯ ²¤x00
00000000 ¤¯²2 + ¯ ²2 + ¯ ²b + ¯ ²y + ¯ ²¤y0

f3 = 1:0;

H1SI21

I = 1; 2i = 1; 2; 3hSI i = (s1 eiÁ1 ; s2 eiÁ2 )0000®±2 + ®0 ±2¤ + ® ±b00®±1 + ®0 ±1¤ + ® ±a00®±2 + ®0 ±2¤ + ® ±b0000000000¯²2 + ¯ 0 ²¤2 + ¯ ²b + ¯ ²y + ¯ ²¤y000000000
0 ¤¡(¯²1 + ¯ ²1 + ¯ ²a + ¯ ²x + ¯ ²¤x )01
A

100 A0Model with DN symmetry
Normal Hierarchy

Inverted Hierarchy

Fogli

Valle

Constrains
PMNSµ12; ¢m212 ; ¢m223 z 1¾ range
mem¹= 0:00473;= 0:0588:
m¹m¿

Model with DN symmetry●Correlation between

The electron mass is proportional toIn the diagonal basis of the charged leptons,Thus, the electron mass andarecorrelated via the CP violating phases.

:Model with DN symmetry
Normal Hierarchy

The size ofis
constrained by theelectron mass aslong as couplings
are not tuned.

Constrains

Summary
●

●

●

●

●

The partial-mass-degenerate limit may be a goodstarting point for understanding the observed fermion
mass spectra and mixing patterns.
Such a mass region would be tested by 0nbb
experiments and CMB observations.
The limit suggests an O(2) (or D_N) flavor symmetry.
We have proposed a simple model for the lepton sector
and shown that the model can reproduce experimental
results without making coupling constants hierarchical.
We have found a novel correlation between the electronmass and.

Future works●

Something distinguishable resulting from the partial
mass-degeneracy.

●

Inclusion of the quark sector.

●

CP violation and masses of the other 1st generations.

●●

Enlarging to O(2) including a new gauge boson andgauge anomalies.
Mass hierarchy among the 3rd generations:

Backup slides

Neutrino mass hierarchy

Fixed:

Neutrino mass hierarchy

Fixed:

Neutrino mass hierarchy

Fixed:

On the experimental sideSuch a mass region would
be excluded or confirmed
by 0nbb experiments andCMB observations.
EXO(90% C.L.)
KamLAND-Zen(90% C.L.)
WMAP9(95% C.L.)

WMAP

What and How?
Example: (Majorana) neutrino masses and lepton # violation.
A Majorana neutrino mass term can be constructed.

H0 H0ºLjºLifij
Li Lj HH¤ºfij=(ºLi H 0 ¡ `Li H + )2¤º[ S.Weinberg, PRL43(1979) ]Setting neutrino masses to zero, the lepton # is restoredlim

mº !0U (1)LThe observed tiny neutrino masses can be regarded as
breaking terms of U (1)L broken at a high-energy scale.