Non-Zero $\theta_{13}$ and $\delta_{CP}$ in a Neutrino Mass Model with $A_4$ Symmetry

In this paper, we consider a neutrino mass model based on $A_4$ symmetry. The spontaneous symmetry breaking in this model is chosen to obtain tribimaximal mixing in the neutrino sector. We introduce $Z_2 \times Z_2$ invariant perturbations in this model which can give rise to acceptable values of $\theta_{13}$ and $\delta_{CP}$. Perturbation in the charged lepton sector alone can lead to viable values of $\theta_{13}$, but cannot generate $\delta_{CP}$. Perturbation in the neutrino sector alone can lead to acceptable $\theta_{13}$ and maximal CP violation. By adjusting the magnitudes of perturbations in both sectors, it is possible to obtain any value of $\delta_{CP}$.


Introduction
The discovery of neutrino oscillations has triggered a lot of experimental and theoretical effort to understand the physics of lepton masses and mixing. Since flavor mixing occurs due to the mismatch between the mass and flavor eigenstates, neutrinos need to have small non-degenerate masses [1,2]. During the past two decades, many neutrino oscillation experiments have been performed and the values of oscillation parameters are determined to a very good precision [3][4][5].
Neutrino oscillation probabilities depend only on the mass-squared differences and the mixing angles. Hence these parameters are determined in the neutrino oscillation experiments. The experimental data has shown two large mixing angles and one small mixing angle. This pattern is different from the case of quark mixing where all angles are small and the mixing matrix is close to identity. The lepton mixing matrix, called Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix, is approximately equal to the tribimaximal (TBM) ansatz proposed in ref. [6]. In this ansatz, the mixing angles have values tan 2 θ 12 = 1 2 , θ 13 = 0 • , and sin 2 θ 23 = 1 2 . The TBM form of the PMNS matrix is where |U e3 | = sin θ 13 = 0. Many recent experiments [7][8][9] have shown that the value of θ 13 is not only non-zero but is relatively high [10]. The values of other mixing angles also have small deviations from the TBM values. Since θ 13 is non-zero, the possibility of a CP violating phase (δ CP ) in the lepton mixing matrix must be considered seriously. The neutrino oscillation experiments have found two mass-squared differences with very different magnitudes. The smaller masssquared difference, denoted ∆m 2 21 = m 2 2 − m 2 1 , is positive and is of the order of 10 −5 eV 2 .
The larger mass-squared difference, ∆m 2 31 = m 2 3 − m 2 1 , is of the order of 10 −3 eV 2 , but its sign is not known. This leads to two possible mass hierarchies for neutrinos: normal hierarchy (NH) in which ∆m 2 31 is positive and m 1 < m 2 < m 3 and inverted hierarchy (IH) where ∆m 2 31 is negative and m 3 < m 1 < m 2 . Finding the sign of ∆m 2 31 is a major goal in many experiments like INO [11,12], ICECube-PINGU [13,14], and long baseline experiments [15,16]. The values of mixing angles and mass-squared differences from the global analysis of data is summarized in Table 1 Table 1: The values of mass-squared differences and mixing angles from the global fits [18]. The numbers in the parenthesis are upper/lower uncertainties at (1σ, 2σ, 3σ) confidence level.
To accommodate the small masses of neutrinos in comparison to charged leptons and quarks, a novel mechanism involving Majorana nature of neutrinos, called seesaw mechanism, was introduced in [19][20][21][22]. In this mechanism, the right handed partners of neutrinos are introduced with Majorana masses at high scale. The neutrinos, in addition, have Dirac masses of the order of charged lepton masses. The most general neutrino mass matrix is a 6 × 6 matrix in the space of three left-handed and three right-handed neutrino fields. A diagonalization of this matrix leads to the generation of small Majorana masses for left-handed neutrinos. A common approach to obtain the observed mixing pattern is to constrain the structure of interaction Lagrangian, which gives rise to the mass matrix, using a discrete non-abelian flavor symmetry [23][24][25][26][27][28][29][30][31]. Many such models are constructed using discrete, non-abelian groups like A 4 [29,[32][33][34][35] and S 4 [36][37][38]. In particular, it was shown in [33][34][35] that models based on A 4 symmetry can lead to the prediction of tribimaximal mixing. Being the smallest group with an irreducible triplet representation, A 4 has been popular group for neutrino mass models since its introduction in ref. [32].
In the wake of θ 13 measurement, it is necessary to modify the models predicting TBM pattern [39][40][41][42]. Two major approaches to incorporate the necessary modifications are vacuum misalignment and symmetry breaking via perturbation terms. All models based on discrete symmetry groups require a special vacuum alignment condition to obtain tribimaximal mixing. A deviation from this, that is , a vacuum misalignment can lead to deviations from TBM pattern [43,44]. Another way to generate deviations from TBM pattern is to add symmetry breaking terms which break the symmetry completely or partially [35,45]. It is common to have different residual symmetries in charged lepton and neutrino sectors after such a perturbation.
In this paper, we will consider modifications of a model based on A 4 group proposed in [35]. TBM pattern is obtained in this model by breaking A 4 symmetry spontaneously to Z 3 in the charged lepton sector and to Z 2 in the neutrino sector. We first introduce a Z 2 × Z 2 invariant complex perturbation in the charged lepton sector only. This perturbation leads to non-zero value for θ 13 , small deviations in the values of θ 12 and θ 23 , but does not lead to any CP violation. If a real Z 2 × Z 2 perturbation is introduced in the neutrino sector only, viable values of θ 13 and maximal CP violation are obtained. By introducing perturbations in both the charged lepton and the neutrino sectors, it is possible to obtain any value of δ CP by adjusting their relative strengths.

The A 4 Model
A 4 is the group of even permutations on four elements and is the smallest group with a three dimensional irreducible representation which makes it a popular group in neutrino mass modelling. This group has three 1-dimensional irreducible representations and one 3dimensional irreducible representation.There are two popular approaches to study the three dimensional irreducible representation: the Ma-Rajasekaran (M-R) approach [32] which makes all the Z 2 × Z 2 elements diagonal and the Altarelli-Feruglio (A-F) approach [34] in which the Z 3 elements are diagonal. We will use the M-R convention in our discussion. A 4 has four classes denoted by C1, C2, C3, and C4. The 3 × 3 matrix representations of the A 4 elements in each of these classes are: The Z 3 elements in this group are In this section, we will discuss the details of a type-I seesaw model based on A 4 group proposed in ref. [35]. We limit ourselves to the leptonic sector of the model. The fields in this sector are the three left-handed SU (2) gauge doublets, three right-handed chargedlepton gauge singlets, and three right-handed neutrino gauge singlets. They are assigned to various irreducible representations of the A 4 group. In addition, there are four Higgs doublets, φ i (i = 1,2,3) and φ 0 , and three scalar singlets χ i (i=1,2,3). The assignments of the fields under various groups, are given in Table 2. By using the Clebsh-Gordon decomposition of A 4 tensor products, the complete G SM ⊗ A 4 invariant (G SM is the standard model gauge symmetry) Yukawa Lagrangian for the leptonic sector can be written as [46] The individual terms of this equation are given by where ω is the cube root of unity, (2.6) and where C is the charge conjugation matrix. Here, L CL Dirac contributes to the Dirac mass matrix in the charged lepton sector, L N Dirac contributes to the Dirac mass matrix in the neutrino sector and L N Majorana contributes to the Majorana mass matrix of the right handed neutrinos. L Yukawa has an additional U (1) X symmetry [35]. Under this symmetry the fields D iL , l iR and φ i have quantum numbers X = 1 and all other fields have X = 0. This symmetry forbids the Yukawa terms of the formD L ν Rφi . These terms are invariant under G SM × A 4 and contribute to the Dirac mass matrix of the neutrinos. Without the contribution of these terms, this matrix retains the simple form needed to obtain the tribimaximal mixing. Spontaneous symmetry breaking leads to the following scalar VEVs: v i for φ i , w i for χ i , and v 0 for φ 0 . With these VEVs, we obtain the different mass terms to be The charged lepton mass matrix M 0 l can be put in a diagonal form by the transformation The Majorana mass matrix M R is diagonalized by an orthogonal transformation The PMNS matrix is now obtained to be tribimaximal up to phases on both sides.
The vacuum alignment for scalar fields spontaneously breaks the A 4 symmetry in the charged lepton sector (coupling only with φ i ) to Z 3 subgroup. In the neutrino sector (coupling with χ and φ 0 ), the residual symmetry is Z 2 . The Lagrangian lacks a common symmetry as there is no subgroup between Z 2 and Z 3 . A novel feature of the model is that the diagonalizing matrix is completely determined by the symmetry, but the lepton masses are given by the arbitrary coupling constants h i (i = 0, 1, 2, 3). The seesaw mechanism generates small masses for the left handed neutrinos given by M

Perturbation in Charged Lepton Sector
In the model discussed till now, the PMNS matrix has the tribimaximal form with zero θ 13 and no CP violation. To generate non-zero values for these, we add small perturbations to the above model. We first introduce a symmetry breaking term in the charged lepton sector which is invariant under the subgroup Z 2 × Z 2 . In order to construct such a perturbation, it is required to know the breaking pattern of A 4 irreducible representations into Z 2 × Z 2 irreducible representations. The group Z 2 × Z 2 is the normal subgroup of A 4 with four elements. It has one trivial singlet representation1(1, 1, 1, 1) and three non-trivial singlet representations, viz.1 (1, 1, −1, −1),1 (1, −1, 1, −1) and1 (1, −1, −1, 1). The breaking of A 4 triplet into Z 2 × Z 2 irreducible representations can be readout from the diagonal matrix elements of Z 2 × Z 2 in the M-R basis, shown in eq.(2.3). This is given as The general Z 2 × Z 2 invariant perturbation can be written as h 1DL (3) whereD L , φ are the three-dimensional reducible representations of Z 2 × Z 2 and l R 's are trivial singlets. For the perturbation to be Z 2 × Z 2 invariant , the matrices M 1 , M 2 and M 3 must commute with the matrices given in eq. (2.3). This is satisfied by any diagonal matrix. It can be observed that introducing a multiplicative factor in the i th row of charged lepton mass matrix in eq. (2.9) will introduce a reciprocal factor in the i th column of its diagonalizing matrix U ω . The U e3 element of the PMNS matrix in the TBM form is zero because the 11 and 13 elements of U ω are equal. The perturbation terms in eq. (3.2) can disturb this balance and lead to non-zero U e3 . The value of U e3 (and hence θ 13 ) depends on the elements of M 1 , M 2 , and M 3 . In order to obtain a simple form for the perturbed charged lepton mass matrix M l , we choose M i s of the form M i = diag(z, 0, ω i−1 z) where z is a complex number with |z| 1. After spontaneous symmetry breaking, the resulting M l = M 0 l + ∆M l , where M 0 l is given in eq. (2.9) and Such a ∆M l can arise from higher order effects of the theory. The form of ∆M l is similar to the form of ∆M u,d given in eq. (4.3) of [35]. In generating these higher order terms, the Higgs VEVs are unaffected. To simplify the phenomenological analysis, we parameterized all the six higher order terms in terms of a single number z. Note that there is no residual symmetry left in the charged lepton sector after the spontaneous symmetry breaking. The perturbed matrix elements of M l introduce reciprocal factors in the respective columns of U ω . Requiring U ω to be unitary, we get z to be We will retain the solution with + sign in order to keep |z| < 1. The perturbation strength is of the order s which we take to be small. Using the parametrization s = sin α, we can transform U ω to (3.5) The PMNS matrix becomes A similar structure for the PMNS matrix is discussed in refs. [48][49][50]. From the above equation, we can compute the perturbed values of the mixing angles to be sin 2 θ 13 = 2 3 sin 2 α = 2s 2 3 sin 2 θ 23 = 2 + cos 2α + √ 3 sin 2α 2(2 + cos 2α) The sine squared values of mixing angles in this scheme are plotted in Figure 1. The perturbation parameter s ∼ 0.19 leads to a very good fit for θ 13 . Such a value of s also gives sin 2 θ 23 very close to the central value and sin 2 θ 12 within 2σ range. Compared to their respective TBM values, sin 2 θ 12 changes very little (∼ 5%), whereas sin 2 θ 23 receives an appreciable correction (∼ 14%).
We introduced perturbations in both the first and third rows of M l . We chose these perturbations to be related to each other. This enabled us to keep the perturbation s at the level of 10 − 20%. In principle, it is possible to choose the perturbing matrix M i = diag (z, 0, 0). Such a perturbation modifies only the first row of M l . Parametrizing z in terms of s as in eq. (3.4), we can obtain the modified values of the mixing angles. With s = sin α, these values are In this case, the amount of perturbation should be double that of the previous case to obtain an acceptable value of θ 13 . Given that we obtained viable values of θ 13 we check if a CP violating phase δ CP is also generated. However, we find that the Jarlskog invariant J of the PMNS matrix in eq. (3.6) is zero. Hence, no CP violation can be generated by the perturbations considered here. So we look for other possible sources of CP violation and also non-zero θ 13 in this model.

Perturbation in Neutrino Sector
In the previous section it was shown that a Z 2 × Z 2 invariant perturbation in the charged lepton sector can give rise to viable θ 13 but no CP violation. In this section, we add a similar perturbation in the neutrino sector and study its influence on θ 13 and δ CP . As in the case of the charged lepton sector, the perturbing matrix should be diagonal to satisfy the Z 2 × Z 2 symmetry. We will derive expressions for θ 13 and δ CP as a function of the two perturbations and show that it is possible to obtain any value of δ CP . It is shown that perturbation only in the neutrino sector leads to maximal CP violation.
We observe that the diagonalizing matrix in the neutrino sector is a rotation matrix of angle π/4. A small imbalance in the degeneracy of 11 and 33 elements of M R in eq. (2.9) shifts the rotation angle slightly away from π/4 [35]. Such an imbalance can be introduced by a Z 2 × Z 2 invariant perturbation in the neutrino sector. We choose this perturbation to be [35,48,49] . (4.1) The mass M is a A 4 invariant soft term in the lagrangian. The perturbation in eq. (4.1) can be introduced as an A 4 breaking but Z 2 × Z 2 preserving soft term in the lagrangian.
The perturbed Majorana mass matrix becomes This matrix can be diagonalized by a rotation of angle 'x', where tan 2x = M /aM . We will denote the perturbation in the neutrino sector by the dimensionless parameter ζ = aM/M (≡ cot 2x). The form of PMNS matrix after the combined perturbations in the charged lepton and the neutrino sectors is We recall that the perturbation in the charged lepton sector s = sin α. The Jarlskog invariant of this matrix can be found to be √ 3 cos 2x/18 which vanishes for x = π/4. We obtain CP violation due to the deviation of the angle 'x' from π/4 through the perturbation in the neutrino sector. Expanding the expressions for the mixing angles up to order ζ 2 and s 2 , we get sin 2 θ 23 = 2 + cos 2α sin 2x + √ 3 sin 2x sin 2α 4 + 2 cos 2α sin 2x From these values and the Jarlskog invariant, we obtain sin δ CP to be sin δ CP = cos 2x(2 + cos 2α sin 2x) 1 − cos 2 2α sin 2 2x 4 + 4 cos 2α sin 2x + (−1 + 2 cos 4α) sin 2 2x . (4.7) The expression in eq. (4.7) is exact. We can obtain a simpler equation by expanding it in ζ and s and keeping only the leading powers in the numerator and the denominator, The value of δ CP goes to zero as ζ tends to zero, corresponding to no perturbation in the neutrino sector. For perturbation only in the neutrino sector, we have s = 0 and δ CP = ±π/2, depending on the sign of ζ. The ν e appearance data of T2K prefers δ CP to be in the lower half plane. From eq. (4.7), this indicates that ζ should be positive. The best fit value of this data is equal to -π/2 which prefers that perturbation in the charged lepton sector is extremely small. The value of δ CP depends on the relative strengths of the perturbations, s in the charged lepton sector and ζ in the neutrino sector. This dependence is plotted in figure 2. From this figure, we note that ζ ≥ 2s if δ CP ≥ π/4 and δ CP quickly becomes very small for ζ < s. Figure 3 shows the variation of mixing angles with respect to ζ where the bands for 1σ and 2σ bounds are also drawn. The value of sin 2 θ 13 ≈ 0.025 near ζ ≈ 0.36. For this value of ζ, the change in sin 2 θ 12 is negligibly small(∼ 3%). Eventhough the value of ζ is moderately large, the parameter a = ζM /M quantifying the perturbation in the neutrino sector is quite small because M M . The value of sin 2 θ 23 remains 0.5 if the perturbation in the charged lepton sector is zero.
In the present scenario, there is a tension between obtaining a large δ CP and a value of sin 2 θ 23 close to the best fit experimental value. This value of sin 2 θ 23 is 15% larger than TBM value of 0.5. In order to obtain this large a deviation, we need a value of s ≈ 0.19 in the charged lepton sector. For s ≈ 0.19, the constraint on sin 2 θ 13 in eq. (4.4) leads to very small values of ζ and hence of δ CP . A large CP violation, on the other hand, requires ζ > 2s, which keeps the value of sin 2 θ 23 close to the TBM value of 0.5, as can be seen from eq. (4.6). Current experiments T2K and NOνA can improve the precision on sin 2 θ 23 . If the central value comes closer to 0.5, then it is possible to have large CP violation. Otherwise, the CP violation is constrained to remain small in this scenario.   Figure 3: The plot of sine squared vales of mixing angles for maximal δ CP through a Z 2 × Z 2 invariant perturbation with lines for 1σ and 2σ range.

Summary and Conclusion
We consider the phenomenology of a model with A 4 symmetry which predicts the tribimaximal form for the PMNS matrix. In this model, we have introduced Z 2 × Z 2 invariant perturbations in both the charged lepton and the neutrino sectors. We find that perturbations in the charged lepton sector alone (ζ = 0) can lead to acceptable values of θ 13 but do not give any CP violation. But, perturbations purely in the neutrino sector (s = 0) give rise to viable values of θ 13 and maximal CP violation. Any desired value of the CP violating phase δ CP can be obtained by choosing the appropriate values for the perturbations in the charged lepton and neutrino sectors. However, there is a tension between the requirement to obtain a large CP violation and the need to have the value of sin 2 θ 23 close to its best fit value. The current experiments may be able to settle this issue. The final Lagrangian has no overall residual symmetry even though the neutrino sector has a residual Z 2 symmetry. It will be interesting to explore whether there could be some consequences due to residual symmetry in neutrino sector.