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Physics Letters B 595 (2004) 425-431

www. elsevier. com/locate/physletb

Almost degenerate neutrinos and bi-large lepton mixing

Sin Kyu Kang

School of Physics, Seoul National University, Seoul 151-734, South Korea Received 5 April 2004; received in revised form 20 May 2004; accepted 25 May 2004 Available online 22 June 2004 Editor: T. Yanagida

Abstract

A scenario is proposed for bi-large lepton mixing in the framework of nearly threefold degenerate Majorana neutrinos. In our proposal, we impose Z3 symmetry in the neutrino sector at a high energy scale to account for the threefold degenerate neutrinos and the maximal mixing between v^ and vT. In order to obtain the atmospheric neutrino mass splitting while keeping the maximal mixing between v^ and vT, we introduce a small perturbation to the neutrino mass matrix without breaking Z3 symmetry. On the other hand, the solar neutrino mixing arises due to the non-diagonal charged lepton mass matrix, and the desirable large mixing and mass splitting for the solar neutrino oscillation can be obtained by radiative corrections. © 2004 Elsevier B.V. All rights reserved.

PACS: 12.15.Ff; 14.60.Pq; 11.30.Hv

Keywords: Degenerate neutrinos; Bi-large lepton mixing; Radiative splitting

Thanks to the accumulating data from the atmospheric and solar neutrinos experiments [1-3], we are now convinced that neutrinos oscillate among three active neutrinos. Interpreting each experiment in terms of two-flavor mixing the oscillation of atmospheric neutrinos requires maximal mixing or nearly so between vM and vT, whereas for the oscillation of solar neutrinos the mixing angle between v^ and non-electron neutrino is not maximal but large. The current refined analysis of the solar and atmospheric neutrino experimental results gives [4],

sin220atm > 0.85, 2.2 X 10-3 < Am2to < 3.0 x 10-3 eV2 (1a), (1)

0.27 < tan2 6W < 0.72, 5 x 10-5 < Am2ol < 2 x 10-4 eV2 (3a). (2)

Combining those results with non-observation of the disappearance of Ve for sin2 26 < 0.2 and Am2 < 10-3 eV2 in the CHOOZ experiment [5], the neutrino mixing matrix U defined via va = J23=1 UajVj (a = e, ¡1, t) is simply

E-mail address: skkang@phya.snu.ac.kr (S.K. Kang).

0370-2693/$ - see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2004.05.060

given to a good approximation by

/cos0 - sin0 0 \

sinff v/2 sinfl

where 0 is the solar neutrino mixing angle and we have taken the atmospheric neutrino mixing is maximal and Ue3 = 0. The result indicates that the mixing pattern of neutrinos is quite different from the quark mixing matrix Vckm which involves only small mixing angles. Thus, it is worthwhile to find the origin of the mixing pattern of neutrinos as well as the tiny mass splittings for the atmospheric and solar neutrino oscillations.

In this Letter, we will propose a scenario to achieve the so-called bi-large mixing pattern [6], i.e., giving maximal mixing for atmospheric neutrinos and large mixing angle for solar neutrinos deviated from bi-maximal [7], in the framework of nearly threefold degenerate Majorana neutrinos by exploring mixing matrices in both neutrino and charged lepton sectors. In this case, each neutrino mass is bounded by the recent observation of the WMAP combined with other recent high precision cosmological data, mVi < 0.23 eV [8]. The exact threefold degenerate neutrino scenario can be simply realized by considering the effective Majorana neutrino mass matrix with some symmetric structure at a high energy scale. Several people have considered the threefold degenerate Majorana neutrino mass patterns based on some dynamical group structures [9-11]. In our proposal, the threefold degeneracy as well as the maximal mixing between vM and vt can be achieved by taking a simple pattern of neutrino mass matrix which is Z3 invariant. However, the degeneracy should be broken appropriately for the realistic case accounting for the neutrino oscillations. As will be shown later, the neutrino mass splitting for the atmospheric neutrinos can be obtained by adding a perturbation to the neutrino mass matrix without breaking the Z3 symmetry, whereas the mass splitting for the solar neutrinos can be achieved through radiative corrections to the neutrino mass matrix which are generated from non-diagonal charged lepton mass matrix. So, the reason why the Z3 symmetry is required only for the neutrino sector is to generate the mixing and mass splitting for the solar neutrino oscillation through the charged lepton sector which is not Z3 invariant. In fact, taking non-diagonal charged lepton mass matrix in the Z3 invariant neutrino basis is equivalent to break the symmetry in the neutrino sector explicitly at the tree level in the charged lepton flavor basis [12], and thus renormalization group effects through the charged lepton Yukawa interactions on neutrino masses at low energy scale can be generated.

For our purpose, let us assume that the neutrino mixing matrix U is the result of two successive rotations given by [13]

/cose -ml

U = U¡

sin# sintf

V2 v/2

cosfl+l cos 0 — 1

cos 0 — 1 cosfl+l

We also assume that U1 and U2 are the mixing matrices which diagonalize the charged lepton mass matrix and the light Majorana neutrino mass matrix at a high energy scale, respectively. Then, our scheme suggests that the mixing between vM and vt for the atmospheric neutrino oscillation originates in neutrino mass matrix itself and the mixing for the solar neutrino oscillation originates from transformation of the charged lepton mass matrix. Such two successive rotations in the lepton sector has been considered in grand unified theories (GUT) [14] mainly because the charged lepton sector is related with quark sector in GUT, but achieving our aim in GUT is beyond the scope of our Letter. A natural question is at this stage what forms of the charged lepton and neutrino mass matrices can be diagonalized by the mixing matrices U1 and U2, respectively, leading to the desirable neutrino mass splittings and the hierarchy of the charged leptons.

First of all, let us consider what form of the neutrino mass matrix can be diagonalized by the mixing matrix U2 = UV which indicates the maximal mixing between vM and vt and lead to threefold degenerate neutrinos. One interesting ansatz to achieve the maximal mixing between vM and vt as well as threefold degenerate neutrinos is to

take the neutrino mass matrix to be /1 0 0 \

M°v = m o 0 0 1 . (5)

\0 1 0/

The mass matrix M° is invariant under the Z3 transformation [15]:

/-I -Jl -Jl\

V /£. -1 1 /

W 8 4 4 /

Z3 = 1, ZVM0ZT = M0. (6)

However, the above form of the neutrino mass matrix reflects the exact degenerate three neutrinos and thus is not realistic. To account for the neutrino oscillation, particularly the atmospheric neutrino oscillation, we introduce a perturbation into the mass matrix M0 so as to lead to the atmospheric mass splitting. The simple way to achieve our purpose while keeping the symmetry in Mj° is to add universal non-vanishing diagonal terms to Mj° which are responsible for the atmospheric mass splitting [15]. Then, the neutrino mass matrix we suggest at high energy scale is given by

' 10 0 \ / 10 0 \ / m0 - £ 0 0 \ M0 = m0 0 0 1 ) - £ ( 0 10) = ( 0 -£ m0 I , (7)

0/ \0 0 1/ \ 0 m0 -£,

where we have taken negative sign of the parameter e in order to keep the hierarchy m^ > m^ > m\. Here, we note that the mass matrix (7) is given in the flavor basis where the charged lepton mass matrix is not diagonal and it is still invariant under the Z3 symmetry. By diagonalizing the above form of the neutrino mass matrix M° with the help of Uv, then, we obtain the three neutrino mass eigenvalues [m^m2,m3] = [m0 - e,m0 - e, -(m0 + e)] and the atmospheric neutrino mass scale is given by Am2tm = m2 - m2 ~ 4m0e.

Let us briefly discuss on the form of the charged lepton mass matrix before considering the mass splitting and large mixing for the solar neutrino oscillation. In our scheme, Ul (= U1) is a consequence from our basic assumption for the forms of the mixing matrices U and Uv, so the Hermitian matrix MlMj is determined to be diagonalized by Ui: MlMj = Ui ■ Diag[m2e,m2x,m2t] ■ uj. But, the charged lepton mass matrix itself is generally diagonalized by bi-unitary transformation, MD = uJm1Ur , which makes the determination of the form of Ml ambiguous. If we further assume that UR = 1, then we obtain a simple form of the charged lepton mass matrix which is a consequence of the mixing matrix U1 given in Eq. (4) and UR = 1:

/ me 0 0 \ | /(c - 1)me

Ml = Ui ■ Diag[me,mM,mr]=( 0 m/À 0 + -Xme

0 0 mx) 1 V -Xme

c-r-mi

where c = cos 6 and X = sin 6 / sfl. In this case, we note that the breaking of Z3 symmetry imposed in the neutrino sector can be simply parameterized in terms of the power of X (cos 0 ~ 1 - X2) and the charged lepton masses.

Based on the above lepton mass matrices, let us consider how the large mixing angle and mass splitting for solar neutrino oscillation can be obtained. As mentioned earlier, the mixing for solar neutrino originates from the mixing matrix of the charged lepton mass matrix given by U1. Due to the non-diagonal charged lepton mass matrix which is not invariant under the Z3 symmetry, non-trivial radiative corrections to the neutrino mass matrix can occur at a low energy scale. As is well known, the renormalized neutrino mass matrix at a low energy scale is generally given by [16,17]

Mv = Mv° + i(/.Mv° + Mv0./), (9)

where the parameter for the radiative corrections I in the neutrino basis is related with the form in the charged lepton flavor basis as follows,

lij =E ULUlßjIaß

(i, j = 1, 2, 3) (a, ß = e, ß, t).

In the charged lepton flavor basis, it is known that the charged lepton Yukawa interactions lead to the radiative corrections simply given by Iaß ~ (Yaß/16n2) log(MX/M) with the high energy scale MX and the weak scale M. Since flavor independent corrections do not lead to the mass splitting, we focus on the flavor dependent corrections. One of possible flavor dependent corrections is attributed to the charged lepton Yukawa interactions. Assuming that the tau Yukawa coupling is the most dominant contributions, the typical size of the correction is I ~ 1tt ~ 10-5 in the standard model. In the minimal supersymmetric standard model (MSSM), 1tt a m2 / cos2 ß, and it is increased with tanß. Let us examine whether those radiative corrections mediated by the charged Yukawa interactions can lead to the desirable mass splitting for solar neutrino oscillations while keeping the bi-large mixing pattern. Assuming that 1tt = eT is dominant contributions, with the help of Eq. (10), the neutrino mass matrix Eq. (7) becomes

/mo - e + s—m0eT m0eT ~^m0eT \

JÏ m0a + %)

+ 0(££t).

Rotating the neutrino mass matrix Eq. (11) by the mixing matrix U2 given in Eq. (4), we obtain the following form of the mass matrix in the leading order (i.e., ignoring O(ssT)),

(ff20(l + s-t£t) -<

-jmoSr 0

~2m Ofir

ff20(l + - e

-»20 (1 + - ej

The above matrix Eq. (12) is diagonalized by the mixing matrix

s c ,0 0

—s ' 0^ 0

where c' = cos 0' and s' = sin 0'. Then, the three neutrino mass eigenvalues are approximately given by

mVx = m0 mV2 = m0 mV3 = -m0

c/2( l + ^£r 1 +2c's'seT +i'z( 1 + —er

s'2{ 1 + '—St ) - 2c's'seT + c'z( 1 + —er

„' 2

1 + 2'r

— 8,

— 8,

— 2e.

We also find that the mixing angle 0' can be determined in terms of the mixing angle 0, tan 0' sin 0

tan2 0'- 1 cos2 0' Finally, the neutrino mixing matrix at a low energy scale is given by

( 1 — sin 0

sine j_

v/2 v/2

sin^ V v/2

c' s' 0^ —s' c' 0 0 0 1/

( c' + ss'

^c'-s') j,(SS> + c>) -Jj

s — sc

is near the cosmological bound and £T ~ 1.5+0 5 x 10 3, we can get Am2ol = m22 - m2vx ~ 7.5+2255 x 10 5 eV2 which is consistent with the recent measurement from the solar neutrino experiments. In addition, Am2tm = 2.5 x

Then the solar neutrino mixing angle is given by

2 (s' - sc')2

This relation implies that the value of tan2 0soi increases with that of sin 0.

Our numerical analysis shows that 0.3 < tan2 0sol < 0.7 for 0.25 < sin0 < 0.36, and when m0 — 0.22 eV which

^t — 1.5+2.2 x 10-3, we can get Am2ol = m2vi

s. In addition, Amatm

10-3 eV2 can be achieved by taking e — 3 x 10-3 eV. However, we notice that the typical value eT — 1.5 x 10-3 cannot be achieved in the SM, while it can be in the MSSM for tan j — 10. We also see that the maximal mixing for atmospheric neutrino oscillation holds and there is no sever correction to Ami;tm. In addition, the radiative corrections to the neutrino mass matrix may induce a non-vanishing contribution to Ue3, but its size is at most of the order of O(eeT) which is ignored in Eqs. (12)-(14) in this framework. Thus, we conclude that the bi-large mixing pattern can be obtained in our nearly degenerate neutrino framework.

For sin0 = 0.32, the magnitude of the mixing matrix Ul (= U1) in Eq. (4) is given by

/ 0.9497 0.226 0.226 \ Ui — I -0.226 0.9749 -0.025 I . (18)

-0.226 -0.025 0.9749

We see that the mixing matrix is similar to the CMK matrix for the quark sector except (1, 3) and (3, 1) components. In particular, the result giving tan2 0soi = 0.5 which can be achieved by taking sin0 ~ 0.32 in our scheme leads to the so-called "tri/bi-maximal" mixing pattern of three neutrinos with \Ue2\ = sin0 = l/\/3 which is close to the best fit value of the solar mixing angle sin2 0sol — 0.3 [18]:

/ JL ( S 1 v/3 0

s/6 v/3 v/2

V s v/3 v/2 /

In passing, the authors in Ref. [18] have also proposed some textures of the neutrino mass matrices at low energy scale that account for the "tri/bi-maximal" mixing pattern and the two mass splittings of three neutrinos.

In summary, a scenario has been proposed for bi-large mixing pattern of nearly threefold degenerate neutrinos. In this scenario, Z3 symmetry in the neutrino sector has been imposed at a high energy scale to account for the threefold degenerate neutrino and the maximal mixing between vM and vT. In order to obtain the atmospheric neutrino mass splitting while keeping the maximal mixing between vM and vT, we have introduced a small perturbation to the neutrino mass matrix without breaking Z3 symmetry. On the other hand, the solar neutrino mixing arises due to the non-diagonal charged lepton mass matrix, and the mass splitting and large mixing for the solar neutrino oscillation can be obtained by radiative corrections generated from the charged lepton Yukawa interactions at a low energy scale particularly in the MSSM, while keeping the maximal mixing for the atmospheric neutrino oscillation.

Acknowledgement

This work was supported in part by BK21 program of the Ministry of Education in Korea and in part by KOSEF Grant No. R01-2003-000-10229-0.

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