Deviation from Tribimaximal mixing using $A_{4}$ flavour model with five extra scalars

A modified neutrino mass model with five extra scalars is constructed using $A_{4}$ discrete symmetry group. The resultant mass matrix is able to give necessary deviation from Tribimaximal mixing which reproduces the current neutrino masses and mixing data with good accuracy. The model gives testable prediction for the future measurements of the neutrinoless double-beta decay parameter $|m_{\beta \beta}|$. The analysis is consistent with latest cosmological bound $\Sigma m_{i}\leq$ 0.12 eV.


I. INTRODUCTION
In the last few decades, the neutrino experiments have confirmed neutrino oscillations and mixings through the observation of solar and atmospheric neutrinos, indicating their masses thereby providing an important solid clue for a new physics beyond the Standard Model (SM) of particle physics. At present, the neutrino oscillation experiments [1][2][3][4][5][6][7] have measured the oscillation parameters viz: mass squared differences (∆m 2 21 and ∆m 2 31 ) and mixing angles (θ 12 , θ 23 and θ 13 ) to a good accuracy. The bounds on the absolute neutrino masses scale are also greatly reduced by direct neutrino mass experiments [8], the neutrinoless double beta decay experiments (0νββ) [9][10][11] and cosmological observation [12]. However, the current data is still unable to explain several key issues such as the octant of θ 23 , the neutrino mass ordering, CP violating phase, etc.
The oscillation data reveals certain pattern of neutrino mixing matrix. Out of the several approaches to explain the observed pattern, the Tribimaximal mixing (TBM) [13,14] used to be very favourable. The A 4 flavour symmetry model proposed by Altarelli and Feruglio [15,16] can accommodate TBM mixing scheme in a neutrino mass model. The TBM mixing pattern has the form However, the recently observed non-zero θ 13 disfavours TBM and leads to the modification of several mass models constructed with TBM [17][18][19][20][21][22]. As a result, the neutrino mixing patterns like TM1 [23] and TM2 [24,25] which are proposed with slight deviation from TBM, gain momentum. Currently, they can predict the observed pattern and mixing angles with good consistency.
In this present work, we propose a model with five extra SM singlet scalars to explain neutrino parameters in their experimental ranges. The present model is constructed in the basis where charged lepton is diagonal. The deviation from TBM and non-zero value of θ 13 are obtained as a consequence of specific Dirac mass matrix which is constructed using antisymmetric part arising from the product of two A 4 triplets. Here, we present a detailed analysis on the neutrino oscillation parameters and its correlation among themselves and with neutrinoless double-beta decay parameter |m ββ |.
The paper is organized as follows: In Section 2, we present the description of the model with the particle contents in the underlying symmetry group. In Section 3, we give the detailed numerical analysis and the results in terms of correlation plots. Section 4 deals with summary and conclusion. The Appendix provides brief description of A 4 discrete group.

II. DESCRIPTION OF THE MODEL
We extend the SM by adding five extra scalars namely ξ 1 , ξ 2 , ξ 3 , φ T and φ S which are transformed as 1, 1 ′′ , 1 ′ , 3 and 3 respectively under A 4 group. The SM lepton doublet l are assigned to the triplet representation under A 4 , right-handed charged lepton e c , µ c , τ c , and right-handed neutrino field ν c are assigned to the A 4 representations 1, 1 ′′ , 1 ′ and 3, respectively. The right-handed neutrino field ν c contributes to the effective neutrino mass matrix through Type-I see-saw mechanism. The A 4 symmetry is supplemented by additional Z 3 × Z 2 group to restrict additional terms otherwise allowed by A 4 symmetry.
The transformation properties of the fields used in the model are given in Table I.
The Yukawa Langrangian terms for the leptons which are invariant under A 4 × Z 3 × Z 2 × SU(2) L transformation, are given in the equation: where Λ is the model cutoff high scale. The Yukawa mass matrices can be derived from Eq. (1) by using the vacuum expectation value given in Table II . The charged lepton mass The Majorana neutrino mass matrix has the structure The Dirac mass matrix is in the form The effective neutrino mass matrix is obtained by using Type-I see-saw mechanism, where ).
In order to explain smallness of active neutrino masses, we consider the heavy neutrino with mass eigenvalues However, in this case, the mass matrix m ν can be diagonalized by U T BM and this possibility would violate the currently observed neutrino oscillation data especially the non-zero values of θ 13 . Therefore, we need to consider the non-zero values for b, c, d and e to get θ 13 = 0.
Hence, the unitary matrix that can diagonalize the resultant mass matrix, should deviate from the U T BM .

III. NUMERICAL ANALYSIS AND RESULTS
As the light neutrino mass matrix m ν obtained in Eq. (5) is in the basis where charged lepton mass matrix is diagonal, the Pontecorvo-Maki-Nakagawa-Sakata leptonic mixing matrix (U P M N S ) which is necessary for the diagonalization of m ν becomes a unitary matrix U.   Therefore, the light neutrino mass matrix m ν is diagonalized as: where m diag = diag(m 1 , m 2 , m 3 ) is the light neutrino mass matrix in the diagonal form.

Parameters
The three neutrino mass eigenvalues can be written as, The upper bound on the sum of neutrino masses ( m i = m 1 + m 2 + m 3 ) obtained by the Planck is 0.12 eV [12].
The PMNS matrix U can be parametrized in terms of neutrino mixing angles and Dirac CP phase δ. Following the PDG convention [27], U takes the form where θ ij (for ij = 12, 13, 23) are the mixing angles (with c ij = cos θ ij and s ij = sin θ ij . P = diag(e iα , e iβ , 1) contains two Majorana CP phases α and β, while P φ = diag(e iφ 1 , e iφ 2 , e iφ 3 ) consists of three unphysical phases φ 1,2,3 that can be removed via the charged-lepton field rephasing [28]. The neutrino mixing angles θ 12 , θ 23 and θ 13 in terms of the elements of U are given below: The Jarlskog invariant is given by the phase redefinition invariant quantity,  In order to show that the model is consistent with the present neutrino oscillation data, we vary the free parameters of our model a, b, c, d and e to fix the neutrino oscillation observables θ 13 , θ 23 , θ 12 , ∆m 2 21 , ∆m 2 31 and δ to their experimental values. The experimental values of neutrino oscillation parameters used in our analysis are given in Table III. The allowed regions of the model parameters that can satisfy current oscillation data for both NH and IH are shown in Fig. 1 and Fig. 2, respectively in the form of correlation plots.
The values of θ 13 obtained in the allowed regions of our model parameters for both NH and IH are shown in Fig. 3 and Fig. 5, respectively while Fig. 4 and Fig. 6 give the variation of θ 23 in NH and IH, respectively. The model predictions of the neutrino oscillation parameters in 3σ range are shown in Fig. 7 and Fig. 8  The additional predictions for the effective Majorana mass |m ββ | vs Jarlskog invariant J in 3σ range for both NH and IH are shown in Fig. 9(a) and Fig. 10(a)   correlation between δ and |m ββ | are depicted in Fig. 9(b) for NH and Fig. 10(b) for IH. And, the effective Majorana mass |m ββ | is given by The The predicted range of |m ββ | for both NH and IH can be tested in the near future.
The detailed studies on A 4 symmetry can be found in Ref [35].