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Neutrino Flavour Models and Impact on LFV
Luca Merlo
26.06.2012, WHAT IS ʋ? INVISIBLES12 and Alexei Smirnov Fest
2
News on neutrino mixingsImpact on neutrino flavour models (discrete symmetries)
Implica;ons for LFV transi;ons in supersymmetric models
and correla;on with the muon g-‐2 discrepancy
Digression: a couple of alterna;ve aDempts
Outline
based on: Altarelli, Feruglio, LM & Stamou, arXiv:1205.4670Altarelli, Feruglio & LM, arXiv:1205.5133Bazzocchi & LM, arXiv:1205.5135
Luca Merlo, Neutrino Flavour Models and Impact on LFV
based on: Alonso, Gavela, D.Hernandez & LM, arXiv:1206.3167Altarelli, Feruglio, Masina & LM, to appear
3
�m2
sol
= (7.54+0.26�0.22)⇥ 10�5 eV2
�m2
atm
= (2.43+0.07�0.09)[2.42
+0.07�0.10]⇥ 10�3 eV2
sin2 ✓12
= 0.307+0.018�0.016
sin2 ✓23
= 0.398+0.030�0.026[0.408
+0.035�0.030]
sin2 ✓13
= 0.0245+0.0034�0.0031[0.0246
+0.0034�0.0031]
� = ⇡(0.89+0.29�0.44)[0.90
+0.32�0.43]
✓13
Very recent global fit: Fogli et al. 1205.5254 (see also )[Tortola et al. 1205.4018]
Recent Results of Global Fits
Luca Merlo, Neutrino Flavour Models and Impact on LFV
[Talks by Walter & Wang & Schwetz]
(Only 3 ac;ve neutrinos...)
4
In the past:large atmospheric angleonly upper bound on the reactor angle
sin2 ✓23 =1
2
sin2 ✓13 = 0
mu-‐tau symmetry
Neutrino Mass PaZerns
This suggests a fundamental structure of nature!
Luca Merlo, Neutrino Flavour Models and Impact on LFV
4
sin2 ✓23 =1
2sin2 ✓13 = 0 sin2 ✓12 =
1
3✓12 = 35.26�
TRI-‐BIMAXIMAL (TB) [Harrison, Perkins & ScoD 2002; Zhi-‐Zhong Xing 2002]
In the past:large atmospheric angleonly upper bound on the reactor angle
sin2 ✓23 =1
2
sin2 ✓13 = 0
mu-‐tau symmetry
1�3�
2
5 +p5
1
3sin2 ✓120
TBGR
3�
1
2
Neutrino Mass PaZerns
This suggests a fundamental structure of nature!
Luca Merlo, Neutrino Flavour Models and Impact on LFV
4
sin2 ✓23 =1
2sin2 ✓13 = 0 tan ✓12 =
1
�
� ⌘ 1 +p5
2
✓12 = 31.72�[Kajiyama, Raidal & Strumia 2007]GOLDEN RATIO (GR)
sin2 ✓23 =1
2sin2 ✓13 = 0 sin2 ✓12 =
1
3✓12 = 35.26�
TRI-‐BIMAXIMAL (TB) [Harrison, Perkins & ScoD 2002; Zhi-‐Zhong Xing 2002]
In the past:large atmospheric angleonly upper bound on the reactor angle
sin2 ✓23 =1
2
sin2 ✓13 = 0
mu-‐tau symmetry
1�3�
2
5 +p5
1
3sin2 ✓120
TBGR
3�
1
2
Neutrino Mass PaZerns
This suggests a fundamental structure of nature!
Luca Merlo, Neutrino Flavour Models and Impact on LFV
5
sin2 ✓23 =1
2sin2 ✓13 = 0 sin2 ✓12 =
1
2✓12 = 45�
[Vissani 1997; Barger et al. 1998]BIMAXIMAL (BM)
1
2
BM
2
5 +p5
1
3sin2 ✓120
TBGR
1�3� 3�
Neutrino Mass PaZerns
Luca Merlo, Neutrino Flavour Models and Impact on LFV
5
sin2 ✓23 =1
2sin2 ✓13 = 0 sin2 ✓12 =
1
2✓12 = 45�
[Vissani 1997; Barger et al. 1998]BIMAXIMAL (BM)
✓Exp
12 ⇡ ✓BM
12 � �
⇡/4 ⇡ ✓12 + �Maybe related to the Quark-‐Lepton Complementarity:[Smirnov; Raidal; Minakata & Smirnov 2004]
[Altarelli, Feruglio and LM 2009, Adelhart, Bazzocchi and LM 2010, Meloni 2011]
1
2
BM
2
5 +p5
1
3sin2 ✓120
TBGR
1�3� 3�
Neutrino Mass PaZerns
Luca Merlo, Neutrino Flavour Models and Impact on LFV
6
sin2 ✓13 = 0.0245+0.0034�0.0031[0.0246
+0.0034�0.0031]
sin2 ✓13BM
0TB GR
1�3� 3�
0.05
Need of Correc`ons!!
Luca Merlo, Neutrino Flavour Models and Impact on LFV
6
sin2 ✓13 = 0.0245+0.0034�0.0031[0.0246
+0.0034�0.0031]
sin2 ✓13BM
0TB GR
1�3� 3�
0.05
Need of Correc`ons!!
Luca Merlo, Neutrino Flavour Models and Impact on LFV
Are these predic6ve pa8erns completely ruled out?
6
me = m(0)e + �me m⌫ = m(0)
⌫ + �m⌫
Such correc;ons can arise from the charged lepton and/or from the neutrino sectors:
sin2 ✓13 = 0.0245+0.0034�0.0031[0.0246
+0.0034�0.0031]
sin2 ✓13BM
0TB GR
1�3� 3�
0.05
Need of Correc`ons!!
Luca Merlo, Neutrino Flavour Models and Impact on LFV
Are these predic6ve pa8erns completely ruled out?
6
me = m(0)e + �me m⌫ = m(0)
⌫ + �m⌫
mdiage = m(0)
e
mdiag⌫ = �UT
⌫ U0T⌫ m⌫ U
0⌫ �U⌫
U0⌫ = {UTB , UGR, UBM}mdiag
⌫ = U0T⌫ m(0)
⌫ U0⌫
(mdiage )2 = �U†
e m†e me �Ue
�U =
0
@1 c12 ⇠ c13 ⇠
�c⇤12 ⇠ 1 c23 ⇠�c⇤13 ⇠ �c⇤23 ⇠ 1
1
A
Such correc;ons can arise from the charged lepton and/or from the neutrino sectors:
in the basis in which the LO masses sa;sfy to
then the NLO correc;ons are encoded in
sin2 ✓13 = 0.0245+0.0034�0.0031[0.0246
+0.0034�0.0031]
sin2 ✓13BM
0TB GR
1�3� 3�
0.05
Need of Correc`ons!!
Luca Merlo, Neutrino Flavour Models and Impact on LFV
Are these predic6ve pa8erns completely ruled out?
7
ce12 ⇡ ce23 ⇡ ce13 c⌫12 ⇡ c⌫23 ⇡ c⌫13
In typical TB (GR) models, the correc;ons are democra;c in all the angles:
⇠e ⇡ ⇠⌫ ⌘ ⇠
A4: Altarelli & Feruglio 2005T’: Feruglio, Hagedorn, LM & Lin 2007S4: Bazzocchi, LM & Morisi 2009
Typical Tri-‐Bimaximal
Luca Merlo, Neutrino Flavour Models and Impact on LFV
7
ce12 ⇡ ce23 ⇡ ce13 c⌫12 ⇡ c⌫23 ⇡ c⌫13
In typical TB (GR) models, the correc;ons are democra;c in all the angles:
⇠e ⇡ ⇠⌫ ⌘ ⇠
�
⇠ ' 0.075
To maximize the success rate for all the three mixing angles inside the 3 :
A4: Altarelli & Feruglio 2005T’: Feruglio, Hagedorn, LM & Lin 2007S4: Bazzocchi, LM & Morisi 2009
Typical Tri-‐Bimaximal
Luca Merlo, Neutrino Flavour Models and Impact on LFV
Luca Merlo, Neutrino Flavour Models and Impact on LFV 8
9
ce12 ⇡ ce23 ⇡ ce13⇠⌫ � ⇠e
c⌫12 = c⌫23 = 0 c⌫13 6= 0
In special TB models, the correc;ons are specific in certain flavour direc;ons:
A4: Lin 2009
Special Tri-‐Bimaximal
Luca Merlo, Neutrino Flavour Models and Impact on LFV
9
ce12 ⇡ ce23 ⇡ ce13⇠⌫ � ⇠e
c⌫12 = c⌫23 = 0 c⌫13 6= 0
In special TB models, the correc;ons are specific in certain flavour direc;ons:
A4: Lin 2009
Special Tri-‐Bimaximal
⇠⌫
⇠⌫ ' 0.18
�
To maximize the success rate for all the three mixing angles inside the 3 :
Luca Merlo, Neutrino Flavour Models and Impact on LFV
Luca Merlo, Neutrino Flavour Models and Impact on LFV 10
sin
2 ✓23 =
1
2
+
1p2
sin ✓13 cos �CP
contours of constant sin2 ✓231�
1�
2�
3�
Luca Merlo, Neutrino Flavour Models and Impact on LFV
[General context: D. Hernandez & Smirnov 2012]
neglec;ng the subleading correc;ons:
10
11
⇠⌫ ⌧ ⇠e
c⌫12 ⇡ c⌫23 ⇡ c⌫13
ce12, ce13 6= 0 ce13 = 0
Also in BM models, the correc;ons are specific in certain flavour direc;ons: S4: Altarelli, Feruglio and LM 2009 Adelhart, Bazzocchi and LM 2010
Bimaximal
Luca Merlo, Neutrino Flavour Models and Impact on LFV
11
⇠⌫ ⌧ ⇠e
c⌫12 ⇡ c⌫23 ⇡ c⌫13
ce12, ce13 6= 0 ce13 = 0
Also in BM models, the correc;ons are specific in certain flavour direc;ons:
⇠e ' 0.17
�
To maximize the success rate for all the three mixing angles inside the 3 :
S4: Altarelli, Feruglio and LM 2009 Adelhart, Bazzocchi and LM 2010
(Similar results for the self-‐complementarity when the correc;ons come from the neutrino sector instead of the charged lepton sector.)[Bazzocchi & LM, arXiv:1205.5135]
Bimaximal
⇠e
Luca Merlo, Neutrino Flavour Models and Impact on LFV
Luca Merlo, Neutrino Flavour Models and Impact on LFV 12
sin
2 ✓12 =
1
2
+ sin ✓13 cos �CP
contours of constant sin2 ✓233�
1�
2�1�
Luca Merlo, Neutrino Flavour Models and Impact on LFV
neglec;ng the subleading correc;ons:
12
13
Which is the meaning of ?
How can we achieve these flavour structures?
⇠
Luca Merlo, Neutrino Flavour Models and Impact on LFV
14
Basic Points on Model Building
Luca Merlo, Neutrino Flavour Models and Impact on LFV
[Talk by King; Alterna;ve way: talk by Ma]
14
Flavour Symmetries to introduce these flavour structures
Basic Points on Model Building
Luca Merlo, Neutrino Flavour Models and Impact on LFV
[Talk by King; Alterna;ve way: talk by Ma]
14
Flavour Symmetries to introduce these flavour structures
Flavour Symmetries cannot be exact: the Yukawas do not show any symmetry
Basic Points on Model Building
Luca Merlo, Neutrino Flavour Models and Impact on LFV
[Talk by King; Alterna;ve way: talk by Ma]
14
Flavour Symmetries to introduce these flavour structures
Flavour Symmetries cannot be exact: the Yukawas do not show any symmetryStar;ng from a Yukawa Lagrangian invariant under a Flavour Symmetry, masses and
mixings arise only through a symmetry breaking mechanism:
where are new heavy scalar fields, singlets under SM, called flavons'
LY =(Ye['n])ij
⇤nf
eci H† `j +
(Y⌫ ['m])ij⇤mf
(`i H⇤)(H† `j)
2⇤L
Basic Points on Model Building
Luca Merlo, Neutrino Flavour Models and Impact on LFV
[Talk by King; Alterna;ve way: talk by Ma]
14
Flavour Symmetries to introduce these flavour structures
Flavour Symmetries cannot be exact: the Yukawas do not show any symmetryStar;ng from a Yukawa Lagrangian invariant under a Flavour Symmetry, masses and
mixings arise only through a symmetry breaking mechanism:
where are new heavy scalar fields, singlets under SM, called flavons
Suitable Spontaneous Symmetry Breaking
'
LY =(Ye['n])ij
⇤nf
eci H† `j +
(Y⌫ ['m])ij⇤mf
(`i H⇤)(H† `j)
2⇤L
' ! h'i
h'e,⌫i⇤f
⇡ ⇠e,⌫
Basic Points on Model Building
Luca Merlo, Neutrino Flavour Models and Impact on LFV
[Talk by King; Alterna;ve way: talk by Ma]
14
Flavour Symmetries to introduce these flavour structures
Flavour Symmetries cannot be exact: the Yukawas do not show any symmetryStar;ng from a Yukawa Lagrangian invariant under a Flavour Symmetry, masses and
mixings arise only through a symmetry breaking mechanism:
where are new heavy scalar fields, singlets under SM, called flavons
Suitable Spontaneous Symmetry Breaking
At LO the PMNS can take one of the previous predic;ve paDerns
'
LY =(Ye['n])ij
⇤nf
eci H† `j +
(Y⌫ ['m])ij⇤mf
(`i H⇤)(H† `j)
2⇤L
' ! h'i
h'e,⌫i⇤f
⇡ ⇠e,⌫
Basic Points on Model Building
Luca Merlo, Neutrino Flavour Models and Impact on LFV
[Talk by King; Alterna;ve way: talk by Ma]
14
Flavour Symmetries to introduce these flavour structures
Flavour Symmetries cannot be exact: the Yukawas do not show any symmetryStar;ng from a Yukawa Lagrangian invariant under a Flavour Symmetry, masses and
mixings arise only through a symmetry breaking mechanism:
where are new heavy scalar fields, singlets under SM, called flavons
Suitable Spontaneous Symmetry Breaking
At LO the PMNS can take one of the previous predic;ve paDerns
At NLO, some correc;ons arise and they are propor;onal to the VEV of the flavons:
larger is the VEV and larger are the correc;ons
'
LY =(Ye['n])ij
⇤nf
eci H† `j +
(Y⌫ ['m])ij⇤mf
(`i H⇤)(H† `j)
2⇤L
' ! h'i
h'e,⌫i⇤f
⇡ ⇠e,⌫
Basic Points on Model Building
Luca Merlo, Neutrino Flavour Models and Impact on LFV
[Talk by King; Alterna;ve way: talk by Ma]
15
Are there consequences of so large ?⇠
Luca Merlo, Neutrino Flavour Models and Impact on LFV
16
mSUSY ⇡ (1÷ 10)TeV ⇤SUSY
(g-‐2)µ discrepancydark maDergauge coupling unifica;onhierarchy problem
Low Energy Observables:
ν masses ν oscilla;ons
GUTsflavour symmetriesνc
superheavy gauge bosons
⇤fh'i EnergyMGUTe.w. scale0
'The Flavour symmetry at the high-‐scale affects the low-‐energy observables indirectly:
the flavons do not lead to direct contribu;ons (suppressed by the heavy mass)the son-‐SUSY breaking parameters are governed by the flavour symmetry and its
breaking mechanism
Impact on LFV
Luca Merlo, Neutrino Flavour Models and Impact on LFV
16
mSUSY ⇡ (1÷ 10)TeV ⇤SUSY
(g-‐2)µ discrepancydark maDergauge coupling unifica;onhierarchy problem
Low Energy Observables:
ν masses ν oscilla;ons
GUTsflavour symmetriesνc
superheavy gauge bosons
⇤fh'i EnergyMGUTe.w. scale0
'The Flavour symmetry at the high-‐scale affects the low-‐energy observables indirectly:
the flavons do not lead to direct contribu;ons (suppressed by the heavy mass)the son-‐SUSY breaking parameters are governed by the flavour symmetry and its
breaking mechanism
non-‐universal boundary condi;ons for the son terms
different results w.r.t. CMSSM scenario
Impact on LFV
Luca Merlo, Neutrino Flavour Models and Impact on LFV
17
BR(µ ! e�) < 2.4⇥ 10�12 @95%C.L.
µ ! e�
BR(µ ! e�)
We focus on the radia;ve decay :
Luca Merlo, Neutrino Flavour Models and Impact on LFV
17
BR(µ ! e�) < 2.4⇥ 10�12 @95%C.L.
µ ! e�
Rij =48⇡3↵em
G2Fm
4SUSY
���AijL
���2+
���AijR
���2�
BR(µ ! e�)
We focus on the radia;ve decay :
The normalized BR is defined by:
AijL = aLL (�ij)LL + aRL
mSUSY
mi(�ij)RL
AijR = aRR (�ij)RR + aLR
mSUSY
mi(�ij)LR
MI
Luca Merlo, Neutrino Flavour Models and Impact on LFV
17
BR(µ ! e�) < 2.4⇥ 10�12 @95%C.L.
µ ! e�
Rij =48⇡3↵em
G2Fm
4SUSY
���AijL
���2+
���AijR
���2�
BR(µ ! e�)
We focus on the radia;ve decay :
The normalized BR is defined by:
AijL = aLL (�ij)LL + aRL
mSUSY
mi(�ij)RL
AijR = aRR (�ij)RR + aLR
mSUSY
mi(�ij)LR
MI
aCC0The are loop factors of the SUSY parameters:
aLL = {2, 27}
aRR = {�1.9,�0.6}
aRL = aLR = 0.3
tan� = {2, 25}
Luca Merlo, Neutrino Flavour Models and Impact on LFV
18
(�ij)CC0
(�ij)CC0 =
�m2
CC0
�ij
m2SUSY
The depend on the son parameters:
Luca Merlo, Neutrino Flavour Models and Impact on LFV
18
(�ij)CC0
(�ij)CC0 =
�m2
CC0
�ij
m2SUSY
�Lm ��e ec
�
m2eLL m2
eLR
m2eRL m2
eRR
!✓eec
◆+ ⌫m2
⌫LL ⌫
The depend on the son parameters:
where the son masses are defined by
Luca Merlo, Neutrino Flavour Models and Impact on LFV
18
(�ij)CC0
(�ij)CC0 =
�m2
CC0
�ij
m2SUSY
�Lm ��e ec
�
m2eLL m2
eLR
m2eRL m2
eRR
!✓eec
◆+ ⌫m2
⌫LL ⌫
The depend on the son parameters:
where the son masses are defined by
m2(e,⌫)LL m2
eRR
m2eLR = (m2
eRL)†
and are hermi;an matrices from the Kähler poten;al
from the superpoten;al
generated from the SUSY Lagrangian analy;cally con;nuing all the coupling constants into superspace.
Luca Merlo, Neutrino Flavour Models and Impact on LFV
18
(�ij)CC0
(�ij)CC0 =
�m2
CC0
�ij
m2SUSY
�Lm ��e ec
�
m2eLL m2
eLR
m2eRL m2
eRR
!✓eec
◆+ ⌫m2
⌫LL ⌫
The depend on the son parameters:
where the son masses are defined by
m2(e,⌫)LL m2
eRR
m2eLR = (m2
eRL)†
and are hermi;an matrices from the Kähler poten;al
from the superpoten;al
generated from the SUSY Lagrangian analy;cally con;nuing all the coupling constants into superspace.
Luca Merlo, Neutrino Flavour Models and Impact on LFV
The MI of these mass matrices are governed by ⇠
19
Typical TB (GR) models
Special TB models
BM models
SR ⇠ 12%
⇠ ' 0.075
SR ⇠ 64%
⇠⌫ ' 0.18
SR ⇠ 3.4%
⇠e ' 0.17
Rij =48⇡3↵em
G2Fm
4SUSY
|aLL + aRL|2 O�⇠4�
Rµe ⇡ R⌧e ⇡ R⌧µ
Rij =48⇡3↵em
G2Fm
4SUSY
|aLL + aRL|2 O�⇠⌫4
�
Rµe ⇡ R⌧e ⇡ R⌧µ
Rij =48⇡3↵em
G2Fm
4SUSY
|aLL + aRL|2 ⇥(O �
⇠e2�
ij = 21, 31
O �⇠e4
�ij = 32
Rµe ⇡ R⌧e � R⌧µ
Luca Merlo, Neutrino Flavour Models and Impact on LFV
20
�0 ⇡ 156 GeV
�± ⇡ 306 GeV
& m0 = 200 GeV
M1/2 . 400 GeV
˜L ⇡ [230, 500] GeV
˜R ⇡ [160, 350] GeV
BR(µ ! e�) < 2.4⇥ 10�12
tan� = 15Typical TB
BMSpecial TB
Luca Merlo, Neutrino Flavour Models and Impact on LFV
21
�aµ
⌘ aexpµ
� aSM
µ
= 302(88)⇥ 10�11
m0, M1/2 2 [200, 5000]GeV
tan� 2 [2, 15]
BR(µ ! e�) < 2.4⇥ 10�12
& BR(µ ! e�) (g � 2)µ
Typical TB
Special TB BM
Luca Merlo, Neutrino Flavour Models and Impact on LFV
22
Intermediate Conclusions
Luca Merlo, Neutrino Flavour Models and Impact on LFV
22
Discrete symmetries can accommodate neutrino mixing paDerns and
TB, GR & BM can s;ll be taken as star;ng point
Intermediate Conclusions
Luca Merlo, Neutrino Flavour Models and Impact on LFV
22
Discrete symmetries can accommodate neutrino mixing paDerns and
TB, GR & BM can s;ll be taken as star;ng point
LFV analysis gives strong constraints
No possibility to sa;sfy and BR(µ ! e�) �aµ
Intermediate Conclusions
Luca Merlo, Neutrino Flavour Models and Impact on LFV
22
Discrete symmetries can accommodate neutrino mixing paDerns and
TB, GR & BM can s;ll be taken as star;ng point
LFV analysis gives strong constraints
No possibility to sa;sfy and
Are TB, GR & BM the flavour structure of nature? or only accidents?
BR(µ ! e�) �aµ
Intermediate Conclusions
Luca Merlo, Neutrino Flavour Models and Impact on LFV
22
Discrete symmetries can accommodate neutrino mixing paDerns and
TB, GR & BM can s;ll be taken as star;ng point
LFV analysis gives strong constraints
No possibility to sa;sfy and
Are TB, GR & BM the flavour structure of nature? or only accidents?
The new data on have put severe doubts on their naturalness:✓13
BR(µ ! e�) �aµ
Intermediate Conclusions
Luca Merlo, Neutrino Flavour Models and Impact on LFV
22
Discrete symmetries can accommodate neutrino mixing paDerns and
TB, GR & BM can s;ll be taken as star;ng point
LFV analysis gives strong constraints
No possibility to sa;sfy and
Are TB, GR & BM the flavour structure of nature? or only accidents?
The new data on have put severe doubts on their naturalness:
-‐ All need large correc;ons in their simplest versions
-‐ The special TB (the most successful) needs dynamical tricks
✓13
BR(µ ! e�) �aµ
Intermediate Conclusions
Luca Merlo, Neutrino Flavour Models and Impact on LFV
22
Discrete symmetries can accommodate neutrino mixing paDerns and
TB, GR & BM can s;ll be taken as star;ng point
LFV analysis gives strong constraints
No possibility to sa;sfy and
Are TB, GR & BM the flavour structure of nature? or only accidents?
The new data on have put severe doubts on their naturalness:
-‐ All need large correc;ons in their simplest versions
-‐ The special TB (the most successful) needs dynamical tricks
-‐ Alterna;ves: Other paDerns?
✓13
BR(µ ! e�) �aµ
Intermediate Conclusions
Luca Merlo, Neutrino Flavour Models and Impact on LFV
[Toorop, Feruglio, Hagedorn 2011 (1°&2°); ecc...]
22
Discrete symmetries can accommodate neutrino mixing paDerns and
TB, GR & BM can s;ll be taken as star;ng point
LFV analysis gives strong constraints
No possibility to sa;sfy and
Are TB, GR & BM the flavour structure of nature? or only accidents?
The new data on have put severe doubts on their naturalness:
-‐ All need large correc;ons in their simplest versions
-‐ The special TB (the most successful) needs dynamical tricks
-‐ Alterna;ves: Other paDerns?
Anarchy? Something in between?
✓13
BR(µ ! e�) �aµ
Intermediate Conclusions
Luca Merlo, Neutrino Flavour Models and Impact on LFV
[Toorop, Feruglio, Hagedorn 2011 (1°&2°); ecc...]
23
Are these paDerns only numerical accidents? If Yes what? Anarchy
Anarchy?
0.001 0.01 0.10.00
0.01
0.02
0.03
0.04
r
0.01 0.1 1.0.00
0.01
0.02
0.03
0.04
0.05
sin q13
0.0001 0.01 1. 100. 10 000.0.00
0.01
0.02
0.03
0.04
0.05
tan2q12
0.0001 0.01 1. 100. 10 000.0.00
0.01
0.02
0.03
0.04
0.05
tan2q23
1.¥10-6 0.00001 0.0001 0.001 0.01 0.10.00
0.01
0.02
0.03
0.04
0.05
J
-150 -100 -50 0 50 100 1500.000
0.005
0.010
0.015
0.020
0.025
0.030
d
-1.0 -0.5 0.0 0.5 1.00.00
0.02
0.04
0.06
0.08
0.10
Sind
0.001 0.01 0.10.00
0.01
0.02
0.03
0.04
r
0.01 0.1 1.0.00
0.01
0.02
0.03
0.04
0.05
sin q13
0.0001 0.01 1. 100. 10 000.0.00
0.01
0.02
0.03
0.04
0.05
tan2q12
0.0001 0.01 1. 100. 10 000.0.00
0.01
0.02
0.03
0.04
0.05
tan2q23
1.¥10-6 0.00001 0.0001 0.001 0.01 0.10.00
0.01
0.02
0.03
0.04
0.05
J
-150 -100 -50 0 50 100 1500.000
0.005
0.010
0.015
0.020
0.025
0.030
d
-1.0 -0.5 0.0 0.5 1.00.00
0.02
0.04
0.06
0.08
0.10
Sind
0.001 0.01 0.10.00
0.01
0.02
0.03
0.04
r
0.01 0.1 1.0.00
0.01
0.02
0.03
0.04
0.05
sin q13
0.0001 0.01 1. 100. 10 000.0.00
0.01
0.02
0.03
0.04
0.05
tan2q12
0.0001 0.01 1. 100. 10 000.0.00
0.01
0.02
0.03
0.04
0.05
tan2q23
1.¥10-6 0.00001 0.0001 0.001 0.01 0.10.00
0.01
0.02
0.03
0.04
0.05
J
-150 -100 -50 0 50 100 1500.000
0.005
0.010
0.015
0.020
0.025
0.030
d
-1.0 -0.5 0.0 0.5 1.00.00
0.02
0.04
0.06
0.08
0.10
Sind
Luca Merlo, Neutrino Flavour Models and Impact on LFV0.001 0.01 0.10.00
0.01
0.02
0.03
0.04
r
0.01 0.1 1.0.00
0.01
0.02
0.03
0.04
0.05
sin q13
0.0001 0.01 1. 100. 10 000.0.00
0.01
0.02
0.03
0.04
0.05
tan2q12
0.0001 0.01 1. 100. 10 000.0.00
0.01
0.02
0.03
0.04
0.05
tan2q23
1.¥10-6 0.00001 0.0001 0.001 0.01 0.10.00
0.01
0.02
0.03
0.04
0.05
J
-150 -100 -50 0 50 100 1500.000
0.005
0.010
0.015
0.020
0.025
0.030
d
-1.0 -0.5 0.0 0.5 1.00.00
0.02
0.04
0.06
0.08
0.10
Sind
[Hall, Murayama & Weiner 1999; de Gouvea & Murayama 2012]
24Luca Merlo, Neutrino Flavour Models and Impact on LFV
SU(5)⇥ U(1)Consider a simple U(1) as flavour symmetry, in a SU(5) inspired context: 10 = (5, 3, 0) 5 = (2, 1, 0)
Anarchy?: BeZer Hierarchy![Altarelli, Feruglio, Masina & LM to appear]
1.¥10-81.¥10-71.¥10-60.000010.0001 0.001 0.01 0.10.00
0.01
0.02
0.03
0.04
r
0.001 0.01 0.1 1.0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
sin q13
0.0001 0.01 1. 100. 10 000.0.00
0.01
0.02
0.03
0.04
0.05
0.06
tan2q12
0.0001 0.01 1. 100. 10 000.0.00
0.01
0.02
0.03
0.04
0.05
0.06
tan2q23
1.¥10-6 0.00001 0.0001 0.001 0.01 0.10.00
0.02
0.04
0.06
J
-150 -100 -50 0 50 100 1500.00
0.01
0.02
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d
1.¥10-81.¥10-71.¥10-60.000010.0001 0.001 0.01 0.10.00
0.01
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r
0.001 0.01 0.1 1.0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
sin q13
0.0001 0.01 1. 100. 10 000.0.00
0.01
0.02
0.03
0.04
0.05
0.06
tan2q12
0.0001 0.01 1. 100. 10 000.0.00
0.01
0.02
0.03
0.04
0.05
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tan2q23
1.¥10-6 0.00001 0.0001 0.001 0.01 0.10.00
0.02
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0.06
J
-150 -100 -50 0 50 100 1500.00
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d
1.¥10-81.¥10-71.¥10-60.000010.0001 0.001 0.01 0.10.00
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r
0.001 0.01 0.1 1.0.000
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sin q13
0.0001 0.01 1. 100. 10 000.0.00
0.01
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0.03
0.04
0.05
0.06
tan2q12
0.0001 0.01 1. 100. 10 000.0.00
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tan2q23
1.¥10-6 0.00001 0.0001 0.001 0.01 0.10.00
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r
0.001 0.01 0.1 1.0.000
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sin q13
0.0001 0.01 1. 100. 10 000.0.00
0.01
0.02
0.03
0.04
0.05
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tan2q12
0.0001 0.01 1. 100. 10 000.0.00
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tan2q23
1.¥10-6 0.00001 0.0001 0.001 0.01 0.10.00
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J
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d
25
10 = (5, 3, 0) 5 = (2, 0, 0) 1 = (1,�1, 0)
SU(5)⇥ U(1)Consider a simple U(1) as flavour symmetry, in a SU(5) inspired context:
Anarchy?: BeZer Hierarchy![Altarelli, Feruglio & Masina 2002; Altarelli, Feruglio, Masina & LM to appear]
1.¥10-6 0.00001 0.0001 0.001 0.01 0.10.00
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sin q13
0.0001 0.01 1. 100. 10 000.0.000
0.005
0.010
0.015
0.020
0.025
0.030
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tan2q12
0.0001 0.01 1. 100. 10 000.0.00
0.01
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tan2q23
1.¥10-6 0.00001 0.0001 0.001 0.01 0.10.00
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sin q13
0.0001 0.01 1. 100. 10 000.0.000
0.005
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0.015
0.020
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0.030
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tan2q12
0.0001 0.01 1. 100. 10 000.0.00
0.01
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1.¥10-6 0.00001 0.0001 0.001 0.01 0.10.00
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sin q13
0.0001 0.01 1. 100. 10 000.0.000
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tan2q12
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sin q13
0.0001 0.01 1. 100. 10 000.0.000
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0.015
0.020
0.025
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tan2q12
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-150 -100 -50 0 50 100 1500.000
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26
Discrete symmetries can accommodate neutrino mixing paDerns and
TB, GR & BM can s;ll be taken as star;ng point
LFV analysis gives strong constraints
No possibility to sa;sfy and
Are TB, GR & BM the flavour structure of nature? or only accidents?
The new data on have put severe doubts on their naturalness:
-‐ All need large correc;ons in their simplest versions
-‐ The special TB (most successful) needs dynamical tricks
-‐ Alterna;ves: Other paDerns?
Anarchy? Something in between?
New correla;ons?
✓13
BR(µ ! e�) �aµ
Intermediate Conclusions
Luca Merlo, Neutrino Flavour Models and Impact on LFV
[Toorop, Feruglio, Hagedorn 2011 (1°&2°);ecc...]
The minimisa`on of the scalar poten`al, that explains the VEV of the flavons in a par;cularly predic;ve MLFV scenario (2 RH neutrinos), links the neutrino spectrum, the mixing angles and the Majorana phase: main responsible Majorana Nature
[Alonso, Gavela, D.Hernandez & LM 1206.3167]
[Gavela, Hambye, D.Hernandez & P.Hernandez 2009]
New Correla`ons?
The minimisa`on of the scalar poten`al, that explains the VEV of the flavons in a par;cularly predic;ve MLFV scenario (2 RH neutrinos), links the neutrino spectrum, the mixing angles and the Majorana phase: main responsible Majorana Nature
[Alonso, Gavela, D.Hernandez & LM 1206.3167]
Gfl ⇠ SU(3)`L ⇥ SU(3)ER ⇥O(2)N
�Lmass = `L�YEER + `L�Y⌫(N1, N2)T + ⇤(N1N
c1 +N2N
c2 ) + h.c.
YE ⇠ ( 3 , 3 , 1) Y⌫ ⇠ ( 3 , 1 , 2)promo;ng
[Gavela, Hambye, D.Hernandez & P.Hernandez 2009]
New Correla`ons?
The minimisa`on of the scalar poten`al, that explains the VEV of the flavons in a par;cularly predic;ve MLFV scenario (2 RH neutrinos), links the neutrino spectrum, the mixing angles and the Majorana phase: main responsible Majorana Nature
[Alonso, Gavela, D.Hernandez & LM 1206.3167]
Gfl ⇠ SU(3)`L ⇥ SU(3)ER ⇥O(2)N
�Lmass = `L�YEER + `L�Y⌫(N1, N2)T + ⇤(N1N
c1 +N2N
c2 ) + h.c.
YE ⇠ ( 3 , 3 , 1) Y⌫ ⇠ ( 3 , 1 , 2)promo;ng
(y2 � y02)pm⌫2m⌫1 sin 2✓ cos 2↵ = 0
tg2✓ = sin 2↵y2 � y02
y2 + y022
pm⌫2m⌫1
m⌫2 �m⌫1
The scalar poten;al is extremized when:
[Gavela, Hambye, D.Hernandez & P.Hernandez 2009]
New Correla`ons?
The minimisa`on of the scalar poten`al, that explains the VEV of the flavons in a par;cularly predic;ve MLFV scenario (2 RH neutrinos), links the neutrino spectrum, the mixing angles and the Majorana phase: main responsible Majorana Nature
[Alonso, Gavela, D.Hernandez & LM 1206.3167]
Gfl ⇠ SU(3)`L ⇥ SU(3)ER ⇥O(2)N
�Lmass = `L�YEER + `L�Y⌫(N1, N2)T + ⇤(N1N
c1 +N2N
c2 ) + h.c.
YE ⇠ ( 3 , 3 , 1) Y⌫ ⇠ ( 3 , 1 , 2)promo;ng
(y2 � y02)pm⌫2m⌫1 sin 2✓ cos 2↵ = 0
tg2✓ = sin 2↵y2 � y02
y2 + y022
pm⌫2m⌫1
m⌫2 �m⌫1
The scalar poten;al is extremized when:
2 family case:
↵ = ±⇡/4
✓12 large m⌫2 ⇡ m⌫1
[Gavela, Hambye, D.Hernandez & P.Hernandez 2009]
New Correla`ons?
The minimisa`on of the scalar poten`al, that explains the VEV of the flavons in a par;cularly predic;ve MLFV scenario (2 RH neutrinos), links the neutrino spectrum, the mixing angles and the Majorana phase: main responsible Majorana Nature
[Alonso, Gavela, D.Hernandez & LM 1206.3167]
Gfl ⇠ SU(3)`L ⇥ SU(3)ER ⇥O(2)N
�Lmass = `L�YEER + `L�Y⌫(N1, N2)T + ⇤(N1N
c1 +N2N
c2 ) + h.c.
YE ⇠ ( 3 , 3 , 1) Y⌫ ⇠ ( 3 , 1 , 2)promo;ng
(y2 � y02)pm⌫2m⌫1 sin 2✓ cos 2↵ = 0
tg2✓ = sin 2↵y2 � y02
y2 + y022
pm⌫2m⌫1
m⌫2 �m⌫1
The scalar poten;al is extremized when:
2 family case:
↵ = ±⇡/4
✓12 large m⌫2 ⇡ m⌫1
3 family case:
↵ = ±⇡/4
m⌫2 ⇡ m⌫1IH: ✓12
large, but nega;ve
[Gavela, Hambye, D.Hernandez & P.Hernandez 2009]
New Correla`ons?
The minimisa`on of the scalar poten`al, that explains the VEV of the flavons in a par;cularly predic;ve MLFV scenario (2 RH neutrinos), links the neutrino spectrum, the mixing angles and the Majorana phase: main responsible Majorana Nature
[Alonso, Gavela, D.Hernandez & LM 1206.3167]
Gfl ⇠ SU(3)`L ⇥ SU(3)ER ⇥O(2)N
�Lmass = `L�YEER + `L�Y⌫(N1, N2)T + ⇤(N1N
c1 +N2N
c2 ) + h.c.
YE ⇠ ( 3 , 3 , 1) Y⌫ ⇠ ( 3 , 1 , 2)promo;ng
(y2 � y02)pm⌫2m⌫1 sin 2✓ cos 2↵ = 0
tg2✓ = sin 2↵y2 � y02
y2 + y022
pm⌫2m⌫1
m⌫2 �m⌫1
The scalar poten;al is extremized when:
2 family case:
↵ = ±⇡/4
✓12 large m⌫2 ⇡ m⌫1
3 family case:
↵ = ±⇡/4
m⌫2 ⇡ m⌫1IH: ✓12
large, but nega;ve
[Gavela, Hambye, D.Hernandez & P.Hernandez 2009]
New Correla`ons?
work in progress
Thanks for your aZen`on
Backup Slides
29
30
Typical Tri-‐Bimaximal
sin2 ✓23 =1
2+Re(ce23) ⇠ +
1p3
⇣Re(c⌫13)�
p2Re(c⌫23)
⌘⇠
sin2 ✓12 =1
3� 2
3Re(ce12 + ce13) ⇠ +
2p2
3Re(c⌫12) ⇠
sin ✓13 =1
6
���3p2 (ce12 � ce13) + 2
p3⇣p
2 c⌫13 + c⌫23
⌘��� ⇠
ce12 ⇡ ce23 ⇡ ce13
c⌫12 ⇡ c⌫23 ⇡ c⌫13⇠e ⇡ ⇠⌫ ⌘ ⇠
31
Special Tri-‐Bimaximal
ce12 ⇡ ce23 ⇡ ce13⇠⌫ � ⇠e
c⌫12 = c⌫23 = 0 c⌫13 6= 0
�CP ⇡ arg c⌫13
sin ✓13 =
�����
r2
3
c⌫13 ⇠⌫+
ce12 � ce13p2
⇠e
�����
sin
2 ✓12 =
1
3
+
2
9
|c⌫13 ⇠⌫ |2 �2
3
Re(ce12 + ce13) ⇠e2
sin
2 ✓23 =
1
2
+
1p3
|c⌫13 ⇠⌫ | cos �CP +Re(ce23) ⇠e2
32
Bimaximal
⇠⌫ ⌧ ⇠e
c⌫12 ⇡ c⌫23 ⇡ c⌫13
ce12, ce13 6= 0 ce13 = 0
�CP = ⇡ + arg (ce12 � ce13)
sin ✓13 =1p2|ce12 � ce13| ⇠e
sin2 ✓12 =1
2� 1p
2Re(ce12 + ce13) ⇠
e
sin2 ✓23 =1
2
33
M
TB⌫ =
0
@x y y
y z x+ y � z
y x+ y � z z
1
A mu-‐tau symmagic sym
sin2 ✓TB12 = 1/3
sin2 ✓TB23 = 1/2
sin ✓TB13 = 0
UTB =
0
@
p2/3 1/
p3 0
�1/p6 1/
p3 �1/
p2
�1/p6 1/
p3 +1/
p2
1
A
[A4: Adhikary; Altarelli; Aris;zabal Sierra; Babu; Bazzocchi; Bertuzzo; Di Bari; Branco; Brahmachari; Chen; Choubey; Ciafaloni; Csaki; Delaunay; Felipe; Feruglio; Frampton; Frigerio; Ghosal; Grimus; Grojean; Grossmann; Hagedorn; He; Hirsch; Honda; Joshipura; Kaneko; Keum; King; Koide; Kuhbock; Lavoura; Lin; Ma; Machado; Malinsky; Matsuzaki; de Medeiros Varzielas; Meloni; LM; Mitra; Molinaro; Morisi; Nardi; Parida; Paris; Petcov; Pleitez; Picariello; Rajasekaran; Riazzudin, Romao; Serodio; Skadhauge; Tanimoto; Torrente-‐Lujan; Urbano; Valle; Villanova del Moral; Volkas; Yin; Zee; ...;
S4, T’, Δ(3n2): de Adelhart Toorop; Altarelli, Bazzocchi; Chen; Ding; Hagedorn; Feruglio; Frampton; Kephart; King; Lam; Lin; Luhn; Ma; Mahanthappa; Matsuzaki; de Medeiros Varzielas; LM; Morisi; Nasri; Ramond; Ross;
...]
Discrete Symmetries:
In the basis of diagonal charged leptons:
Typical Tri-‐Bimaximal
Luca Merlo, Neutrino Flavour Models and Impact on LFV
34
h'`ih'⌫i
Gf
= A4 ⇥Gaux
G` = Z3 G⌫ = Z2
Mdiage MTB
⌫
A4 is the group of even permuta;ons of 4 objects isomorphic to the group of the rota;ons which leave a regular tetrahedron invariant (Subgroup of SO(3)).It has 12 elements and 4 representa;ons: 3, 1, 1’, 1’’
me ⌧ mµ ⌧ m⌧
-‐ keeps separated the two sectors-‐ explains the hierarchy
U = U †eU⌫ = UTB
[Altarelli & Feruglio 2005]
The Altarelli-‐Feruglio Model
Luca Merlo, Neutrino Flavour Models and Impact on LFV
35
M⌫ =
0
B@a+ 2
3b � b3 � b
3
� b3
23b a� b
3
� b3 a� b
323b
1
CA v2u
Me = diag(ye t2, yµ t, y⌧ ) vd u
me
mµ=
mµ
m⌧= t ⇡ 0.05
Mdiag⌫ = v2udiag(a+ b, a, �a+ b)
r ⌘ �m2sol
�m2atm
⇡ 1
35
h'T i⇤
= (u, 0, 0)
h'Si⇤
= cb(u, u, u)
h⇠i⇤
= cau
h✓i⇤
= t
vacuum alignment:
we = ye�2
�3ec (⇤T ⌅)hd + yµ
�
�2µc (⇤T ⌅)
0 hd + y�1
�⇥ c (⇤T ⌅)
00 hd
w� = xa�
�
hu⇤hu⇤
�L+ xb
✓⇥S
�
hu⇤hu⇤
�L
◆�/⇤
Expansion in
[Altarelli & Feruglio 2005] MaDer fields Higgs Flavons
The Altarelli-‐Feruglio Model
Luca Merlo, Neutrino Flavour Models and Impact on LFV
36
�m2
eLL
�ij' � 1
8⇡2
�3m2
0 +A20
�X
k
(
ˆY †⌫ )ik log
✓⇤
Mk
◆(
ˆY⌫)kj
Y⌫ = k U† + ...
m⌫ =v2
2Y T⌫ M�1Y⌫
M�1 =2
|k|2v2mdiag⌫
�m2
eLL
�ij' � |k|2
8⇡2
�3m2
0 +A20
� Ui2 log
m2
m1U⇤j2 + Ui3 log
m3
m1U⇤j3
�+ ...
⇢(g) Y †⌫ Y⌫ ⇢(g)† = Y †
⌫ Y⌫ !⇥⇢(g), Y †
⌫ Y⌫
⇤= 0 ! Y †
⌫ Y⌫ / 1 ! Y⌫ is unitary
With RH NeutrinoWhen RH neutrinos are present in the spectrum, their RGE are important:
If the RH neutrinos transform as 3dim irreducible representa;ons then
Wri;ng the usual type I See-‐Saw rela;on:
Very predic`ve rela`on: it only depends on the LO mixing paZern and neutrino spectrum Luca Merlo, Neutrino Flavour Models and Impact on LFV
37
�m2
eLL
�µe
/ 1
3
log
✓m2
m1
◆
�m2
eLL
�⌧e
/ 1
3
log
✓m2
m1
◆
�m2
eLL
�⌧µ
/ 1
3
log
✓m2
m1
◆� 1
2
log
✓m3
m1
◆
�m2
eLL
�µe
/ � 1p10
log
✓m2
m1
◆
�m2
eLL
�⌧e
/ � 1p10
log
✓m2
m1
◆
�m2
eLL
�⌧µ
/ 5 +
p5
20
log
✓m2
m1
◆� 1
2
log
✓m3
m1
◆
�m2
eLL
�µe
/ 1
4
r3
2
log
✓m2
m1
◆
�m2
eLL
�⌧e
/ 1
4
r3
2
log
✓m2
m1
◆
�m2
eLL
�⌧µ
/ 3
8
log
✓m2
m1
◆� 1
2
log
✓m3
m1
◆
TB paZern
GR paZern
BM paZern
Expressing all the neutrino masses in terms of the lightest one, these quan;;es depend on only 1 parameter
Luca Merlo, Neutrino Flavour Models and Impact on LFV
38
To get EDM, MDM and the LFV transi;ons we should calculate diagrams as:
A Good analy;cal approach is the Mass Inser;on approxima;on:
Mass Inser`on Approxima`on
Luca Merlo, Neutrino Flavour Models and Impact on LFV
39
(�ij)CC0
(�ij)CC0 =
�m2
CC0
�ij
m2SUSY
�Lm ��e ec
�
m2eLL m2
eLR
m2eRL m2
eRR
!✓eec
◆+ ⌫m2
⌫LL ⌫
L �Z
d
2✓ ye e
c`hd !
Zd
2✓
�ye + xe m0 ✓
2�e
c`hd
L �Z
d2✓ d2✓ ¯ !Z
d2✓ d2✓�1 + km2
0 ✓2✓2
�¯
The depend on the son parameters:
where the son masses are defined by
m2(e,⌫)LL m2
eRR
m2eLR = (m2
eRL)†
and are hermi;an matrices from the Kähler poten;al
from the superpoten;al
generated from the SUSY Lagrangian analy;cally con;nuing all the couplings constants into superspace:
Luca Merlo, Neutrino Flavour Models and Impact on LFV
40
(m2eLL)K =
0
@1 O(⇠n) O(⇠n)
O(⇠n) 1 O(⇠n)O(⇠n) O(⇠n) 1
1
Am20
Ye =
0
@ye ye O(⇠n) ye O(⇠n)
yµ O(⇠n) yµ yµ O(⇠n)y⌧ O(⇠n) y⌧ O(⇠n) y⌧
1
A
m2eRL =
0
@ye ye O(⇠n) ye O(⇠n)
yµ O(⇠n) yµ yµ O(⇠n)y⌧ O(⇠n) y⌧ O(⇠n) y⌧
1
A m0 vd
L �Z
d2✓�Ye +Ae m0✓
2�ij
eci `j hd
L �Z
d2✓ d2✓�1 + km2
0 ✓2✓2� ¯ + ¯ 'n
⇤nf
!The flavour is encoded into the soh masses:
Non-‐canonical kine;c terms
same flavour structure but different coefficients Luca Merlo, Neutrino Flavour Models and Impact on LFV
41
m2R(mW ) ' m2
R(⇤f ) + 1.5M21 (⇤f )
Mi(mW ) ' ↵i(mW )
↵i(⇤f )Mi(⇤f )
↵i(⇤f ) =1
25
|µ|2 ' 1 + 0.5 tan2 �
tan2 � � 1m2
0 +0.5 + 3.5 tan2 �
tan2 � � 1M2
1/2 �1
2m2
Z
Mi(⇤f ) ⌘ M1/2
Many parameters:
All of them are not independent:
SUGRA context:
M1,M2, µ, tan�,m2L,m
2R, A0
m2L(mW ) ' m2
L(⇤f ) + 0.5M22 (⇤f ) + 0.04M2
1 (⇤f )
SUSY Parameters
m2L(⇤f ) = m2
R(⇤f ) = A0 ⌘ m0
tan� ⇡ 100 ⌘ y⌧
Luca Merlo, Neutrino Flavour Models and Impact on LFV
42
& m0 = 5000 GeV
BR(µ ! e�) < 2.4⇥ 10�12
tan� = 15
Special TB
Typical TB
BM
Luca Merlo, Neutrino Flavour Models and Impact on LFV
43
Anarchy vs. Hierarchy?
Luca Merlo, Neutrino Flavour Models and Impact on LFV
H
Amt
Aâ10
Non-SeeSaw
0.1 0.2 0.3 0.4 0.5 0.60.000
0.005
0.010
0.015
l
100âP
HAAmt
Amt
A
SeeSaw
0.1 0.2 0.3 0.4 0.5 0.60.000
0.002
0.004
0.006
0.008
0.010
l
100âP
44
Scalar Poten`al
Luca Merlo, Neutrino Flavour Models and Impact on LFV
Tr⇣YEY†
E
⌘Tr
�Y⌫Y†
⌫
�det (YE)
Tr⇣YEY†
E
⌘2Tr
⇣YEY†
EY⌫Y†⌫
⌘
Tr�Y⌫Y†
⌫
�2Tr
�Y⌫�2Y†
⌫
�2
V =� µ2 ·X2 +�X2
�†�X2 + (µD det (YE) + h.c.)+
+ �E Tr⇣YEY†
E
⌘2+ gTr
⇣YEY†
EY⌫Y†⌫
⌘+
+ hTr�Y⌫Y†
⌫
�2+ h0 Tr
�Y⌫�2Y†
⌫
�2
gTr⇣
YEY†EY⌫Y†
⌫
⌘
/gn
(m2e +m2
µ)(y2+ y02)(m⌫2 +m⌫1)+
+ (m2µ �m2
e)
h
(m⌫2 �m⌫1)(y2+ y02) cos 2✓+
+ (y2 � y02)2pm⌫2m⌫1 sin 2↵ sin 2✓
io
operators:
scalar poten;al:
the mixingterm for 2 family case:
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