Phenomenological study of extended seesaw model for light sterile neutrino

We study the zero textures of the Yukawa matrices in the minimal extended type-I seesaw (MES) model which can give rise to $\sim$ eV scale sterile neutrinos. In this model, three right handed neutrinos and one extra singlet $S$ are added to generate a light sterile neutrino. The light neutrino mass matrix for the active neutrinos, $ m_{\nu}$, depends on the Dirac neutrino mass matrix ($ M_{D} $), Majorana neutrino mass matrix ($ M_{R} $) and the mass matrix ($ M_{S} $) coupling the right handed neutrinos and the singlet. The model predicts one of the light neutrino masses to vanish. We systematically investigate the zero textures in $ M_{D} $ and observe that maximum five zeros in $ M_{D} $ can lead to viable zero textures in $ m_{\nu} $. For this study we consider four different forms for $ M_R $ (one diagonal and three off diagonal) and two different forms of $(M_{S})$ containing one zero. Remarkably we obtain only two allowed forms of $ m_{\nu} $ ($m_{e\tau} = 0 $ and $m_{\tau\tau}=0$) having inverted hierarchical mass spectrum. We re-analyze the phenomenological implications of these two allowed textures of $m_\nu$ in the light of recent neutrino oscillation data. In the context of the MES model, we also express the low energy mass matrix, the mass of the sterile neutrino and the active-sterile mixing in terms of the parameters of the allowed Yukawa matrices. The MES model leads to some extra correlations which disallow some of the Yukawa textures obtained earlier, even though they give allowed one-zero forms of $m_\nu$. We show that the allowed textures in our study can be realized in a simple way in a model based on MES mechanism with a discrete Abelian flavor symmetry group $Z_8 \times Z_2$.


Introduction
Neutrino oscillation experiments have established the fact that neutrinos have tiny mass and they change from one flavor to another during their propagation. This requires the Standard Model (SM) of particle physics to be extended in order to generate their masses. The standard 3-flavor neutrino oscillation scenario has six key parameters. These are the two mass squared differences (∆m 2 i1 , = m 2 i − m 2 1 , i = 2, 3 ) which control the oscillations of the solar and atmospheric neutrinos respectively, three mixing angles θ ij (i, j = 1, 2, 3; i < j) and a Dirac CP phase, δ 13 . Global analysis of three flavor neutrino oscillation data from [1][2][3] give us the best fit values and the allowed 3σ ranges of these parameters. In 3-flavor paradigm, there are two more CP violating phases if neutrinos are Majorana particles. But as Majorana phases do not appear in the neutrino oscillation probability, they are not measurable in the oscillation experiments. Apart from these phases another major unknown is the absolute value of the neutrino mass since oscillation experiments are only sensitive to the mass squared differences. Planck data provide an upper bound on sum of neutrino masses to be ≤ 0.23 eV [4] at 95% C.L. The sensitivity for the neutrino masses in the upcoming Karlsruhe Tritium Neutrino experiment (KATRIN) is expected to be around 200 meV (90% C.L.) [5].
Another interesting aspect of neutrino oscillation experiments is the search for the existence of a light sterile neutrino. As sterile neutrinos are SM singlets they do not take part in the weak interactions. But they can mix with the active neutrinos. Therefore, sterile neutrinos can be probed in neutrino oscillation experiments. The oscillation results from LSND experiment showed the evidence of at least one sterile neutrino having mass in the ∼ eV scale [6][7][8]. The latest data of MiniBooNE experiment [9] also have some overlap with the allowed regions of the LSND experiment and hence support the existence of the sterile neutrino hypothesis. The recently observed Gallium anomaly can also be explained by the sterile neutrino hypothesis [10]. Another evidence of eV sterile neutrino comes from the reactor antineutrino flux studies. This shows the deficit in the observed and predicted event rate of electron antineutrino flux and the ratio is 0.943 ± 0.023 at 98.6% C.L. [11]. Recent analysis of the Planck data shows the possibility of light sterile neutrino in the eV scale if one deviates slightly from the base ΛCDM model [4]. In short, the scenario with a light sterile neutrino is quite riveting at present and many future experiments are proposed to confirm/falsify this [12]. Although it is possible to have a better fit of neutrino oscillation data with more than one light sterile neutrino [13][14][15], the 3+1 scheme i.e., three active neutrinos and one sterile neutrino in the sub-eV and eV scale respectively, is considered to be minimal. There are three different ways to add sterile neutrino in SM mass patterns and these are, (i) 3+1 scheme in which three active neutrinos are of sub-eV scale and sterile neutrino is of eV scale [16,17], (ii) 2+2 scheme in which two different pairs of neutrino mass states differ by eV 2 but this scheme was disfavored by solar and atmospheric data [18], and (iii) 1+3 scheme in which three active neutrinos are in eV scale and sterile neutrino is lighter than active neutrinos. This scenario is however disfavored from cosmology [19,20]. Hence, we focus on the 3+1 scenario in our study.
Flavor symmetry models giving rise to eV sterile neutrinos have been studied in the literature [21][22][23]. These models might require modifications to usual seesaw framework [24,25]. In the explicit seesaw models the eV scale sterile neutrinos with their mass suppressed by Froggatt -Nielsen mechanism can be naturally accommodated in non Abelian A 4 flavor symmetry [22,26,27]. S 3 bimodel or schizophrenic models for light sterile neutrinos are also widely studied [28,29]. In order to have a theoretical understanding of the origin of eV sterile neutrino as well as admixtures between sterile and active neutrinos, the authors of Refs. [22,26,27] have studied an extension to the canonical type-I seesaw model. This model is known as "minimal extended type -I seesaw" (MES) model. In the MES model a fermion singlet, S, is added along with three right handed neutrinos. This extension results into an eV scale sterile neutrino naturally, without imposing tiny mass scale or Yukawa term for this neutrino.
In this paper, for the first time we study the various possible textures of the Dirac neutrino mass matrix, M D , Majorana neutrino mass matrix, M R and the mass matrix M S that originate from the Yukawa interaction between right handed neutrinos with the gauge singlet within the framework of MES model and classify the allowed possibilities. Several papers have studied the consequences of imposing zeros in the neutrino mass matrix in standard three neutrino [30][31][32][33][34][35][36][37][38][39][40] and the 3+1 framework [41][42][43][44][45]. The more natural study would be to explore the zeros in the Yukawa matrices that appear in the Lagrangian rather than light neutrino mass matrix, m ν . It has been noted by many authors [46][47][48][49][50] that the zeros of the Dirac neutrino mass matrix M D and the right handed Majorana mass matrix M R are the progenitors of zeros in the effective Majorana mass matrix m ν through type -I seesaw mechanism. We also seek extra correlations connecting the parameters of the active and sterile sector which can put further constraints on the allowed possibilities. This motivates us to look for zeros in various neutrino mass matrices in the MES model which can lead to viable texture zeros in neutrino mass matrix. We classify different structures of M D , M R and M S that can give allowed textures for the light neutrino mass matrix m ν . Interestingly the only allowed form of m ν that we obtain are the two one zero textures -namely m eτ = 0 and m τ τ = 0 which are phenomenologically allowed and have the inverted hierarchical mass spectrum. For a m ν originating from ordinary seesaw mechanism both these textures are viable. However, in the MES model, because of extra correlations connecting active and sterile sector, not all Yukawa matrices that give m eτ = 0 or m τ τ = 0 for m ν are allowed. We study these additional correlations and tabulate the allowed textures. We also include a discussion on the impact of NLO corrections in this model. In this context it is also important to study the origin of zero textures. Here, we show that it is possible to obtain various zero entries in lepton mass matrices with an Abelian discrete symmetry group Z 8 × Z 2 . An alternative approach to obtain lepton mixing is discussed in [51] by considering non-Abelian symmetry group. We follow the method discussed in [52] to obtain Abelian discrete symmetry group which can generate viable zero textures in m ν . Their method is based on type -I seesaw and we extend it to apply on MES model. The paper is organized in the following manner. In the next section a brief review of the MES model is given. In Section 3 and its subsections we list the various forms of M D , M R and M S that lead to viable textures in m ν . In Section 4 we discuss the implication of the allowed forms of one zero textures in m ν obtained in Section 3. The following Section 5 discusses the results obtained from the comparison of low energy and high energy neutrino mass matrices and the extra correlations connecting active and sterile sector. Symmetry realizations for the allowed zero textures are discussed in Section 6. The summary of our findings and conclusions are presented in Section 7.

Minimal extended type I seesaw mechanism
In this section we describe the basic structure of MES model. Here, the fermion content of the SM is extended by three right handed neutrinos together with a gauge singlet field S. One can get a natural eV-scale sterile neutrino without inserting any small Yukawa coupling in this model [22,26]. The Lagrangian containing the neutrino masses is given by, Here, M D , M R are the (3 × 3) Dirac and Majorana mass matrices respectively and M S is a (1 × 3) coupling matrix between right handed neutrinos with the gauge singlet. In the basis (ν L , ν c R , S c ), the (7 × 7) neutrino mass matrix can be expressed as, Considering the hierarchical mass spectrum of these mass matrices i.e. M R M S > M D , in analogy of type -I seesaw, the right handed neutrinos are much heavier compared to the electroweak scale and thus they will decouple at the low scale. Therefore, Eq.(2.2) can be block diagonalized using seesaw mechanism and the effective neutrino mass matrix in the basis (ν L , S c ) can be written as, Note that the rank of M 4×4 ν is three (see [26]) and hence one of the light neutrino remains massless.
Considering the case that M S > M D , one can apply seesaw approximation once again on Eq.(2.3) to obtain the active neutrino mass matrix as 1 , whereas the mass of the sterile neutrino is given by, Note that the zero textures of fermion mass matrices in the context of type -I seesaw mechanism studied in [46][47][48]50], leading to viable texture zeros in m 3×3 ν can be different from that of MES model because of the presence of the first term of Eq.(2.4). The activesterile neutrino mixing matrix is given by, where R 3×1 governs the strength of active-sterile mixing and can be expressed as, Additionally in our formalism we assume |V τ 4 | = 0, which is allowed by the current active sterile neutrino mixing data. Figure 1: Allowed mass spectrum in 3+1 scheme for normal (SNH) and inverted (SIH) mass hierarchy.
As the sterile neutrino mass (∼ eV) is heavier than active neutrinos, therefore, the mass pattern in the active sector can be arranged in two different ways. We denote 3+1 scenario as (SNH) when the three active neutrinos follow normal hierarchy (m 1 < m 2 m 3 ) and the second choice is (SIH) when the three active neutrinos follow inverted hierarchy m 3 m 1 ≈ m 2 ) as shown in Fig(1). These masses can be expressed in terms of the mass squared differences obtained from oscillation experiments as given in Table(  (∆m 2 32 ) are the solar and atmospheric mass squared differences and ∆m 2 41 (∆m 2 43 ) is the active sterile mass squared difference. The allowed ranges of these three mass squared differences are given in Table(2). fit values along with 3σ ranges of neutrino oscillation parameters used in our numerical analysis are given in Table(2). In the next section we systematically explore the various zero texture structures of M D , M R and M S which can give rise to viable zero textures of m 3×3 ν .

Formalism
In our formalism, the charge lepton mass matrix, M l , is considered to be diagonal.  Table 2: The latest best-fit and 3σ ranges of active ν oscillation parameters from [3]. The current constraints on sterile neutrino parameters are from the global analysis [53][54][55]. Here R ν is the solar to atmospheric mass squared difference ratio.
These three non-diagonal forms of M R correspond to L e − L µ , L e − L τ and L µ − L τ flavor symmetry respectively. Such forms of M R in the context of zero textures in type-I seesaw model have been considered for instance in [56]. M S = (s 1 , s 2 , s 3 ) being a 1 × 3 matrix can have one zero or two zeros. In [26], an A 4 based model was considered with 2 zeros in M S and 3 zeros in M D to obtain the m 3×3 ν as given by Eq. (2.4). But, in our analysis we find that mass matrices with 5 zeros in M D and two zeros in M S do not lead to any viable textures in m ν . The only allowed possibility therefore is one zero in M S result in three possible structures. We find that the maximum number of zeros of M D that can give phenomenologically allowed zero textures in m ν is five. The possible combinations of M D , M R and M S that lead to phenomenologically viable textures of m ν are discussed in the following subsections.

5 zeros in M D and diagonal M R
First let us assume M R to be diagonal. As M D is a non-symmetric 3 × 3 matrix, 5 zeros can be arranged in 9 C 5 = 126 ways. Thus considering 126 cases of M D together with 3 cases of M S and 1 case of M R , we obtain total 378 possible structures of m ν . Out of all possible combinations of these matrices the only allowed texture that we obtain is the one zero texture in m ν with m eτ = 0. Here, we have three possible forms of M S and these are, The various forms of M D which lead to viable texture m eτ = 0 are presented below: (3.6) Here, Z 12 , Z 13 and Z 23 are the permutation matrices that exchange first and second columns, first and third columns and second and third columns respectively. Therefore, we observe that out of 126 cases only 12 above forms of M The form of M R that we consider here corresponds to flavor symmetry L e − L µ as given in Eq.(3.2). Among the 378 possibilities we obtain two allowed one zero textures of m ν , namely m eτ = 0 and m τ τ = 0. We observe that out of total 126 forms of M D , only four structures give rise to m eτ = 0 while eight structures give rise to m τ τ = 0. We list them below: (3.7)

5 zeros in
The form of M R that we consider in this subsection corresponds to flavor symmetry L e − L τ as given in Eq. (3.2). In this case also we observe that out of total 126 cases of M D , only four structures of M D give rise to m eτ = 0 and eight forms of M D give rise to texture m τ τ = 0. We list them below. Note that these forms of M D are different from those obtained in the earlier subsection.

Textures leading to
3.4 5 zeros in M D and non-diagonal M R corresponding to L µ − L τ flavor symmetry The form of M R that we consider here corresponds to flavor symmetry L µ − L τ as given in Eq. (3.2). Here also we observe that out of 126 cases of M D only four structures of M D give rise to texture M eτ = 0 and 8 forms of M D give rise to texture M τ τ = 0. But these forms of M D are different from those obtained in the earlier two subsections: (3.13)

Structures leading to
Note that in general the entries of the Yukawa matrices M D , M R and M S are complex (of the form pe iθ ). However some of the phases can be absorbed by redefinition of the leptonic fields. For the case when M R is diagonal, the number of un-absrobed phases is two -one each in M D and M S whereas for the off-diagonal M R only one phase remains in M S . In this section we do not explicitly write the phases. However in section 5 where we discuss specific cases, the phases are explicitly included.

Active neutrino mass matrix with one zero texture
The (3 × 3) light neutrino mass matrix being symmetric, there are 6 possible cases of one zero textures with a vanishing lowest mass and these are studied in details in Refs. [57][58][59][60].
In the above section we observed that in context of MES model only viable textures of m ν that we obtain are m eτ = 0 and m τ τ = 0. According to the recent studies [59][60][61], both these textures are ruled out for normal hierarchy when the lowest mass m 1 is zero but they can be allowed for the inverted hierarchy even when then lowest mass m 3 is zero 2 . This kind of mass pattern can be obtained completely from group theoretical point of view if one assumes that Majorana neutrino mass matrix displays flavor antisymmetry under some discrete subgroup of SU(3) as discussed in [62,63]. In this section we re-analyse the textures m eτ = 0 and m τ τ = 0 for the inverted hierarchical mass spectrum assuming m 3 = 0 in the light of recent neutrino oscillation data as given in Table(2). In our analysis we find that correlations among various oscillation parameters become highly constrained as compared to the earlier studies. This is due to the recent constraints on the 3σ ranges of the mass squared differences and θ 13 as compared to earlier results in [58][59][60] 3 .
In three neutrino paradigm, low energy Majorana neutrino mass matrix can be diagonalized as, Here, U = U.P (P = diag(1, e iα , e i(β+δ 13 ) )) is a lepton mixing matrix in the basis where M l is diagonal. The Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix U has 3 mixing angles and a CP violation phase δ 13 . The elements of neutrino mass matrix can be calculated from Eq.(4.1) are, where, a, b = e, µ and τ and m i (i = 1, 2, 3) are given in Table(1). We express elements of m ν as m ab in the text.
Imposing the condition of zero texture for IH with m 3 = 0 in the above equation we get, which can be simplified to obtain the mass ratio Let us define the ratio of the two mass squared differences as, (4.7) 2 We also observed that both these textures are disallowed for NH with the most recent data. 3 The latest constraint on |∆m 2 31 | comes from T2K and NOνA including both appearance and disappearance modes [64][65][66][67]. Whereas reanalysis of KamLAND data shows decrease in the value of ∆m 2 21 and sin 2 θ12 as discussed in [3].
The R ν defined above can be calculated either using the current neutrino mass squared differences as given in Table(2) or by calculating |q|. If the value of R ν calculated using |q| falls in the allowed 3σ range of R ν from the current data, then we say the texture under consideration is allowed by the current data. As given in Table(2) we vary the Dirac CP phase δ 13 from 0 • < δ 13 < 360 • while the relevant Majorana phase α in the range 0 • < α < 180 • and find the correlations among different parameters, specially the predictions for α and δ 13 .
We also study the effective Majorana neutrino mass, m ee , governing neutrinoless double beta decay (0νββ) for these allowed textures. In three flavor paradigm this can be written as, where c ij (s ij ) = cos θ ij (sin θ ij ), (i < j, i, j = 1, 2, 3). From the above equation we understand that m ee depends on the Majorana phases but not on the Dirac phase. Various experiments such as CUORE [68], GERDA [69], SuperNEMO [70], KamLAND-ZEN [71] and EXO [72] are looking for signatures for neutrinoless double beta decay (0νββ). The current experiments provide bounds on the effective Majorana mass m ee from the non-observation of 0νββ. For instance, the combined results from KamLAND-ZEN and EXO-200 [71] give the upper bound on the effective Majorana neutrino mass as m ee < (0.12 -0.25) eV where the range signifies the uncertainty in the nuclear matrix elements. The future experiments can improve this limit by one order of magnitude. Below we discuss the various correlations that we obtain for the allowed textures.
Imposing the condition of zero texture with vanishing lowest mass (m 3 = 0) for IH, we get, The mass ratio m 2 m 1 should be greater than 1. For this to happen cos δ 13 should be negative. We find that due to the interplay of the terms O(s 13 ) and O(s 2 13 ) the phase δ 13 is restricted to the range [85 • − 95 • ] and [265 • − 275 • ]. The effective mass, m ee as function of Majorana phase α is constrained due to very small allowed range of α (5 • < α < 10 • , 170 • < α < 175 • ) as shown in Eq(4.8). The allowed range of m ee for this texture is 0.046 eV < m ee < 0.05 eV and which can be probed in future experiments. Also, this texture predicts Dirac CP phase ∼ 270 • which is in agreement with the indications from the current ongoing oscillation experiments like T2K and NOνA. There is however no constrain on the values of the neutrino mixing angles θ 13 and θ 23 seen in right panel of Fig.2 for this texture.

Case II: m τ τ = 0
The Majorana mass matrix element m τ τ in 3-flavor case can be written as, Imposing the condition of texture zero with vanishing lowest mass(m 3 = 0) for IH, we get,  Since this mass ratio m 2 m 1 is always greater than 1 from oscillation data, we find that cos δ 13 should be negative for this texture as well. As can be seen from Fig 3 that δ 13 is constrained in the range 140 • < δ 13 < 220 • . We observe that, due to the more constrained values of mass squared differences and θ 13 from present data, as considered in our analysis, the atmospheric mixing angle θ 23 is restricted to be below maximal. In the earlier analysis [58][59][60] there was no preferred octant of θ 23 . The values of θ 23 > 45 • are disallowed for this texture as can be seen in Fig 3. The effective mass, m ee , being function of unknown Majorana phase α as seen in Eq(4.8) is constrained due to very small allowed range of α (80 • < α <110 • ). The allowed range of m ee for this texture is 0.014 eV < m ee < 0.018 eV which is smaller compared to the case m eτ =0 where a vanishing element is off-diagonal.
The allowed values of the effective mass m ee for diagonal texture m τ τ are on the lower side having no overlap with non diagonal texture zero m eτ . Thus, m ee can be used to distinguish between diagonal and off-diagonal one texture zero classes with a vanishing neutrino mass. Note that allowed ranges of δ 13 and m ee are more constrained in our analysis as compared to references [58,59] again due to the recent improved constraints on the mass squared differences and θ 13 at 3σ.

Comparison of low and high energy neutrino mass matrix elements
In this section we obtain the light neutrino neutrino mass matrix (m ν ) (Eq. For an illustration we will discuss three specific cases. In case I and II we discuss m eτ = 0 assuming diagonal structure of M R and in the case III we talk about m τ τ = 0 by considering the off diagonal form of M R . Note that here we consider the complex phases in our calculation. We compare high energy mass matrix with low energy mass matrix after the decoupling of the eV sterile neutrino as discussed in section II. Here m ab , a, b = e, µ, τ are the low energy neutrino mass matrix elements. The eigen values of m 3×3 ν will give the masses of the three active neutrinos. Note that, only allowed hierarchy in our case is IH and hence m 3 = 0 and m s = m 4 = ∆m 2 43 . From Eq.(5.4) we get, We find that the lhs of Eq.(5.5) lies in the range (0.63 -3.06) whereas rhs lies in (3.9 -5.9) in their 3σ range. This shows that there is no overlapping between lhs and rhs of Eq.(5.5) and hence disallowed from current neutrino oscillation data. We observe that out of 12 forms of M and using them in Eqs(2.4, 2.5 and 2.7) we get the texture m eτ = 0, The sterile mass and active sterile mixing becomes It can be seen from the above equations that From Eq.(5.7) we get the following relation between the light neutrino mass matrix elements, (5.10) To obtain Eq. However the values of α and m ee which are predicted by the two cases are similar. The prediction of the texture with m eτ = 0 is 6 • < α < 13 • and 167 • < α < 174 • while the MES model predicts a slightly constrained range 11.7 • < α < 13 • and 167 • < α < 168.1 • . In this case we also obtain another correlation for sterile neutrino mass from this model of the form, In the lower panels of Fig.4, we have plotted the prediction of m s as given by Eq.(5.11) by varying V e4 and V µ4 within their allowed range as given in table (2). This is obtained when both the conditions i.e., m eτ = 0 and Eq.(5.10) is satisfied simultaneously. From the figures we see that the prediction of m s by this model is consistent with data coming from the SBL experiments.
• Case III : Considering the cases for the off-diagonal forms of M R given in Eqs.
D . All these forms of M D and the corresponding forms of M R and M S lead to neutrino mass matrix with m τ τ = 0. We found that, all these M D 's lead a correlation of the form, which is satisfied by current oscillation data. The rhs of Eq.
From the above matrices we find the following correlation,  In Table(3) and Table(4) we summarize the allowed cases that we obtained in our study for texture m eτ = 0 and m τ τ = 0 respectively.

NLO correction for MES model
Same as Case I Table 3: The various forms of M D , M R and M S which leads to a phenomenologically allowed m eτ = 0.
texture zeros. The NLO correction term can be calculated following the standard algorithm given in [77]. To calculate the NLO term, let us rewrite equation (2.3) in the form, where, In the second line we use the form of M L , M D and M R as given by equation (5.18) to obtain the final form given by equation (5.19). We see that the contribution of the NLO terms of equation ( 4 Hence, there exist a parameter space where we can safely neglect NLO correction terms in our analysis compared to leading order terms and consider the texture zero even with the inclusion of the NLO term. 5 Thus, all the model predictions corresponding to leading order terms remain unchanged. Note that similar conclusions can also be obtained for the texture m τ τ = 0.

Symmetry realization
Singular one zero neutrino mass matrices can be realized using a discrete Abelian flavor symmetry within the context of MES mechanism. Earlier in [52] authors studied the possibilities to enforce zero textures in arbitrary entries of the fermion mass matrices by means of Abelian symmetries in the context of type -I seesaw mechanism. We adopt the same approach to probe the zero textures of m ν in the context of MES mechanism. We observe that one zero textures of m ν with a vanishing mass can be realized by Z 8 ×Z 2 symmetry. To realize the texture structures we extend the SM particle composition by three right handed neutrinos (ν eR , ν µR , ν τ R ) as required in MES model and two more Higgs doublets (φ , φ ) in addition to the SM one (φ). Few SU (2) L scalar singlets (χ i , i = 1, 2) are required to realize diagonal M R whereas two singlets λ i , i = 1, 2 helps in realizing one zero texture structure of M S . Note that the model that we discuss here to get the zero texture structure is general, flexible and in no way unique. The additional discrete group Z 2 is introduced to restrict some of the unwanted terms in the Lagrangian. For illustration, we present the detailed symmetry realization of our two viable textures of m ν (m eτ , m τ τ = 0). The particle assignments for (m eτ = 0 which is allowed by current data (case II) under the action of Z 8 × Z 2 symmetry are given in Table (5). Table 5: Here,D L l denote SU (2) L doublets and l R , ν l R (l = e, µ, τ ) are the right-handed (RH) SU (2) L singlet for charged lepton and neutrino fields respectively. Also, φ, φ and φ are the Higgs doublets.
According to the charge assignments of the leptonic field given in Table (5) the bilinears D L l l R ,D L l ν l R and ν T l R C −1 ν l R relevant for M l , M D and M R transform as, where ω = e πi/4 , ω 8 = 1 . We introduce three SU (2) L doublet Higgs (φ, φ ,φ ). One of these Higgs doublet φ, is invariant under Z 8 while the other two fields transforms as: here allφ = iτ 2 φ * . The Higgs fields acquires the vacuum expectation values φ o = 0 and results in the M l and M D of the following form, Here For the right-handed Majorana mass matrix (M R ) and for the mass matrix M S , we introduce few SU (2) L scalar singlets and their transformation under Z 8 × Z 2 is given in the Table (6). Thus the mass matrices M R and M S becomes, (ω 4 , 1) λ 2 (ω 2 , -1) Table 6: Here, scalar singlet χ 1 and χ 2 give M R whereas λ 1 and λ 2 give M S .
We also give the transformation to the singlet field S as (ω 6 , -1) under (Z 8 × Z 2 ) which will prevent the term of the form S c S as demand by the MES model will still give the correct form of M S . Using the minimal extended type I seesaw given in Eqn (2.4) with the mass matrices M D , M R and M S as discussed above leads to effective neutrino mass matrix m ν with a texture zero at (1,3) position. The fields descriptions are same as given in Table(5).
Similarly, one can assign the various fields transformation under the action of (Z 8 × Z 2 ) to obtain the texture with m τ τ = 0. The form of M  Table 7. Here, no extra scalar singlet is needed to obtain the mass structure of M R which has L e − L µ symmetry and for M S we need two scalar singlets (λ 1 , λ 2 ) which transform under Z 8 × Z 2 as (ω 2 , 1) and (ω 7 , -1) respectively. We also give transformation to singlet field S as (ω, 1) under (Z 8 × Z 2 ) which will prevent the term S c S. Note that symmetry realization of this texture is more economical than the m eτ = 0 texture.

Conclusions
In this paper we have studied the low energy phenomenology of the minimal extended type I seesaw model which can accommodate an eV scale light sterile neutrino [22,26]. This model is motivated by the recent experimental evidences which support the existence of light sterile neutrinos in addition to three active neutrinos. In this model, apart from three right handed neutrinos, an extra gauge singlet S is added to the SM. Under the minimal extended seesaw mechanism, this model give rise to three active neutrinos in the sub-eV scale with one of the active neutrinos having vanishing mass and one sterile neutrino in the eV scale. In this model the Dirac mass matrix, M D , is an arbitrary 3 × 3 complex matrix, the Majorana mass matrix M R is a 3 × 3 complex symmetric matrix and M S which couples the right handed neutrinos and the singlet S is a 1 × 3 matrix.
We obtain different textures of M D , M R and M S that give rise to phenomenologically allowed zero textures in the low energy neutrino mass matrix, m ν . The maximum number of zeros in M D that results in viable m ν are found to be five. Thus, there are 126 different possible structures of M D to be probed. We consider four possible structures of M R with one diagonal and three non diagonal forms. The maximum number of zeros in M S is one as two zeros do not result in phenomenologically viable textures of m ν . This leads to three possible structures of M S . After analyzing all the different combinations we obtain only two viable one zero textures of m ν (m eτ = 0 and m τ τ = 0) with different possible structures of M D , M R and M S . We study these textures of m ν in the light of the current oscillation data. Both these textures have inverted hierarchical mass spectrum and we get constraints on observables like effective Majorana neutrino mass m ee and Dirac CP phase δ 13 . For the texture m eτ = 0, we obtain the allowed values of Dirac CP phase δ 13 is around ±90 • . Note that δ 13 ∼ −90 • is favored by current neutrino oscillation experiments. For m τ τ = 0, δ 13 lies between (150 • -240 • ). The allowed range for the effective Majorana mass is different for both these textures. It can thus be used to distinguish between the two textures. Also, in our study we observed that due to improved constraints on the mass squared differences and θ 13 the texture m τ τ = 0 disfavours higher octant of θ 23 .
Next we studied the predictions of the MES model for the Yukawa matrices that gave viable forms of m ν and check whether any extra correlations can come from the model. This is expected since in the framework of this model both the active and sterile neutrino masses as well as the active sterile mixing depend on the parameters of the Yukawa matrices M D , M R and M S . Thus, there may be additional relations between different observables, which are the predictions of the model. We find that some of the Yukawa matrices which can generate allowed one zero textures m eτ = 0 and m τ τ = 0 in the active neutrino mass matrix, m ν , cannot satisfy the extra correlations coming from the predictions of the MES model. Our analysis reveals that due to these additional correlations among the 126 × 4 × 3 = 1512 possible combinations of M D , M R and M S , only 6 combinations giving m eτ = 0 and other 6 combinations giving m τ τ = 0 are allowed from the current oscillation data. The 6 allowed combinations which give m eτ = 0, reveal severe restrictions on the values of θ 23 and θ 13 due to the extra correlations in the MES model and only the lower octant of θ 23 and relatively higher values of θ 13 remains allowed. In addition an interesting correlation is obtained connecting the mass of the sterile neutrino to the active sterile mixing parameters which also involves the light neutrino masses and mixing. Thus this correlation connects the active and the sterile sector. For m eτ = 0 the prediction for the sterile neutrino mass obtained from the MES model is in complete agreement with what is obtained from global analysis. The texture, m τ τ = 0 also predicts a correlation for sterile neutrino mass. This however is in marginal agreement with the global analysis. We also explored the consequences of NLO correction terms in our analysis and depicted the parameter space in M D , M R and M S for which the NLO corrections can be neglected as compared to the leading order term. Finally, working within the framework of MES mechanism, we present simple discrete Abelian symmetry models Z 8 × Z 2 leading to the two phenomenologically allowed zero textures of m ν .
In conclusion, we analyzed the low energy prediction of the minimal extended seesaw model that can give an eV scale sterile neutrino. We emphasize that this task is performed for the first time in this paper. The results described in our analysis shows the compatibility of this model to the neutrino oscillation data. We also find correlations that can be tested in future experiments. This kind of study is indispensable to test the viability of a given model in the context of present and forthcoming neutrino oscillation experiments.