A new texture of neutrino mass matrix with three constraints

We present a new texture of neutrino mass matrix having three complex relations among its elements and study in detail the phenomenological implications. A characteristic feature of the resulting neutrino mass matrix is that the atmospheric neutrino mixing angle is predicted to lie in a very narrow region near $45^{\circ}$. We illustrate how such a form of the neutrino mass matrix can be realized using the non-Abelian flavor symmetry $A_4$ in the framework of type-I+II seesaw mechanism.


Neutrino Mass Matrix
In the present work we propose the following texture of the neutrino mass matrix in the flavor basis: The number of free parameters in the above neutrino mass matrix are reduced from 12 to 6 by imposing three independent relations among the neutrino mass matrix elements: The fourth relation M 33 = 2M 13 , derivable from the first three, is not independent. First two relations 2 and 3 are expressions of µ − τ anti-symmetry [13]. But, a mass matrix with µ − τ anti-symmetry should also have vanishing (1,1) and (2,3) entries. Since, µ − τ anti-symmetry is not allowed experimentally, we need non-zero M 11 and M 23 to explain the neutrino masses [13]. The third relation enhances the predictive power of this texture by imposing an additional relation on the mass matrix elements apart from the two relations of the µ − τ anti-symmetry.
To analyse the phenomenological implications of present neutrino mass model, we reconstruct the neutrino mass matrix in the flavor basis (i.e. in the basis where the charged lepton mass matrix is diagonal) assuming neutrinos to be Majorana particles. In this basis, a complex symmetric neutrino mass matrix can be diagonalized by a unitary matrix V ′ as where M diag ν = diag(m 1 , m 2 , m 3 ). The unitary matrix V ′ can be parametrized as where [14] U =   c 12 c 13 s 12 c 13 s 13 e −iδ −s 12 c 23 − c 12 s 23 s 13 e iδ c 12 c 23 − s 12 s 23 s 13 e iδ s 23 c 13 s 12 s 23 − c 12 c 23 s 13 e iδ −c 12 s 23 − s 12 c 23 s 13 e iδ c 23 c 13   with s ij = sin θ ij and c ij = cos θ ij and Here, P ν is the diagonal phase matrix with two Majorana-type CP-violating phases α, β and one Diractype CP-violating phase δ. The phase matrix P l contains unphysical phases that depend on the phase convention. The matrix V is called the neutrino mixing matrix or the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix [15]. Using Eqs. (5) and Eq. (6), the neutrino mass matrix can be written as 2 The simultaneous existence of relations (2) and (3) between the elements of the neutrino mass matrix implies e i(ϕµ+ϕµ) M ν(µµ) + e i(ϕτ +ϕτ ) M ν(τ τ ) = 0 (10) or where In this notation, the relations (2) and (3) between the neutrino mass matrix elements yield two complex equations viz.
A more elegant version of these equations can be rewritten as where with (i = 1, 2, 3). The two complex Eqs. (17) and (18) involve nine physical parameters: m 1 , m 2 , m 3 , θ 12 , θ 23 , θ 13 , and three CP-violating phases (α, β, and δ). In addition, there are three unphysical phases (ϕ e , ϕ µ , ϕ τ ) which enter in the mass ratios as two phase differences. The masses m 2 and m 3 can be calculated from the mass-squared differences ∆m 2 21 and |∆m 2 32 | using the relations where m 2 > m 3 for an inverted neutrino mass ordering (IO) and m 2 < m 3 for the normal neutrino mass ordering (NO). Using the experimental inputs of the two mass-squared differences and the three mixing angles we can constrain the other parameters. Simultaneously solving Eqs. (17) and (18) for two mass ratios, we obtain 21) and m1 m3 The magnitudes of the two mass ratios in Eqs. (21) and (22) are given by  Table 1: Current neutrino oscillation parameters from global fits [16]. Here ∆m 2 3l ≡ ∆m 2 31 > 0 for normal ordering and ∆m 2 3l ≡ ∆m 2 32 < 0 for inverted ordering while the CP-violating Majorana phases α and β are given by As ∆m 2 21 and |∆m 2 32 | are known experimentally, the values of a mass ratio (r 13 or r 12 ) from Eq.(23) or Eq. (24) can be used to calculate m 1 . For example, by inverting Eq.(23), we obtain Also, the mass ratios in Eqs. (23) and (24) can be used to obtain the expression for the parameter R ν , which we define as the ratio of mass squared differences (∆m 2 ij = m 2 i − m 2 j ): where m 1 > m 3 for an IO and m 1 < m 3 for the NO. For Eqs. (2) and (3) to be consistent with the present neutrino oscillation data, the parameter R ν should lie within its experimentally allowed range. We solve the three constraints on the neutrino mass matrix (Eqs. (2)- (4)) by generating sufficiently large number of random points for our free parameters. The points in the parameter space that satisfy the constraints of Eqs. (2) and (3) are then required to satisfy the third relation in Eq. (4) with the identical accuracy level determined by the experimental error on the neutrino oscillation parameters. The current experimental best fit values and the corresponding 1σ and 3σ errors used in the numerical analysis are given in Table 1. We also have an upper bound on the sum of neutrino masses: Planck satellite data [17] combined with WMAP, CMB and BAO experiments limit the sum of neutrino masses m i ≤ 0.12 eV at 95% confidence level (CL). In the present work, we assume a more conservative limit of m i ≤ 1 eV. The results of this analysis are shown as correlation plots in figures 1 and 2. We have depicted the points in the allowed parameter space at the 3σ confidence level. A characteristic feature of this texture is that the atmospheric neutrino mixing angle θ • 23 is predicted to lie in a very narrow region near 45 • (Fig.1 for NO 4 and Fig.2 for IO). Also, the Dirac-type CP violating phase δ is correlated with θ • 23 ( Fig. 1(a) for NO and Fig. 2(a) for IO). As a generic prediction of such textures, the two Majorana phases show sharp correlations with one other (Fig.1(c) for NO and 2(c) for IO).
Another notable prediction of this model is a lower bound on the effective mass relevant to neutrinoless double beta decay (|M ee | > 0.046 eV) for inverted neutrino mass ordering ( Fig.2(b)) which is testable in the currently running and forthcoming neutrinoless double beta decay experiments [18].
We also depict the correlation plots of the neutrino masses (m 1 and m 3 ) in Fig.1(d) (for NO) and Fig.  2(d) (for IO). For the present neutrino mass model, the sum of neutrino masses ( m i ) remains above for 0.08 eV for NO and 0.1 eV for IO which implies that the neutrino mass matrix realized in present work predicts a quasi-degenerate neutrino mass spectrum.

Symmetry Realization
To obtain the desired form of the form of the neutrino mass matrix from A 4 symmetry, we extend the standard model (SM) by adding two additional SU (2) L doublet Higgs fields to the SM and three SU (2) L triplet Higgs fields (we do not discuss the Higgs phenomenology in this work). The additional triplet Higgs fields lead to non-zero neutrino masses via the type-II seesaw mechanism [19]. We also add three right handed neutrino fields that contribute to the effective neutrino mass matrix via the type-I seesaw mechanism [20]. The transformation properties of various leptonic and Higgs fields under A 4 flavor symmetry and SM gauge group SU (2) L are given in Table-2. The above transformation properties lead to the following A 4 invariant Yukawa Lagrangian: For the φ i Higgs fields, the vacuum expectation values (VEVs) along the direction φ o = v φ (1, 1, 1) T 5 lead to the following charged lepton mass matrix in the symmetry basis: The Dirac neutrino mass matrix has a structure similar to the charged lepton mass matrix i.e.
The right-handed neutrino mass matrix has the following form: The type-I seesaw contribution to the effective neutrino mass matrix after redefining some parameters takes the following form: For the type-II seesaw contribution to the effective neutrino mass matrix, the A 4 and SU (2) L triplet Higgs fields are assumed to have the VEV alignment ∆ o = v ∆ (0, −1, 1) T . Such an alignment of VEVs has been achieved in Refs. [21,22] by allowing specific terms in the scalar potential which break A 4 softly. This leads to the following form of the neutrino mass matrix via the type-II seesaw mechanism: which can be redefined as: The complete effective neutrino mass matrix in the symmetry basis has the following form: In the present basis (symmetry basis) the charged lepton mass matrix is non-diagonal. Next, we make the transformation M l = U † L m l U R , which leads to a diagonal charged lepton mass matrix with and U R = I, where I is a 3 × 3 unit matrix. In this basis, where the charged lepton mass matrix is diagonal, the neutrino mass matrix takes the following form:  By redefining the elements of M ν in above equation, we obtain the following form of the neutrino mass matrix in the flavor basis: Since the above neutrino mass matrix is realized at the seesaw scale, there are also renormalization group corrections which need to be incorporated at the electroweak scale. However, these details are beyond the scope of present work.