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FTUAM-11-46 IFT-UAM/CSIC-11-27

Neutrino masses in SU(5) U(1)F with adjoint flavonsEnrico Nardi1,2,3, Diego Restrepo4, and Mauricio Velasquez4

1 INFN, Laboratori Nazionali di Frascati,C.P. 13, I00044 Frascati, Italy2 IFT-UAM/CSIC, Nicolas Cabrera 15, C.U. Cantoblanco, 28049 Madrid, Spain

3 Departamento de Fsica Teorica, Universidad Autonoma de Madrid,

C.U. Cantoblanco, 28049 Madrid, Spain4 Instituto de Fsica, Universidad de Antioquia, A.A.1226, Medelln, Colombia

May 29, 2011

Abstract

We discuss a SU(5) U(1)F supersymmetric model for neutrino masses and mixings thatincludes three heavy singlet neutrinos, two flavons, and a single mass scale: the unificationscale. The breaking of the Abelian flavor symmetry U(1)F leaves Rparity as an exact accidentalsymmetry. By assigning flavons to the adjoint representation ofSU(5) and assuming universalityfor all the fundamental couplings, the coefficients of the effective Yukawa and Majorana massoperators are calculable, and are fixed in terms of group theoretical quantities. We discuss howthese coefficients yield a quantitative improvement of the naive order-of-magnitude estimates ofthe neutrino mass matrix.

1 Introduction

The standard model (SM) is a very successful framework for describing particle physics phenomena.

However, it suffers from some serious theoretical problem, among which: neutrinos are massless, there

is no candidate for the dark matter, and the conditions for baryogenesis are not fulfilled. Some of

these problems are solved in extensions of the SM, like the SM plus the seesaw mechanism [?,?,?,?,?],

or the supersymmetric SM (SSM).

In contrast to the SM, the SSM does not have accidental lepton (L) and baryon-number (B)

symmetries. This situation leads to other severe phenomenological problems, like fast proton decay.

The standard solution to this is to impose a discrete symmetry, Rparity, to forbid all dangerous

operators. Similarly to the SM, also the SSM does not provide any explanation for the strong

hierarchy in the fermion Yukawa couplings. One way to explain the flavor puzzle and the suppression

of the fermion masses with respect to the electroweak breaking scale is to impose Abelian flavor

symmetries, that we generically denote as U(1)F, that are broken by SM-singlets commonly denoted

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as flavons. Besides the Yukawa couplings, these symmetries can also suppress, but often not forbid

completely, the SSM B and L violating terms. Along these lines, consistent models can be build

in which small neutrino masses can be accommodated (for a review see [ ?]). Due to the fact that

Rparity is not an exact simmetry, the LSP can provide a candidate for decaying dark matter [?].

A different possibility arises when, due to an enlarged gauge symmetry, Rparity arises as anaccidental symmetry, like it happens in the SM for B and L. R-parity conservation can be for

example enforced by an extended gauge symmetry together with supersymmetry (that requires a

holomorphic superpotential) as in the model studied in [?], or solely by the gauge symmetry thanks

to a suitable choice of the U(1)Fcharges, as in ref. [?]. In this paper we focus on this second

possibility, and we work in the framework of a unified SU(5) U(1)F model. The U(1)F gaugecharges automatically lead to conserved matter parity, forbidding L = 1 and B = 0 operatorsat all orders. However, non-vanishing L = 2 Majorana masses for heavy singlet neutrinos remain

allowed, and thus the seesaw mechanism can be embedded in the model. More in detail, following [ ?]

we chose the F-charges in such a way that operators with even Rparity have an overall F-charge

that is an integer multiple of the charge of the U(1)F breaking scalar field (that, without loss of

generality, we set equal to 1). In contrast, all the Rparity breaking operators, that have an overall

half-odd-integer Fcharge, are forbidden. To allow for the L = 2 Majorana masses while forbidding

L = 1 operators, the Fcharges of the right handed neutrinos must then be halfodd-integers.

Differently from the SM case [?], in SU(5) GUTs it is rather difficult to implement this kind of

horizontal symmetries, because there is less freedom in choosing the Fcharges (see for example [?]).

However, if the flavons that break the horizontal symmetry are assigned to the adjoint representation

of SU(5) [?, ?, ?], charges that were forbidden in the singlet flavon case become allowed, under theassumption that certain representations for the Froggatt-Nielsen (FN) messengers fields [?] do not

exist. In contrast to the non-unified SU(3) SU(2) U(1) U(1)F model, where the singletnature of the flavons is mandatory, in SU(5) U(1)F assigning the flavons to the adjoint has theadditional bonus that non-trivial group theoretical coefficients concur to determine the coefficients

of the FN effective operators [?, ?, ?]. Therefore, assuming that at the fundamental level all the

Yukawa couplings obey to some principle of universality [?], the order one coefficients that determine

quantitatively the structure of the mass matrices become calculable quantities.

2 The Model

We assume that at the fundamental level all the Yukawa couplings are universal, and that all the

heavy messengers states carrying U(1)F charges have the same mass, as it would happen if the

masses are generated by the vacuum expectation values (vev) of some singlet scalar. With these

assumptions, the only free parameter of the model is the ratio between the vacuum expectation

value of the flavons and the mass of the heavy vectorlike FN fields. This parameter is responsible

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for the fermion mass hierarchy, and all the remaining features of the mass spectrum are calculable in

terms of group theoretical coefficients. More precisely, in our model the flavor symmetry is broken by

vevs of scalar fields in the 24dimensional adjoint representation ofSU(5), where the subscriptsrefer to the values 1 of the U(1)F charges that set the normalization for all the other charges. The

vevs + = = Va with Va = V diag(2, 2, 2,3,3)/60 are also responsible for breaking theGUT symmetry down to the electroweakcolor gauge group. The order parameter for the flavor

symmetry is then = V /M where M is the common mass of the heavy FN vectorlike fields. This

symmetry breaking scheme has two important consequences: power suppressions in appear with

coefficients related to the different entries in Va, and the FN fields are not restricted to the 5, 5, or

10, 10, multiplets as is the case when the U(1)F breaking is triggered by singlet flavons [?].

The model studied in [?] adopted this same scheme, and yields a viable phenomenology, since

it produces quark masses and mixings and charged lepton masses that are in agreement with the

data. The U(1)F charge assignments of the model yield U(1)F mixed anomalies, that are canceled

trough the Green-Schwartz mechanism [?]. The values of the charges are determined only modulo

an overall rescaling, that may be chosen in order to forbid baryon and lepton number violating

couplings. With the choice of charges adopted in [?], both L = 1 and L = 2 violating operators

were forbidden, and thus the seesaw mechanism could not be embedded in the model. In order to

avoid this unpleasant feature, in this work we explore the possibility of forbidding just the L = 1

operators while allowing the L = 2 seesaw operator for neutrino masses. We will show that by

means of a suitable choice of the F charges, the seesaw mechanism with three right handed neutrinos

can be implemented, and one can obtain neutrino masses and mixings in agreement with oscillation

data, while L = 1 and B = 0 (and thus Rparity violating) operators are forbidden at all ordersby virtue of the F-charges. We will also show that the lightest SU(2) singlet neutrino mass is of the

right order to allow to generate the baryon asymmetry through leptogenesis.

2.1 Charge assignments

There are some requirements that the F charges have to satisfy in order to yield a viable phenomeno-

plogy. In order to allow the Higgsino term at tree level, we must require

5d + 5u = 0 , (1) e

where 5d, 5u denote the F-charges of the d, u Higgs chiral multiplets, and we will use a similar

notation for all the other F-charges and fields. It is easy to see that with the constraint (1) the

overall charge of the Yukawa operators for the charged fermion masses 10I5J5d and 10I10J5u,

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that are even under Rparity, are invariant under the charge redefinitions [?]:

5I 5I + an (2) e10I 10I an

3

5d 5d 2a

n3

5u 5u +2an

3,

where I = 1, 2, 3 is a generation index, and an is an arbitrary shift that redefines the charges.

Assuming 5u = 0, then the anomalous solution that was chosen in ref. [?] can be written as

5u = 5d =0 , 5I = 2

I 7 , 10I = 3 I . (3) e

Starting from a set of integer charges, redefining this set by means of the shift eq. ( 2) with

an = 32

2n5

+ 1

, (4) e

where n is an integer, it is easy to see that the Rparity violating operators 10I5J5K and 5I5u have

halfoddinteger charges, and hence are forbidden at all orders.

To generate neutrino masses, we now introduce three heavy multiplets with halfoddinteger

Fcharges, that we assume belonging to the adjoint representation 24. The SU(2) U(1) SU(3)neutral member of these multiplets will play in our model the same role that is usually played by

heavy SU(2) singlet Majorana neutrinos. 1 The halfoddinteger charges of the new states, after the

charges of the other fields have been shifted according to eqs. (2), can be parametrized as

NI =2mI + 1

2, (5) e

where mI are integers. The new effective superpotential terms that give rise to the seesaw are

Wseesaw = YIJ 5I5u NJ +

12

MIJR NINJ . (6) e

For the Dirac operators in eq. (6) we have the following set of Fcharges:

5I + 5u + NJ =2I 7 + an + 2an/3 + NJ

=2I 9 + n + mJ .

Explicitly:

F(5I5u NJ) =

7 n + m1 7 n + m2 7 n + m35 n + m1 5 n + m2 5 n + m31 n + m1 1 n + m2 1 n + m3

. (7) e

1We have veryfied that with the minimal choice of only two heavy neutrino supermultiplets, in our framework it isnot possible to explain the neutrino oscillation data.

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For the mass operators of the adjoint neutrinos, we have the following (integer) Fcharges

NI + NJ =1 + mI + mJ ,

F(NINJ) =

1 + 2m1 1 + m1 + m2 1 + m1 + m31 + m1 + m2 1 + 2m2 1 + m2 + m3

1 + m1 + m3 1 + m2 + m3 1 + 2m3

. (8) e

The Dirac neutrino mass matrix and the right-handed neutrino mass matrix can thus be obtained

just in terms of the dimensional parameter MR and of various suppression factors, corresponding to

powers of the small factor , that are determined by the values of the charges in eqs. (7) and (8).

The light neutrino mass matrix is then obtained from the seesaw formula [?, ?]

Mv2 sin2 YM1R YT , (9) e

where v = 175 GeV.

In our model, the order one coefficients of the light neutrino mass operator can be computed with

the same technique introduced in [?] for computing the down-quark and charged lepton masses.

The details of the mass spectrum are thus determined by nonhierarchical computable group

theoretical coefficients that only depend on the way the heavy FN states are assigned to SU(5)

representations. The effective couplings appearing in eq. (6) are suppressed by the symmetry breaking

parameter as

YIJ |5I+5u+NJ| , MIJR M |NI+NJ| . NOTE THE M ! (10) e

where in the second relation M is the mass of the FN messenger fields. Since we have two flavon

multiplets with opposite charges, the horizontal symmetry allows for operators with charges of

both signs, and hence the exponents in eq. (10) must be given in terms of the absolute values of

the sum of charges. Our assumption of universality for the fundamental Yukawa couplings implies

that the matrices in eq.(10) have no arbitrary O(1) coefficients of unspecified origin. Note thatthis condition excludes the simple (and often used) charge assignments in which there are two zero

eigenvalues in the light neutrino mass matrix, as in [?, ?].

2.2 Computing the Dirac and Majorana effective operators

In this section we analyze the contributions of different effective operators to Y and to MR, showing

that a phenomenologically acceptable structure, able to reproduce (approximately) the correct mass

ratios and to give reasonable neutrino mixing angles can be obtained.

We assume that a large number of vectorlike FN fields exist in various SU(5) representations.

Since we assign the heavy Majorana neutrinos to the adjoint N = Nab , the possible FN field represen-

tations R can be identified starting from the following tensor products involving the representations

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of the fields in the external lines (see the diagrams in Fig. 1):

5 5u = 1 24 , (11) e5 = 5 45 70 , (12)

N = 1S 24S 24A 75S 126A 126A 200S , (13)where the subscripts S,A in the last line denote the symmetric or antisymmetric nature of the cor-

responding representations. We assume that all FN fields transform nontrivially under SU(5), and

thus that no singlet exists and, for simplicity, we restrict ourselves to representations with dimen-

sionality less than 100, which gives the following possibilities R = 24, 5, 45, 70. Since the mass

M of these fields is assumed to be larger than GUT, the contributions to the operators in eq. (6)

can be evaluated by means of insertions of effective pointlike propagators. As in [?] we denote the

contractions of two vectorlike fields in the representation R, R as

Rabc...de... Rpq...lmn...

= i

MSabc...pq...de...lmn... (14)

where all the indices are SU(5) indices, and Sis the appropriate group index structure. The struc-tures Sfor [5a 5b],

45abc 45

nlm

and

70abc 70

nlm

(and for several other SU(5) representations) can be

found in Appendix A of [?]. In addition we need the following contractions

i M

24ab 24lm

S

= (SS)a lbm =5

2

am

lb +

al

mb

ab lm , (15) ei M

24ab 24

lm

A

= (SA)a lbm =5

2

am

lb al mb

. (16) e

These two expressions are obtained by imposing the traceless condition for the adjoint (SS,A)a lam =(SS,A)a lb l = 0 and the normalization factor is fixed by the requirement that the (subtracted) singletpiece ab

lm in eq. (15) provides the proper singlet contraction, that is, by inserting the singlet in the

diagram of fig.1(b) one must obtain the operator

5a5au

Njl lj with coefficient one. The verticesinvolve 5u or the adjoint with the external fermions 5 and N, or with the FN representations R

in the internal lines. The vertices have the general form iVwhere is universal for all vertices.Including symmetry factors, the relevant field contractions V = R 5u R or V = R R, withR, R = 5, 24, 45, 70, are:

5a24ab5b 5a24cb45bac 5a24cb70bac (17)

45cab24

bd45

dac

1

245

cab24

dc45

bad (18)

70cab24

bd70

dac

1

270

cab24

dc70

bad (19)

45cab24

bd70

dac 24

ac24

cb (24S,A)

ba . (20)

There are two inequivalent ways of contracting the indices for the vertices involving the 24 with

pairs of 45 and 70 [?]. They are distinguished in eqs. (18) and (19) by an up- (24) or down (24)

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arrow-label. As explained in [?], this can be traced back to the fact that these representations are

contained twice in their tensor products with the adjoint. We write the SU(5) U(1)F breakingvevs as

= V 160

diag(2, 2, 2,3,3) (21) e

where the factor 1/60 gives the usual normalization of the SU(5) generators Tr(RaRb) = (1/2)ab,and the coefficients of the neutrino couplings 0u N and of the Majorana neutrinos mass terms N N

are obtained by projecting the representations 5, 5u and N onto the relevant field components

according to

= 55 = a5 5a NOTE THE MINUS SIGN!! (22)0u = 5

5u

= 5b 5bu

(23)

N =160

diag(2N11 , 2N22 , 2N33 , 3N44 , 3N55 ). NOTE THE 3!! (24)

The assumption of a unique heavy mass parameter M for the FN fields and of universality of

the fundamental scalar-fermion couplings yield a remarkable level of predictivity. In particular, for

the vertices involving we can always reabsorb V V. This leaves just an overall power of common to all effective Yukawa operators that involve one insertion of the Higgs multiplet 5u (see

the diagrams in Figs. 1) and no at all for the contributions to MR, (see the diagrams in Figs. 2).

The contributions to Y and MR at different orders can be computed using the vertices Vgivenin eqs. (17)-(20) and the relevant group structures S in eqs. (15), (16) and in Appendix A of [?],that account for integrating out the heavy FN fields. Additionally, the multiplets 5, N, and 5uin the external legs of the diagrams must be projected on the relevant components according to

eqs. (22)-(24) and the flavons have to be projecte onto the vacuum according to eq. (21).

We have evaluated the Y including the contributions up to O(2) that are diagramatically de-picted in Figs. 1: (a) at O(0), (b)(c) at O(1), and (d)(f) at O(2), and MR including contribu-tions up to O(3) corresponding to the diagrams in Figs. 2: (a) at O(), (b) at O(2), and (c) atO(3). At each specific order i the results can be written as Y(i) = i+1 i yi for i = 0, 1, 2 andM

(i)R = V

i+3 i ri+1 for i = 1, 2, 3, where = 1/

60 is the normalization factor for and the

two external Ns in the adjoint, V = M, is defined in (21), and yi and ri are the nontrivial group

theoretical coefficients. Our results for yi are given in Table 1 (where we have followed the notationof [?]), while the results for ri are given in Table 2.

We have searched for all possible Fcharge assignments such that the absolute value of individual

charges remains smaller than 10, and we have examined the resulting neutrino mass matrices.

We have found some promising possibilities. If we choose, for example, n = 6, m1 = 2, m2 = 1,and m3 = 6 (eq. (38) of Appendix A), we obtain the Fcharges shown in Table 3, which can beobtained from the set given in eq. (3) through the redefinitions eqs. (2) with a6 = 21/10.

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(a) (b) (c)

(d) (e) (f)

Figure 1: Diagrams contributing to Y at different orders. The lowest order coefficient corresponding

to diagram (a) is y0 = 3. Diagrams (b)(c) contribute at O(1

) and yield the coefficients y1 in thesecond column in Table 1. Diagrams (d)(f) contribute at O(2) and give the coefficients y2 in thefourth column of the table.Is the following last sentence correct ? There isL in diag. (f) andR in all the other diagrams. I suggest to changeL

to R, and simply omit the final sentence. I think thatL = 24 cannot enter in (f) right ? .L stands for either 24, 5, 45 f

N

24

24 N N

24

24 24

24 N

(a) (b) (c)

Figure 2: Diagrams contributing to MR at different orders. The lowest order coefficient r1 is obtainedfrom diagram (a), r2 from (b), and r3 from (c). f

According to eq. (7), this set ofFcharges gives the following orders of magnitude for the various

entries of the Yukawa coupling matrix:

Y

2

7

3 1 5

7 4

. (25) e

Neglecting terms ofO(4) and higher, and including the coefficients yi, we have

Y y1() y2()

2 0y3()

3 y0 00 0 y1()

. (26)

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1 y1 2 y2

[5u] [5u]

O(; 24S) 15 O(2; 24S, 24S) 75O(; 24A) 0 O(

2; 24A, 24S) 0

O(2; 24S, 24A) 0O(2; 24A, 24A) 0

[5u] [5u]

O(; 5) 9 O(2; 5, 24S) 45O(2; 5, 24A) 0

O(; 45) 75 O(2; 45, 24S) 300

O(2; 45, 24A) 0

O(; 70) 225 O(2; 70, 24S) 900O(2; 70, 24A) 0

[5u]

O(2; 5, 5) 81O(2; 5, 45) 225

O(2; 5, 70) 675O(2; 45, 5) 225

O(2; 45, 45) 1425O(2; 45, 45) 525O(2; 45, 70) 1125

O(2; 70, 5) 675O(2; 70, 45) 1125O(2; 70, 70) 4725O(2; 70, 70) 1350

RO(; R) 174 RO(2; R) 5826

Table 1: Operators contributing to Y =

i Y(i) at O() and O(2) and values of the corresponding

coefficients yi = Y(i) / ( i+1 i). The coefficient at O(1) is y0 = 3. t

Similarly, by using eq. (8) we have

MR M

5 2 3

2 6

3 6 11

. (27) e

Neglecting terms ofO(4) and higher, and taking into account the coefficients ri, we have

MR V 3 0 r2() r3()

2

r2() r1 0r3()

2 0 0

, (28) e

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1 r1 2 r2

3 r3

[] [] []

O(; 24) 30 O(2; 24S) 150 O(3; 24S, 24S) 750O(2; 24A) 0 O(

3; 24A, 24S) 0

O(3; 24S, 24A) 0

O(3; 24A, 24A) 0

RO(; R) 30 RO(2; R) 150 RO(3; R) 750

Table 2: Operators contributing to MR =

i MiR at O(), O(2) and O(3), and values of the

corresponding coefficients ri = M(i1)R / (V

i+2 i1). t

where we have substituted M = V.

According eq. (9), the light neutrino mass matrix is then

Mv2 sin2

V

2

r1 r3

0 0 y21r1

0 y20r3 y0y1r2y21r1 y0y1r2 1r3 y21r22

(29) e

where V GUT, and we have neglected terms ofO()2 and higher. This matrix correspondsto the two zerotexture type of neutrino mass matrix discussed in [?].

It is remarkable to note that while both Y and MR are hierarchical, that is the first one has a

hierarchy between its eigenvalues of

O() and the second of

O(2), M is not. In fact at leading

order it does not depend at all on the hierarchical parameter but only on the group theoretical

coefficients. This is interesting because it is commonly believed that a hierarchical patter for the

heavy Majorana neutrinos MR1 MR2 MR3 could be reconciled with quasi degenerate lightneutrinos only at the cost of a severe fine tuning in the Yukawa couplings. Our results show that this

is not true, and that under certain conditions (specific values for the charges) a symmetry like U(1)F

can enforce precisely this result. In our case it is precisely this fact that opens up the possibility of

explaining non-hierarchical neutrino masses and large mixing angles, although the whole scenario is

defined at the fundamental level in terms of the rather small hierarchical parameter .

Let us comment at this point that, as it is discussed in [?], corrections from sets of higherorder diagrams to the various entries in Y and MR are generally quite sizable, although suppressed

by higher powers of . This is because at higher orders the number of diagrams contributing to the

various operators proliferate, and the individual group theoretical coefficients also become generically

much larger, as can be seen in tables 1 and 2. By direct evaluation of higher orders, the related

effects were estimated in [?] to be typically of a relative order 20% 30%. To take into accountthe possible effects of these corrections, in all our final numbers we will then allow for a 25%numerical uncertainty.

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If we allow for all the possible contributions at O() listed in Table 1, the coefficient y1 = 174for Y, turns out to be too large in order to reproduce the neutrino oscillation data. We can then

assume that only some contributions are presend, for example only those that reproduce the diagrams

structure [5u], while representattions that would produce the structure [ 5u] are absent. This

can be easily obtained by assuming that no FN fields exist in the representations 7039/10, 7041/10;4539/10, 4541/10; 539/10, 541/10. Doing so, we obtain a coefficient that is more than ten times smaller:

y1 = 15 which we will use henceforth.Here we should say which contributions are left for y2 and which is the value of the

new coefficent.

y2: 7019/10, 709/10 ??

Clearly, the absence of these representations implies that several contributions to the higher

order term Y(2) are also absent, and this results in the much smaller value y2 = 75?? IS THIS

CORRECT ? - PLEASE PUT THE CORRECT NUMBER. In contrast, the contributions

to MR, that arise only from insertions of the 24, are not affected.

By using y0 = 3, y1 = 15, y2 = 75?? and the numbers in table 2 we then have

M2.3 v2

GUT2

0 0 0.980 1.0 0.990.98 0.99 0.99

. (30)

(Killed and corrected 2)

Where we have taken tan 1. With v = 174GeV and GUT 1016 GeV The numericalvalue of the prefactor is 2.3 v

2

GUT2 0.007 2eV. The athmospheric mass scale 0.05 eV can then be

reproduced for an acceptable value of the universal coupling 2.7.With the charges given in table 3, the Yukawa matrix for the charged lepton mass operator

LE YIJE 5105d (31) ehas entries of the following orders of magnitude:

YE =

3 4 5

2 3

3 2

0 0 0 2 0

0 2

(32) e

from which the mixing matrix for the charged leptons is

UE 1

2 02 1

212

2 12

12

(33) e

51 52 53 101 102 103 5u = 5d N1 N2 N329

10- 910

3110

1310

310

710

75

52

12

112

Table 3: Fcharges obtained with n = 6 in eq. (4), and m1 = 2, m2 = 1, and m3 = 6 in eq. (5). t

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From which the PMNS matrix UPMNS = UUE is

Allowing for such an uncertainty in the entries of the matrix in eq. (??) we find that it is possible

to fit the neutrino oscillation data, with the exception of sin2 12 0.5 for which a particularly largecorrections is needed.

Alternativly, if only FN fields in the representations 7039/10, 7041/10; 4539/10, 4541/10 are for-

bidden, a similar solution can be also obtained with the same degree of uncertainty but with 0.36(Check this value of ! Note that we do not have to worry much about the precise

value, .1 or 2 are all OK.)From eq. (28) we can estimate the masss of the lightest heavy Majorana neutrino

M1 2 1010 GeV , (34)

which is of the right order of magnitude to allow for thermal leptogenesis [?].

UNTIL HERE

3 Conclusions

In an anomalous SU(5) U(1)F model with 8 Fcharges used to explain the charged fermion hi-erarchy, a shift parameter is left which can be used to fix the Fcharge of the Rparity breaking

operators to be half-odd-integer in order to have Rparity as a remnant of the U(1)F symmetry.Then we can further choose proper right handed neutrinos with halfoddinteger Fcharges to im-

plement the seesaw mechanism. As a bonus, the light right handed neutrino is in the proper range

to generate leptogenesis.

A Selected neutrino mass solutionsss

in Table 4 we summarize the solutions suitable to explain the neutrino oscillation data that we have

found after explore the full range of possible Fcharges. The numerical results were obtained fixing

y1 as in eq. (??). The equations referenced in the Table are shown below, in terms of the adimensionalmatrix

hYR1 YT , (35)

where R = MR/, and where we have neglected terms ofO(2)

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Eq. a N1 N2 N3 m232/m

221

(36) 3910

112

52

32

-

(37) 2110 72 32 12 13.2(38) 21

1052

12

112

14

(39) 310 12 32 72 13Table 4: Charge assignments resulting in normalized neutrino mass matrix with more that five 1s.The ratio of the squared difference of eigenvalues in the last column could be compared with the3range: (m232/m

221)exp = (24.8 39.0) [?] t

h =1

1 1 11 1 11 1 1

1

y20

r3r1y0y2

r23

r1y0y2r23

r1y0y2r23

r21y22

r33

r21y22

r33

r1y0y2r23

r2

1y2

2r33

r2

1y2

2r33

1

0.00017 0.013 0.013

0.013 1.0 1.0

0.013 1.0 1.0

(36) e

Note that this texture is rather similar to the usually obtained in the context of one-flavon horizontal

models as in [?].

h =1

0 0 1

0 1 11 1 1

1

0 0 y0y1r20

y20

r1y0y1

r2y0y1r2

y0y1r2

r1y21r22

1

0 0 0.98

0 1.0 0.980.98 0.98 0.97.

(37) e

1

0 0 y2

1r3

0y20

r1 r2y0y1

r1r3y21

r3r2y0y1

r1r3

r22y21

r1r23

1

0 0 0.980 1.0 0.99

0.98 0.99 0.99

(38) e

h =1

1 1 1

1 1 1

1 1 0

1

r1y21r22

r1y21r22

y0y1r2

r1y21r22

r1y21r22

y0y1r2

y0y1r2

y0y1r2

0

1

0.98. 0.98. 1.00.98. 0.98. 1.0

1.0 1.0 0

(39) e