neutrinos and the flavour puzzlevietnam.in2p3.fr/2013/inauguration/transparencies/gavela.pdf[alonso,...
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![Page 1: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/1.jpg)
Neutrinos and the Flavour Puzzle
(Alonso, Gavela, D.Hernandez, Merlo, Rigolin)
Belén Gavela
(Alonso, Gavela, Isidori, Maiani)
![Page 2: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/2.jpg)
Cabibbo’s dream
(Alonso, Gavela, D.Hernandez, Merlo, Rigolin)
Belén Gavela
(Alonso, Gavela, Isidori, Maiani)
![Page 3: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/3.jpg)
3
Neutrino light on flavour ?
![Page 4: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/4.jpg)
Neutrinos lighter because Majorana?
![Page 5: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/5.jpg)
the size of ν Yukawa couplings is alike
Pilar Hernandez drawings
Within seesaw,to that for other fermions:
![Page 6: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/6.jpg)
0.8 0.5 ~ 9º -0.4 0.5 -0.7
-0.4 0.5 +0.7
VPMNS =
~1 λ λ3
λ ~1 λ2
λ3 λ2 ~1
VCKM = λ~0.2
Leptons
Quarks
Why so different?
![Page 7: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/7.jpg)
0.8 0.5 ~ 9º -0.4 0.5 -0.7
-0.4 0.5 +0.7
VPMNS =
~1 λ λ3
λ ~1 λ2
λ3 λ2 ~1
VCKM = λ~0.2
Leptons
Quarks
Perhaps also because νs may be Majorana?
![Page 8: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/8.jpg)
•Dynamical Yukawas
![Page 9: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/9.jpg)
Yukawa couplings are the source of flavour in the SM
YD
YU
QL
QL
UR
DR
H
H
![Page 10: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/10.jpg)
YE
YN
L
L
NR
ER
H
H
(i.e. Seesaw type-I)
Yukawa couplings are a source of flavour in the ν-SM
![Page 11: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/11.jpg)
May they correspond to dynamical fields
(e.g. vev of fields that carry flavor) ?
![Page 12: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/12.jpg)
Instead of inventing an ad-hoc symmetry group,
why not use the continuous flavour group
suggested by the SM itself?
![Page 13: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/13.jpg)
We have realized that the different pattern for
quarks versus leptons
may be a simple consequence of the
continuous flavour group of the SM (+ seesaw)
(Alonso, Gavela, D.Hernandez, Merlo, Rigolin)
(Alonso, Gavela, Isidori, Maiani)
![Page 14: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/14.jpg)
We have realized that the different pattern for
quarks versus leptons
may be a simple consequence of the
continuous flavour group of the SM (+ seesaw)
Our guideline is to use:
- maximal symmetry - minimal field content
(Alonso, Gavela, D.Hernandez, Merlo, Rigolin)
(Alonso, Gavela, Isidori, Maiani)
![Page 15: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/15.jpg)
Global flavour symmetry of the SM
* QCD has a global -chiral- symmetry in the limit of massless quarks. For n generations:
* In the SM, fermion masses and mixings result from Yukawa couplings. For massless quarks, the SM has a global flavour symmetry:
SM
fermions
fermions
[Georgi, Chivukula, 1987]
Quarks
Gflavour
![Page 16: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/16.jpg)
This continuous symmetry of the SM
QL DR
H Y spurion
is phenomenologically very successful and
at the basis of Minimal Flavour Violation
in which the Yukawa couplings are only spurions
D’Ambrosio+Giudice+Isidori+Strumia; Cirigliano+Isidori+Grinstein+Wise
Gflavour
![Page 17: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/17.jpg)
This continuous symmetry of the SM
QL DR
H Y spurion
is phenomenologically very successful and
at the basis of Minimal Flavour Violation
in which the Yukawa couplings are only spurions
D’Ambrosio+Giudice+Isidori+Strumia; Cirigliano+Isidori+Grinstein+Wise
Gflavour
Υαβ+ Υδγ Qα γµQβ Qγ γµ Qδ Λf2
- -
![Page 18: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/18.jpg)
One step further
(Alonso, Gavela, D.Hernandez, Merlo, Rigolin, 2012 -2013)
(Alonso, Gavela, Isidori, Maiani, 2013)
![Page 19: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/19.jpg)
Quarks
![Page 20: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/20.jpg)
< Y > Λf
< Y > Λf
QL DR
H <Yd >
For this talk:
each YSM -- >one single field Y YSM ~
QL UR
H <Yu >quarks:
Gflavour= SU(3)QL x SU(3)UR x SU(3)DR ....Anselm+Berezhiani 96; Berezhiani+Rossi 01... Alonso+Gavela+Merlo+Rigolin 11...
![Page 21: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/21.jpg)
< Y > Λ< Y > Λf
QL DR
H <Yd >
For this talk:
each YSM -- >one single field Y YSM ~
QL UR
H <Yu >quarks:
Gflavour= SU(3)QL x SU(3)UR x SU(3)DR .... Yd ~ (3,1,3)
Yu ~ (3,3,1)
-- “bifundamentals”
![Page 22: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/22.jpg)
QL DR
H <Yd >
Yd ~ (3,1,3)
QL UR
H <Yu >
¿V(Yd, Yu)?
Gflavour= SU(3)QL x SU(3)UR x SU(3)DR ....
Yu ~ (3,3,1)
- -
![Page 23: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/23.jpg)
QL DR
H <Yd >
Yd ~ (3,1,3)
QL UR
H <Yu >
Gflavour= SU(3)QL x SU(3)UR x SU(3)DR ....
Yu ~ (3,3,1)
- -
* Does the minimum of the scalar potential justify the observed masses and mixings?
![Page 24: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/24.jpg)
V(Yd, Yu)
* Invariant under the SM gauge symmetry
* Invariant under its global flavour symmetry Gflavour
Gflavour= U(3)QL x U(3)UR x U(3)DR
![Page 25: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/25.jpg)
V(Yd, Yu)
* Invariant under the SM gauge symmetry
* Invariant under its global flavour symmetry Gflavour
Gflavour= U(3)QL x U(3)UR x U(3)DR There are as many independent invariants I as physical variables
V(Yd, Yu) = V( I(Yd, Yu))
![Page 26: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/26.jpg)
Minimization
a variational principle fixes the vevs of the Fields
This is an homogenous linear equation;if the rank of the Jacobian ,
Maximum:then the only solution is:
Less than Maximum:then the number of equations reduces to a number equal to the rank
is:
masses, mixing angles etc.
![Page 27: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/27.jpg)
Boundariesfor a reduced rank of the Jacobian,
there exists (at least) a direction for whicha variation of the field variables does
not vary the invariants
that is a Boundary of the I-manifold[Cabibbo, Maiani,1969]
Boundaries Exhibit Unbroken Symmetry [Michel, Radicati, 1969]
(maximal subgroups)
![Page 28: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/28.jpg)
Bi-fundamental Flavour Fields
[Feldmann, Jung, Mannel;Jenkins, Manohar]
For quarks: 10 independent invariants (because 6 masses+ 3 angles + 1 phase) that we may choose as
quark case
![Page 29: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/29.jpg)
Bi-fundamental Flavour Fields
masses and mixing[Feldmann, Jung, Mannel; Jenkins, ManoharAlonso, Gavela, Isidori, Maiani 2013]
only masses
quark case
![Page 30: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/30.jpg)
Jacobian Analysis: Mixing
the rank is reduced the most for:
VCKM= PERMUTATION
no mixing: reordering of states
(Alonso, Gavela, Isidori, Maiani 2013)
![Page 31: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/31.jpg)
Quark Natural Flavour Pattern
giving a hierarchical mass spectrum without mixing
a good approximation to the observed Yukawas to order (λC)2
Summarizing, a possible and natural breaking pattern arises:
Gflavour (quarks)
< YD > < YU >
![Page 32: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/32.jpg)
And what happens for leptons ?
Any difference with Majorana neutrinos?
![Page 33: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/33.jpg)
Global flavour symmetry of the SM + seesaw
* In the SM, for quarks the maximal global symmetry in the limit of massless quarks was:
SM
quarks
* In SM +type I seesaw, for leptons
the maximal leptonic global symmetry in the limit of massless light leptons is
-> degenerate heavy neutrinos
Gflavour
![Page 34: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/34.jpg)
Bi-fundamental Flavour Fields
Very direct results using the bi-unitary parametrization:
Physical parameters=Independent Invariants
* m e, μ, τ= v yE
mν =Y v2 Y Τ M
*But the relation of Yν with light neutrino masses is through
![Page 35: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/35.jpg)
Bi-fundamental Flavour Fields
Very direct results using the bi-unitary parametrization:
Physical parameters=Independent Invariants
* m e, μ, τ= v yE
*But the relation of Yν with light neutrino masses is through
![Page 36: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/36.jpg)
Bi-fundamental Flavour Fields
Very direct results using the bi-unitary parametrization:
*But the relation of Yν with light neutrino masses is through
Physical parameters=Independent Invariants
* m e, μ, τ= v yE
UR is relevant for leptons
![Page 37: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/37.jpg)
mν ~Yν v2 Yν Τ = y1 y2 v2 0 11 0
UPMNS = 1 1-1 1
eiπ/4 0 0 e-iπ/4
Degenerate neutrino masses {
M M
* For instance for two generations: O(2)NR
e.g. two families
Generically, O(2) allows : - one mixing angle maximal - one relative Majorana phase of π/2 - two degenerate light neutrinos
![Page 38: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/38.jpg)
Now for three generations and
considering all
possible independent invariants
easier using the bi-unitary parametrization as we did for quarks
![Page 39: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/39.jpg)
Leptons
UL and eigenvalues
UR and eigenvalues
New
Inva
rian
ts w
rt Q
uark
s
Gflavour (leptons) = 3 3 3
Number of Physical parameters = number of Independent Invariants 15 invariants for
![Page 40: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/40.jpg)
Leptons
UL and eigenvalues
UR and eigenvalues
New
Inva
rian
ts w
rt Q
uark
s
Gflavour (leptons) = 3 3 3
Number of Physical parameters = number of Independent Invariants 15 invariants for
![Page 41: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/41.jpg)
Leptons
UL and eigenvalues
UR and eigenvalues
New
Inva
rian
ts w
rt Q
uark
s
Gflavour (leptons) = 3 3 3
Number of Physical parameters = number of Independent Invariants 15 invariants for
![Page 42: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/42.jpg)
Jacobian Analysis: Mixing
same as for VCKM
O(3) vs U(3)
![Page 43: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/43.jpg)
...which is now not trivial mixing...
Jacobian Analysis: Mixing
...in fact it allows maximal mixing:
![Page 44: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/44.jpg)
...which is now not trivial mixing...
...in fact it leads to one maximal mixing angle:
Jacobian Analysis: Mixing
and maximal Majorana phase
mν2=mν3
![Page 45: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/45.jpg)
...in fact it leads to one maximal mixing angle:
...which is now not trivial mixing...
Jacobian Analysis: Mixing
θ23 =45o; Majorana Phase Pattern (1,1,i)
& at this level mass degeneracy: mν2 = mν3
related to the O(2) substructure [Alonso, Gavela, D. Hernández, L. Merlo;[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin]
![Page 46: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/46.jpg)
if the three neutrinos are quasidegenerate, perturbations:
This very simple structure is signaled by the extrema of the potential and
has eigenvalues (1,1,-1)
and is diagonalized by a maximal θ = 45º
![Page 47: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/47.jpg)
if the three neutrinos are quasidegenerate, perturbations:
3 degenerate light neutrinos+ a maximal Majorana phase
This very simple structure is signaled by the extrema of the potential and
has eigenvalues (1,1,-1)
and is diagonalized by a maximal θ = 45º
![Page 48: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/48.jpg)
=
Generalization to any seesaw model
the effective Weinberg Operator
shall have a flavour structure that breaks U(3)L to O(3)
then the results apply to any seesaw model
v2 Cd=5
Cd=5
mν
![Page 49: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/49.jpg)
First conclusion:
* at the same order in which the minimum of the potential
does NOT allow quark mixing,
it allows:
- hierarchical charged leptons
- quasi-degenerate neutrino masses
- one angle of ~45 degrees
- one maximal Majorana phase and the other one trivial
![Page 50: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/50.jpg)
Perturbations can produce a second large angle
if the three neutrinos are quasidegenerate, perturbations:
produce a second large angle and a perturbative one together with mass splittings
degenerate spectrum
![Page 51: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/51.jpg)
Perturbations can produce a second large angle
if the three neutrinos are quasidegenerate, perturbations:
only thisvanishes with the
perturbations
this angle does not vanish
withvanishing
perturbations
produce a second large angle and a perturbative one together with mass splittings
degenerate spectrum
![Page 52: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/52.jpg)
Perturbations can produce a second large angle
if the three neutrinos are quasidegenerate, perturbations:
produce a second large angle and a perturbative one together with mass splittings
degenerate spectrum
![Page 53: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/53.jpg)
accommodation of angles requires degenerate spectrum at reach in future neutrinoless double β exps.!
rough estimate m~0.1eV
![Page 54: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/54.jpg)
Slide from Laura Baudis talk presenting the new Gerda data at Invisibles13 workshop 3 weeks ago
~ 2 1026 yr
![Page 55: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/55.jpg)
![Page 56: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/56.jpg)
Where do the differences in Mixing originated?
in the symmetries of the Quark and Lepton sectors
for the type I seesaw employed here;
in general
![Page 57: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/57.jpg)
Where do the differences in Mixing originate?
From theMAJORANA vs DIRAC nature of fermions
![Page 58: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/58.jpg)
Conclusions• Spontaneous Flavour Symmetry Breaking is a predictive
dynamical scenario
• Simple solutions arise that resemble nature in first approximation
• The differences in the mixing pattern of Quarks and Leptons arise from their Dirac vs Majorana nature (U vs. O symmetries). O(2) singled out -> O(3).
• A correlation between large angles and degenerate spectrum emerges. Explicitly, for neutrinos we find: fixed Majorana phases (1,1,i) , θ23 =45o, θ12 large, θ13 small and deg. ν’s
• This scenario will be tested in the near future by 0ν2β experiments (~.1eV).... or cosmology!!!
![Page 59: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/59.jpg)
![Page 60: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/60.jpg)
Back-up slides
![Page 61: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/61.jpg)
Fundamental Fields
May provide dynamically the perturbations
In the case of quarks they can give the right corrections:
under study in the lepton sector
[Alonso, Gavela, Merlo, Rigolin]
We set the perturbations by hand. Can we predict them also dynamically?
![Page 62: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/62.jpg)
YD
YU
QL
QL
UR
QLQL
DR
H
H
(1,1,3)
(1,3,1)(3,1,1)
Use the flavour symmetry of the SM with masless fermions:
replace Yukawas by fields:
(3,1,1)
(3,1,1)
(3,1,3)
(3,3,1)
< >
< > Spontaneous breaking of flavour symmetry dangerous
Gf = U(3)Q x U(3)U x U(3)D L RR
![Page 63: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/63.jpg)
Flavour Symmetry Breaking
To prevent Goldstone Bosons the symmetry can beGauged
[Grinstein, Redi, VilladoroGuadagnoli, Mohapatra, Sung
Feldman]
![Page 64: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/64.jpg)
Jacobian Analysis: Masses
IU2
IU33
U�2�2�U�1�
U�3�
U�2��U�1�2
U�2��U�1� U�1�4
U�2��U�1�
[U(3)xU(3) broken to]
(Alonso, Gavela, Isidori, Maiani 2013)
![Page 65: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/65.jpg)
Jacobian Analysis: Eigenvalues
IU2
IU33
U�2�2�U�1�
U�3�
U�2��U�1�2
U�2��U�1� U�1�4
U�2��U�1�
[U(3)LxU(3)ER broken to]
![Page 66: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/66.jpg)
Jacobian Analysis: [40 years ago...][Cabibbo, Maiani]Breaking of
-
![Page 67: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/67.jpg)
Lepton Natural Flavour PatternSummarizing, a possible and natural
breaking pattern:
brings along hierarchical charged leptons
and (at least) two degenerate neutrinosand maximal angle and Majorana phase
θ23 =45o; Majorana Phase Pattern (1,1,i)& Mass degeneracy: mν2 = mν3
![Page 68: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/68.jpg)
Boundaries Exhibit Unbroken Symmetry
Extra-Dimensions Example
The smallest boundaries are extremal points of any function
[Michel, Radicati, 1969]
![Page 69: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/69.jpg)
YD
YU
QL
QL
UR
QLQL
DR
H
H
Gf = SU(3)Q x SU(3)U x SU(3)D L RR
(3,1,1) (1,1,3)
(1,3,1)(3,1,1)
~ (3,1,3)
~ (3,3,1)
The non-abelian part of the flavour symmetry of the SM:
broken by Yukawas:
![Page 70: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/70.jpg)
Some good ideas:
Minimal Flavour Violation:
- Use the flavour symmetry of the SM in the limit of massless fermions quarks: Gflavour= U(3)QL x U(3)UR x U(3)DR - Assume that Yukawas are the only source of flavour in the SM and beyond Υαβ+ Υδγ Qα γµQβ Qγ γµ Qδ Λflavour2
... agrees with flavour data being aligned with SM
... allows to bring down Λflavour --> TeV
D’Ambrosio+Giudice+Isidori+Strumia; Cirigliano+Isidori+Grinstein+Wise
QL DR
H Y spurion
(Chivukula+ Georgi)
![Page 71: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/71.jpg)
Some good ideas:
Minimal Flavour Violation:
- Use the flavour symmetry of the SM in the limit of massless fermions quarks: Gflavour= U(3)QL x U(3)UR x U(3)DR - Assume that Yukawas are the only source of flavour in the SM and beyond Υαβ+ Υδγ Qα γµQβ Qγ γµ Qδ Λflavour2
... agrees with flavour data being aligned with SM
... allows to bring down Λflavour --> TeV
(Chivukula+Georgi 87; Hall+Randall; D’Ambrosio+Giudice+Isidori+Strumia; Cirigliano+Isidori+Grisntein+Wise; Davidson+Pallorini; Kagan+G. Perez + Volanski+Zupan,... )
QL DR
H Y spurion
Lalak, Pokorski, Ross; Fitzpatrick, Perez, Randall; Grinstein, Redi, Villadoro
(Chivukula+ Georgi)
![Page 72: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/72.jpg)
YD
YU
QL
QL
UR= (c, s, t)R
QLQL
DR = (d, s, b)R
H
H
Gf = U(3)Q x U(3)U x U(3)D L RR
(1,1,3)
(1,3,1)(3,1,1)
Use the flavour symmetry of the SM with masless fermions:
which is broken by Yukawas:
(3,1,1)
(3,1,1)
~ (3,1,3)
~ (3,3,1)
![Page 73: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/73.jpg)
YD
YU
QL
QL
UR
QLQL
DR
H
H
(1,1,3)
(1,3,1)(3,1,1)
Use the flavour symmetry of the SM with masless fermions:
replace Yukawas by fields:
(3,1,1)
(3,1,1)
(3,1,3)
(3,3,1)
< >
< >
Gf = U(3)Q x U(3)U x U(3)D L RR
![Page 74: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/74.jpg)
Flavour FieldsThe Yukawa Operator has to be explicitly flavour invariant
at high energies
A single and therefore “bi-fundamental” field
(d=5)
(QL)!(DR)"
H(YD)!"
![Page 75: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/75.jpg)
Bi-fundamental Flavour Fields
Physical parameters=Independent Invariants
These are (proportional to):
3 masses in de up sector, 3 masses in de down sector, 4 mixing parameters in VCKM
![Page 76: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/76.jpg)
Yd Yu
<Yd> <Yu>
![Page 77: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/77.jpg)
for the sub-Jacobian which involves only masseswe can identify the shape of the I-manifold
Jacobian Analysis
(Alonso, Gavela, Isidori, Maiani 2013)
![Page 78: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/78.jpg)
Renormalizable Potential
![Page 79: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/79.jpg)
Invariants at the Renormalizable Level
![Page 80: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/80.jpg)
Renormalizable Potential
with the definition
the potential
which contains 8 parametersmass spectrum
![Page 81: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/81.jpg)
Renormalizable Potential
with the definition
the potential
which contains 8 parameters
mixing
![Page 82: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/82.jpg)
Tr(Yu Yu+ Yd Yd+)
e.g. for the case of two families:
mq
θ
V
at the minimum:
same conclusion for 3 families
Berezhiani-Rossi; Anselm, Berezhiani; Alonso, Gavela, Merlo, Rigolin
;
-> NO MIXING
![Page 83: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/83.jpg)
Von Neumann Trace Inequality
Renormalizable Potential, mixing three families
So the Potential selects:
“normal”Hierarchy
“inverted”Hierarchy
No mixing, independently of the mass spectrum
coefficient in the potential
![Page 84: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/84.jpg)
Tr(Yu Yu+ Yd Yd+)
e.g. for the case of two families:
mq
θ
V
at the minimum:
same conclusion for 3 families
Berezhiani-Rossi; Anselm, Berezhiani; Alonso, Gavela, Merlo, Rigolin
;
-> NO MIXING
![Page 85: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/85.jpg)
Using Casas-Ibarra parametrization Yν= UPMNS mν1/2 R MN1/2
Tr(YE YE+ Yν Yν+) = Tr( mi1/2 U+ ml
2 U mi1/2 R+ ΜΝ R)
diagonal eigenvalues
with ω real,
* In degenerate limit of heavy neutrinos MN1=MN2=M
ch ω -i sh ω
i sh ω ch ω ( )R = with ω real,
it follows that:
2 families, leptons; let us analyze the mixing invariant
complex orthogonal; it encodes our ignorance of the high energy theory
![Page 86: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/86.jpg)
for 2 generations, the mixing terms in V(YE, Yν) is :
Tr(YE YE+ Yν Yν+)
Tr(Yu Yu+ Yd Yd+)
Leptons
Quarks
cosθ sinθ -sinθ cosθ
e-iα 0 0 eiαwhere UPMNS=
Tr(YE YE+ Yν Yν+)
![Page 87: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/87.jpg)
e.g., for 2 generations, the mixing terms in V(YE, Yν) is :
Tr(Yu Yu+ Yd Yd+)
This mixing term unphysical if either “up” or “down” fermions degenerate
Mixing physical even with degenerate neutrino masses,if Majorana phase non-trivial
Leptons
Quarks
Tr(YE YE+ Yν Yν+)
![Page 88: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/88.jpg)
Tr(YE YE+ Yν Yν+)
α= π/4 or 3π/4
Minimisation
*
*
Large angles correlated with degenerate masses
Maximal Majorana phase
e.g., for 2 generations, the mixing terms in V(YE, Yν) is :
tgh 2ω
(for non trivial sin2ω)
sin2ω
![Page 89: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/89.jpg)
The flavour symmetry is Gf =
which adds a new invariant for the lepton sector. In total:
Tr ( YE YE+ )
Tr ( Yν Yν+ )
mixingTr ( YE YE+ Yν Yν+ )
Tr ( YE YE+ )2
Tr ( Yν Yν+ )2
Tr ( Yν+Yν YνT Yν* ) <-- Ο(2)N
Example: 2 families; consider the renormalizable set of invariants:
![Page 90: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/90.jpg)
The flavour symmetry is Gf =
which adds a new invariant for the lepton sector. In total:
Tr ( YE YE+ )
Tr ( Yν Yν+ )
mixingTr ( YE YE+ Yν Yν+ )
Tr ( YE YE+ )2
Tr ( Yν Yν+ )2
<-- Ο(2)N
Example: 2 families; consider the renormalizable set of invariants:
Tr ( Yν+Yν (Yν+ Yν)T )
![Page 91: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/91.jpg)
Tr(YE YE+ Yν Yν+)
α= π/4 or 3π/4
Minimisation of
Large angles correlated with degenerate masses
Maximal Majorana phase
e.g., for 2 generations, the mixing terms in V(YE, Yν) is :
tgh 2ω
cosθ sinθ -sinθ cosθ
e-iα 0 0 eiαwhere UPMNS=
![Page 92: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/92.jpg)
Jacobian
![Page 93: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/93.jpg)
Jacobian Analysis: Mixing
What is the symmetry in this boundary?
which is extended if the eigenvalues are degenerate
[Alonso, Gavela, G. Isidori, L. Maiani]
![Page 94: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/94.jpg)
Renormalizable Potential
![Page 95: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/95.jpg)
hU
hD
I
II
III
IV
Renormalizable Potential, masses
![Page 96: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/96.jpg)
Renormalizable Potential, Stability
This region’s sizenonetheless
depends on the restof paramters (λ,g)
hU
hD
I
II
III
IV
![Page 97: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/97.jpg)
Renormalizable Potential
defining
the potential reads:
9 parameters
![Page 98: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/98.jpg)
hΝ
hE
I
II
III
IV
VII
VI
VVIII
Renormalizable Potential: Masses
![Page 99: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/99.jpg)
Renormalizable Potential
defining
the potential reads:
9 parameters
![Page 100: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/100.jpg)
Renormalizable Potential: Mixing
One maximal angle again
but not quite in the right place
The solution with a maximal θ23 , may arise in a Non-Renormalizable
Potential or could be a Local Minimaof the Renormalizable Potential
![Page 101: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/101.jpg)
O(2), SU(n), O(n) .... ?
* What is the role of the neutrino flavour group?
Leptons: Gflavour = U(2)L x U(2)ER x ?
Inmediate results using for both quark and leptons Y= UL ydiag UR
![Page 102: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/102.jpg)
To analyze this in general, use common parametrization for quarks and leptons:
Y = UL ydiag. UR
* Quarks, for instance: UR unphysical, UL --> UCKM
YD = UCKM diag(yd, ys, yb) ; YU = diag(yu, yc, yt)
* Leptons: YE = diag(ye, yµ, yτ) ; Yν = UL ydiag. UR
UPMNS diagonalize UL yνdiag. UR v2 URT yνdiag. ULT
* What is the role of the neutrino flavour group?
mν ~Yν v2 Yν Τ = M M
![Page 103: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/103.jpg)
U(n)
* What is the role of the neutrino flavour group?
![Page 104: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/104.jpg)
U(n)
* What is the role of the neutrino flavour group?
U(3)L x U(3)ER x U(2)NRi.e.:
or: U(3)L x U(3)ER x U(3)NR
![Page 105: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/105.jpg)
* What is the role of the neutrino flavour group?
e.g. generic seesaw
M
with M carrying flavour M spurion
More invariants in this case:
Tr ( YE YE+ )
Tr ( Yν Yν+ )Tr ( YE YE+ Yν Yν+ )Tr ( YE YE+ )2
Tr ( Yν Yν+ )2
Tr ( MN MN+ ) Tr ( MN MN+ ) 2 Tr ( MN MN+ Yν+Yν )
e.g. U(n)NR ... leptons
Result: no mixing for flavour groups U(n)
![Page 106: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/106.jpg)
SU(n)
![Page 107: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/107.jpg)
* What is the role of the neutrino flavour group?
e.g. generic seesaw
with M carrying flavour M spurion
More invariants in this case:
Tr ( YE YE+ )
Tr ( Yν Yν+ )Tr ( YE YE+ Yν Yν+ )Tr ( YE YE+ )2
Tr ( Yν Yν+ )2
Tr ( MN MN+ ) Tr ( MN MN+ ) 2 Tr ( MN MN+ Yν+Yν)
e.g. SU(n)NR ... leptons
* Tr ( Yν Yν+ YΕ YΕ+ ) = Tr ( UL yνdiag. 2 UL + yldiag. 2) At the minimum:
UL=1
* Tr ( MN MN+ Yν Yν+ ) = Tr ( UR yνdiag. 2 UR + Midiag. 2) UR=1
M
![Page 108: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/108.jpg)
Y = UL ydiag. UR
* Quarks, for instance: UR unphysical, UL --> UCKM
YD = UCKM diag(yd, ys, yb) ; YU = diag(yu, yc, yt)
Tr ( Yu Yu+ Yd Yd+ ) = Tr ( UL yudiag. 2 UL + yddiag. 2)
UL=U CKM ~1 at the minimum
NO MIXING
same conclusion for 3 families of quarks:
![Page 109: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/109.jpg)
O(n)
![Page 110: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/110.jpg)
Can its minimum correspond naturally to the observed masses and mixings?
i.e. with all dimensionless λ’s ~ 1
and dimensionful µ´s Λf<=
![Page 111: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/111.jpg)
* 3 generations: for the largest fraction of the parameter space, the stable solution is a degenerate spectrum
( )yuyc
yt( )y
yy
instead of the observed hierarchical spectrum, i.e.
~
( )yuyc
yt( )0
0y~
(at leading order)
Spectrum for flavons Σ in the bifundamental:Y --> one single field Σ
![Page 112: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/112.jpg)
ie, the u-part:
Spectrum: the hierarchical solution is unstable in most of the parameter space.
Stability:
µ2/µ2~
Hierarchical dominates
Degenerate dominates
![Page 113: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/113.jpg)
ie, the u-part:
Spectrum: the hierarchical solution is unstable in most of the parameter space.
Stability:
µ2/µ2~
Hierarchical dominates
Degenerate dominates
Nardi emphasized this solution (and extended the analysis to include also U(1) factors)
![Page 114: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/114.jpg)
![Page 115: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/115.jpg)
Y --> one single field Σ
The invariants can be written in termsof masses and mixing
* two families:
< Σd > = Λf . diag (yd ) ; < Σu > = Λf .VCabibbo diag(yu )
VCabibbo =
<Tr ( Σu Σu+ )> = Λf 2 (yu2 +yc2) ; <det ( Σu )> = Λf 2 yuyc
<Tr ( Σu Σu+ Σd Σd+ )>=Λf4 [( yc2 - yu2) ( ys2 - yd2) cos2θ +...]/2
(Alonso, Gavela, Merlo, Rigolin, arXiv 1103.2915)
![Page 116: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/116.jpg)
Non-degenerate masses No mixing !
Notice also that ~ (Jarlskog determinant)
Y --> one single field Σ
(Alonso, Gavela, Merlo, Rigolin, arXiv 1103.2915)
![Page 117: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/117.jpg)
Non-degenerate masses No mixing !
e.g. adding non-renormalizable terms... NO
Y --> one single field Σ
(Alonso, Gavela, Merlo, Rigolin, arXiv 1103.2915)
![Page 118: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/118.jpg)
Non-degenerate masses No mixing !
e.g. adding non-renormalizable terms...
Y --> one single field Σ
* To accomodate realistic mixing one must introduce wild fine tunnings of O(10-10) and nonrenormalizable terms of dimension 8
* Without fine-tuning, for two families the spectrum is degenerate
NO
(Alonso, Gavela, Merlo, Rigolin)
![Page 119: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/119.jpg)
Y --> one single field Σ three families
* at renormalizable level: 7 invariants instead of the 5 for two families
Interesting angular dependence:
![Page 120: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/120.jpg)
The real, unavoidable, problem is again mixing:
* Just one source:
Tr ( Σu Σu+ Σd Σd+ ) = Λf4 (P0 + Pint)
Sad conclusions as for 2 families: needs non-renormalizable + super fine-tuning
![Page 121: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/121.jpg)
*a good possibility for the other angles :
Yukawas --> add fields in the fundamental of the flavour group
![Page 122: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/122.jpg)
1) Y -- > one single scalar Y ~ (3, 1, 3)
2) Y -- > two scalars χ χ+ ~ (3, 1, 3)
×
χ ×
Ψ ×3) Y -- > two fermions ΨΨ ~ (3, 1, 3)_
Y
![Page 123: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/123.jpg)
1) Y -- > one single scalar Y ~ (3, 1, 3)
2) Y -- > two scalars χ χ+ ~ (3, 1, 3)
χ ~ (3, 1, 1)
×
χ ×
Ψ ×3) Y -- > two fermions ΨΨ ~ (3, 1, 3)_
Y
![Page 124: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/124.jpg)
1) Y -- > one single scalar Y ~ (3, 1, 3)
2) Y -- > two scalars χ χ+ ~ (3, 1, 3)
χ ~ (3, 1, 1)
Y ×
χ ×
Ψ ×3) Y -- > two fermions ΨΨ ~ (3, 1, 3)_
d=5 operator
d=6 operator
d=7 operator
![Page 125: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/125.jpg)
✝
χ × Y --> quadratic in fields χ
Y ~< χ χ >Λf
2
Holds for 2 and 3 families !
Automatic strong mass hierarchy and one mixing angle already at the renormalizable level
![Page 126: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/126.jpg)
✝
χ × 2) Y --> quadratic in fields χ
Y ~< χ χ >Λf
2
i.e. YD ~ χLd (χRd )+ ~ (3, 1, 1) (1, 1, 3) ~ (3, 1, 3)
Λf2
![Page 127: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/127.jpg)
( a , b , c .......)
![Page 128: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/128.jpg)
Y --> quadratic in fields χ
f f
![Page 129: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/129.jpg)
Y --> quadratic in fields χ
f f
![Page 130: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/130.jpg)
Towards a realistic 3 family spectrum
e.g. replicas of χL , χR , χRu d
???
Suggests sequential breaking:
SU(3)3 SU(2)3 ............ mt, mb mc, ms, θC
Y --> quadratic in fields χ
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* From bifundamentals: <Yu> =
<Yd> =
00
00
* From fundamentals χ : yc, ys and θC
![Page 132: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/132.jpg)
Towards a realistic 3 family spectrum
e.g. replicas of χL , χR , χRu d
???
Suggests sequential breaking:
SU(3)3 SU(2)3 ............ mt, mb mc, ms, θC
Y --> quadratic in fields χ
Maybe some connection to: Berezhiani+Nesti; Ferretti et al., Calibbi et al. ??
![Page 133: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/133.jpg)
* At leading (renormalizable) order:
* The masses of the first family and the other angles from non-renormalizable terms or other corrections or replicas ?
without unnatural fine-tunings
....and analogously for leptonic mixing ?
<Yu>
<Yd>
i.e. for quarks, a possible path:
![Page 134: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/134.jpg)
Towards a realistic 3 family spectrum
i.e. combining d=5 and d =6 Yukawa operators
Y --> linear + quadratic in fields
![Page 135: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/135.jpg)
Low M, large Y is typical of seesaws with approximate Lepton Numberconservation
U(1)LN
Wyler+Wolfenstein 83, Mohapatra+Valle 86, Branco+Grimus+Lavoura 89, Gonzalez-Garcia+Valle 89, Ilakovac+Pilaftsis 95, Barbieri+Hambye+Romanino 03, Raidal+Strumia+Turzynski 05, Kersten+Smirnov 07, Abada+Biggio+Bonnet+Gavela+Hambye 07, Shaposhnikov 07, Asaka+Blanchet 08, Gavela+Hambye+D. Hernandez+ P. Hernandez 09
(-> ~ degenerate heavy neutrinos)
LHC is more competitive for concrete seesaw models:
These models separate the flavor and the lepton number scale
![Page 136: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/136.jpg)
For instance, in the minimal seesaw I, Lepton number scale and flavour scale linked:
MT
M
Y
YT
seesaw I
mν =Y v2 Y Τ M UlN ~
YvM
UlN
l+
N
![Page 137: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/137.jpg)
* What is the role of the neutrino flavour group?
e.g. seesaw with approximately conserved lepton number e.g. O(2)NR ... leptons
MT
M
![Page 138: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/138.jpg)
* What is the role of the neutrino flavour group?
e.g. seesaw with approximately conserved lepton number e.g. O(2)NR ... leptons
![Page 139: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/139.jpg)
< YE > < Yν >
< YE > < Yν >
me
mµ
mτ
![Page 140: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/140.jpg)
α= π/4 or 3π/4y2-y’2
y2-y’2
*In the O(2)model used before: tgh 2ω = and y2-y’2
y2-y’2
*If we had used instead a flavor SU(2)model sinh 2ω =0 -->NO MIXING
![Page 141: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/141.jpg)
* e-µ, µ-τ etc. oscillations and rare decays studied: Gavela, Hambye, Hernandez2 ; .....
![Page 142: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/142.jpg)
Gavela, Hambye, Hernandez2 ;
![Page 143: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/143.jpg)
* Alonso + Li, 2010, MINSIS report: possible suppresion of µ-e transitions for large θ13
Gavela, Hambye, Hernandez2 ;
cancellationsfor large θ13
![Page 144: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/144.jpg)
* e-µ, µ-τ etc. oscillations and rare decays studied: Gavela, Hambye, Hernandez2 09 ; .....
* Alonso + Li, 2010: possible suppresion of µ-e transitions ->important impact of νµ - ντ at a near detectors
![Page 145: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/145.jpg)
* e-µ, µ-τ etc. oscillations and rare decays studied: Gavela, Hambye, Hernandez2 09; .....
* Alonso + Li, 2010: possible suppresion of µ-e transitions ->important impact of νµ - ντ at a near detectors
![Page 146: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/146.jpg)
Eboli, Gonzalez-Fraile, Gonzalez-Garcia
For type III version of our 2 N model, signals observable at LHC up to Λ~ 500 GeV for 30 fb-1
Nee
Neµ
Neτ
α α <-- Majorana phase
![Page 147: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/147.jpg)
Some good ideas:
“Partial compositeness”:
D.B. Kaplan-Georgi in the 80s proposed a composite Higgs:
* Higgs light because the whole Higgs doublet is multiplet of goldstone bosons They explored SU(5)--> SO(5). Explicit breaking of SU(2)xU(1) symmetry via external gauged U(1) Nowadays SO(5)--> SO(4) and explicit breaking via SM weak interaction (Contino, Nomura, Pomarol; Agashe, Contino, Pomarol; Giudice, Pomarol, Ratazzi, Grojean; Contino, Grojean, Moretti; Azatov, Galloway, Contino...) SO(6) --> SO(5) to get also DM (Frigerio, Pomarol, Riva, Urbano)
(Kaplan, Georgi, Dimopoulos, Banks, Dugan, Galison)
![Page 148: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/148.jpg)
~1
Anarchy: alive with not so small θ13 and not θ23 not maximal
~1
~1
~1
~1 ~1
~1~1
~1
mν ~
no symmetry in the lepton sector, just random numbers
(Hall, Murayama, Weiner; Haba, Murayama; De Gouvea, Murayama... Going towards hierarchy: Altarelli, Feruglio, Masina, Merlo)
- Does not relate mixing to spectrum- Does not address both quarks and leptons
![Page 149: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/149.jpg)
*3 families with O(2)NR :
- 3 light + 2 heavy N degenerate: bad θ12 quadrant. It cannot accomodate data!
- 3 light + 3 heavy N : OK for θ23 maximal and spectrum
*What about the other angles?
experimentally sin2θ23= 0.41 +-0.03 or 0.59+-0.02 Gonzalez-Garcia, Maltoni, Salvado, Schwetz Sept. 2012
( O(2) )0 0
3x3
![Page 150: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/150.jpg)
*3 families with O(2)NR :
- 3 light + 2 heavy N degenerate: bad θ12 quadrant. It cannot accomodate data!
- 3 light + 3 heavy N : OK for θ23 maximal and spectrum
*What about the other angles?
experimentally sin2θ23= 0.41 +-0.03 or 0.59+-0.02 Gonzalez-Garcia, Maltoni, Salvado, Schwetz Sept. 2012
Moriond this morning, T2K best fit point sin22θ23=1.00 -0.068 90%CL -> 45º !
![Page 151: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/151.jpg)
BSM electroweak
* HIERARCHY PROBLEM Fine-tuning issue: if BSM physics, why Higgs so light
Interesting mechanisms to solve it: SUSY, strong-int. light Higgs, extra-dim….
In practice, none without further fine-tunings
![Page 152: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/152.jpg)
BSM electroweak
* HIERARCHY PROBLEM Fine-tuning issue: if BSM physics, why Higgs so light
Interesting mechanisms to solve it: SUSY, strong-int. light Higgs, extra-dim….
In practice, none without further fine-tunings
* FLAVOUR PUZZLE: ~no theoretical progress
New B physics data AND neutrino masses and mixings
Understanding of the underlying physics stalled since 30 years. BSM theories tend to make it worse.
![Page 153: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/153.jpg)
BSM electroweak
* HIERARCHY PROBLEM Fine-tuning issue: if BSM physics, why Higgs so light
Interesting mechanisms to solve it: SUSY, strong-int. light Higgs, extra-dim….
In practice, none without further fine-tunings
* FLAVOUR PUZZLE : no progress
New B physics data AND neutrino masses and mixings
Understanding of the underlying physics stalled since 30 years. BSM theories tend to make it worse.
Λelectroweak ~ 1 TeV ?
Λf ~ 100’s TeV ???
![Page 154: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/154.jpg)
The FLAVOUR WALL for BSM
i.e susy MSSM:
competing with SM at one-loop
i.e susy MSSM:
ii) FCNC
< 1 loop in SM ---> Best (precision) window of new physics
i) Typically, BSMs have electric dipole moments at one loop
![Page 155: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/155.jpg)
The FLAVOUR WALL for BSM
i.e susy MSSM:
competing with SM at one-loop
i.e susy MSSM:
ii) FCNC
< 1 loop in SM ---> Best (precision) window of new physics
i) Typically, BSMs have electric dipole moments at one loop
conversion (MEG, µ2e...)
![Page 156: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/156.jpg)
What happens if we add
non-renormalizable terms to the potential?
In fact one should consider as many invariants as physical variables
![Page 157: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/157.jpg)
Just TWO heavy neutrinos
Raidal, Strumia, TurszynskiGavela, Hambye, Hernandez2
MM
Lepton number scale and flavour scale distinct
seesaw I with
![Page 158: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/158.jpg)
Just TWO heavy neutrinos
MM
mν = Y v2 Y´ Τ M UlN ~ Y
M--> Lepton number conserved conserved if either Y or Y’ vanish:
Raidal, Strumia, TurszynskiGavela, Hambye, Hernandez2
![Page 159: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/159.jpg)
Gavela, Hambye, Hernandez2
Raidal, Strumia, Turszynski
--> One massless neutrino and only one Majorana phase α
Just TWO heavy neutrinos
MM
![Page 160: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/160.jpg)
Comparing the scales reached by
Neutrino Oscillations vs µ-e experiments vs LHC
e.g. in Seesaw type I scales (heavy singlet fermions)
* ν-oscillations: 10-3eV - MGUT ~ 1015 GeV, because interferometry
* µ-e conversion: 2MeV - 6000 GeV
* LHC: ~ # TeV
![Page 161: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/161.jpg)
The flavour symmetry is Gf =
adds a new invariant for the lepton sector, in total:
Tr ( YE YE+ )
Tr ( Yν Yν+ )
mixingTr ( YE YE+ Yν Yν+ )
Tr ( YE YE+ )2
Tr ( Yν Yν+ )2
Tr ( Yν σ2Yν+ ) 2 <-- Ο(2)N
Ο(2)N is simply associated to Lepton Number
![Page 162: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/162.jpg)
Leptons
Just TWO heavy neutrinos
MM
(Alonso, Gavela, D. Hernandez, Merlo, Rigolin)
![Page 163: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/163.jpg)
The flavour symmetry is Gf =
Leptons
Just TWO heavy neutrinos
MM
(Alonso, Gavela, D. Hernandez, Merlo, Rigolin)
![Page 164: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/164.jpg)
The flavour symmetry is Gf =
Just TWO heavy neutrinos
MM
(Alonso, Gavela, D. Hernandez, Merlo, Rigolin)
![Page 165: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/165.jpg)
Jacobian Analysis: Mixing
What is the symmetry in this boundary?
related to the O(2) substructure
[Alonso, Gavela, D. Hernández, L. Merlo;[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin]
![Page 166: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/166.jpg)
In many BSM the Yukawas do notcome from dynamical fields:
![Page 167: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/167.jpg)
Some good ideas: D.B. Kaplan-Georgi in the 80’s proposed a light SM scalar because being a (quasi) goldstone boson: composite Higgs
(D.B. Kaplan, Georgi, Dimopoulos, Banks, Dugan, Galison.......Contino, Nomura, Pomarol; Agashe, Contino, Pomarol; Giudice, Pomarol, Ratazzi, Grojean; Contino, Grojean, Moretti; Azatov, Galloway, Contino... Frigerio, Pomarol, Riva, Urbano...)
![Page 168: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/168.jpg)
Some good ideas: D.B. Kaplan-Georgi in the 80’s proposed a light SM scalar because being a (quasi) goldstone boson: composite HiggsFlavour “Partial compositeness” D.B Kaplan 91:
mq= v YSM
QL UR
H YSM
(D.B Kaplan 91; Redi, Weiler; Contino, Kramer, Son, Sundrum; da Rold, Delauney, Grojean, G. Perez; Contino, Nomura, Pomarol, Agashe, Giudice, Perez, Panico, Redi, Wulzer...)
A sort of “seesaw for quarks” (nowadays sometimes justified from extra-dim physics )
![Page 169: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/169.jpg)
“Partial compositeness”:
QL UR
H
YSM = YΔL ΔR/M2
YMΔL ΔR
mq= v YSM
A sort of “seesaw for quarks” (nowadays sometimes justified from extra-dim physics )
(D.B Kaplan 91; Redi, Weiler; Contino, Kramer, Son, Sundrum; da Rold, Delauney, Grojean, G. Perez; Contino, Nomura, Pomarol, Agashe, Giudice, Perez, Panico, Redi, Wulzer...)
Some good ideas: D.B. Kaplan-Georgi in the 80’s proposed a Higgs light because being a (quasi) goldstone boson: composite Higgs
![Page 170: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/170.jpg)
“Partial compositeness”:
QL UR
H
YSM = YΔL ΔR/M2
YMΔL ΔR
mq= v YSM
(D.B Kaplan 91; Redi, Weiler; Contino, Kramer, Son, Sundrum; da Rold, Delauney, Grojean, G. Perez; Contino, Nomura, Pomarol, Agashe, Giudice, Perez, Panico, Redi, Wulzer...)
Neutrino masses:
L
H H
Ld=5 Weinberg operator
A sort of “seesaw for quarks” (nowadays sometimes justified from extra-dim physics )
Some good ideas: D.B. Kaplan-Georgi in the 80’s proposed a Higgs light because being a (quasi) goldstone boson: composite Higgs
![Page 171: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/171.jpg)
“Partial compositeness”:
QL UR
H
YSM = YΔL ΔR/M2
YMΔL ΔR
mq= v YSM
(D.B Kaplan 91; Redi, Weiler; Contino, Kramer, Son, Sundrum; da Rold, Delauney, Grojean, G. Perez; Contino, Nomura, Pomarol, Agashe, Giudice, Perez, Panico, Redi, Wulzer...)
Neutrino masses:
A sort of “seesaw for quarks” (nowadays sometimes justified from extra-dim physics )
YM
YxL
H H
mν= Y v2/M YT
type I
L
Some good ideas: D.B. Kaplan-Georgi in the 80’s proposed a Higgs light because being a (quasi) goldstone boson: composite Higgs
![Page 172: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/172.jpg)
“Partial compositeness”:
QL UR
H
YSM = YΔL ΔR/M2
YMΔL ΔR
mq= v YSM
(D.B Kaplan 91; Redi, Weiler; Contino, Kramer, Son, Sundrum; da Rold, Delauney, Grojean, G. Perez; Contino, Nomura, Pomarol, Agashe, Giudice, Perez, Panico, Redi, Wulzer...)
Neutrino masses:
A sort of “seesaw for quarks” (nowadays sometimes justified from extra-dim physics )
M
YL
H H
mν= Y µ v2/M2
µtype II
L
Some good ideas: D.B. Kaplan-Georgi in the 80’s proposed a Higgs light because being a (quasi) goldstone boson: composite Higgs
![Page 173: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/173.jpg)
For instance, in discrete symmetry ideas:
The Yukawas are indeed explained in terms of dynamical fields. And they do not need to worry about goldstone bosons.
In spite of θ13 not very small, there is activity.For instance, combine generalized CP (Bernabeu, Branco, Gronau 80s) with discrete Z2 groups in the neutrino sector : maximal θ23, strong constraints on values of CP phases (Feruglio, Hagedorn and Ziegler 2013; Holthausen, Lindner and Schmidt 2013)
But:- Discrete approaches do not relate mixing to spectrum- Difficulties to consider both quarks and leptons
They were popular mainly because they can lead easily to large mixings (tribimaximal, bimaximal...)
![Page 174: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/174.jpg)
Some good ideas:
- Use the flavour symmetry of the SM in the limit of massless fermions quarks: Gflavour= U(3)QL x U(3)UR x U(3)DR
U(2) (Pomarol, Tomasini; Barbieri, Dvali, Hall, Romanino...).... U(2)3 (Craig, Green, Katz; Barbieri, Isidori, Jones-Peres, Lodone, Straub..
..Sala) Sequential ideas (Feldman, Jung, Mannel; Berezhiani+Nesti; Ferretti et al.,
Calibbi et al. ...)
QL DR
H Y
Hybrid dynamical-non-dynamical Yukawas:
U(2)0 0 1
spurion
(Chivukula+ Georgi)
Minimal Flavour Violation:
![Page 175: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/175.jpg)
For this talk:
each YSM -- >one single field Y YSM ~
< Y > Λfl
Alonso, B.G., D. Hernandez, L. Merlo, Rigolin
Can it shed light on why quark and neutrino mixings are so different?
![Page 176: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/176.jpg)
QL DR
H < > <-->YD
Assume that the Yukawa couplings correspond to dynamical fields at high energies .......
YSM ~ < φ > or YSM ~ 1/< φ > or ...... < (φ χ)n>
(Alonso+Gavela+Merlo+Rigolin 11) ...
[Cabibbo,Michel,+Radicati, Cabibbo+Maiani ...
C. D. Froggat, H. B. NielsenAnslem+Berezhiani, Berezhiani+Rossi]
![Page 177: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/177.jpg)
For this talk:
each YSM -- >one single field Y YSM ~
< Y > Λf
Anselm+Berezhiani 96; Berezhiani+Rossi 01... Alonso+Gavela+Merlo+Rigolin 11...
transforming under the SM flavour group
![Page 178: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/178.jpg)
=
Generalization to any seesaw model
the effective Weinberg Operator
shall have a flavour structure that breaks U(3)L to O(3)
then the results apply to any seesaw model
![Page 179: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/179.jpg)
This did not need any ad-hoc discrete symmetries, but simply using the in-built continuous flavour symmetry of the SM + seesaw, U(3)5 x O(3)
Also, note that often people working with “flavons” invents a “texture” that goes well with data, and then tries to designa potential that leads to it. In our case, the inevitable potential minima encompass the different patterns of quarks and leptons.
![Page 180: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/180.jpg)
Some good ideas, based on continuous symmetries:
Frogatt-Nielsen ‘79: U(1)flavour symmetry
- Yukawa couplings are effective couplings,- Fermions have U(1)flavour charges
< φ > n Q H qR , Y ~ < φ > nn Λ Λ
e.g. n=0 for the top, n large for light quarks, etc.
--> FCNC ?
QL DR
H φ’s
![Page 181: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/181.jpg)
![Page 182: Neutrinos and the Flavour Puzzlevietnam.in2p3.fr/2013/Inauguration/transparencies/Gavela.pdf[Alonso, Gavela, D. Hernández, L. Merlo, S. Rigolin] if the three neutrinos are quasidegenerate,](https://reader036.vdocuments.site/reader036/viewer/2022063021/5fe3d5878c58a470cf163af6/html5/thumbnails/182.jpg)
In some BSM theories, Yukawas do correspond to dynamical fields:
- for instance in discrete symmetry scenarios
- also with continuous symmetries