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

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.

(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)]

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)

(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)

(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.

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)

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)

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.

(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)]

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)

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)

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

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.

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

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).　

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)

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 .

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-06-05.

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/0605074

(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)

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.

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)

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.

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

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.

(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

Many open questions still remains We need further consideration Thank you