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Transcript

arX

iv:h

ep-p

h/04

0215

5v2

23

Mar

200

4

DFPD-04/TH/05

CERN-PH-TH/2004-027

ROMA1-TH/1368-04

Can Neutrino Mixings Arise from the

Charged Lepton Sector?

Guido Altarelli 1,

CERN, Department of Physics, Theory Division,

CH-1211 Geneve 23, Switzerland

Ferruccio Feruglio 2,

Dipartimento di Fisica ‘G. Galilei’, Universita di Padova and

INFN, Sezione di Padova, Via Marzolo 8, I-35131 Padova, Italy

Isabella Masina 3

Enrico Fermi Center, Via Panisperna 89/A, I-00184 Roma, Italy and

INFN, Sezione di Roma, P.le A. Moro 2, I-00185 Roma, Italy

Abstract

The neutrino mixing matrix U is in general of the form U = U†eUν , where

Ue arises from the diagonalization of charged leptons and Uν is from the

neutrino sector. We discuss the possibility that Uν is nearly diagonal (in the

lagrangian basis) and the observed mixing arises with good accuracy from Ue.

We find that the fact that, in addition to the nearly maximal atmospheric

mixing angle θ23, the solar angle θ12 is definitely also large while at the same

time the third mixing angle θ13 is small, makes the construction of a natural

model of this sort considerably more complicated. We present an example of

a natural model of this class. We also find that the case that Uν is exactly

of the bimixing type is severely constrained by the bound on θ13 but not

excluded. We show that planned experimental searches for θ13 could have a

strong impact on bimixing models.

1e-mail address: [email protected] address: [email protected] address: [email protected]

1 Neutrino Mixings from the Charged Lepton

Sector

The observed neutrino mixing matrix U = U †eUν , in the limit of vanishing sin θ13 =

s13, has the approximate form:

U =

c s 0

s/√

2 −c/√

2 1/√

2

−s/√

2 c/√

2 1/√

2

, (1)

where s and c stand for sin θ12 and cos θ12 respectively and we took the atmospheric

angle θ23 as exactly maximal. The effective mass matrix of light neutrinos is in

general given by:

mν = U∗mdiagν U † . (2)

Starting from the lagrangian basis, where all symmetries of the theory are specified,

we want to investigate whether it is possible to obtain the observed mixings in a

natural way from the diagonalization of the charged lepton mass matrix by Ue while

Uν is nearly diagonal. The possible deviations from maximal θ23 and from s13 = 0

can be omitted in eq. (1) and attributed to small effects from Uν that will be in

general not exactly zero. One might think that given the rather symmetric role of

Ue and Uν in the formula U = U †eUν one way or the other should be equivalent. But

we will show that this is not so. Actually now that we know that also the solar angle

θ12 is large, this tends to clash with a small θ13, in the case of mixings dominated

by Ue.

In terms of Ue the charged lepton mass matrix me (defined as RmeL from right-

handed (R) and left-handed (L) charged lepton fields in the lagrangian basis) can

be written as:

me = Vemdiage U †

e . (3)

Indeed Ldiag = UeL and Rdiag = VeR are the transformations between the lagrangian

and the mass basis for the R and L fields. Assuming that U ∼ U †e , given that

mdiage = Diag[me, mµ, mτ ] we can write:

me = Vemdiage U = Ve

cme sme 0

s/√

2mµ −c/√

2mµ mµ/√

2

−s/√

2mτ c/√

2mτ mτ/√

2

. (4)

We will come back later on the matrix Ve that determines the right-handed mixings

of charged leptons. For the time being it is already interesting to consider the matrix

m†eme which is completely fixed by Ue:

m†eme = Ue(m

diage )2U †

e . (5)

1

Neglecting for simplicity the electron mass, we find, for U †e = U :

m†eme = U †(mdiag

e )2U =1

2(m2

τ +m2µ)

s2 −cs −s(1 − 2λ4)

−cs c2 c(1 − 2λ4)

−s(1 − 2λ4) c(1 − 2λ4) 1

,

(6)

where we definedm2

τ − m2µ

m2τ + m2

µ

= 1 − 2λ4 (7)

so that approximately λ4 ∼ m2µ/m

2τ . The problem with this expression for m†

eme is

that all matrix elements are of the same order and the vanishing of s13 as well as the

hierarchy of the eigenvalues arise from precise relations among the different matrix

elements. For example, the result s13 = 0 is obtained because the eigenvector with

zero eigenvalue is of the form e1 = (c, s, 0)T and the crucial zero is present because

the first two columns are proportional in eq. (6). These features are more difficult

to implement in a natural way than matrices with texture zeros or with a hierarchy

of matrix elements. Only if the solar angle θ12 is small, that is s is small, then the

first row and column are nearly vanishing and s13 is automatically small.

Consider, for comparison, the case where we do not make the hypothesis that all

the mixings are generated by the charged leptons, but rather that Ue ∼ 1. To make

the comparison more direct, let us assume that the neutrino mass spectrum is of

the normal hierarchy type with dominance of m3: mdiagν ∼ m3Diag[0, ξ2, 1], where

ξ2 = m2/m3 is small and m1 is neglected. In this case, the effective light neutrino

mass matrix is given by (note the crucial transposition of U , which in eq. (1) is real,

with respect to eq. (6)):

mν = U∗mdiagν U † ∼ m3

2

s2ξ2 −csξ2/√

2 csξ2/√

2

−csξ2/√

2 (1 + c2ξ2)/2 (1 − c2ξ2)/2

csξ2/√

2 (1 − c2ξ2)/2 (1 + c2ξ2)/2

. (8)

In this case, no matter what the value of s is, the first row and column are of order

ξ2. By replacing terms of order ξ2 by generic small terms of the same order, s13

remains of order ξ2. We can also replace the terms of order 1 in the 23 sector

by generic order 1 quantities provided that we have a natural way of guaranteeing

that the subdeterminant 23 is suppressed and remains of order ξ2. As well known

this suppression can be naturally induced through the see-saw mechanism either

by dominance of a single right-handed Majorana neutrino [1] or by a lopsided [2]

neutrino Dirac matrix. Natural realizations of this strategy have been constructed,

for example, in the context of U(1)F flavour models [3].

We now come back to the expression for the charged lepton mass matrix me

in eq. (4) where the matrix Ve appears. This matrix describing the right-handed

mixings of charged leptons is not related to neutrino mixings. In minimal SU(5)

2

the relation me = mTd holds between the charged lepton and the down quark mass

matrices. In this case Ve describes the left-handed down quark mixings: Ve = Ud.

The CKM matrix, as well known, is given by VCKM = U †uUd. Given that the quark

mixing angles are small, either both Uu and Ud are nearly diagonal or they are nearly

equal. Thus one possibility is that Ud is nearly diagonal. In this case, for Ve = Ud,

me is approximately given by eq. (4) with Ve ∼ 1. Neglecting the electron mass and

setting λ2 = mµ/mτ we obtain:

me ≈ mdiage U = mτ

0 0 0

s/√

2λ2 −c/√

2λ2 λ2/√

2

−s/√

2 c/√

2 1/√

2

. (9)

This matrix is a generalization of lopsided models with all three matrix elements in

the third row of order 1 (unless s is small: for small solar angle we go back to the

situation of normal lopsided models). We recall that lopsided models with the 23

and 33 matrix elements of order 1 provide a natural way to understand a large 23

mixing angle. In fact from the matrix relation

0 0 0

0 0 0

0 s23 c23

1 0 0

0 c23 s23

0 −s23 c23

=

0 0 0

0 0 0

0 0 1

, (10)

we see that in lopsided models one automatically gets a large 23 mixing from Ue.

In the generalized case of eq. (9), while the natural prediction of a large 23 mixings

remains, the relation s13 = 0 does not arise automatically if the entries of the matrix

are replaced by generic order 1 terms in the third row and of order λ2 in the second

row. If we call v3 the 3-vector with components of order 1 in the third row and λ2vλ

the vector of the second row, we can easily check that to obtain s13 = 0 it is needed

that vλv3 = 0 and also that vλ is orthogonal to a vector of the form (c, s, 0).

In democratic models all matrices Uu, Ud, Ue are nearly equal and the smallness

of quark mixings arises from a compensation between U †u and Ud. This sort of

models correspond, for Ve = Ud, to Ve = Ue = U † and a symmetric matrix me:

me = U †mdiage U . In this case we obtain a matrix exactly equal to that in eq. (6)

for m†eme except that squared masses are replaced by masses. As discussed in the

case of eq. (6), we need fine-tuning in order to reproduce the observed hierarchy

of mass and to obtain s13 = 0 unless the solar angle s is small. Note in fact, that

in the democratic model of [4], the vanishing of s13 is only accommodated but not

predicted.

2 A Natural Class of Models

We now attempt to identify a set of conditions that make possible the construction

of an explicit model where the mixing in the lepton sector is dominated by the

3

charged lepton contribution. One obvious condition is a dynamical or a symmetry

principle that forces the light neutrino mass matrix to be diagonal in the lagrangian

basis. The simplest flavour symmetries cannot fulfill this requirement in a simple

way. For instance, a U(1) symmetry can lead to a nearly diagonal neutrino mass

matrix, of the form:

mν =

ξ2p ξp+1 ξp

ξp+1 ξ2 ξ

ξp ξ 1

m , (11)

where ξ < 1 is a U(1) breaking parameter, p ≥ 1 and all matrix elements are defined

up to unknown order one coefficients. The problem with this matrix is that the ratio

between the solar and the atmospheric squared mass differences, close to 1/40, is

approximately given by ξ4 and, consequently, a large atmospheric mixing angle is

already induced by mν itself. If we consider a discrete symmetry like S3, mν can be

of the general form:

mν =

1 0 0

0 1 0

0 0 1

m + α

0 1 1

1 0 1

1 1 0

m , (12)

where α is an arbitrary parameter. In this case we need both the extra assumption

|α| ≪ 1 and a specific symmetry breaking sector to lift the mass degeneracy [4]. A

stronger symmetry like O(3) removes from eq. (12) the non-diagonal invariant, but

requires a non-trivial symmetry breaking sector with a vacuum alignment problem

in order to keep the neutrino sector diagonal while allowing large off-diagonal terms

for charged leptons [5].

A simple, though not economical, possibility to achieve a diagonal neutrino mass

matrix, is to introduce three independent U(1) symmetries, one for each flavour:

F=U(1)F1×U(1)F2

×U(1)F3×..., where F denotes the flavour symmetry group. The

Higgs doublet giving mass to the up-type quarks is neutral under F. Each lepton

doublet is charged under a different U(1) factor, with the same charge +1. In the

symmetric phase all neutrinos are exactly massless. Flavour symmetry breaking is

obtained by non-vanishing vacuum expectation values (VEVs) of three flavon fields,

also charged under a separate U(1) factor, with charge -2. In this way only diagonal

neutrino mass terms are induced. If the VEVs in the flavon sector are similar, we

expect neutrino masses of the same order, and the observed hierarchy between the

solar and atmospheric squared mass differences requires a modest adjustment of the

flavon vev’s and/or of the coefficients of the lepton violating operators.

A second condition can be identified by considering a mass matrix for the charged

leptons which is very close, but slightly more general than the one of eq. (9):

me =

O(λ4) O(λ4) O(λ4)

x21λ2 x22λ

2 O(λ2)

x31 x32 O(1)

m , (13)

4

where xij (i = 2, 3) (j = 1, 2) is a matrix of order one coefficients with vanishing

determinant:

x21x32 − x22x31 = 0 . (14)

The eigenvalues of me in units of m are of order 1, λ2 and λ4, as required by

the charged lepton masses. Moreover, the eigenvalue of order λ8 of m†eme has an

eigenvector:

(c, s, O(λ4))s

c= −x31

x32+ O(λ4) . (15)

In terms of the lepton mixing matrix U = U †e , this means θ13 = O(λ4) and θ12 large,

if x31 ≈ x32. When the remaining, unspecified parameters in me are all of order one,

also the atmospheric mixing angle θ23 is large. Notice that, by neglecting O(λ4)

terms, the following relation holds for the mass matrix me/m (cfr. eq. (10)):

0 0 0

x21λ2 x22λ

2 O(λ2)

x31 x32 O(1)

c −s 0

s c 0

0 0 1

=

0 0 0

0√

x221 + x2

22λ2 O(λ2)

0√

x231 + x2

32 O(1)

, (16)

wheres

c= −x31

x32. (17)

Therefore the natural parametrization of the unitary matrix Ue that diagonalizes

m†eme in this approximation is:

Ue = Ue12U

e23 , (18)

where Ueij refers to unitary transformation in the ij plane. Using U = U †

e , we

automatically find the leptonic mixing matrix in the standard parametrization U =

U23U13U12 (neglecting phases), with U23 = Ue†23, U13 = 1 and U12 = Ue†

12. Had

we used the standard parametrization also for Ue, we would have found three non-

vanishing rotation angles θeij with non-trivial relations in order to reproduce θ13 = 0.

This successful pattern of me, eq. (13), has two features. The first one is the

hierarchy between the rows. It is not difficult to obtain this in a natural way. For

instance, we can require a U(1) flavour symmetry acting non-trivially only on the

right-handed charged leptons, thus producing the required suppressions of the first

and second rows. The second one is the vanishing determinant condition of eq. (14).

We can easily reproduce this condition by exploiting a see-saw mechanism operating

in the charged lepton sector.

To show this we add to the field content of the standard model additional vector-

like fermion pairs (La, Lca) (a = 1, ...n) of SU(2) doublets, with hypercharges Y =

(−1/2, +1/2). The Lagrangian in the charged lepton sector reads:

L = kinetic terms + ηijeci ljhd + λiae

ciLahd + µajL

calj + MaL

caLa + h.c. (19)

5

where li and eci (i = 1, 2, 3) are the standard model leptons, doublet and singlet

under SU(2), respectively, and hd denotes the Higgs doublet. We assume a diagonal

mass matrix for the extra fields. We expect Ma, µaj ≫ 〈hd〉 and in this regime there

are heavy fermions that can be integrated out to produce a low-energy effective

Lagrangian. The heavy combinations are Lca and

La +µaj

Malj (a = 1, ...n) . (20)

These fields are set to zero by the equations of motion in the static limit and we

should express all remaining fermions in term of the three combinations that are

orthogonal to those in eq. (20), and which we identify with the light degrees of

freedom. To illustrate our point it is sufficient to work in the regime |µaj | < |Ma|and expand the relevant quantities at first order in |µaj/Ma|. To this approximation

the light lepton doublets l′i are:

l′i = li −µai

MaLa . (21)

The effective lagrangian reads:

L = kinetic terms + (ηij −λiaµaj

Ma

) eci l′jhd + h.c. , (22)

and the mass matrix for the charged leptons is:

me = 〈hd〉(ηij −λiaµaj

Ma) . (23)

This result is analogous to what obtained in the neutrino sector from the see-saw

mechanism. There is a term in me coming from the exchange of the heavy fields

(La, Lca), which play the role of the right-handed neutrinos, and there is another

term that comes from a single operator and that cannot be interpreted as due to

the exchange of heavy modes. In the regime 1 > |µ/M | > |η/λ| the “see-saw”

contribution dominates. Moreover, if the lower left block in me is dominated by a

single exchange, for instance by (L1, Lc1), then

[me21 me22

me31 me32

]

=〈hd〉M1

[λ21µ11 λ21µ12

λ31µ11 λ31µ12

]

, (24)

and the condition of vanishing determinant in eq. (14) is automatically satisfied.

Additional vector-like leptons are required by several extensions of the stan-

dard model. For instance, a grand unified theory based on the E6 gauge symmetry

group with three generations of matter fields described by three 27 representations

of E6, includes, beyond the standard model fermions, two SU(5) singlets and an

SU(5) vector-like (5, 5) pair per each generation. In such a model a “see-saw”

6

mechanism induced by the exchange of heavy (5, 5) fields is not an option, but

a necessary ingredient to recover the correct number of light degrees of freedom.

We should still show that it is possible to combine the above conditions in a nat-

ural and consistent framework. In the Appendix A we present, as an existence

proof, a supersymmetric SU(5) grand unified model possessing a flavour symmetry

F=U(1)F0×U(1)F1

×U(1)F2×U(1)F3

. The first U(1)F0factor is responsible for the

hierarchy of masses and mixing angles in the up-type quark sector as well as for the

hierarchy between the rows in the charged lepton mass matrix. The remaining part

of F guarantees a diagonal neutrino mass matrix and, at the same time, dominance

of a single heavy (5, 5) pair in the lower left block of me. Notice that, at variance

with most of the other existing models [6], this framework predicts a small value for

θ13, of order λ4 which is at the border of sensitivity of future neutrino factories.

3 Corrections to Bimixing from Ue

Even when the neutrino mass matrix Uν is not diagonal in the lagrangian basis,

the contribution from the charged lepton sector can be relevant or even crucial

to reproduce the observed mixing pattern. An important example arises if the

neutrino matrix Uν instead of being taken as nearly diagonal, is instead assumed of

a particularly simple form, like for bimixing:

Uν =

1/√

2 1/√

2 0

1/2 −1/2 1/√

2

−1/2 1/2 1/√

2

. (25)

This configuration can be obtained, for instance, in inverse hierarchy models with

a Le − Lµ − Lτ U(1) symmetry, which predicts maximal θν12, large θν

23, vanishing

θν13 and ∆m2

sol = 0. After the breaking of this symmetry, the degeneracy between

the first two neutrino generations is lifted and the small observed value of ∆m2sol

can be easily reproduced. Due to the small symmetry breaking parameters, the

mixing angles in eq. (25) also receive corrections, whose magnitude turns out [7]

to be controlled by ∆m2sol/∆m2

atm: θν13 . 1 − tan2 θν

12 ∼ ∆m2sol/(2∆m2

atm) ∼ 0.01.

These corrections are too small to account for the measured value of the solar angle.

Thus, an important contribution from Ue is necessary to reconcile bimixing with

observation.

In this section we will reconsider the question of whether the observed pattern

can result from the corrections induced by the charged lepton sector. Though not

automatic, this appears to be at present a rather natural possibility [8, 9] - see

also the recent detailed analysis of Refs. [10] and [11]. Our aim is to investigate

the impact of planned experimental improvements, in particular those on |Ue3|, on

bimixing models. To this purpose it is useful to adopt a convenient parametrization

7

of mixing angles and phases. Let us define

U =

1 0 0

0 c23 s23

0 −s23 c23

c13 0 s13eiδ

0 1 0

−s13e−iδ 0 c13

c12 s12 0

−s12 c12 0

0 0 1

, (26)

where all the mixing angles belong to the first quadrant and δ to [0, 2π]. The

standard parameterization for U reads: U = U× a diagonal U(3) matrix accounting

for the two Majorana phases of neutrinos (the overall phase is not physical). Since

in the following discussion we are not interested in the Majorana phases, we will

focus our attention on U .

It would be appealing to take the parameterization (26) separately for Ue and Uν ,

by writing s12, se12, sν

12 etc to distinguish the mixings of the U , Ue and Uν matrices,

respectively. However, as discussed in the Appendix B, even disregarding Majorana

phases, U is not just determined in terms of Ue and Uν , with the latter defined to be

of the form (26). The reason is that, by means of field redefinitions Ue and Uν can

be separately but not simultaneously written respectively as Ue and Uν× a diagonal

U(3) matrix. Without loss of generality we can adopt the following form for U :

U = U †e Uν = U †

ediag(−e−i(α1+α2),−e−iα2 , 1)Uν︸ ︷︷ ︸

=U

×phases (27)

where Ue, Uν have the form (26), the phases α1, α2 run from 0 to 2π and we

have introduced two minus signs in the diagonal matrix for later convenience. This

expression for U is not due to the Majorana nature of neutrinos and a similar result

would also hold for quarks. More technical details on the parametrization (27) can

be found in the Appendix B.

Assume now that Uν corresponds to bimixing: sν13 = 0, sν

12 = cν12 = 1/

√2 and

sν23 = cν

23 = 1/√

2. Clearly, our discussion holds true irrespectively of the light

neutrino spectrum. It is anyway instructive to explicitate the mass matrices, e.g. in

the case of inverted hierarchy

mν =

0 1 1

1 0 0

1 0 0

∆m2

atm√2

, me = Ve

mee−i(α1+α2) −se

12mee−iα2 −se

13meeiδe

se12mµe

−i(α1+α2) mµe−iα2 −se

23mµ

se13mτe

−i(α1+α2+δe) se23mτe

−iα2 mτ

(28)

where we have set ∆m2sol = 0 since, as already mentioned, the corrections induced

by setting it to the measured value are negligible in the present discussion.

We then expand U of eq. (27) at first order in the small mixings of Ue, se12, se

13

and se23

1:

U11 = −e−i(α1+α2)

√2

− se12e

−iα2 + se13e

iδe

21To this approximation any ordering of the three small rotations in Ue gives exactly the same

results, and our conclusions are independent on the adopted parametrization.

8

U12 = −e−i(α1+α2)

√2

+se12e

−iα2 + se13e

iδe

2

U13 =se12e

−iα2 − se13e

iδe

√2

U23 = −e−iα21 + se

23eiα2

√2

U33 =1 − se

23e−iα2

√2

. (29)

The smallness of the observed s13 implies that both se12 and se

13 must be at most of

order s13. As a consequence, the amount of the deviation of s12 from 1/√

2 is limited

from the fact that it is generically of the same order as s13. Note that, instead, the

deviation of the atmospheric angle s23 from 1/√

2 is of second order in se12 and se

13,

so that it is natural to expect a smaller deviation as observed. From eqs. (29) we

obtain the following explicit expressions for the observable quantities 2:

tan2 θ23 = 1 + 4se23 cos(α2) (30)

δsol ≡ 1 − tan2 θ12 = 2√

2(se12 cos(α1) + se

13 cos(δe + α2 + α1)) (31)

|Ue3| =1√2(se

122 + se

132 − 2 cos(δe + α2)s

ese13)

1/2 (32)

tan δ =se12 sin(α1) − se

13 sin(δe + α2 + α1)

se12 cos(α1) − se

13 cos(δe + α2 + α1), (33)

to be compared with the experimental data. According to [12] the 3-σ windows are

|Ue3| ≤ 0.23 and 0.36 ≤ δsol ≤ 0.70.

Notice that the sign of δsol is not necessarily positive, so that only a part (say

half) of the parameter space in principle allowed for the phases is selected. With the

correction to δsol going in the good direction, one roughly expects |Ue3| ∼ δsol/4 ≈0.1 − 0.2. Hence, at present it is not excluded that charged lepton mixing can

transform a bimixing configuration into a realistic one but there are constraints

and, in order to minimize the impact of those constraints, |Ue3| must be within a

factor of 2 from its present upper limit. On the other hand, an upper limit on |Ue3|smaller than δsol/4 would start requiring a fine-tuning. Indeed, in order to reduce

|Ue3| significantly below 0.1−0.2 a cancellation must be at work in eq. (32), namely

δe+α2 should be close to 0 or 2π and se12 and se

13 should be of comparable magnitude.

In addition, to end up with the largest possible δsol/4, eq. (31) would also suggest

a small value for α1.

The above considerations can be made quantitative by showing, for different

upper bounds on |Ue3|, the points of the plane [se12, s

e13] which are compatible with

the present 3 σ window for the solar angle. This is shown in fig. 1, where the

2Eqs. (30,31,32) have been independently derived also in Ref. [11].

9

three plots correspond to different choices for α1. A point in the plane [se12, s

e13] is

excluded if there is no value of α2+δe for which (32) and (31) agree with experiment.

Regions in white are those excluded by the present bound on |Ue3|. With increasingly

stronger bounds on |Ue3|, the allowed regions, indicated in the plots with increasingly

darkness, get considerably shrinked. For |Ue3| ≤ 0.05 only |α1| < π/2 is allowed.

Notice also that at present the two most natural possibilities se12 ≫ se

13 and se12 ≪

se13 are allowed but, with |Ue3| < 0.1, they are significantly constrained and with

|Ue3| ≤ 0.05 ruled out completely. Below the latter value for |Ue3|, a high level of

degeneracy between se12 and se

13 together with a small value for α1 and δe + α2 are

required.

0 0.1 0.2 0.3 0.40

0.1

0.2

0.3

0.4

0 0.1 0.2 0.3 0.40

0.1

0.2

0.3

0.4

0 0.1 0.2 0.3 0.40

0.1

0.2

0.3

0.4

α1 = 0 α1 = π/2 (or 3π/2) α1 = π

se12 se

12 se12

se13 se

13 se13

.23

.23

.23

.1

.1

.05

.01

Figure 1: Taking an upper bound on |Ue3| respectively equal to 0.23, 0.1, 0.05, 0.01,

we show (from yellow to red) the allowed regions of the plane [se12, s

e13]. We do not

display the region outside se12, s

e13 ≤ 0.4, where our approximation, linear in se

12 and

se13, breaks down. Each plot is obtained by setting α1 to a particular value, while

leaving α2 + δe free. We keep the present 3 σ window for δsol [12].

Summarising, planned improvements in the sensitivity to |Ue3| - which could

reach the 0.05 level [13] -, could have a crucial impact on bimixing models. They

could either disfavour it as unnatural (in the sense that a dynamical principle or a

symmetry acting also on the charged lepton mass matrix would have to be invoked)

or, if |Ue3| were to be found, support bimixing models.

4 Conclusion

We have considered the possibility that the observed pattern of neutrino mixings is

dominantly determined by the charged lepton sector, that is by Ue, with a nearly

diagonal neutrino matrix Uν . Of course, one can always choose an ad hoc basis

where this is true: the point is to decide whether this formal choice can be naturally

10

justified in the physical basis where the symmetries of the lagrangian are naturally

specified. We find that in presence of two large mixing angles θ12 and θ23 with the

third angle θ13 being small, the construction of a natural model with dominance of

Ue is made much more difficult than in the case of only the atmospheric angle θ23

large. We have discussed the conditions that must be fulfilled and presented a model

where indeed all observed mixings arise from Ue. The stated difficulty is reflected

in the relatively complicated symmetry structure required and in the fact that we

were led to implement in the charged lepton sector a see-saw mechanism as a trick

to naturally obtain an approximately vanishing subdeterminant. While the see-saw

mechanism is a common and natural occurrence in the neutrino sector, its presence

in the charged lepton sector is much more special.

We have also studied the case where the matrix Uν is of the bimixing type, which

is naturally obtained, for example, in inverse hierarchy models with a Le −Lµ −Lτ

U(1) symmetry, and the observed pattern of mixings is obtained by adding correc-

tions from Ue. We have shown that the smallness of θ13 imposes strong constraints

on the maximum deviation of the solar angle from maximality, while the deviations

of the atmospheric angle from the maximal value can be naturally smaller. Planned

experimental searches for |Ue3| could thus have a strong impact in supporting or

disfavouring bimixing.

Acknowledgements

We gratefully acknowledge Andrea Romanino for a stimulating conversation that

attracted our interest on the subject and anticipated some of the conclusions that are

substanciated here. I.M. thanks the Dept. of Physics of Rome1 for kind hospitality.

F.F. and I.M thanks the CERN Theory Division where they were Visitors for periods

in summer and respectively in fall 2003 and beginning 2004. This project is partially

supported by the European Programs HPRN-CT-2000-00148 and HPRN-CT-2000-

00149.

Appendix A

Here we sketch a supersymmetric SU(5) grand unified model with a flavour symme-

try F=U(1)F0×U(1)F1

×U(1)F2×U(1)F3

. The field content and the transformation

properties under F are collected in tables 1 and 2.

11

101 102 103 5l1 5l

2 5l3 5H 5H 51 52 51 52

F0 4 2 0 0 0 0 0 0 0 0 0 0

F1 2 2 2 1 0 0 0 0 2 0 -2 0

F2 2 2 2 0 1 0 0 0 2 0 -2 0

F3 2 2 2 0 0 1 0 0 0 2 0 -2

Table 1. Matter fields and flavour charges. Fields are denoted by their transfor-

mation properties under SU(5). Higgs doublets are in (5H , 5H).

θ0 θ1 θ2 θ3 θ4 θ5 θ6 θ7

F0 -1 0 0 0 0 0 0 0

F1 0 -2 0 0 -3 0 0 -4

F0 0 0 -2 0 0 -3 0 -4

F0 0 0 0 -2 0 0 -3 -4

Table 2. Flavon fields and their flavour charges.

The superpotential is given by:

W = yij10i10j5H +ci

Λ(5l

i5H)(5li5H)+ηij10i5

lj 5H +λia10i5a5H +µaj5a5

lj +Ma5a5a ,

(34)

where Λ denotes a large mass scale, close to the cut-off of the theory, and all the

coupling constants are proportional to powers of 〈θi〉/Λ that can be read off from

tables 1 and 2. Here for simplicity we assume that all flavon fields acquire a similar

VEV, 〈θi〉/Λ ≈ λ. In this case we have:

ci = ciλ , ηij = ηijλ(9−2i) , λia = λiaλ

(6−2i+a) , µaj = µajλ(3−a) , (35)

where hatted quantities (and Ma) do not depend on λ. We also have µ13 = µ21 =

µ22 = 0.

We get for the up-type quarks a mass matrix of the form:

mu =

λ8 λ6 λ4

λ6 λ4 λ2

λ4 λ2 1

λ〈5H〉 . (36)

After eliminating the heavy degrees of freedom, in the regime 1 > |µaj/Ma| >

|ηia/λij|, the mass matrix for charged leptons is dominated by the see-saw contribu-

tion and is given by:

me = −

λ11µ11

M1

λ4 λ11µ12

M1

λ4 λ12µ23

M2

λ4

λ21µ11

M1λ2 λ21µ12

M1λ2 λ22µ23

M2λ2

λ31µ11

M1

λ31µ12

M1

λ32µ23

M2

λ3〈5H〉 . (37)

12

We see that the first and second column are dominated by the M1 contribution, thus

producing the desired vanishing determinant. Actually, in the considered regime,

the first two column are exactly proportional and the electron mass vanishes. This

can be corrected by sub-dominant terms, arising for instance from the neglected

contributions. The neutrino mass matrix is diagonal, with masses given by

mi = ci〈5H〉2λ

Λ. (38)

If we assume dimensionless coefficients η and λ of order one, the condition 1 >

|µaj/Ma| > |ηia/λij| requires to choose the dimensionful parameters µai and Ma in

the window 1 < µai/Ma < 1/λ2.

Appendix B

We show here that (27) is a general way of writing U . Similar parametrizations have

also been used in [9, 14]. Starting from the basis of the (unknown) flavour symmetry

L = νT mνν + ecT mee + e†νW (39)

then

mν = U∗ν mdiag

ν U †ν me = Vem

diage U †

e (40)

where diag stands for a diagonal matrix with real non-negative elements and Uν ,

Ue, Ve are unitary matrices. The MNS mixing matrix U is then given by U = U †eUν .

We are going to exploit the fact that any unitary matrix can in general be written

unambiguously as:

U = eiφ0 diag(ei(φ1+φ2), eiφ2 , 1)︸ ︷︷ ︸

≡Φ

U diag(ei(φ3+φ4), eiφ4 , 1)︸ ︷︷ ︸

≡Φ′

, (41)

where φi (i=0,...,4) run from 0 to 2π and U is the standard parameterization for the

CKM mixing matrix, namely

U =

1 0 0

0 c23 s23

0 −s23 c23

c13 0 s13eiδ

0 1 0

−s13e−iδ 0 c13

c12 s12 0

−s12 c12 0

0 0 1

, (42)

where all the mixing angles belong to the first quadrant and δ to [0, 2π]. If we adopt

such a parameterization for both Ue and Uν we obtain, in the basis where mass

matrices are diagonal:

U = ei(φν

0−φe

0) Φ′∗

e U †e Φ∗

eΦν U †ν Φ′

ν . (43)

13

Now, by redefining properly left and right-handed charged lepton fields, we can get

rid of all the phases at the left of U †e . The phases in between U †

e and Uν cannot be

eliminated, but we can always define their product as:

Φ∗eΦν ≡ diag(−e−i(α1+α2),−e−iα2 , 1) . (44)

Also the phases on the right of Uν cannot be eliminated and contribute to the

Majorana phases. As a result of all the above redefinitions, eq. (43) becomes:

U = U †e diag(−e−i(α1+α2),−e−iα2 , 1) U †

ν Φ′ν , (45)

which is precisely the parameterization used in the text (see eq. (27)).

14

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16

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