Admixture of quasi-Dirac and Majorana neutrinos with tri-bimaximal mixing

We propose a realization of the so-called bimodal/schizophrenic model proposed recently. We assume S4, the permutation group of four objects as flavor symmetry giving tri-bimaximal lepton mixing at leading order. In these models the second massive neutrino state is assumed quasi-Dirac and the remaining neutrinos are Majorana states. In the case of inverse mass hierarchy, the lower bound on the neutrinoless double beta decay parameter m_ee is about two times that of the usual lower bound, within the range of sensitivity of the next generation of experiments.


I. INTRODUCTION
Charged particles are Dirac fermions while electrically neutral fermions, like neutrinos, can be either Dirac or Majorana. Neutrinoless double beta decay 0νββ experiments will confirm (if observed) the Majorana nature of neutrinos [1]. Experiments for 0νββ currently under construction will have sensitivity in the range of the inverse hierarchy mass spectrum [2][3][4][5]. Recently, it has been observed in [6] that if the second massive neutrino is of Dirac type (and so does not participate to the 0νββ decay) in the case of inverse mass hierarchy, the lower bound on the 0νββ parameter m ee is about two times that of the usual bound. In reference [6], they forbid the Majorana mass for the second neutrino at tree level by means of a flavor symmetry.
The parameter m ee can be written as combination of neutrino masses, namely m ee = 3 i=1 U 2 ei m νi where U is the lepton mixing matrix. In the inverse hierarchy case, when three neutrinos are of Majorana type, we have |m ee | ≈ |(cos 2 θ 12 + e iα sin 2 θ 12 )m atm | > m atm 3 ≈ 17 meV.
If the second massive neutrino is of Dirac type, that is m ν2 = 0 in m ee we have |m ee | ≈ | cos 2 θ 12 m atm | > 2m atm 3 ≈ 34 meV.
Such a value is in the range of sensitivity of the next generation of experiments and could be ruled out very soon. A four component spinor ψ is a Majorana spinor if ψ = ψ c where ψ c is the charge conjugate of ψ. The Dirac mass term for a massive spin 1/2 fermion is given by where ψ = (χ, σ 2 φ * ) and χ, φ are two component spinors. Assuming , a four component Dirac mass term (3) is equivalent to two Majorana mass terms of equal mass and opposite parity [7,8] For an arbitrary number of Majorana neutrinos the neutrino mass matrix is given by In general the eigenvalues of the mass matrix M can have different signs and we can assign a signature matrix diag(+, +, ..., −, −, ..). For two neutrino states we can have diag(+, −) or diag(+, +). In the former case, if the absolute value of the masses is the same, the two neutrino types make up a Dirac neutrino. When the two neutrinos are active-sterile we have the so-called quasi-Dirac neutrino [9] and when they are active-active we have the so called pseudo-Dirac neutrino [10]. In Ref. [6] the second massive neutrino state has a quasi-Dirac mass 1 , while the first and third neutrinos get a Majorana mass a la seesaw. Since each flavor state is an admixture of quasi-Dirac and Majorana states, they call such a case schizophrenic. For recent studies on this subject see also [11][12][13]. There are several models in the literature for exact tri-bimaximal [14] based on the group of permutation of four objects S 4 as flavor symmetry [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29]. Here we study the schizophrenic case assuming the S 4 group with extra abelian symmetries as flavor symmetry. Breaking S 4 into different Z 2 subgroups respectively in the charged lepton and neutrino sectors we obtain tri-bimaximal mixing at tree-level. The difference between our model and the model of Ref. [6] is that they assume the permutation of three objects S 3 flavor symmetry instead of S 4 and they obtain tri-bimaximal mixing only assuming the charged lepton mass matrix to be diagonal, while in our model the charged lepton mass matrix is diagonal at tree-level by means of S 4 .
The Letter is organized as follow: in section II we present the model, in section III we give the neutrino and charged lepton mass matrices, in section IV we study the problem of the vacuum alignments and we give our conclusions.

II. THE MODEL
We extend the Standard Model (SM) with a G f = S 4 × Z 3 × Z ′ 3 × Z ′′ 3 flavor symmetry where S 4 is the permutation group of four objects, Z 3 , Z ′ 3 , Z ′′ 3 are abelian groups characterized respectively by ω 3 = 1, ω ′ 3 = 1 and ω ′′ 3 = 1. In order to simplify the study of the S 4 -alignments of the scalar fields we assume supersymmetry, therefore all the fields are assumed to be superfields. We also add three right-handed neutrinos and eight scalar isosinglets called flavons. We assume ν c 2 to be a singlet of S 4 and ν c 1 , ν c 3 to form a doublet ν c D of S 4 . The SU L (2) doublet L and singlet l c are both triplets 3 1 of S 4 . The matter content of the model is given in table I. 31 11 2 11 31 11 2 11 11 2 11 Since ν c 2 is charged under Z ′ 3 the mass term ν c 2 ν c 2 is forbidden. The scalar flavons take vacuum expectation value (vev) along the following direction of S 4 (see section IV) When the scalar flavons take such vevs, the elements ST 2 , S 2 T S, T S, S 3 T 2 leave invariant the charged leptons while the elements T ST, T ST S 2 , S, S 3 leave invariant the neutrino sector. Here S and T are generators of S 4 , see the Appendix A. The different breaking in the charged lepton and neutrino sectors gives (at tree-level) tri-bimaximal mixing. The scalar S 4 singlets ξ ν , χ l ,χ l , σ andσ take vevs different from zero.

III. MASS MATRICES
From the superpotential w ν and the vevs alignments given in eq. (7) the Dirac couplings for the neutrinos are proportional to the following S 4 contractions Then the Dirac neutrino mass matrix is given by where The right-handed Majorana neutrino mass matrix is given by where M 1 = M 3 . The neutrino mass matrix is diagonalized by the tri-bimaximal mixing matrix, see eq. (10). One neutrino has a quasi-Dirac mass 2 m ν2 ≡ m D ν2 (see eq. (11)) and two neutrinos have Majorana masses Note that the masses m ν3 and m ν1 are proportional one to each other, so the atmospheric mass spliting arises from the M 1 and M 3 mass splitting. Assuming Yukawa couplings of order one and the following value for the scales where the scalar fields take vev then the neutrino masses m ν1 , m ν2 and m ν3 are at the eV scale with M R ∼ 10 5 GeV . As a particular example, taking y 1ν = 0.2200, y 2ν = 0.6345, y ϕ = 1, y σ = −0.2300, we have giving about ∆m 2 sol ≈ 7.5 · 10 −5 eV 2 and ∆m 2 atm ≈ 2.4 · 10 −3 eV 2 in agreement with data. We observe that the next to leading order term ν c 2 ν c 2 ξ 2 ν /M Λ is allowed giving a contribution to M R of order 10 −5 GeV that is negligible. 2 Next to leading order terms as well as loop corrections generate a negligible mass term for ν c 2 then we have a quasi-Dirac state instead of a Dirac one.
The charged lepton mass matrix is given from the superpotential w l . It is not difficult to show that the resulting mass matrix is diagonal. This arises from the S 4 symmetry and the masses are given as 3 If ϕ l1 and ϕ l2 are free, we have three combinations of free parameters and we can fit the charged lepton masses as given below that are of order of the mass of the τ , in agreement with the assumption in eq. (14). In the limit m e,µ → 0 from eqs. (21) and (22) we have in agreement with the vev alignment given in eq. (7). The mass of the muon m µ arises from a small deviation the alignment ϕ l ∼ (− √ 3(1 + ǫ), 1). Such a deviation can arise from next to leading order terms in the scalar superpotential as well as by assuming S 4 soft breaking terms in the superpotential. While the electron mass m e arises by means of a fine-tuning of the coupling y 1l . We can easily accommodate the three charged lepton masses in our model, in particular m µ ≪ m τ arises from the alignment ϕ l ∼ (− √ 3, 1).

IV. VACUUM ALIGNMENTS
In the previous sections we showed that assuming the alignments in eq. (7) we obtain tri-bimaximal neutrino mixing and diagonal charged lepton mass matrix. Here we show that the alignment of the flavon fields can arise from the minimization of the superpotential.
The superpotential invariant under S 4 × Z 3 × Z ′ 3 × Z ′′ 3 for the flavon fields of table (I) is given by w = λ 1 ϕ l ϕ l ϕ l + λ 2 ϕ l ϕ l χ l + λ 3 χ l χ l χ l + λ 4 χ lχl + λ 5χlχlχl + + λ 6 ϕ ν ϕ ν ϕ ν + λ 7 ϕ ν ϕ νσ + λ 8 σσ + λ 9σσσ + λ 10 σσσ + where the terms proportional to µ φ and µ ξ break softly the auxiliary Z ′ 3 symmetry while the Z 3 and Z ′′ 3 are preserved in the superpotential. We denote the vevs of the flavon fields as below We show below that r 1 = r 2 = r 3 = r, v 1 = 0, v 2 = v, u 1 = − √ 3u and u 2 = u is a possible solution of the minimization of the superpotential. Then we have to solve the set of equations where we have assumed r 1 = r 2 = r 3 = r, v 1 = 0, v 2 = v, u 1 = − √ 3u and u 2 = u. It is easy to show that such a system admits a solution with r, v and u different from zero and fixed by the coupling constants of the superpotential in eq. (24).
In summary, we present a realization of the so-called bimodal/schizophrenic ansatz, that is one of the massive neutrino state is of Dirac-type and the remaining two are Majorana. Then each flavor state is an admixture of Dirac and Majorana states giving distinct predictions for the neutrinoless double beta decay rate. The model consist of a supersymmetric extension of the SM based on the S 4 × Z 3 3 flavor symmetry, where we add three right-handed neutrinos, the second of them transforming as a singlet of S 4 and the other two as a doublet of S 4 , and eight scalar singlets of the SM. The model also gives tri-bimaximal mixing for neutrinos at leading order. As was pointed out in [6] this kind of models can be ruled out very soon by neutrinoless double beta decay experiments.

V. ACKNOWLEDGMENTS
The 24 elements of S 4 belong to five classes (A2) The elements of C 2,4 define two different sets of Z 2 subgroups of S 4 , the ones of the class C 4 a set of Z 3 abelian discrete symmetries and those belonging to C 5 a set of Z 4 abelian discrete symmetries. The S 4 irreducible representations are two singlets, 1 1 , 1 2 , one doublet, 2, and two triplets, 3 1 and 3 2 . We adopt the following basis for the doublet representation and for the triplet representations 3 1 and 3 2 respectively. Clearly the generators (S + , T ) and (S − , T ) define the two triplet representations 3 1 , 3 2 respectively. All the product rules can be straightforwardly derived. We remind the reader to the product rules reported in [30] (see also [31]). The product of S 4 representation: where we introduced the notation [µ × µ] for the symmetric and {µ × µ} for the anti-symmetric part of the product µ × µ. Note that ν × µ = µ × ν for all representations µ and ν. For the irreducible representations: The explicit products for 1 1 representation with any µ representation: and the product of 1 2 with the any µ representation: The products of µ × µ: for 2 For 2 × 3 1 :