Neutrino%Flavour%Models% and%Impact%on%LFV · 3 m2 sol =(7.54 +0.26 0.22) ⇥ 10 5 eV2 m2 atm...
Transcript of Neutrino%Flavour%Models% and%Impact%on%LFV · 3 m2 sol =(7.54 +0.26 0.22) ⇥ 10 5 eV2 m2 atm...
![Page 1: Neutrino%Flavour%Models% and%Impact%on%LFV · 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](https://reader031.fdocuments.net/reader031/viewer/2022012001/608c9b6661838c7ce379ed6b/html5/thumbnails/1.jpg)
Neutrino Flavour Models and Impact on LFV
Luca Merlo
26.06.2012, WHAT IS ʋ? INVISIBLES12 and Alexei Smirnov Fest
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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
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�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...)
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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
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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
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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
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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
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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
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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
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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?
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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?
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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?
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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
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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
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Luca Merlo, Neutrino Flavour Models and Impact on LFV 8
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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
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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
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Luca Merlo, Neutrino Flavour Models and Impact on LFV 10
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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
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⇠⌫ ⌧ ⇠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
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⇠⌫ ⌧ ⇠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
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Luca Merlo, Neutrino Flavour Models and Impact on LFV 12
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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
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Which is the meaning of ?
How can we achieve these flavour structures?
⇠
Luca Merlo, Neutrino Flavour Models and Impact on LFV
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Basic Points on Model Building
Luca Merlo, Neutrino Flavour Models and Impact on LFV
[Talk by King; Alterna;ve way: talk by Ma]
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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]
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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]
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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]
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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]
![Page 30: Neutrino%Flavour%Models% and%Impact%on%LFV · 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](https://reader031.fdocuments.net/reader031/viewer/2022012001/608c9b6661838c7ce379ed6b/html5/thumbnails/30.jpg)
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]
![Page 31: Neutrino%Flavour%Models% and%Impact%on%LFV · 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](https://reader031.fdocuments.net/reader031/viewer/2022012001/608c9b6661838c7ce379ed6b/html5/thumbnails/31.jpg)
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]
![Page 32: Neutrino%Flavour%Models% and%Impact%on%LFV · 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](https://reader031.fdocuments.net/reader031/viewer/2022012001/608c9b6661838c7ce379ed6b/html5/thumbnails/32.jpg)
15
Are there consequences of so large ?⇠
Luca Merlo, Neutrino Flavour Models and Impact on LFV
![Page 33: Neutrino%Flavour%Models% and%Impact%on%LFV · 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](https://reader031.fdocuments.net/reader031/viewer/2022012001/608c9b6661838c7ce379ed6b/html5/thumbnails/33.jpg)
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
![Page 34: Neutrino%Flavour%Models% and%Impact%on%LFV · 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](https://reader031.fdocuments.net/reader031/viewer/2022012001/608c9b6661838c7ce379ed6b/html5/thumbnails/34.jpg)
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
![Page 35: Neutrino%Flavour%Models% and%Impact%on%LFV · 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](https://reader031.fdocuments.net/reader031/viewer/2022012001/608c9b6661838c7ce379ed6b/html5/thumbnails/35.jpg)
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
![Page 36: Neutrino%Flavour%Models% and%Impact%on%LFV · 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](https://reader031.fdocuments.net/reader031/viewer/2022012001/608c9b6661838c7ce379ed6b/html5/thumbnails/36.jpg)
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
![Page 37: Neutrino%Flavour%Models% and%Impact%on%LFV · 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](https://reader031.fdocuments.net/reader031/viewer/2022012001/608c9b6661838c7ce379ed6b/html5/thumbnails/37.jpg)
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
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18
(�ij)CC0
(�ij)CC0 =
�m2
CC0
�ij
m2SUSY
The depend on the son parameters:
Luca Merlo, Neutrino Flavour Models and Impact on LFV
![Page 39: Neutrino%Flavour%Models% and%Impact%on%LFV · 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](https://reader031.fdocuments.net/reader031/viewer/2022012001/608c9b6661838c7ce379ed6b/html5/thumbnails/39.jpg)
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
![Page 40: Neutrino%Flavour%Models% and%Impact%on%LFV · 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](https://reader031.fdocuments.net/reader031/viewer/2022012001/608c9b6661838c7ce379ed6b/html5/thumbnails/40.jpg)
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
![Page 41: Neutrino%Flavour%Models% and%Impact%on%LFV · 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](https://reader031.fdocuments.net/reader031/viewer/2022012001/608c9b6661838c7ce379ed6b/html5/thumbnails/41.jpg)
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 ⇠
![Page 42: Neutrino%Flavour%Models% and%Impact%on%LFV · 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](https://reader031.fdocuments.net/reader031/viewer/2022012001/608c9b6661838c7ce379ed6b/html5/thumbnails/42.jpg)
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
![Page 43: Neutrino%Flavour%Models% and%Impact%on%LFV · 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](https://reader031.fdocuments.net/reader031/viewer/2022012001/608c9b6661838c7ce379ed6b/html5/thumbnails/43.jpg)
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
![Page 44: Neutrino%Flavour%Models% and%Impact%on%LFV · 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](https://reader031.fdocuments.net/reader031/viewer/2022012001/608c9b6661838c7ce379ed6b/html5/thumbnails/44.jpg)
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
![Page 45: Neutrino%Flavour%Models% and%Impact%on%LFV · 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](https://reader031.fdocuments.net/reader031/viewer/2022012001/608c9b6661838c7ce379ed6b/html5/thumbnails/45.jpg)
22
Intermediate Conclusions
Luca Merlo, Neutrino Flavour Models and Impact on LFV
![Page 46: Neutrino%Flavour%Models% and%Impact%on%LFV · 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](https://reader031.fdocuments.net/reader031/viewer/2022012001/608c9b6661838c7ce379ed6b/html5/thumbnails/46.jpg)
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
![Page 47: Neutrino%Flavour%Models% and%Impact%on%LFV · 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](https://reader031.fdocuments.net/reader031/viewer/2022012001/608c9b6661838c7ce379ed6b/html5/thumbnails/47.jpg)
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
![Page 48: Neutrino%Flavour%Models% and%Impact%on%LFV · 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](https://reader031.fdocuments.net/reader031/viewer/2022012001/608c9b6661838c7ce379ed6b/html5/thumbnails/48.jpg)
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
![Page 49: Neutrino%Flavour%Models% and%Impact%on%LFV · 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](https://reader031.fdocuments.net/reader031/viewer/2022012001/608c9b6661838c7ce379ed6b/html5/thumbnails/49.jpg)
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
![Page 50: Neutrino%Flavour%Models% and%Impact%on%LFV · 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](https://reader031.fdocuments.net/reader031/viewer/2022012001/608c9b6661838c7ce379ed6b/html5/thumbnails/50.jpg)
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
![Page 51: Neutrino%Flavour%Models% and%Impact%on%LFV · 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](https://reader031.fdocuments.net/reader031/viewer/2022012001/608c9b6661838c7ce379ed6b/html5/thumbnails/51.jpg)
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...]
![Page 52: Neutrino%Flavour%Models% and%Impact%on%LFV · 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](https://reader031.fdocuments.net/reader031/viewer/2022012001/608c9b6661838c7ce379ed6b/html5/thumbnails/52.jpg)
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...]
![Page 53: Neutrino%Flavour%Models% and%Impact%on%LFV · 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](https://reader031.fdocuments.net/reader031/viewer/2022012001/608c9b6661838c7ce379ed6b/html5/thumbnails/53.jpg)
23
Are these paDerns only numerical accidents? If Yes what? Anarchy
Anarchy?
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Luca Merlo, Neutrino Flavour Models and Impact on LFV0.001 0.01 0.10.00
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[Hall, Murayama & Weiner 1999; de Gouvea & Murayama 2012]
![Page 54: Neutrino%Flavour%Models% and%Impact%on%LFV · 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](https://reader031.fdocuments.net/reader031/viewer/2022012001/608c9b6661838c7ce379ed6b/html5/thumbnails/54.jpg)
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]
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![Page 55: Neutrino%Flavour%Models% and%Impact%on%LFV · 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](https://reader031.fdocuments.net/reader031/viewer/2022012001/608c9b6661838c7ce379ed6b/html5/thumbnails/55.jpg)
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]
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sin q13
0.0001 0.01 1. 100. 10 000.0.000
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![Page 56: Neutrino%Flavour%Models% and%Impact%on%LFV · 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](https://reader031.fdocuments.net/reader031/viewer/2022012001/608c9b6661838c7ce379ed6b/html5/thumbnails/56.jpg)
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...]
![Page 57: Neutrino%Flavour%Models% and%Impact%on%LFV · 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](https://reader031.fdocuments.net/reader031/viewer/2022012001/608c9b6661838c7ce379ed6b/html5/thumbnails/57.jpg)
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?
![Page 58: Neutrino%Flavour%Models% and%Impact%on%LFV · 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](https://reader031.fdocuments.net/reader031/viewer/2022012001/608c9b6661838c7ce379ed6b/html5/thumbnails/58.jpg)
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?
![Page 59: Neutrino%Flavour%Models% and%Impact%on%LFV · 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](https://reader031.fdocuments.net/reader031/viewer/2022012001/608c9b6661838c7ce379ed6b/html5/thumbnails/59.jpg)
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?
![Page 60: Neutrino%Flavour%Models% and%Impact%on%LFV · 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](https://reader031.fdocuments.net/reader031/viewer/2022012001/608c9b6661838c7ce379ed6b/html5/thumbnails/60.jpg)
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?
![Page 61: Neutrino%Flavour%Models% and%Impact%on%LFV · 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](https://reader031.fdocuments.net/reader031/viewer/2022012001/608c9b6661838c7ce379ed6b/html5/thumbnails/61.jpg)
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?
![Page 62: Neutrino%Flavour%Models% and%Impact%on%LFV · 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](https://reader031.fdocuments.net/reader031/viewer/2022012001/608c9b6661838c7ce379ed6b/html5/thumbnails/62.jpg)
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
![Page 63: Neutrino%Flavour%Models% and%Impact%on%LFV · 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](https://reader031.fdocuments.net/reader031/viewer/2022012001/608c9b6661838c7ce379ed6b/html5/thumbnails/63.jpg)
Thanks for your aZen`on
![Page 64: Neutrino%Flavour%Models% and%Impact%on%LFV · 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](https://reader031.fdocuments.net/reader031/viewer/2022012001/608c9b6661838c7ce379ed6b/html5/thumbnails/64.jpg)
Backup Slides
29
![Page 65: Neutrino%Flavour%Models% and%Impact%on%LFV · 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](https://reader031.fdocuments.net/reader031/viewer/2022012001/608c9b6661838c7ce379ed6b/html5/thumbnails/65.jpg)
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 ⇡ ⇠⌫ ⌘ ⇠
![Page 66: Neutrino%Flavour%Models% and%Impact%on%LFV · 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](https://reader031.fdocuments.net/reader031/viewer/2022012001/608c9b6661838c7ce379ed6b/html5/thumbnails/66.jpg)
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
![Page 67: Neutrino%Flavour%Models% and%Impact%on%LFV · 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](https://reader031.fdocuments.net/reader031/viewer/2022012001/608c9b6661838c7ce379ed6b/html5/thumbnails/67.jpg)
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
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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
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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
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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
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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
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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
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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
![Page 74: Neutrino%Flavour%Models% and%Impact%on%LFV · 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](https://reader031.fdocuments.net/reader031/viewer/2022012001/608c9b6661838c7ce379ed6b/html5/thumbnails/74.jpg)
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
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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
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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
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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
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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
![Page 79: Neutrino%Flavour%Models% and%Impact%on%LFV · 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](https://reader031.fdocuments.net/reader031/viewer/2022012001/608c9b6661838c7ce379ed6b/html5/thumbnails/79.jpg)
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: