Analyses of scalar potential and lepton flavor violating decays in a model with $A_4$ symmetry

We have considered a model, originally proposed by Ma and Wegman, where the mixing pattern in neutrino sector is explained with three Higgs doublets, six Higgs triplets and $A_4$ symmetry. The mixing pattern is explained with the help of vacuum expectation values (VEVs) of the above mentioned doublets and triplets. In order to study about the VEVs of the scalar fields, we construct the full invariant scalar potential of this model. After minimizing this scalar potential, we have found that two Higgs triplets can acquire zero VEVs. In order to generate non-zero VEVs to all the six Higgs triplets, we have added two more Higgs doublets to the model. Thereafter we have demonstrated that the current neutrino oscillation data can be consistently explained in our model. To study some phenomenological implications of this model, we have worked out on the branching ratios for lepton flavor violating decays.


Introduction
Neutrino sector can give hints about physics beyond the standard model (SM) [1]. The masses of neutrinos are tiny as compared to other fermion masses [2]. In order to explain the tiny masses for neutrinos, one has to extend the SM. In addition to the masses of neutrinos, mixing pattern in neutrino sector can also give a hint to physics beyond the SM. From the global fits to neutrino oscillation data [3], the three neutrino mixing angles are found approximately close to the tri-bimaximal mixing (TBM) [4]. To understand this mixing pattern in neutrino sector, the SM should be extended with additional symmetries and particle content [5].
In this work, we consider the Ma-Wegman (MW) model [6], where A 4 symmetry [7] is introduced to explain the neutrino mixing pattern. For early works on A 4 symmetry, see Refs. [8]. In the MW model, the scalar sector contains three Higgs doublets and six Higgs triplets. Due to the presence of scalar Higgs triplets, neutrinos acquire masses via type II seesaw mechanism [9] in this model. The above mentioned scalar fields and lepton doublets are charged under A 4 symmetry in such a way that a realistic neutrino mixing pattern can be explained. A unique feature of the MW model is that the six Higgs triplets, which are responsible for obtaining the neutrino mixing pattern, are charged under all possible irreducible representations of A 4 symmetry. See Ref. [7] for an introduction to A 4 symmetry. In Appendix A we have summarized product rules among the irreducible representations of A 4 symmetry.
The A 4 symmetry in the MW model is spontaneously broken when the neutral component of doublet and triplet Higgs fields acquire VEVs. The VEVs of the triplet Higgses generate a mixing mass matrix for neutrino fields. After diagonalizing this mass matrix, one can obtain neutrino masses and mixing angles. In the work of MW model [6], this diagonalization has been done after making some assumptions on the VEVs of triplet Higgs fields, and thereby, it is concluded that neutrino masses can have normal ordering. This problem of diagonalizing the neutrino mass matrix of the MW model has been revisited in Ref. [10]. In the work of Ref. [10], after relaxing some of the assumptions made in Ref. [6] and also after using some approximation procedure [11], diagonalization has been done for the neutrino mass matrix of the MW model. Thereafter, it is concluded that both normal and inverted orderings for neutrino masses are possible in the MW model, apart from explaining the mixing pattern in neutrino sector.
As described above, the VEVs of scalar triplet Higgses are responsible for generating the neutrino masses and mixing angles in the MW model. One obtains the VEVs of scalar fields after minimizing the invariant scalar potential among these fields. The scalar potential in the MW model contains both the doublet and triplet Higgs fields. Minimization for this scalar potential has not been done before. On the other hand, minimization for the invariant scalar potential containing only the Higgs doublets has been done in Ref. [7], where it is shown that there exist a parameter region in which the three Higgs doublets of this model acquire the same VEV. This is known as the vacuum alignment of the Higgs doublets [12], which is necessary to achieve in order to diagonalize the charged lepton mass matrix, and thereby to explain the mixing pattern in neutrino sector.
In this work, in order to see the implications of scalar potential on neutrino masses and mixing pattern, we write the full invariant scalar potential containing the three doublet and six triplet Higgses of the MW model. After minimizing this scalar potential, we have found that the two triplets, which are charged under the non-trivial singlet representations of A 4 symmetry, acquire zero VEVs. It is to remind here that in our previous work of Ref. [10], we assumed the VEVs of all triplet Higgses be non-zero and later showed that neutrino oscillation data can be explained in the MW model. Moreover, it is stated before that we followed a specific diagonalization procedure in our previous work of Ref. [10]. Now, in this work, after finding that two Higgs triplets can acquire zero VEVs, with the diagonalization procedure of Ref. [10], we have found that the current neutrino oscillation data cannot be consistently explained. To alleviate the above mentioned problem, we add two more Higgs doublets to the MW model. After doing this, we show that at the minimum of the scalar potential, all the six Higgs triplets can acquire non-zero VEVs.
As a result of this, we demonstrate that the neutrino oscillation data can be fitted in this model for both normal and inverted neutrino mass orderings. While doing the above mentioned minimization, we also address the problem on vacuum alignment of the Higgs doublets. We show that sufficient parameter region exist in this model, where the vacuum alignment of the necessary Higgs doublets can be achieved.
After analyzing the scalar potential, it is worth to study some phenomenological consequences of our model. We argue below that the scalar fields of our model can drive lepton flavor violating (LFV) processes such as → γ and → 3 . Here, and are charged leptons belonging to different families. None of the above mentioned LFV decays have been observed in experiments, and as a result of that, upper limits on the branching ratios of these processes have been obtained [13]. See Refs. [14], for related studies on LFV processes in neutrino mass models. In our model, the above mentioned LFV decays are driven by the scalar fields which are charged under the A 4 symmetry. Hence, one can expect that these decays carry imprints of A 4 symmetry. In this work, one of our interests is to study signatures of A 4 symmetry in LFV decays. In a related direction to this, see Ref. [15].
The scalar triplet Higgses of our model drive LFV decays, since the Yukawa couplings for lepton doublets are flavor violating in a type II seesaw framework [16,17]. We compute branching ratios for the decays → 3 and → γ in our model. The decays → 3 are driven by doubly charged scalar triplets at tree level, whereas, the decays → γ are driven by doubly and singly charged scalar triplets at 1-loop level. The above mentioned LFV decays can also be driven by scalar fields of doublet Higgses, however, the contribution from these scalars has been neglected in this work. We comment about this contribution later. While computing the branching ratios for the above mentioned decays, one needs to know the mass eigenstates of the doubly and singly charged scalar triplets. These we obtain from the invariant scalar potential of our model, which we have described above. The branching ratios of the LFV decays in our work depend on Yukawa couplings and the masses of above mentioned scalar fields. We have found that for some decays the branching ratios are vanishingly small, if we assume degenerate masses for triplet scalar fields. Another fact we have found is that, due to the presence of A 4 symmetry, some of the couplings between charged scalar triplets and leptons can depend on one another. As a result of this, branching ratios for some LFV decays can depend on each other. The above mentioned facts are some of the signatures of A 4 symmetry in our model. Since the Yukawa couplings depend on neutrino oscillation observables, numerically we study the variation of these branching ratios in terms of neutrino mixing angles and the CP violating Dirac phase δ CP .
The paper is organized as follows. In the next section, we briefly describe the MW model and present essential results from our earlier work [10] on this model. In Sec.
3, we construct the full invariant scalar potential of this model and give our analysis on the minimization of this potential. We study the implication of this analysis on the neutrino mixing pattern by taking into account of the results of our previous work [10]. We demonstrate that by adding two additional Higgs doublets, one can explain the neutrino mixing pattern consistently in our model. In Sec. 4, we study the LFV decays of our model. In Sec. 5, we describe future directions based on the phenomenology of our model. We conclude in the last section. In Appendix A, we have given the product rules of A 4 symmetry, which are useful for making invariant terms in our scalar potential.
In Appendix B, we have listed all different quartic terms of the scalar potential, which contain only the Higgs triplets.

The MW model and essential results from it
In this section, we describe the MW model [6]. As stated in the previous section, the method of diagonalizing the neutrino mass matrix of this model has been improved in Ref. [10]. The essential results, related to neutrino masses and mixing angles, from the work of Ref. [10] are also presented in this section. These results are used in our study on LFV decays, which is presented in Sec. 4.
The relevant fields of the MW model, along with their charge assignments under the electroweak and A 4 symmetries are tabulated in Table 1. With the charge assignments of Table 1: Fields in the lepton sector of the MW model [6]. Here, i = 1, 2, 3 and j = 4, 5, 6.
Here, i, j, k = 1, 2, 3. h ijk are Yukawa couplings, whose form is determined by A 4 symmetry, which can be seen in Ref. [7]. Assuming that the three Higgs doublets acquire the same VEV, after the electroweak symmetry breaking, we get a mixing mass matrix for charged leptons. This mass matrix can be diagonalized with the following transformations on the charged lepton fields [7].
In the neutrino sector, the invariant Lagrangian is Here, y 1 , y 2 , y 3 , y are dimensionless Yukawa couplings. Also, L c i , where i = 1, 2, 3, are charge conjugate doublets of L i . The above invariant Lagrangian can be obtained from the product rules of A 4 symmetry, which are given in Appendix A. After giving VEVs to neutral component of ξ k , from Eq. (3), we get mixing mass matrix for neutrino fields, which is given below [6].
Here, ξ 0 i = v i , i = 1, · · · , 6. As stated in the previous section, the masses for neutrinos are very small. In order to obtain small masses for neutrinos, using the above relations, we can take either the Yukawa couplings or the VEVs of Higgs triplets to be small. In this work, we choose the VEVs of Higgs triplets to be small so that the Yukawa couplings can be O (1). With this choice, we can notice that LFV decays in this model are unsuppressed, and as explained in the previous section, study of LFV decays is another topic of interest in this work.
The matrix in Eq. (5) can be diagonalized after assuming b − c, e, f to be small and also after applying the following transformation on the neutrino fields [10].
In the unitary matrix U [18,19], the parameters are complex and the real and imaginary parts of these are assumed [10] to be less than or of the order of sin θ 13 ∼ 0.15 [3], where θ 13 is a neutrino mixing angle. Here, one can notice that U is unitary only up to first order in parameters. From the above equation, we can see that U gives a perturbation to U T BM , which can produce deviation from TBM mixing pattern. There are other ways to parametrize these perturbations. However, in this work we stick to the above mentioned parametrization, which is suggested in Refs. [18,19]. Now, while diagonalizing the neutrino mass matrix in Eq. (5), terms which are of the order of sin 2 θ 13 ∼ m 2 s m 2 a ∼ 10 −2 [3] have been neglected [10]. Here, m s and m a are the square-root of solar and atmospheric mass-square differences among the neutrino fields, respectively. The central values for these mass-square differences are given below [3].
. (8) Here, m 1,2,3 are neutrino mass eigenvalues and NO(IO) represents normal(inverted) ordering. In order to fit the above mass-square differences, the neutrino mass eigenvalues can be taken as follows.
As described previously, terms of the order of or higher than that of sin 2 θ 13 ∼ m 2 s m 2 a are neglected in the diagonalization of M ν of Eq. (5) [10]. As a result of this, the neutrino mass eigenvalues in terms of model parameters have been found to be [10] The above expressions are valid in NO and IO cases. The relations for other model parameters containing in M ν depend on the neutrino mass ordering. Expressions for these are given below [10].

.
IO : Now, after diagonalizing the mass matrix M ν , one can get expressions for neutrino mixing angles. The procedure for this is explained below. After comparing the transformations for charged leptons and neutrinos of Eqs. (2) and (7), the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix can be written as The PMNS matrix is parametrized in terms of three neutrino mixing angles and δ CP , in accordance with the PDG convention [13]. Using this parametrization in Eq. (12) and after solving the relations of this matrix equation, the leading order expressions for the three neutrino mixing angles and δ CP are found to be [10] sin Here, Re( ij ) and Im( ij ) are real and imaginary parts of ij , i, j = 1, · · · , 3.
Using the above expressions in Eq. (5), we can see that all Yukawa couplings of the MW model can be determined in terms of neutrino oscillation observables and the VEVs of Higgs triplets. Here, one can notice that the coupling y can be obtained from either d or e. The fact is that v 4 and v 5 are not independent parameters. As a result of this, we can consider the following two cases in order to determine y.
In case I(II), v 4 (v 5 ) is independent parameter and v 5 (v 4 ) is determined in terms of v 4 (v 5 ).
An interesting point is that if we choose v 4 as an independent parameter, the coupling y do not depend on neutrino mixing angles and δ CP . On the other hand, in case II, y depends on neutrino mixing angles and δ CP . The above mentioned cases can make a difference in the branching ratios for LFV decays of this model, which is presented in Sec.

Analysis of scalar potential
In the MW model [6], three Higgs doublets and six Higgs triplets exist. From the previous section, we have seen that the VEVs of Higgs triplets generate masses and mixing angles for neutrino fields. The VEVs for these fields arise after minimizing the scalar potential of this model. Hence, in this section, we write the full invariant scalar potential of the MW model. Thereafter, we analyze the implications of this potential on neutrino mixing.

Scalar potential of the MW model
The invariant scalar potential in the MW model can be written as Here, V 0 (Φ) is a potential which depends only on the Higgs doublets, whose form is already given in Ref. [7]. V 1 (Φ, ξ) contains terms involving both Higgs doublets and triplets.
V Q (ξ) contains exclusively the quartic interaction terms among the Higgs triplets. In the minimization of the scalar potential, quartic terms in V Q (ξ) give negligibly small corrections, due to the following reasons. From precision electroweak tests [13], ρ parameter gives a constraint on VEV of triplet Higgs to be less than about 1 GeV. In the MW model, since three Higgs doublets exist, we can choose the VEVs of Higgs doublets to be around 100 GeV. Hence, while doing the minimization, terms in V Q (ξ) are at least suppressed by 10 −4 as compared to that in V 1 (Φ, ξ). In our work, as stated in the previous section, we choose VEVs of Higgs triplets to be much smaller than 1 GeV, say around 0.1 eV, in order to explain the small neutrino masses. Shortly below, we give arguments for making triplet Higgs VEVs to be as small as 0.1 eV. For the above mentioned reasons, we can notice that terms in V Q (ξ) give negligibly small contributions to the minimization of the potential.
Hence, we omit these terms in our analysis. However, for the sake of completeness, we present all the invariant terms of V Q (ξ) in Appendix B. In order to write the invariant , we follow the work of Ref. [20]. In Ref. [20], invariant scalar potential under the electroweak symmetry is given, for the case of one doublet and triplet Higgses. To write the terms in V 1 (Φ, ξ) + V Q (ξ), we generalize the potential given in Ref.
[20], by including three Higgs doublets, six Higgs triplets and A 4 symmetry. In order to make the scalar potential invariant under A 4 symmetry, we follow the product rules of A 4 symmetry, which are given in Appendix A.
To write the scalar potential of the MW model, we define the following quantities.
Here, f (ξ) is a function depending on the Higgs triplet fields. Now, we have [7] In the above, m 2 has mass-square dimension and λ parameters are dimensionless. The invariant terms in V 1 (Φ, ξ) can be written as In the above equation, m 2 0,1,2,3 have mass-square dimensions, µ parameters have mass dimensions and λ parameters are dimensionless. Here,Φ k = iσ 2 Φ * k , k = 1, 2, 3. We assume λ parameters to be O(1). As stated before, φ 0 give mass scale for scalar triplet Higgses. Since we want these scalar triplet Higgses to be produced in the LHC experiment, we take m 2 0,1,2,3 ∼ m 2 T ∼ (100 GeV) 2 . Now, after minimizing Eq. (19) with respect to triplet Higgses, naively we expect the VEVs of these Here, µ T represents any of the µ parameters of Eq. (19). After using the above mentioned choices of the parameters, we can see that the VEVs of triplet Higgses can be as small as 0.1 eV, provided the µ parameters are suppressed to around 0.1 eV. By suppressing the µ parameters, one can realize the hierarchy in the VEVs of doublet and triplet Higgses. See Ref. [16], for a loop induced mechanism in order to explain the smallness of µ parameters.
The minimization of V 0 (Φ) has been done in Ref. [7] and it is shown that φ 0 i = v can be achieved for i = 1, 2, 3. In this work, doublet Higgses have interactions with triplet Higgses. Since we are taking VEVs of triplet Higgses to be around 0.1 eV, the contribution from V 1 (Φ, ξ) is negligibly small in comparison to V 0 (Φ) . Hence, in this work, we get φ 0 i ≈ v for i = 1, 2, 3. As stated previously, this is known as the vacuum alignment of Higgs doublets, which is necessary in order to diagonalize the charged lepton mass matrix, which is described around Eq. (2). Now, after minimizing the V 1 (Φ, ξ) with respect to neutral components of ξ 2 and ξ 3 , we get From the above equations, we get v 2 = v 3 = 0. This implies b = c = 0. Using this in Eq.
(11), we get 13 = 0. Thereafter, relations in Eq. (13) can be solved for δ CP = π, which is allowed for the case of NO by the current neutrino oscillation data [3]. As a result of this, at leading order, we get the following constraint relation The above constraint relation cannot be satisfied in the allowed 3σ regions for sin 2 θ 23 and sin 2 θ 13 [3].
The problem described in the previous paragraph arises due to the fact that ξ 2 and ξ 3 , which transform as 1 and 1 respectively under A 4 , acquire zero VEVs. On the other hand, the other Higgs triplets ξ 1 and ξ j , which transform as 1 and 3 respectively under A 4 , can acquire non-zero VEVs. Let us mention here that in Ref. [19] a model with ξ 1 and ξ j is presented in order to explain neutrino mixing pattern. It is shown that the model of Ref. [19] can consistently explain neutrino mixing pattern and can predict normal ordering of masses for neutrinos. Hence, one can see that the MW model, for the case of ξ 0 2,3 = 0, effectively reduces to that of Ref. [19], as far as neutrino mixing is concerned. As a result of this, even if ξ 0 2,3 = 0, the MW model can explain neutrino mixing pattern but may only predict normal mass ordering for neutrinos. In this regard, it is worth to see if the MW model can be modified in such a way that it can explain both normal and inverted mass orderings for neutrinos. In our earlier work [10], we had shown that the above mentioned orderings are possible in the MW model, provided ξ 0 2,3 = 0. So to solve the above mentioned problem, one needs to find a mechanism which can give ξ 0 2,3 = 0 in the MW model.
One can notice that, because of the vacuum alignment of Higgs doublets, the tri-linear couplings µ 2 , µ 3 of Eq. (19) do not contribute to ξ 0 2 and ξ 0 3 after minimizing the scalar potential. Whereas, the other tri-linear couplings µ 1 , µ can contribute to the VEVs of rest of the Higgs triplets, even with the vacuum alignment of Higgs doublets. One cannot break the vacuum alignment of Higgs doublets in the MW model, since it will affect the diagonalization of charged lepton mass matrix, which in turn has an effect on the mixing pattern in neutrino sector. Hence, in order to give non-zero VEVs to ξ 2,3 , one can introduce additional tri-linear couplings involving these fields. We know that ξ 2,3 are charged under 1 , 1 of A 4 symmetry. With 1 , 1 , the following are the only two singlet combinations, which can represent tri-linear terms in the potential: 1 ×1 ×1 , 1 ×1 ×1 .
Hence, in the additional tri-linear couplings containing ξ 2,3 , the Higgs doublets should be charged under 1 and 1 of A 4 symmetry. As a result of this, we propose additional Higgs doublets Φ 5 and Φ 6 which transform as 1 and 1 respectively under A 4 symmetry. Now, one can see that the following terms can exist in the scalar potential, which can give non-zero VEVs to ξ 2 and ξ 3 : However, the Higgs doublets Φ 5 and Φ 6 can give rise to extra terms in the scalar potential with the fields Φ i , i = 1, 2, 3. These extra terms can affect the vacuum alignment of Higgs doublets of the MW model. We study these topics in the next subsection.

Extension of the MW model with two additional Higgs doublets
As described previously, in order to get non-zero VEVs to ξ 2 and ξ 3 , we add the Higgs doublets Φ 5 and Φ 6 to the MW model. Since these Higgs doublets are charged under 1 and 1 of A 4 symmetry, with the charge assignments given in Table 1, one can notice that they do not generate Yukawa couplings for charged leptons and neutrinos. However, the Higgs doublets Φ 5,6 can have interactions with the other Higgs doublets Φ 1,2,3 and also with the Higgs triplets of this model. As a result of this, the scalar potential of the MW model, which is given in Eq. (16), will change to Here, V 0 (Φ) and V 1 (Φ, ξ) contain terms between Φ 5,6 and already existing scalars of the MW model. Their forms are given below.
The VEVs of Higgs triplets can be found after minimizing the potential V 1 (Φ, ξ) + V 1 (Φ, ξ). Expressions for these are given below.
From Eqs. (28) and (29), we can notice that the VEVs for ξ 2 and ξ 3 can be non-zero due to the contribution from µ parameters. In fact, using Eqs.

LFV decays
In this section, we compute the branching ratios for LFV decays in the scenario where we extend the MW model with the Higgs doublets Φ 5,6 . As described in Sec. 1, LFV decays can be of the following two types: → 3 , → γ. In our scenario, decays of the form to compute the branching ratios for these decays, one needs to obtain the mass eigenstates for doubly and singly charged scalars. Below we present these mass eigenstates.
It is to be noted that neutral fields from doublet Higgses, other than the standard model Higgs, can also contribute to the above mentioned LFV decays [7]. Most of these decays are suppressed due to smallness of charged lepton Yukawa couplings. However, there are some decays, whose amplitudes are proportional to tau Yukawa coupling, can give appreciable contribution [7], provided the masses of the neutral fields are low. To study the contribution of neutral fields to LFV decays in our scenario, we have to diagonalize the mixing masses among the neutral fields of the Higgs doublets. It is to remind here that since the VEVs of triplet Higgses are very small, the mixing between neutral fields of doublets and triplets can be neglected. In this work, we assume that the masses for the above mentioned neutral fields are high enough that their contribution to LFV decays are suppressed. In this regard, let us mention that in Ref. [21], LFV decays driven by neutral scalar fields are studied. The work done in Ref. [21] is based on some flavor models [22] containing A 4 symmetry, where neutral flavon fields induce LFV decays.

Mass eigenstates of doubly and singly charged scalars
The doubly charged scalars belong to the Higgs triplets of our model. The masses for these fields can be obtained from the scalar potential of this model, which is given in the previous section. Since six triplet Higgses exist in the model, one can expect mixing masses among the doubly charged scalars. These mixing masses are given below.
From Eq. (33) we can notice that there is no mixing between ξ ++ 1,2,3 and ξ ++ 4,5,6 . However, from the quartic terms of the potential, which are given in Appendix B, there may be mixing between the above mentioned doubly charged scalars. One can expect this mixing to be proportional to square of the VEVs of Higgs triplets, which in our case is very small. Hence, we neglect the above mentioned mixing. After diagonalizing the matrices X, Y of Eq. (33), we get mass eigenstates for doubly charged scalars, which are defined below.
Here, i = 1, 2, 3 and ξ (m)++ k , where k = 1, · · · , 6, are the mass eigenstates of the doubly charged scalars. The unitary matrices U ++ , V ++ diagonalize X, Y as In analogy to doubly charged scalars, mass eigenstates for singly charged scalars can be obtained. Singly charged scalars belong to both doublet and triplet Higgses. In our scenario, due to smallness of VEVs of Higgs triplets, we can neglect the mixing among singly charged scalars between doublet and triplet Higgses. Singly charged scalars of doublet Higgses can drive LFV decays → γ through charged lepton Yukawa couplings.
One can expect this contribution to be small unless these decays are induced by tau Yukawa coupling. To simplyfy our analysis we assume the masses for the singly charged scalars of doublet Higgses are high enough that their contribution to LFV decays is suppressed. As a result of this, in this model, the above mentioned LFV decays are dominantly driven by singly charged scalars of triplet Higgses. For these reasons, below we present the mass eigenstates for singly charged scalars from triplet Higgses. These scalars can have mixing masses, which can be written as 6 |v| 2 , )|v| 2 , y 13 = (y 12 ) * , y 23 = y 12 .(39) From Eq. (38), in analogy to doubly charged scalars, we can notice that there is no mixing between ξ + 1,2,3 and ξ + 4,5,6 at leading order. Now, we can define the mass eigenstates for singly charged scalars as Here, i = 1, 2, 3 and ξ (m)+ k , where k = 1, · · · , 6, are the mass eigenstates of singly charged scalars. The unitary matrices U + , V + diagonalize X , Y as

Branching ratios of → 3
In this subsection, we compute the branching ratios for decays of the form → 3 . Since Here, C is the charge conjugation matrix and (m) j is a mass eigenstate of charged lepton.
From the above equation, we can notice that some of the couplings between doubly charged scalars and charged leptons are related to one another. This is a result due to A 4 symmetry of the model. This result has implications on the branching ratios of the decays of the form → 3 . We will explain these implications shortly later.
Using the couplings in Eq. (43), we compute the branching ratios for → 3 , after neglecting the masses of final state charged leptons. Branching ratios for τ decays are found to be Here, G F is the Fermi constant and Br(τ → µνν) = 0.1739 [13]. Moreover, the indices i, j, k = 1, 2 are for electron and muon fields. S = 1(2) if j = k ( j = k ). In the above equation, one should use f j,k 1n = f k,j 1n and f j,k 2n = f k,j 2n . These relations follow from the Lagrangian of Eq. (43). The branching ratio for µ → 3e is In this work, we have assumed Br(µ → eνν) = 100%.
As stated before, some relations exist among the couplings in the Lagrangian of Eq.
On the other hand, in the limit where the masses for all doubly charged scalars are degenerate, Eqs. (44) and (45) imply that the branching ratios for following decays go to zero: τ →ēee, τ →ēeµ, τ →μµµ, τ →μeµ, µ →ēee. Relations among the branching ratios described in Eqs. (46) and (47) are due to the A 4 symmetry of our model. In the work of Ref. [21], which is based on A 4 symmetry, a similar kind of relations among various branching ratios for → 3 have been derived. Since the flavor models considered in Ref. [21] are different from our model, the relations for branching ratios given in Ref. [21] are different from Eqs. (46) [23]. In order to satisfy this constraint in our work, we study the branching ratio of µ →ēee. From Eq.
(45), we can see that Br(µ →ēee) depends on masses of doubly charged scalars and on couplings between doubly charged scalars and charged leptons. These couplings, which can be seen from Eq. (43), depend on neutrino Yukawa couplings and the unitary matrices which diagonalize the mixing masses for doubly charged scalars. Hence, the masses for doubly charged scalars and the above mentioned unitary matrices are determined from the parameters of the scalar potential. On the other hand, the neutrino Yukawa couplings are determined from the VEVs of Higgs triplets and neutrino oscillation observables. This fact can be seen from Eqs. (6), (14) and (13). From these equations, one can notice that the neutrino Yukawa couplings depend on θ 13 and θ 23 , but not on θ 12 .
As described above, Br(µ →ēee), in our work, depend on neutrino oscillation observables, VEVs of Higgs triplets and parameters of scalar potential. It is interesting to see the variation of Br(µ →ēee) in terms of neutrino oscillation observables. Hence, we have fixed VEVs of Higgs triplets and parameters of scalar potential to some specific values in our analysis. The details of our analysis have been described below.
To simplify our numerical analysis, we take all the independent Higgs triplet VEVs to be same as v T . It is to remind here that the VEVs v 4 , v 5 are not independent. It is discussed in Sec. 2 that the Yukawa coupling y can be determined in terms of v 4 or v 5 . As a result of this, from Eq. (15), we can see that in case I(II) v 4 (v 5 ) is independent parameter.
As for the masses of doubly charged scalars, they are determined after diagonalizing the X, Y matrices of Eq. (33). Since there are several λ parameters exist in X, Y , for the sake of illustration, we choose all these λ parameters to be 0.1. We take the mass-square parameters of X, Y as m 2 1 = m 2 2 = m 2 3 = m 2 0 = (850 GeV) 2 . We have taken the VEVs for doublet Higgses as v = v 5 = v 6 = 174/ √ 5 GeV. With the above mentioned parameters, we have found the masses for all doubly charged scalars to be slightly above 850 GeV.
These mass values for doubly charged scalars satisfy the lower bound on them, which is obtained by the LHC experiment [24].
In Figs. 1 and 2, we have given the plots for branching ratios of µ →ēee, in the cases of NO and IO respectively. As already described above, the neutrino Yukawa couplings of our model, up to the leading order, do not depend on the mixing angle θ 12 . Hence, in Figs. 1 and 2, Br(µ →ēee) is plotted against sin 2 θ 13 , sin 2 θ 23 and δ CP . The allowed ranges and best fit values for the neutrino mixing angles and δ CP , which are used in this work, are tabulated in Table 2. In Figs. 1 and 2, in the plot between Br(µ →ēee) and sin 2 θ 13 , we have fixed the best fit values for sin 2 θ 23 and δ CP , which are given in Table   2. Similar kind of things have been done in other plots of Figs. 1 and 2. In the plots of both these figures, we have taken the lightest neutrino mass to be zero and the other neutrino masses are computed from Eqs. (8) and (9). In Figs. 1 and 2, we have taken v T to be 0.08 eV and 0.14 eV respectively. If we decrease v T below than the above mentioned   for the cases I and II respectively. δ CP is expressed in degrees. In these plots, v T = 0.14 eV. In the top-left plot, sin 2 θ 23 and δ CP are fixed to the best fit values of Table 2. In the top-right plot, sin 2 θ 13 and δ CP are fixed to the best fit values of Table 2. In the bottom plot, sin 2 θ 13 and sin 2 θ 23 are fixed to the best fit values of Table 2. In all these plots, lightest neutrino mass is taken to be zero and the other neutrino masses are computed from Eqs. (8) and (9). For details related to masses of doubly charged scalars, see the text.

Branching ratios of → γ
As stated before, decays of the form → γ are driven by both doubly and singly charged triplet scalars. Interaction terms between doubly charged scalars and charged leptons, which are given in Eq. (43), drive → γ at 1-loop level. In addition to this contribution, singly charged triplet scalars interacting with charged leptons and neutrinos also contribute to → γ at 1-loop level. To obtain these interaction terms, which involve singly charged scalars, we use the transformations for left-handed charged leptons and neutrinos of Eqs. (2) and (7) Table 2: Values of the neutrino oscillation parameters [3], which are used in this work.
this, we get the following interaction terms for singly charged triplet scalars with charged leptons and neutrinos.
In the above equation, ν , a , Using the amplitude in Eq. (49), we find the branching ratios for the decays of the form → γ, where we have neglected the mass of . Expressions for these are given below.
Among the various decays of the form → γ, branching ratio for µ → eγ is severely constrained and we have Br(µ → eγ) < 4.2 × 10 −13 [25]. From the expression given for do not depend on the mixing angle θ 12 , at the leading order. Since it is interesting to study variation of Br(µ → eγ) with respect to neutrino oscillation observables, we have fixed VEVs of Higgs triplets and parameters of the scalar potential to some specific values, which will be described below. It should be noticed that both Br(µ → eγ) and Br(µ →ēee) are determined by a common set of parameters, since doubly charged triplet Higgses contribute to both of the above observables. In addition to this common set of observables, Br(µ → eγ) is determined by parameters related to singly charged triplet Higgses.
We have computed Br(µ → eγ) in our model for the cases of NO and IO, which are presented in Figs. 3 and 4 respectively. While computing the Br(µ → eγ), we have used the same set of parameters which are described for the computation of Br(µ →ēee).
Now the additional parameters which govern the decay µ → eγ are due to the singly charged triplet scalar fields. The masses and couplings of these singly charged scalars are determined after diagonalizing the mass matrices for these, which are given in Eq.
(38). There is a common set of parameters in the mass matrices for singly and doubly charged scalar fields. This common set of parameters is same as what we have used for the computation of Br(µ →ēee). The additional λ parameters in the mass matrices of singly charged triplet scalars are taken to be 0.1 in this analysis. As a result of this, the masses for both doubly and singly charged triplets are slightly above 850 GeV. After using the above mentioned parameters for the computation of Br(µ → eγ), from Figs  the cases I and II respectively. δ CP is expressed in degrees. In these plots, v T = 0.08 eV.
The neutrino oscillation parameters in these plots are taken to be same as for Fig. 1. For details related to charged triplet scalar masses, see the text. ratio is two orders lower than that for µ →ēee, whose results can be seen from Figs. 1 and 2. The reason for this suppression in the branching ratio is due to the fact that the decays µ → eγ and µ →ēee take place at 1-loop and tree level respectively. As a result of this, a loop suppression factor of α ∼ 10 −2 exist in the Br(µ → eγ), which gives the above mentioned suppression.
In the upcoming MEG II experiment, the sensitivity to probe Br(µ → eγ) is around 10 −14 [26]. Hence, the parameter region of Figs  the cases I and II respectively. δ CP is expressed in degrees. In these plots, v T = 0.14 eV.
The neutrino oscillation parameters in these plots are taken to be same as for Fig. 2 It is mentioned previously that contribution from the neutral scalar fields to the LFV decays is neglected in this work. Even after including this contribution, it is still an interest to know the results about LFV decays, in the limit where the masses of these fields are heavy enough that the contribution can be neglected. In this work, we have analyzed the above mentioned case. On the other hand, depending on the masses and coupling strengths of these neutral scalar fields, the results mentioned in this work can be altered. It is worth to study this contribution, however, it is stated that only the neutral scalars which interact with tau lepton may give appreciable contribution. Before studying this contribution, one has to diagonalize the mixing masses among the neutral scalar fields, which is an involved work and we postpone it to future.

Future directions and phenomenology of our model
The model presented in this work contains additional scalar fields which are five Higgs doublets and six Higgs triplets. After the electroweak symmetry breaking, the following fields remain in the theory: six doubly charged scalars, ten singly charged scalars, twenty one neutral scalars. One of these neutral scalars can be identified as the Higgs boson, which is discovered in the LHC. All the above mentioned scalars have gauge interactions.
Hence, it is possible to produce them at the LHC, and after production, they can subse-  [27]. Using this bound on m ee , allowed regions for LFV decays in our work can be studied. Apart from the above mentioned bounds, precision electroweak observables [13] can also give additional constraints on the model.
The singly and doubly charged scalars of our model can drive H → γγ at 1-loop level.
Here, H is a neutral scalar of our model, which represents Higgs boson of standard model.
The decay rate for H → γγ in our model depends on the tri-linear couplings of H with singly and doubly charged scalars. These couplings are determined by the parameters of the scalar potential of our model. Since the signal strength for H → γγ at the LHC [13] agrees with the standard model prediction for Higgs boson, there can be additional constraints on the above mentioned tri-linear couplings in our model.
In Sec. 3.2, we have given the minimization conditions for the doublet and triplet Higgses of our model. These conditions can represent a possible minimum for the scalar potential of our model. This minimum may or may not be a global minimum of our scalar potential. We may expect additional conditions to be imposed on the parameters of the scalar potential in order to make this minimum to be global. For related studies in this direction, see Refs. [28].
In this work, we have studied mixing pattern in lepton sector by introducing additional Higgs doublets and triplets. It is interesting to know about masses and mixing pattern of quarks in our framework with A 4 symmetry. In this direction, in Refs. [29], breaking of A 4 symmetry is suggested for obtaining realistic mixing pattern in quark and lepton sectors. Following these ideas, one can study quark masses and mixing pattern in our model.

Conclusions
In this work, we have considered the MW model [6], where the mixing pattern in neutrino sector is explained with three Higgs doublets, six Higgs triplets and with the additional symmetry A 4 . The VEVs of Higgs triplets play a part in explaining the neutrino mixing pattern, apart from the fact that the VEVs of Higgs doublets should be same in order to diagonalize the charged lepton mass matrix. To study the pattern of VEVs of scalar fields of the MW model, in this work, we have constructed the invariant scalar potential of this model. After minimizing this scalar potential, we have found that among the six Higgs triplets two of them acquire zero VEVs. As a result of this, after using the results from the diagonalization procedure of our previous work [10], we have found that the neutrino mixing angles cannot be consistently explained. In order to see if we can get a consistent picture with the diagonalization procedure of our previous work [ To study some phenomenological consequences of the model under consideration, we have computed branching ratios for the LFV decays of the form → 3 and → γ. We have found that A 4 symmetry of this model can bring some relations among the couplings between charged triplet scalars and lepton fields. As a result of this, relations can exist among branching ratios for different decays. Relation shown in Eq. (46) is independent of any assumption on the masses of charged triplet scalars. However, relations in Eqs.
(47), (52) and (53) are valid under some assumptions made on the masses of charged triplet scalars. Apart from this, branching ratios for the LFV decays in our work depend on the neutrino mixing angles θ 13 and θ 23 and also on the CP violating Dirac phase δ CP .
We have plotted branching ratios for these decays in both the cases of NO and IO. From these plots, we have found that the choice of free parameters among the VEVs of Higgs triplets can have implications on the branching ratios for the LFV decays of this model.