A = ; ; A : (:) Hereafter, we will call the basis ( A A A ) dened by Eq.(.) the basis A. We can also take another S basis ( )(we call it the basis ) w

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1 S Symmetry and Neutrino Masses and Mixings Yoshio Koide Deartment of Physics, Uniersity of Shizuoka, 5- Yada, Shizuoka 4-85, Jaan address: koideu-shizuoka-ken.ac.j Abstract ased on a uniersal seesaw mass matrix model with three scalars i, and by assuming an S aor symmetry for the Yukawa interactions, the neutrino masses and mixings are inestigated. Suggested from that the obsered neutrino mixing is nearly the tribimaximal mixing, it is assumed that when the charged letons (e ) are regarded as the states (e e e) of S, the neutrino mass-eigenstates are nearly in the states ( ), where ( ) and are a doublet and a singlet of S, resectiely. Possible structures of the Yukawa interactions are inestigated systematically. Introduction It is generally considered that masses and mixings of the quarks and letons will obey a simle law of nature, so that we exect that we will nd a beautiful relation among those alues. Howeer, een if there is such a simle relation in the quark sector, it is hard to see such a relation in the quark sector, because the relation will be soiled by the gluon cloud. We may exect that such abeautiful relation will be found just in the leton sector. Therefore, in the resent aer, we will conne ourseles to the inestigation of the leton masses and mixings. Here, we would like to emhasize that we should search a model which gies a reasonable descrition of not only the masses, but also the mixings. Esecially, we should direct our attention to the mixing attern rather than to the mass sectrum in the neutrino sector. It is also considered that the mass matrices of the fundamental articles will be goerned by a kind of symmetry. In the resent aer, we take notice of a ermutation symmetry S []. Let us begin with giing a short reiew how useful a descrition based on the S symmetry is in the leton masses and mixings. The obsered neutrino data hae strongly suggested that the neutrino mixing is aroximately described by the so-called tribimaximal mixing [] U = ; ; A : (:) According to the conentional notations, we dene the doublet (, ) and singlet of the ermutation symmetry S as A A A A = A A (:)

2 A = ; ; A : (:) Hereafter, we will call the basis ( A A A ) dened by Eq.(.) the basis A. We can also take another S basis ( )(we call it the basis ) which isdenedby A = A (:4) = ; ; A : (:5) Of course, an S -inariant interaction is inariant under the transformation U A A T,which transforms the basis A into the basis. Since UA =, we can say that the bases A and are dual each other. When we takeaaor basis (e e e )=( e) [and also ( )=( e )],ifthe states ( ) are mass eigenstates, and if their masses satisfy the relation m <m <m (:) the neutrino mixing matrix U of the basis ( ) to the basis (e e e ) = (e ) is gien by the form (.). Here, we would like to emhasize that the condition (.) is essential to obtaining the tribimaximal mixing (.). (Hereafter, since we always discuss the mass matrix form on the basis, we will dro the index.) On the other hand, it is well-known that the obsered charged leton mass sectrum [] satises the relation [4, 5] m e + m + m = ; me + m + m (:7) with remarkable recision. The mass formula (.7) is inariant under any exchange m i $ m j (i j = e ). This, too, suggests that a descrition by S may be useful for a mass matrix model. As an exlanation of the mass formula (.7), the author has roosed a model [5,, 7] with aor scalars i in the framework of the uniersal seesaw model [8]: A fermion mass matrix M f is gien by M f = m f L M ; F mf R (:8) where M F is a mass matrix of hyothetical heay fermions F i (i = ). For examle, for the charged leton sector, we assume m e L = me R = y e diag( ) (:9)

3 ( is a constant with ) and M E /, where m e L and me R are dened by `Lme L E R and `R(m e R )y EL c and i h Li i = h Ri i=, `L=R = ( L=R e L=R ), and L=R = ( + L=R ). If we L=R assume that the acuum exectation alues (VEV) i satisfy the relation + + = ( + + ) (:) we can obtain the relation (.7). Of course, here, we hae assumed that the Yukawa interaction in the charged leton sector is gien by ans inariant form (and also a similar interaction for `R R E L ). H e = y e ; `L L E R + `L L E R + `L L E R (:) The relation among the VEVs i, (.), can read + = (:) in terms of S, where ( ) hae been dened by Eq.(.4). For a Higgs otential model based on an S symmetry which leads to the relation (.), for examle, see Ref. [9]. The S symmetry is again related to the leton masses and mixings. Thus, it is likely that the S symmetry (or a higher symmetry which include S ) lays an essential role on a unied descrition of the leton mass matrices. In the resent aer, we will assume that, in the uniersal seesaw model with three aor scalars, the Yukawa interactions are exactly inariant under the S symmetry, and the S symmetry is broken only by the VEVs i of the three scalars i. Thereby, we will inestigate the masses and mixings of the neutrinos systematically. Another motiation of the resent aer is in a seculation, which has recently been ointed out by rannen [], that the neutrino masses also satisfy a relation similar to the relation (.7) for the charged leton masses, m + m + m = (; m + m + m ) : (:) Of course, we cannot extract the alues of the neutrino mass ratios m =m and m =m from the neutrino oscillation data m solar and m atm unless we hae more information on the neutrino masses, so that we cannot judge whether the obsered neutrino masses satisfy the relation (.) or not. Howeer, we interest in inestigating what constraints are imosed on the mass matrix arameters when we regard the rannen relation (.) as true. We will conclude that the resent S model cannot generally gie the relation (.), excet for a case with a secic relation among the Yukawa couling constants. y the way, the seesaw-tye model (.8) with scalars Li (and Ri ) causes some trouble, for examle, the aor changing neutral currents (FN) roblem, the soiling of the asymtotic freedom of the SU() color, andsoon. Therefore, instead of the model (.8), we may consider a Frogatt-Nielsen [] tye model with six dimensional oerators f Li fi H L fi f Ri, where H L is the conentional SU() L -doublet Higgs scalar, and fi are -family SU() L -singlet scalars.

4 Howeer, in the resent aer,itisessential that the leton masses m fi are gien by a bilinear form m fi =(z fi ) m f (:4) where the sector-deendent arameters z fi are normalized as (z f ) +(z f ) +(z f ) =. At resent, we will not discuss whether the eectie mass matrix form originates in a seesaw model or in a Frogatt-Nielsen model. Since our interest is only in the structure of z fi, for conenience, in the resent aer, we will inestigate a ossible Yukawa interaction form in the framework of the seesaw mass matrix model (.8). The results in the resent aer are also alicable to a Frogatt-Nielsen tye model. Mass eigenalues In general, an S inariant Yukawa interaction with scalars a (a = ) is gien by H = + y y ; +y + + ; + +y + y 4 + (:) where we read = `L ( L e L ), = E R and a = d a =( d+ d a ) for the charged leton sector, and = `L, = N R (or R ) and a = u a = ( u ) for the neutrino sector. For a a u; a examle, the interaction (.) in the charged leton sector corresonds to the case y = y e y = y = y e y = y 4 = r y e: (:) The Yukawa interaction (.) gies the mass matrix m f L for the basis ( ), m f L = y + y y y 4 + y y y + y y 4 ; y y y y ; y A : (:) Hereafter, for simlicity, we conne ourseles to inestigating a case with a symmetric mass matrix form (m f L )T = m f L, i.e. with y = y 4. Then, we hae still 5 arameters, y y y y and =, in the model, so that the model has no redictability. In the resent aer, we do not imose a further symmetry on the model. Alternatiely, we will inestigate what constraints on the mass matrix arameters (or secic relations among those) are required from the henomenological studies. 4

5 Now let us return to the subject on the neutrino Dirac mass matrix m L for the basis ( ) is, in general, gien by the form (.). As we discussed in the reious section, the resent neutrino oscillation date faor to the tribimaximal mixing, so that the neutrino states are aroximately in the mass eigenstates ( ) with m <m <m. Therefore, for conenience, we inestigate a case in the limit of y = y 4 =. The mass matrix with y = y 4 = is diagonalized by a rotation R( )= c s ;s c A (:4) where c =cos and s =sin, and tan = ; (:5) as R T ( )m LR( ) = diag(m m m ): (:) The mass eigenalues m, m and m are gien by m = y + y jy j q + m = y + y jy j q + m = y ; y (:7) where we hae dened y + y > (:8) and the uer and lower signs in jy j (and also jy j) corresond to the cases y > and y <, resectiely. In the reious section, we hae assumed that the VEVs d i of the scalars d i, which coule to the charged letons, satisfy the relation (.). Therefore, we also assume that the VEVs i u of the scalar u i, which coule to the neutrino sector, satisfy the relation ( u ) +( u ) =( u ) u (:9) where we do not always assume h u i i = hd i i. Then, the mass eigenalues (.7) lead to m = y + y jy j u m = y + y jy j u m = y ; y u : (:) 5

6 Note that the mass sectrum is indeendent of the arameters = u u and u =, u and only deends on the arameters y =y and jy j=y. On the other hand, as seen in Eq.(.5), the mixing angle is indeendent of the arameters y i and only deends on the arameter = u u. As we discussed in Sec., the obsered tribimaximal mixing suggests that the neutrino mass eigenstates are ( ). If the mass hierarchy is a normal tye, it demands m <m m, and if it is an inerse tye, it demands m m < m. In order to check those cases, we estimate the dierences among those masses as follows: m ; m = jy j( y + y ) u (:) m ; m = 4 ( y jy j)( y ; y jy j) u (:) m ; m = 4 ( y jy j)( y ; y jy j) u : (:) Since we hae dened the factor ( y +y ) as ositie in Eq.(.8), Eq.(.) means that, for the case of the normal hierarchy withm >m,wemust take the uer signs in Eqs.(.)-(.), i.e. m ; m = 4 ( y + jy j)( y ; y + jy j)u > (:4) m ; m = 4 ( y ;jy j)( y ; y ; jy j)u < (:5) and, for the case of the inerse hierarchy with m Eqs.(.)-(.), i.e. < m, we must take the lower signs in m ; m = 4 ( y ;jy j)( y ; y ; jy j)u < (:) m ; m = 4 ( y + jy j)( y ; y + jy j) u < : (:7) In conclusion, the model gies m >m or m >m according as y > or y <. Since the obsered neutrino mixing is aroximately by the tribimaximal mixing, if we want to build a model with a normal mass hierarchy (or an inerse mass hierarchy), we must seek for a model with m < m < m (or m < m < m ). The conditions for m < m < m and m <m <m are gien by Eqs.(.4)-(.5) and Eqs.(.)-(.7), resectiely. y the way, wehae still two adjustable arameters y =y and y =y to redict the neutrino mass sectrum. In the following sections, we will inestigate two tyical cases by utting assumtions for the couling constants y, y and y. Of course, the assumtions must also be alicable to the charged leton couling constants (.).

7 ase with y = y + y In the mass matrix (.), the y -andy -terms are traceless, while the trace of the y -term is not zero. This suggests that the y -term may be distinguished from the other terms under a higher symmetry. Therefore, by way of trial, we ut the following normalization condition for the couling constants y = y + y + (y + y 4) (:) which is satised by the couling constants (.) in the charged leton sector. Since we hae assumed that y = y 4 = in the neutrino sector, we can exlicitly write the condition (.) as Then, we can rewrite Eqs.(.) as y = y sin y = y cos : (:) h m = ; sin ; i y u h m = ; sin ; i y u h i m = ; sin y u (:) where ; (cos >), and, for <,we substitute ; for in Eq.(.). Note that the case with the condition (.) which leads to Eq.(.) gies the relation m + m + m = (m + m + m ) : (:4) Since these masses (m m m ) are Dirac masses in the neutrino seesaw mass matrix M = m L M ; N (m L )T,ifwetaketheheay Majorana mass matrix M N with the unit matrix form, we obtain the neutrino masses which are roortional to m, m and m, resectiely. Therefore, the neutrino masses will satisfy the relation (.7) which has seculated by rannen []. The exressions (.)-(.) become m ; m = cos ( + sin )y u (:5) m ; m = cos ; sin y u (:) m ; m = cos h ; sin i y u (:7) where jj <=. For a case with a normal hierarchy, as seen in Eq.(.), we should read the uer signs in Eqs.(.5)-(.7), so that we obtain m <m <m for ; << : (:8) 7

8 For a case with an inerse hierarchy, since we should read the lower signs in Eqs.(.5)-(.7), we obtain m <m <m for ; <<; : (:9) Next, let us seek for the numerical alue of which gies the ratio of the obsered alues m solar =(7:9+: ;:5 ) ;5 ev []tom atm =(:7 +:8 ;:5 ) ; ev [], R obs m solar m atm =(:9 :5) ; : (:) From the numerical study of we nd R() m4 () ; m 4 () m 4 () ; m 4 () (:) = ; : ;: +:4 (:) where the sign corresonds to the sign of the exerimental error in Eq.(.), as a normal hierarchical solution. Howeer, we could not nd a solution with an inerse hierarchy. The solution =(: ;: +:4 ) gies m = ;(:7 +: ;:8 )y u m =(:8 :)y u m =(:9 :)y u : (:) The result m < leads to the change of sign m! ; m in Eq.(.) which has been seculated by rannen []. The alues (.) redicts the following neutrino masses m =(:5 :5) ;4 ev m =(8:7 :) ; ev m =(5: ;:5 +:5 ) ; ev: (:4) Generally, when masses m fi = z fi m f (i = ) satisfy the relation (.7) [or (.)], the arameters z fi can always be exressed by the form z f = ; sin f z f = ; sin( f + ) z f = ; sin( f + 4 ) (:5) where we hae taken zf <z f <z f. From the obsered charged leton mass alues, we obtain the numerical alue of e e = 4 ; " =4:74 (" =:7 ): (:) 8

9 Note that, in the limit of "!, the electron mass becomes zero. We consider that the arameter " is a fundamental arameter which goerns the charged leton mass sectrum. Recently, rannen [] has seculated that the alue of is gien by = e + (:7) which redicts =57:74 : (:8) omaring the exression (.) (with the uer signs) with the exression (.5), we nd that the arameter is connected to by the relation = ; : (:9) Therefore, we obtain ; e = ; ; 4 ; " = + " ; : (:) Since the alue of, (.), which is a solution of R() = R obs, is ery close to the alue " = :7 from the obsered charged leton masses, if we regard as = ", we obtain the relation (.7). Thus, the seculation (.7) is accetable henomenologically. Howeer, this does not mean that the resent model gies a basis for rannen's seculation (.7). In the resent model with y = y 4, as discussed in Sec., the mass sectrum of the neutrinos is indeendent of the VEVs u i, and it deends only on the alues y, y and y. On the other hand, the charged leton mass sectrum deends only on the VEVs i d, because we hae assumed the uniersality ofthe Yukawa couling constants. The arameter e (therefore, ") is one which characterizes the VEV sectrum ( d d d ), while the arameters (therefore, ) in the resent model is one which characterizes the structure of the neutrino Yukawa couling constants. Therefore, the arameter is dierent in kind from the arameter e. At resent, it is an oen question whether the coincidence ' " is accidental or not. 4 ase with y + y = y In the reious section, we hae assumed a constraint (.) on the Yukawa couling constants y, y and y. Howeer, the theoretical basis of the constraint is not clear. In the resent section, instead of the constraint (.), we assume another constraint y + y = y + (y + y4) (4:) which is again satised by theyukawa couling constants (.) in the charged leton sector. The condition (4.) means a requirement of the uniersality of the couling constants in an extended meaning: indiidually normalized couling constants of scalars and ( ) are equal to each other. In the neutrino sector, since we hae assumed y = y 4 =, we can denote the condition (4.) as y = y cos y = y sin : (4:) 9

10 Then, the mass eigenalues (.) are exressed as follows : m = [sin( + ) ] jy j u m = [sin( + ) ] jy j u m = cos( + )y u (4:) where sin = we hae again took the condition (.8), i.e. r cos = ( =54:74 ) (4:4) sin( + ) > (; <<; ) (4:5) and the uer and lower signs in Eq.(4.) corresond to the cases y > (a normal hierarchy case) and y < (an inerse hierarchy case), resectiely. From the exression (4.), we nd m + m + m = y u (4:) Therefore, we obtain m + m + m = r y u cos : (4:7) (m + m + m ) m + m + m = cos =; sin : (4:8) Thus, the arameter in the resent model denotes a deiation from the mass formula (.) [(.7)]. Note that if we nd a solution = which gies R() =R obs [R() isgien by Eq.(.) with!, and R obs is gien by Eq.(.)], the alue = ; [ is dened by Eq.(4.4)] is also a solution of R() =R obs. From the exression (4.), it is obious that the solutions and gie the same alues for m and m, but they gie thealues with the oosite signs to each other for m. We list those solutions of R() =R obs in Table, together with the alues of m, m and m. In Table, we also list the redicted alues of the neutrino masses m = m =M N, m = m =M N and m = m =M N (M N is a Majorana mass M N M N = M N = M N of the heay neutrinos N i ). Here, as the inut alue, we hae used m = m atm = :5 ev for the normal hierarchy case, and m = m atm = :5 ev for the inerse hierarchy case. At resent, the numerical alues of m i should not be taken rigidly. Therefore, we hae omitted the error alues from Table. 5 Neutrino mixing matrix

11 As we discussed in Sec., the additional rotation R( ) from the tribimaximal mixing, (.4), deends only on the alue u =u, and it is indeendent of the alues of y, y and y, i.e. of the mass eigenalues. If we want that the neutrino mixing is urely the tribimaximal mixing, we must take u =u =, i.e. u = u. Howeer, such a model in which the VEVs of the u-scalars u i can comletely be unrelated to those of the down-scalars d i is not so attractie to us. We exect that h u i i will hae some relation to hd i i. In order to see the eects of the additional rotation R( ) dened by Eq.(.), we change from the basis ( ) dened by Eq.(.4) into the basis ( ) gien by A = U T e A (5:) where U is the tribimaximal mixing matrix dened by Eq.(.). If = =,i.e. R( ) =, the neutrino mixing matrix U is gien by U = U c s ;s c A = ; c ; s c + s s ; c c ; s s + c A (5:) i.e. tan solar = (5:) c sin atm = ; 4 s (5:4) For conenience, we dene the following z i -arameters (U ) = s : (5:5) h u i i = z u i u h d i i = z d i d (5:) with the normalizations P i (zu i ) = P i (zd i ) =. For the zi d -arameters, from the relation (.7), we obtain z d me = zd m = zd m = me + m + m (5:7) i.e. z d =:47 z d =:78 z d =:974: (5:8) If we assume z u i = z d i (i = ), we obtain zu = :599, z u = :4798 and z u = = from the denition (.4). Then, the rotation angle = ;(=) tan ; ( u =u )=;: is too large to exlain the obsered neutrino mixings [see Eqs.(5.)-(5.5)], so that the case is ruled out.

12 As we discussed in Sec., the S bases A and are dual each other. Although we hae inestigated the neutrino mass matrix form on the basis, as the second best idea instead of the case (z u z u z u )=(z d z d z d ), it is likely that the VEVs ( u u u ) on the basis is gien by z u z u A = A z d z d A = z d z d A (5:9) z u z d z d contrast to the VEVs of the down-tye scalars d on the basis, i.e. (z d zd zd ) T = (zd zd zd )T. When we assume the relation (5.9), we obtain so that we obtain the redictions z u =:5584 z u = ;:897 z u == (5:) tan =: =: (5:) tan solar =:5 (5:) sin atm =:975 (5:) (U ) =:8: (5:4) At resent, the alues (5.){(5.4) cannot be ruled out by the obsered neutrino oscillation data [, ], so that the ossibility (5.9) is accetable. Howeer, of course, the relation (5.9) is only a seculation at resent. oncluding remarks In conclusion, based on a uniersal seesaw mass matrix model with three scalars i, and by assuming an S aor symmetry for Yukawa interactions, we hae inestigated the neutrino masses and mixings. For the VEV alues of f i (f = u d), stimulated from a Higgs otential model [9] for i,wehae assumed the constraint h f i + h f i = h f i (:) where ( ) are dened by Eq.(.4). Since the obsered neutrino mixing is aroximately gien by the tribimaximal mixing (.), we hae inestigated only a simle case with - mixing, where ( ) is a doublet of S. In the case, the mass eigenalues deends only on the alues of the couling constants y, y and y, while the - mixing angle deends only on the alue of h u i=h u i. From the economical oint of iew of the arameter number, wehae inestigate two tyical cases with the constraints y = y + y and y + y = y. The former case leads to a case which satises rannen's relation (.) for the neutrino masses. Although it is ery interesting, the theoretical basis of the constraint y = y + y is not so clear. On the other hand, the later case cannot satisfy the relation (.). Howeer, for a small alue of the arameter, the deiation from the relation (.) is negligibly small. For examle, for the solution = :94 gien in

13 Table, the deiation from the relation (.) is ery small, sin =:, as seen in Eq.(4.8), so that the relation (.) is aroximately satised. In any case, in order to t the resent data from the neutrino oscillation exeriments, the existence of the y -term with a small couling constant. Although wehaegien the "rediction" of the neutrino masses in Secs. and 4, exactly seaking, those are not redictions. Those are results by adjusting the arameter (or ). Since we consider that the mass matrix structures in the letons should be as simle as ossible, we may seculate a concise structure [4] of the Yukawa interaction in the neutrino sector: H = y `N + `N + `N u + `N + `N u + `N ; `N u (:) where we hae required the uniersality of the couling constants as well as in the charged leton sector [howeer, not in the basis ( ), but in the basis ( )]. Then, the neutrino masses are redicted as m = ; m m = m m = + m (:) without an adjustable arameter. The case redicts R = m m = 4 ; 9 4 =:4: (:4) +9 The alue (.4) is somewhat large comaring with the obsered alue (.), but, at resent, the case is not ruled out within three sigma. Again, regarding m as m = m atm,we redict the exlicit neutrino mass alues as follows: m =(5: +:4 ;: ) ;4 ev m =(:5 +:7 ;:5 ) ; ev m =(5: +:5 ;:5 ) ; ev: (:5) The case (.) is also interesting because of the simleness of its structure. y the way, in the neutrino Yukawa interaction (.), by assuming the uniersality ofthe couling constants, we hae taken y = y = y. Howeer, as seen in Eq.(5.), the case gies u < for the choice (5.9) on the basis dened by Eq.(.4). From the discussion in Sec., it seems that the case y = y > gies y <, so that the case leads not to a normal hierarchy case (m > m ), but to an inerse hierarchy case (m < m ). Therefore, it seems that we must choose not the case y = y >, but the case y = ;y <. Howeer, this is not serious

14 roblem. When we redene the bases A and by the alternatie matrices A = ; ; ; A (:) = ; ; ; A (:7) instead of A and dened by Eqs.(.) and (.5), resectiely, the signs of the elds ( ) are changed, so that we obtain >. Then, we also obtain the trimaximal mixing U = ; ; ; A (:8) instead of the form (.). This hase change!; does not aect to the results (5.)-(5.5), because the sign of is also changed together with ( ; )! (; ; ). Although the case with u = which exactly leads to the tribimaximal mixing (.) is attractie to us, at resent, there is no reason for u =. Rather, the case (5.9) is attractie although it is still seculatie. Then, we can redict small deiations (5.), (5.) and (5.4) from the ideal tribimaximal mixing (.). At resent, there is no reason for such a ermuted VEV relation (5.9) between ( u ) and u u (d d d ). Howeer, from the henomenological oint of iew, it will be worth while inestigating quark mass matrices with such a ermuted VEV relation. The study will be gien elsewhere. In conclusion, the resent model (a leton mass matrix model with a bilinear form) based on the S symmetry has gien many interesting features for the mass sectra and mixings. Howeer, the model still includes some adjustable arameters. Further inestigation based on another symmetry which gies stronger constraints on the arameters than those in the S symmetry will be desired. References [] S. Pakasa and H. Sugawara, Phys. Lett. 7, (978) H. Harari, H. Haut and J. Weyers, Phys. Lett. 78, 459 (978) E. Derman, Phys. Re. D9, 7 (979) D. Wyler, Phys. Re. D9, (979). [] P. F. Harrison, D. H. Perkins and W. G. Scott, Phys. Lett. 458, 79 (999) Phys. Lett. 5, 7 () Z. Z. Xing, Phys. Lett. 5, 85 () P. F. Harrison and W. G. Scott, 4

15 Phys. Lett. 55, () Phys. Lett. 557, 7 () E. Ma, Phys. Re. Lett. 9, 8 (). I. Low and R. R. Volkas, Phys. Re. D8, 7 () X.-G. He and A. Zee, Phys. Lett. 5, 87 (). [] S. Eidelman et al. (Particle Data Grou), Phys. Lett. 59, (4). [4] Y. Koide, Lett. Nuoo imento 4, (98) Phys. Lett., (98) Phys. Re. D8, 5 (98). [5] Y. Koide, Mod. Phys. Lett. A5, 9 (99). [] Y. Koide and H. Fusaoka, Z. Phys. 7, 459 (99). [7] Y. Koide and M. Tanimoto, Z. Phys. 7, (99). [8] The seesaw mechanism for charged articles is known as the \uniersal seesaw mechanism": Z. G. erezhiani, Phys. Lett. 9, 99 (98) Phys. Lett. 5, 77 (985) D. hang and R. N. Mohaatra, Phys. Re. Lett. 58, (987) A. Daidson and K.. Wali, Phys. Re. Lett. 59, 9 (987) S. Rajoot, Mod. Phys. Lett. A, 7 (987) Phys. Lett. 9, (987) Phys. Re. D, 479 (987) K. S. abu and R. N. Mohaatra, Phys. Re. Lett., 79 (989) Phys. Re. D4, 8 (99) S. Ranfone, Phys. Re. D4, 89 (99) A. Daidson, S. Ranfone and K.. Wali, Phys. Re. D4, 8 (99) I. Sogami and T. Shinohara, Prog. Theor. Phys. 8, (99) Phys. Re. D47, 95 (99) Z. G. erezhiani and R. Rattazzi, Phys. Lett. 79, 4 (99) P. ho, Phys. Re. D48, 5 (99) A. Daidson, L. Michel, M. L. Sage and K.. Wali, Phys. Re. D49, 78 (994) W. A. Ponce, A. Zeeda and R. G. Lozano, Phys. Re. D49, 4954 (994). [9] Y. Koide, Phys. Re. D7, 579 (). Also, see Y. Koide, Phys. Re. D, 77 (999). []. rannen, htt://brannenworks.com/masses.df, (). []. Frogatt and H.. Nielsen, Nucl. Phys. 47, 77 (979). [] S. N. Ahmed et al., SNO ollaboration, Phys. Re. Lett. 9, 8 (4) T. Araki et al. KamLAND ollaboration, Phys. Re. Lett. 94, 88 (5) A. W. P. Poon, in Proceedings of the XXII International Symosium on Leton and Photon Interactions at High Energies, Usala, Sweden, June { 5 July, 5, edited by R. renner,. P. Heros and J. Rathsman (World Scientic Pub. ),. 78. [] J. Nelson, MINOS collaboration, resented in Neutrino, at Santa Fe. Also see, J. Hosaka et al. the Suer-Kamiokande ollaboration, arxi: he-ex/4. [4] Y. Koide, arxie: he-h/574. 5

16 Table Solution of R()=R obs m =y u m =y u m =y u m [ev] m [ev] m [ev] :94 ;:775 :78 :95 :8 :877 :5 7:59 ;:775 ;:78 :95 :8 :877 :5 ;5:4 : :8 ;:4 :55 :5 : :7 : ;:8 ;:4 :55 :5 :