1. Yoshio Koide University of Shizuoka
Joint Meeting of Pacific Region Particle Physics Communities
November 1, 2006, Honolulu, Hawaii
Neutrino Masses and Mixing Suggested by the Charged Lepton Mass Formula
Yoshio Koide
University of Shizuoka

2. 1 Introduction Why we investigate the lepton masses and mixings?
Today, I would like to confine my talk to the investigation of the lepton masses and mixings. The reason is as follows:
Why we investigate the lepton masses and mixings?
It is generally considered that masses and mixings of the quarks and leptons will obey a simple law of nature, so that we expect that we will find a beautiful relation among those values. However, even if there is such a simple relation in the quark sector, it is hard to see such a relation in the quark sector, because the relation will be spoiled by the gluon cloud. We may expect that such a beautiful relation will be found just in the lepton sector.

3. (1) Charged lepton mass relation
It is well-known that the observed charged lepton mass spectrum satisfies the relation
(1.1)
with remarkable precision.
[Koide, LNC (1982); PLB (1983)]
The mass formula (1.1) is invariant under　any exchange
This suggests that a description by S3 may be useful for the mass matrix model.
[S3: Pakvasa and Sugawara (1978); Harari, Haut and Weyers (1978)]

4. For example, the mass formula (1.1) can be understood
from a universal seesaw model with 3-flavor scalars
[Koide (1990)]
(1.2)
[Universal seesaw: Gerezhiani (1983); Chang, Mohapatra (1987);
Davidson, Wali (1987), Rajpoot (1987), Babu, Mohapatra (1989)]
For the charged lepton sector, we take
(1.3)
where the VEV satisfy the relation
(1.4)

5. (1.5) The VEV relation (1.4) means where are defined by
(1.6)
The Higgs potential which gives the relation (1.5) is, for example,
found in YK, PRD73 (2006), where S3 plays an essential role:
(1.7)

6. (2) Neutrino mass relation [Brannen (2006)]
Recently, Brannen has speculated a neutrino mass relation similar to the charged lepton mass relation (1.1):
(1.8)
Of course, we cannot extract the values of the neutrino mass ratios and from the neutrino oscillation data and unless we have more information on the neutrino masses, so that we cannot judge whether the observed neutrino masses satisfy the relation (1.8) or not.

7. Generally, the masses which satisfy the relations (1. 1) and (1
Generally, the masses which satisfy the relations (1.1) and (1.8) can be expressed as
(1.9)
where
(1.10)
(1.11)
Then, Brannen has also speculated the relation
(1.12)

8. From the observed charged lepton mass values, (1.13) (1.14)
we obtain
(1.13)
Then, the Brannen relation (1.12) gives
(1.14)
which predicts
(1.15)
(1.16)

9. Hereafter, we will refer
Therefore, the speculations by Brannen are favorable to the observed neutrino data.
Hereafter, we will refer
the relation (1.8) as the Brannen's first relation
and the relation (1.12) as the Brannen's second relation
In the present talk, I would like to report
the investigation based on the S3 symmetry.

10. (3) Tribimaximal mixing
Here, I would like to emphasize that the investigation of the mass relations without the investigation of the mixing is meaningless because we have known the existence of the neutrino mixing and CKM mixing.
The present neutrino data have strongly suggested that the neutrino mixing is approximately described by the so-called tribimaximal mixing
(1.17)
[Harrison, Perkins, Scott, PLB (1999)]

11. If the neutrino mass eigenstates are with in contrast to ,
We define the doublet and singlet of S3
　　 (1.18)
If the neutrino mass eigenstates are
with
in contrast to ,
then we can obtain the tribimaximal mixing (1.17):
(1.19)

12. 2 Seesaw Model In the present model, it is essential that
the mass matrix is expressed by a bilinear form
(2.1)
Although we can adopt a Frogatt-Nielsen type model,
for convenience, in the present talk, we adopt
a seesaw-type mass matrix model with 3 scalars
(2.2)

13. in the structure of in the structure of
My Old Model My Revised model
is not diagonal
Neutrino mixing originates Neutrino mixing originates
in the structure of in the structure of

14. Note that although we assume the VEV relation (1.4)
it is not trivial
whether the neutrino masses satisfy
the Brannen's first relation or not
i.e. whether the parameters satisfy
the relation (1.10)
or not
because the present is not diagonal.

15. 3 Mass matrix form under S3
The general form of the S3 invariant Yukawa interaction is
given by
(3.1)

16. For the charged lepton sector, we have already assumed the form (3.2)
　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　(3.2)
　　　　　　　　where and
The form (3.2) corresponds to the case
(3.3)
in the general form (3.1).　

17. The present neutrino oscillation date favor to the tribimaximal
mixing, so that the neutrino mass eigenstates are approximately
in the states with
For convenience, we investigate a case in the limit of
(3.4)
The mass matrix (3.4) is diagonalized by a rotation
(3.5) as
(3.6)

18. Then we obtain the mass eigenvalues
(3.7)
where we have used the relation
(3.8)
Note that the mass spectrum is independent of the parameters
and , and only depends on the parameters
and
On the other hand, as seen in Eq.(3.5), the mixing angle is
independent of the parameters and only depends on
the parameter

19. We have still 3 free parameters , and
even if we assume
We put the following normalization condition
[Condition A] [Condition B]
The case satisfies the Brannen's
first relation (1.8)
but, there is no reasonable ground
The general study under the S3 symmetry will be found
in YK, a preprint US

20. 4 An S3-invariant neutrino interaction with a concise structure
In this talk, I would like to propose an S3-invariant neutrino
interaction with a concise form
(4.1)
where Here, we have assumed
the universality of the coupling constants
of the terms.
YK, preprint US-06-03, hep-ph/

21. (4.2) Here, we have used a relation (4.3) independently of the value .
Then, we obtain
(4.2)
Here, we have used a relation
(4.3)
from the S3 Higgs potential model.
The results satisfy the Brannen's first relation
independently of the value
Comparing the definition of (1.7), we obtain
(4.4)

22. The value is somewhat larger than the
value from the Brannen's second
relation .
The present case predicts
(4.5)
The predicted value is somewhat larger than
(4.6)
However, the case cannot, at present, be ruled
out within three sigma.

23. If , cannot be diagonal on the
basis The mixing angle of the
further rotation between are given by
(4.7)
It is well known that the symmetry is
promising for neutrino mass matrix description.
[Fukuyama-Nishiura (1997), Ma-Raidal (2001), Lam (2001),
Balaji-Grimus-Schwetz (2001), Grimus-Laboura (2001)]
We assume the symmetry for , i.e.
, which leads to
(4.8)

24. Therefore, the present model gives the exact
tribimaximal mixing:
(4.9)
Note that if we require the symmetry for
the fields , the symmetry
will affect the charged lepton sector, too.
Here, we have assumed the symmetry
only for , not for , so that the symmetry
does not affect the charged lepton mass matrix.

25. Why we could explain the neutrino mixing in spite of ?
Charged lepton sector Neutrino sector
is not diagonal
on the basis on the basis
is diagonal
on the basis
We have required
the universality of the coupling constants
on the basis on the basis

26. 5 Conclusion (1) The first relation by Brannen
can be understood from a universal seesaw
mass matrix model with the S3 symmetry,
however, only for a special case.
(2) We have failed to derive the Brannen's
second relation
under the S3 symmetry.

27. (3) We have proposed an S3 invariant neutrino
Yukawa interaction with a concise structure
which predicts the neutrino masses
and tribimaximal mixing under a 2-3 symmetry

28. Many open questions still remains
We need further consideration
Thank you