An extension of tribimaximal lepton mixing

# An extension of tribimaximal lepton mixing

## Abstract

Harrison, Perkins and Scott have proposed simple charged lepton and neutrino mass matrices that lead to the tribimaximal mixing . We consider in this work an extension of the mass matrices so that the leptonic mixing matrix becomes , where is a unitary matrix needed to diagonalize the charged lepton mass matrix and measures the deviation of the neutrino mixing matrix from the bimaximal form. Hence, corrections to arise from both charged lepton and neutrino sectors. Following our previous work to assume a Qin-Ma-like parametrization for the charged lepton mixing matrix in which the CP-odd phase is approximately maximal, we study the phenomenological implications in two different scenarios: and . We find that the latter is more preferable, though both scenarios are consistent with the data within ranges. The predicted reactor neutrino mixing angle in both scenarios is consistent with the recent T2K and MINOS data. The leptonic CP violation characterized by the Jarlskog invariant is generally of order .

## I Introduction

The large values of the solar () and atmospheric () mixing angles may be telling us about some new symmetries of leptons not presented in the quark sector and may provide a clue to the nature of the quark-lepton physics beyond the standard model. If there exists such a flavor symmetry in Nature, the tribimaximal (TBM) (1) pattern for the neutrino mixing will be a good zeroth order approximation to reality :

 sin2θ12=13 ,sin2θ23=12 ,sinθ13=0 . (1)

For example, in a well-motivated extension of the standard model through the inclusion of discrete symmetry, the TBM pattern comes out in a natural way in the work of (2). Although such a flavor symmetry is realized in Nature leading to exact TBM, in general there may be some deviations from TBM. Recent data of the T2K (3) and MINOS (4) Collaborations and the analysis based on global fits (5); (6) of neutrino oscillations enter into a new phase of precise measurements of the neutrino mixing angles and mass-squared differences, indicating that the TBM mixing for three flavors of leptons should be modified. In the weak eigenstate basis, the Yukawa interactions in both neutrino and charged lepton sectors and the charged gauge interaction can be written as

 −L = 12¯¯¯¯¯¯νL Mν (νL)c+¯¯¯¯¯ℓLmℓℓR+g√2W−μ ¯¯¯¯¯ℓLγμνL+H.c. . (2)

When diagonalizing the neutrino and charged lepton mass matrices , one can rotate the neutrino and charged lepton fields from the weak eigenstates to the mass eigenstates . Then we obtain the leptonic unitary mixing matrix from the charged current term in Eq. (2). In the standard parametrization of the leptonic mixing matrix , it is expressed in terms of three mixing angles and three CP-odd phases (one for the Dirac neutrino and two for the Majorana neutrino) (7)

 UPMNS=⎛⎜⎝c13c12c13s12s13e−iδCP−c23s12−s23c12s13eiδCPc23c12−s23s12s13eiδCPs23c13s23s12−c23c12s13eiδCP−s23c12−c23s12s13eiδCPc23c13⎞⎟⎠Pν , (3)

where and , and is a diagonal phase matrix which contains two CP-violating Majorana phases, one (or a combination) of which can be in principle explored through the neutrinoless double beta () decay (8). For the global fits of the available data from neutrino oscillation experiments, we quote two recent analyses: one by Gonzalez-Garcia et al.  (5)

 sin2θ12 = 0.319+0.016 (+0.053)−0.016 (−0.046) ,sinθ13=0.097+0.052−0.050 (≤0.217) , sin2θ23 = 0.462+0.082 (+0.185)−0.050 (−0.124) , (4)

in () ranges, or equivalently

 θ12=34.4∘+1.0∘ (+3.2∘) −1.0∘ (−2.9∘) ,     θ23=42.8∘+4.7∘ (+10.7∘) −2.9∘ ( −7.3∘) ,     θ13=5.6∘+3.0∘ (+6.9∘) −2.9∘ (−5.6∘) , (5)

and the other given by Fogli et al. with new reactor neutrino fluxes  (6):

 sin2θ12 = 0.312+0.017 (+0.052)−0.006 (−0.047) ,sin2θ13=0.025+0.007 (+0.025)−0.007 (−0.020) , sin2θ23 = 0.42+0.08 (+0.22)−0.03 (−0.08) , (6)

corresponding to

 θ12=34.0∘+1.0∘ (+3.2∘) −1.0∘ (−3.0∘) ,     θ23=40.4∘+4.6∘ (+12.7∘) −1.3∘ ( −4.7∘) ,     θ13=9.1∘+1.2∘ (+3.8∘) −1.4∘ (−5.0∘) . (7)

The analysis by Fogli et al. includes the T2K (3) and MINOS (4) results. The T2K Collaboration (3) has announced that the value of is non-zero at C.L. with the ranges

 0.03 (0.04)≤sin22θ13≤0.28 (0.34) , (8)

or

 4.99∘ (5.77∘)≤θ13≤15.97∘ (17.83∘) (9)

for , and the normal (inverted) neutrino mass hierarchy. The MINOS Collaboration found

 sin22θ13≤0.12 (0.20) , (10)

with a best fit of

 sin22θ13=0.041+0.047−0.031 (0.079+0.071−0.053) , (11)

for , and the normal (inverted) neutrino mass hierarchy. The experimental result of non-zero implies that the TBM pattern should be modified. However, properties related to the leptonic CP violation remain completely unknown yet.

The trimaximal neutrino mixing was first proposed by Cabibbo (9)4 (see also (10))

 VC=1√3⎛⎜⎝1ω2ω1111ωω2⎞⎟⎠ , (12)

with being a complex cube-root of unity. This mixing matrix has maximal CP violation with the Jarlskog invariant . However, this trimaximal mixing pattern has been ruled out by current experimental data on neutrino oscillations. In their original work, Harrison, Perkins and Scott (HPS) (1) proposed to consider the simple mass matrices

 M2ℓ=⎛⎜⎝abb∗b∗abbb∗a⎞⎟⎠ ,M2ν=⎛⎜⎝x0y0z0y0x⎞⎟⎠ , (13)

that can lead to the tribimaximal mixing, where and are real parameters,5 and . The mass matrices are diagonalized by the trimaximal matrix for charged lepton fields and the bimaximal matrix defined below for neutrino fields, that is, and . The combination of trimaximal and bimaximal matrices leads to the so-called TBM mixing matrix:

 UTBM=V†C UBM=⎛⎜ ⎜ ⎜ ⎜ ⎜⎝√231√30−1√61√3−i√2−1√61√3i√2⎞⎟ ⎟ ⎟ ⎟ ⎟⎠ with UBM=⎛⎜ ⎜ ⎜⎝1√20−1√20101√201√2⎞⎟ ⎟ ⎟⎠ . (14)

It is clear by now that the tribimaximal mixing is not consistent with the recent experimental data on the reactor mixing angle because of the vanishing matrix element in .

In this work we consider an extension of the tribimaximal mixing by considering small perturbations to the mass matrices and which we will call and , respectively (see Eq. (15) below) so that is no longer in the bimaximal form and deviates from the trimaximal structure, where is the unitary matrix needed to diagonalize the matrix . As a consequence, small perturbations. Hence, the corrections to the TBM pattern arise from both charged lepton and neutrino sectors. Inspired by the T2K and MINOS measurements of a sizable reactor angle , there exist in the literature intensive studies of possible deviations from the exact TBM pattern. However, most of these investigations were focused on the modification of TBM arising from either the neutrino sector (12) or the charged lepton part (13); (14), but not both simultaneously.

The paper is organized as follows. In Sec. II, we set up the model by making a general extension to the charged lepton and neutrino mass matrices. Then in Sec. III we study the phenomenological implications by considering two different scenarios for the charged lepton mixing matrix. Our conclusions are summarized in Sec. IV.

## Ii A simple and realistic extension

In order to discuss the deviation from the TBM mixing, let us consider a simple and general extension of the original proposal by HPS given in Eq. (13), by taking into account perturbative effects on the mass matrices and . The generalized mass matrices and can be introduced as 6

 M′ 2f=⎛⎜ ⎜⎝a+g3b+χ3b∗+χ∗2b∗+χ∗3a+g2b+χ1b+χ2b∗+χ∗1a+g1⎞⎟ ⎟⎠ ,M′ 2ν=m20⎛⎜⎝x′0y′010y′0x′+ρ⎞⎟⎠ , (15)

where and are defined as the hermitian square of the mass matrices and , respectively, with the subscript denoting charged fermion fields (charged leptons or quarks). Due to the hermiticity of and , the parameters are real, while and are complex. The parameters , and represent small perturbations. Note that the (11), (13), (22) elements (i.e., , and ) in are assumed to contain any perturbative effects on the elements , , and in , respectively. For simplicity, it is assumed that is real just as the other elements in and the vanishing off-diagonal elements in remain zeros in .

The parameters and are encoded in (1) as

 a=~m2f13+~m2f23+~m2f33 ,  b=~m2f13+~m2f2ω23+~m2f3ω3 , (16)

where the subscript indicates a generation of charged fermion field, and represents a bare mass of , for example, for charged lepton fields.

We first discuss the hermitian square of the neutrino mass matrix, , in Eq. (15). It can be diagonalized by

 Uν=⎛⎜⎝cosθ0−sinθ010sinθ0cosθ⎞⎟⎠Pν=⎛⎜ ⎜⎝1/√20−1/√20101/√201/√2⎞⎟ ⎟⎠W , (17)

with

 tan2θ=−2y′ρ (18)

and

 Unknown environment 'array% (19)

where the diagonal phase matrix contains two additional phases, which can be absorbed into the neutrino mass eigenstate fields. For a small perturbation , the mixing parameter can be expressed in terms of

 θ=π/4+ϵ   with  |ϵ|≪1 . (20)

is then reduced to

 W=⎛⎜⎝cosϵ0−sinϵ010sinϵ0cosϵ⎞⎟⎠Pν . (21)

The neutrino mass eigenvalues are obtained as

 m21 = m20(x′+ρsin2θ+y′sin2θ),m22=m20,m23=m20(x′+ρcos2θ−y′sin2θ) (22)

and their differences are given by

 Δm221 ≡ m22−m21=m20(1−x′+ρ 1−sin2ϵ2sin2ϵ) , Δm231 ≡ m23−m21=m20 2ρsin2ϵ , (23)

from which we have a relation . It is well known that the sign of is positive due to the requirement of the Mikheyev-Smirnov-Wolfenstein resonance for solar neutrinos. The sign of depends on that of : for the normal mass spectrum and for the inverted one. The quantities (or ) are determined by the four parameters , while the Majorana phases in Eq. (17) are hidden in the squared mass eigenvalues.

We next turn to the hermitian square of the mass matrix for charged fermions in Eq. (15). This modified charged fermion mass matrix is no longer diagonalized by

 V†CM′ 2fVC=⎛⎜ ⎜⎝m2a+η11η12η13η∗12m2b+η22η23η∗13η∗23m2c+η33⎞⎟ ⎟⎠ , (24)

where

 m2a=a+b+b∗ ,m2b=a+b ω+b∗ ω2 ,m2c=a+b ω2+b∗ ω , (25)

corresponding to , respectively, and is composed of the combinations of and . To diagonalize , we need an additional matrix which can be, in general, parametrized in terms of three mixing angles and six phases:

 VfL=⎛⎜⎝c2c3c2s3eiα3s2eiα2−c1s3e−iα3−s1s2c3ei(α1−α2)c1c3−s1s2s3ei(α1−α2+α3)s1c2eiα1s1s3e−i(α1+α3)−c1s2c3e−iα2−s1c3e−iα1−c1s2s3ei(α3−α2)c1c2⎞⎟⎠Pf , (26)

where , and a diagonal phase matrix which can be rotated away by the phase redefinition of left-charged fermion fields. The charged fermion mixing matrix now reads .

Finally, we arrive at the general expression for the leptonic mixing matrix

 UPMNS=U†LUν=Vℓ†LUTBMW . (27)

A simple and general extension of the mass matrices given in Eq. (15) thus leads to two possible sources of corrections to the tribimaximal mixing: measures the deviation of the charged lepton mixing matrix from the trimaximal form and characterizes the departure of the neutrino mixing from the bimaximal one. The charged lepton mass matrix in Eq. (15) or (24) has 12 free parameters. Three of them are replaced by the phases in Eq. (26) which can be eliminated by a redefinition of the physical charged lepton fields. The remaining 9 parameters can be expressed in terms of . ¿From Eqs. (24) and (26) the mixing angles and phases can be expressed as

 θ1 ≃ |η23|~m2τ ,θ2≃|η13|~m2τ ,θ3≃|η12|~m2μ ,α1=arg(η23), α2 ≃ 12arg(η23)+arg(η13),α3≃12[arg(η13)−arg(η23)]+arg(η12) , (28)

with the condition . In the charged fermion sector, there is a qualitative feature that distinguishes the neutrino sector from the charged fermion one. The mass spectrum of the charged leptons exhibits a similar hierarchical pattern to that of the down-type quarks, unlike that of the up-type quarks which show a much stronger hierarchical pattern. For example, in terms of the Cabbibo angle , the fermion masses scale as  , and . This may lead to two implications: (i) the Cabibbo-Kobayashi-Maskawa (CKM) matrix (16) is mainly governed by the down-type quark mixing matrix, and (ii) the charged lepton mixing matrix is similar to that of the down-type quark one. Therefore, we shall assume that (i) and , where is associated with the diagonalization of the down-type (up-type) quark mass matrix and is a unit matrix, and (ii) the charged lepton mixing matrix has the same structure as the CKM matrix, that is, or .

Recently, we have proposed a simple ansatz for the charged lepton mixing matrix , namely, it has the Qin-Ma-like parametrization in which the CP-odd phase is approximately maximal (13). Armed with this ansatz, we notice that the 6 parameters in are reduced to four independent ones . It has the advantage that the TBM predictions of and especially will not be spoiled and that a sizable reactor mixing angle and a large Dirac CP-odd phase are obtained in the mixing . The Qin-Ma (QM) parametrization of the quark CKM matrix is a Wolfenstein-like parametrization and can be expanded in terms of the small parameter  (17). However, unlike the original Wolfenstein parametrization (18), the QM one has the advantage that its CP-odd phase is manifested in the parametrization and is near maximal, i.e., . This is crucial for a viable neutrino phenomenology. It should be stressed that one can also use any parametrization for the CKM matrix as a starting point. As shown in (19), one can adjust the phase differences in the diagonal phase matrix in Eq. (26) in such a way that the prediction of will not be considerably affected.

For , the QM parametrization (17); (13) can be obtained from Eq. (26) by the replacements :

 Vf†L=P∗f⎛⎜ ⎜⎝1−λ2/2λeiδhλ3−λe−iδ1−λ2/2(f+he−iδ)λ2fλ3e−iδ−(f+heiδ)λ21⎞⎟ ⎟⎠+O(λ4) . (29)

On the other hand, for the QM parametrization is obtained by the replacements :

 VfL=⎛⎜ ⎜⎝1−λ2/2λeiδhλ3−λe−iδ1−λ2/2(f+he−iδ)λ2fλ3e−iδ−(f+heiδ)λ21⎞⎟ ⎟⎠Pf+O(λ4) , (30)

where the superscript denotes (down-type quarks) or (charged leptons). From the global fits to the quark mixing matrix given by (20) we obtain

 f=0.749+0.034−0.037,h=0.309+0.017−0.012,λ=0.22545±0.00065,δ=(89.6+2.94−0.86)∘. (31)

Because of the freedom of the phase redefinition for the quark fields, we have shown in (21) that the QM parametrization is indeed equivalent to the Wolfenstein one in the quark sector.

Finally, the leptonic mixing parameters () except Majorana phases can be expressed in terms of five parameters (or ), , the last four being the QM parameters in the lepton sector. If we further assume that all the QM parameters except have the same values in both the CKM and PMNS matrices, then only two free parameters left in the lepton mixing matrix are and . If is fixed to be the same as the CKM one, then there will be only one free parameter in our calculation. In the next section, we shall study the dependence of the mixing angles and the Jarlskog invariant on and .

To make our point clearer, let us summarize the reduction of the number of independent parameters in this work. In the leptonic sector, we start with 16 free parameters (12 from the charged lepton mass matrix and 4 from the neutrino mass matrix ) as shown in Eq. (15). Among the 12 parameters from , three phases can be rotated away by the redefinition of the charged lepton fields. The remaining 9 parameters correspond to three charged lepton masses () and six angles in the charged lepton mixing matrix as shown in Eq. (26), while the 4 parameters from correspond to three neutrino masses () plus one angle ( or ) in the neutrino mixing matrix as shown in Eq. (17) or (21). With our ansatz for discussed before, the 6 angles in are reduced to four QM parameters (). Thus, the number of parameters finally becomes five ( plus (or )), except for the six lepton masses. Under the further assumption of the QM parameters having the same values in both the CKM and PMNS matrices, these five parameters are reduced to only two ones and .

## Iii Neutrino phenomenology

We now proceed to discuss the low energy neutrino phenomenology with the neutrino mixing matrix (see Eq. (17)) characterized by the mixing angle or the small parameter and the charged lepton mixing matrix in which is assumed to have the similar expression as the QM parametrization (17); (13) given by or (see Eq. (29) and Eq. (30), respectively). The lepton mixing matrix thus has the form

 UPMNS=⎧⎨⎩VQMUTBMWfor Vℓ†L=VQM,V†QMUTBMWfor VℓL=VQM. (32)

Therefore, the corrections to the TBM matrix within our framework arise from the charged lepton mixing matrix characterized by the parameters and the matrix specified by the parameter whose size is strongly constrained by the recent T2K data. Indeed, the parameters and in the lepton sector are a priori not necessarily the same as that in the quark sector. Hereafter, we shall use the central values in Eq. (31) of the parameters for our numerical calculations.

In the following we consider both cases:

(i)

With the help of Eqs. (14) and (29), the leptonic mixing matrix corrected by the replacements and , can be written, up to order of and , as

 U(i)PMNS = U†L Uν(π/4+ϵ)=VQMUTBMW (33) = UTBM+ϵ⎛⎜ ⎜ ⎜ ⎜ ⎜⎝−ϵ2√230√23i√2+ϵ2√60−1√6+iϵ2√2−i√2+ϵ2√60−1√6−iϵ22√2⎞⎟ ⎟ ⎟ ⎟ ⎟⎠ + λ⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝−eiδ+λ+hλ2√6eiδ−λ2−hλ2√3−i(eiδ−hλ2)√2−2e−iδ+(12−f−he−iδ)λ√6−e−iδ+(12−f−he−iδ)λ√3i(12+f+he−iδ)λ√2(f+heiδ)λ+2fe−iδλ2√6−(f+heiδ)λ+fe−iδλ2√3i(f+heiδ)λ√2⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠ − λϵ⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝−i(eiδ−hλ2)√2−ϵeiδ+λ+hλ22√60eiδ+λ+hλ2√6−ϵi(eiδ−hλ2)2√2i(12+f+he−iδ)λ√2−ϵ2e−iδ+(f+he−iδ−12)λ2√602e−iδ+(f+he−iδ−12)λ√6+ϵi(f+he−iδ+12)λ2√2i(f+heiδ)λ√2+ϵ(f+heiδ)λ+2fe−iδλ22√60−(f+heiδ)λ+2λ2feiδ√6+ϵi(f+heiδ)λ2√2⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠ .

Note that here contains five independent parameters ( and ).7 By rephasing the lepton and neutrino fields , , and , the PMNS matrix is recast to

 UPMNS=⎛⎜ ⎜⎝|Ue1||Ue2||Ue3|e−i(α1−α3)Uμ1e−iβ1Uμ2ei(α1−α2−β1)|Uμ3|Uτ1e−iβ2Uτ2ei(α1−α2−β2)|Uτ3|⎞⎟ ⎟⎠Pν , (34)

where is an element of the PMNS matrix with corresponding to the lepton flavors and to the light neutrino mass eigenstates. In Eq. (34) the phases defined as ,