Non-zero $\theta_{13}$ and leptonic CP phase with $A_4$ Symmetry

We consider a model based on $A_4$ symmetry to explain the phenomenon of neutrino mixing. The spontaneous symmetry breaking of $A_4$ symmetry leads to a co-bimaximal mixing matrix at leading order. We consider the effect of higher order corrections in neutrino sector and find that the mixing angles thus obtained, come well within the $3\sigma$ ranges of their experimental values. We study the implications of this formalism on the other phenomenological observables, such as CP violating phase, Jarlskog invariant and the effective Majorana mass $|M_{ee}|$. We also obtain the branching ratio of the lepton flavour violating decay $\mu\rightarrow e \gamma$ in the context of this model and find that it can be less than its present experimental upper bound.


I. INTRODUCTION
Neutrinos are the least interacting entities among the standard model particles and exist in three flavours (electron neutrino, muon neutrino and tau neutrino). They change their flavour as they propagate and this phenomenon is known as neutrino oscillation which occurs since the flavour eigenstates of neutrinos are mixture of mass eigenstates. The mixing is described by PMNS matrix [1], which can be parameterized in terms of three mixing angles and three CP violating phases as where c ij = cos θ ij and s ij = sin θ ij , θ 12 , θ 23 and θ 13 are the three mixing angles, δ CP is the Dirac phase and the other two Majorana phases come in P ν P ν = diag(e iρ , e iσ , 1) .
Neutrino oscillation experiments gained a lot of interest as a probe to neutrino mixing and mass spectrum since the oscillation probability depends on mixing angles, Dirac CP phase and the mass square differences (∆m 2 21 and ∆m 2 23 ). Results from earlier experiments indicated that θ 13 is very small, can be zero and the lepton mixing is very close to TBM (Tri-bimaximal mixing) [2], which predicts sin θ 13 = 0, sin 2 θ 23 = 1/2 and tan 2 θ 12 = 1/2. This made it possible to explain the neutrino mixing as TBM type, with small deviation due to perturbation in the charged-lepton or neutrino sector. There are many models which explain TBM mixing pattern on the basis of A 4 symmetry [18] with a certain set of Higgs scalars and vacuum alignments. Recent experimental observations of moderately large θ 13 [3], made neutrino mixing a little far from TBM type, but close to co-bimaximal mixing which predicts non-zero θ 13 (θ 13 = 0, θ 23 = π/4, δ CP = ±π/2) [4]. Supersymmetric models based on A 4 family symmetry, combined with the generalized CP symmetry [5], can also predict trimaximal (TM) lepton mixing, (in which either only the first column or only the second column of the lepton mixing matrix is assumed to take the TBM form), together with either zero CP violation or δ CP = ±π/2. Also models based on S 4 family symmetry and generalized CP symmetry [6] predict trimaximal lepton mixing and the Dirac CP is predicted to be either conserved or maximally broken. In Ref. [7], a minimal extension of the simplest A 4 model has been considered, which not only can induce non-zero θ 13 value, consistent with the recent observations, but also can correlate the CP violation in neutrino oscillation with the octant of the atmospheric mixing angle θ 23 . In this paper, we would like to consider a model based on A 4 symmetry which gives co-bimaximal mixing in neutrino sector at leading order. To accommodate deviations in mixing angles to make them compatible with the experimental results, we include a perturbation in neutrino sector due to higher order corrections, which can be represented as five-dimensional operators. The best-fit values and 3σ ranges of neutrino oscillation parameters taken from Ref. [8] are given in Table I.
The paper is organized as follows. The details of our model is presented in section II.
In sections III and IV, we discuss the vacuum alignment and lepton flavour violating muon decay µ → eγ in the context of the model. In section V, we describe the higher order corrections in neutrino sector and we conclude our discussion in section VI.

Mixing Parameters
Best  The model is based on A 4 group [9], which is the group of even permutation of four objects and is the smallest non-Abelian discrete group with triplet irreducible representation. It has four irreducible representations: 1, 1 , 1 and 3, with the multiplication rule 3 × 3 = 1 + 1 + 1 + 3 + 3 .
In this work, our discussion is limited to the leptonic sector. The particle content of the model includes, in addition to standard model fermions (i.e., the lepton doublets l iL and charged lepton singlets l iR ), three right-handed neutrinos (ν iR ), four Higgs doublets (φ i , φ 0 ) and three Higgs singlets (χ i ). They belong to four irreducible representations of A 4 as given in Table II. Here A 4 symmetry is accompanied by an additional U (1) X symmetry as discussed in Ref. [18], which prevents the existence of Yukawa interactions of the forml iL ν iRφi and l iL l iR φ 0 as l iL , l iR ,φ 0 have quantum number X = 1, and all other fields have X = 0. The phenomenologically disallowed Nambu-Goldstone boson does not arise in this case as U (1) X symmetry does not break spontaneously but explicitly. Thus, the Yukawa Lagrangian for the leptonic sector is given as [11] whereν iR are antiparticles of ν iR and l iL φ i , l iL φ i and (ν iRνiR ) 3 are 1 , 1 and triplet representations of A 4 respectively. As the scalars φ i , φ 0 and χ i get vacuum expectation values v i , v 0 and ω i respectively, the above Lagrangian becomes where M l , M D and M R are charged-lepton, Dirac neutrino and right-handed neutrino mass matrices and have the forms where I is the identity matrix, and For the vacuum alignment v i = v, the charged lepton sector can be diagonalized by the transformation: where U ω is the Cabibbo-Wolfenstein matrix given in Eq. (3). The light neutrino mass is given by the type-I seesaw formula Since M D is proportional to an identity matrix, the neutrino mixing matrix will be the one which diagonalizes the right-handed neutrino mass matrix M R . The Majorana mass matrix M R can be parameterized as in a basis where charged-lepton mass matrix is not diagonal. However, in the charged lepton ω and can be diagonalized by tri-bimaximal (TBM) mixing matrix for D = C = 0, which we don't need as it gives vanishing θ 13 . Even if these conditions are not satisfied some of the off-diagonal elements of M R become zero in TBM basis and one can go to the TBM basis through the transformation where With the condition which can be diagonalized by U R , having the form where s and c stand for sin θ and cos θ respectively and satisfy the relation It should be noted that, this ratio should be real, since ω 1,2 are VEV of real scalar fields χ i .
The condition C = −D can be realized with the vacuum alignment χ i = (ω 1 , ω 2 , −ω 2 ) as discussed in [12]. Thus, the lepton mixing matrix becomes which basically known as co-bimaximal mixing matrix and predicts the mixing angles and CP violating Dirac phase as θ 13 = 0, θ 23 = π/4 and δ CP = ±π/2. Also, the mixing angles θ 12 and θ 13 are not independent and one can express sin 2 θ 12 in terms of sin 2 θ 13 as To illustrate these results, we show in Fig. 1

III. VACUUM ALIGNMENT
The complete scalar potential is given by with The last term in Eq. (24) breaks U (1) X symmetry explicitly and removes Goldstone boson which occurs due to the spontaneous breaking of U (1) X symmetry. In this model, we have the vacuum alignment φ 0 = u, φ i = (v, v, v), and χ i = (w 1 , w 2 , −w 2 ) which is a possible minimum of scalar potential for V (φ i χ i ) = 0. A vanishing V (φ i χ i ) can be achieved in the limit χ i decouples from rest of the field as mentioned in Ref. [18]. The decoupling of χ i requires λ χ → 0, λ φ 0 χ i → 0. To generate an acceptable neutrino mass spectrum λ χ has to be nonzero but can be small. A small but nonzero λ χ will generate a sufficiently small V (φ i χ i ) which will be too small to alter vacuum alignment considerably. In this limit the minimization condition on u is given by The above Eq. (26) has a solution a.
Thus, for this case, i.e., for |u| 2 |v i | 2 minimization conditions on v i are given as Considering λ φ i 4 as real, one can get the solution which is allowed.

b.
Minimization conditions on w i is given by one of the solutions of above set of equations is w 1 = 0, w 3 = −w 2 = 0, which is the vacuum alignment condition for χ i .

IV. EFFECT OF ADDITIONAL HIGGS DOUBLETS ON LEPTON FLAVOUR VIOLATING DECAY µ → eγ
Since | u | 2 v 2 one can neglect the mixing between φ i and φ 0 and the mass-squared matrices in the Re[φ 0 i ], Im[φ 0 i ], and φ ± i bases have the same form [19].

PERTURBATION IN NEUTRINO SECTOR
In this section, we will consider the perturbations to mass matrices due to higher order corrections. Prominent corrections come from five-dimensional operator λ ijνiRνjR χ i χ j which modifies right-handed neutrino mass matrix. Charged lepton and Dirac neutrino masses also receive corrections from λ jkl il φ i l jR χ i and λ jkl ilφ0 ν jR χ i respectively, and here we are neglecting those corrections since they allow the mixing of χ i with other fields.
These corrections will modify the light neutrino mass matrix and the inverse of modified light neutrino mass matrix in TBM basis can be parameterized as Hence, in the charged lepton diagonal basis light neutrino mass matrix can be diagonalized by where with s = sin θ and c = cos θ .
To obtain mixing angles we compare lepton mixing matrix U (39) with PMNS matrix (1), i.e., The mixing angles sin 2 θ 12 , sin 2 θ 23 and sin 2 θ 13 are related to the elements of U as where U ij is the ij th element of the lepton mixing matrix U . Now using Eqs. (3), (13), (39) and (42), we obtain Another important parameter is J CP , the Jarlskog invariant, which is a measure of CP violation, is found to have the value in this model as In standard parametrization, the value of J CP is J CP = 1 8 sin 2θ 12 sin 2θ 23 sin 2θ 13 cos θ 13 sin δ CP .
Comparing Eqs. (46) and (47), we obtain where To show that the model predicts the mixing angles compatible with the observed data,we obtain the allowed parameter space compatible with the 3σ range of the observed data by Several experiments like KamLAND-Ze [13], EXO [14] and GERDA [15] are searching for the neutrino-less double beta decay. These experiments put upper bound on |M ee |, the (1,1) element of neutrino mass matrix, since the half-life of 0νββ decay is proportional to |M ee | 2 .
The expression for |M ee | in the flavor basis is where m 1 , m 2 , and m 3 are light neutrino masses and U 1j 's are elements of first row of the  lepton mixing matrix U , which are given as The lowest upper bound on |M ee | is 0.22 eV came from GERDA phase-I data. Here we study the variation of |M ee | with the lightest neutrino mass m 1 (m 3 ), in the case of normal (inverted) hierarchy as shown in Fig. 4. In our calculation we have used the relations for normal hierarchy and m 1 = m 2 3 + ∆m 2 13 , for inverted hierarchy, and obtained upper limit on m 1 (m 3 ) as 0.071 (0.065 eV) taking into account the cosmological upper bound on Σ i m i as 0.23 eV [16]. Another observable is the kinetic electron neutrino mass in beta decay (m e ), which is probed in direct search for neutrino masses, can be expressed as m e = |U 11 | 2 m 2 1 + |U 12 | 2 m 2 2 + |U 13 | 2 m 2 3 .
In the right panel of Fig. 4, we show the variation of m e with the lightest neutrino mass m 1 (m 3 ) for normal hierarchy (inverted hierarchy) case, and the upper limit on m e is found to be 0.07 (0.08) eV.

VI. CONCLUSIONS
We consider a model based on A 4 symmetry, which gives co-bimaximal form (θ 23 = π/4, δ CP = ±π/2 and θ 13 = 0) for the leading order neutrino mixing matrix. There are four (φ) of φ i behaves exactly as standard model Higgs doublet while neutral component of the other two (φ , φ ) contribute to the lepton flavour violating decays such as µ → eγ. We have considered higher order corrections in neutrino sector coming from five-dimensional operators after spontaneous breaking of A 4 symmetry. The mixing angles, thus obtained are found to be within the 3σ ranges of their experimental values. The CP violating phase δ CP is found to be around the region ±π/2, and the upper limit on the Jarlskog invariant is O(10 −2 ). We also studied the variation of the effective neutrino mass |M ee | with the lightest neutrino mass m 1 (m 3 ) in the case of normal (inverted) hierarchy and found its value to be lower than the experimental upper limit for all allowed values of m 1 (m 3 ).