Radiative generation of realistic neutrino mixing with $A4$

TL;DR

This paper proposes a one-loop radiative model using $A4$ symmetry to generate realistic neutrino mixing angles, including non-zero $ heta_{13}$, with potential dark matter candidate in the inert scalar sector.

Contribution

It introduces a novel scheme to produce non-zero $ heta_{13}$ and deviations in mixing angles via small mass splittings in a scotogenic $A4\times Z_2$ framework, extending previous models.

Findings

Non-zero $ heta_{13}$ achieved through small right-handed neutrino mass splitting.

Model accommodates deviations from TBM, BM, and GR mixing patterns.

Potential dark matter candidate identified among inert scalars.

Abstract

Radiative generation of realistic mixing in neutrino sector is studied at one-loop level in a scotogenic $A4\times Z_2$ symmetric framework. A scheme of obtaining non-zero $\theta_{13}$ through small mass splitting in right-handed neutrino sector is proposed. The model consists of three right-handed neutrinos, two of which were required to be degenerate in masses to yield the common structure of the left-handed neutrino mass matrix that corresponds to $\theta_{13}=0$, $\theta_{23}=\pi/4$ and any $\theta_{12}^0$ in particular the choices specific to the Tribimaximal (TBM), Bimaximal (BM) and Golden Ratio (GR) mixings. Non-zero $\theta_{13}$, deviations of $\theta_{23}$ from maximality and small corrections to the solar mixing angle $\theta_{12}$ can be generated in one stroke by shifting from this degeneracy in the right-handed neutrino sector by a small amount. The lightest among the three $Z_2$ odd inert $SU(2)_L$ doublet scalars present in the model can be a potential dark matter candidate.

Tables

Table 1. Table 1: θ 12 0 subscript superscript 𝜃 0 12 \theta^{0}_{12} for the different popular lepton mixings viz. TBM, BM, and GR mixing.

Model	TBM	BM	GR
${\theta }_{12}^0$	35.3∘	45.0∘	31.7∘

Table 2. Table 2: Fields and their quantum numbers. Here we are concerned with the neutrino sector only.

Leptons	$SU{\left(2\right)}_L$	$A4$	$Z_2$
	$2$	$3$	$1$
$N_{\alpha R}\equiv \left(\begin{array}{c}{N}_{1R}\\ N_{2R}\\ N_{3R}\end{array}\right)$	$1$	$3$	$-1$
Scalars	$SU{\left(2\right)}_L$	$A4$	$Z_2$
	$2$	$3$	$1$
	$2$	$3$	$-1$

Equations

$$\displaystyle U=\left(\begin{array}[]{ccc}c_{12}c_{13}&s_{12}c_{13}&-s_{13}e^{-i\delta}\\ -c_{23}s_{12}+s_{23}s_{13}c_{12}e^{i\delta}&c_{23}c_{12}+s_{23}s_{13}s_{12}e^{i\delta}&-s_{23}c_{13}\\ -s_{23}s_{12}-c_{23}s_{13}c_{12}e^{i\delta}&-s_{23}c_{12}+c_{23}s_{13}s_{12}e^{i\delta}&c_{23}c_{13}\end{array}\right)\;\;,$$

$$U^{0}=\pmatrix{\cos\theta_{12}^{0}&\sin\theta_{12}^{0}&0\cr-\frac{\sin\theta_{12}^{0}}{\sqrt{2}}&\frac{\cos\theta_{12}^{0}}{\sqrt{2}}&-{1\over\sqrt{2}}\cr-\frac{\sin\theta_{12}^{0}}{\sqrt{2}}&\frac{\cos\theta_{12}^{0}}{\sqrt{2}}&{1\over\sqrt{2}}},$$

$${\theta }_{12}$$

$${\theta }_{23}$$

$${\theta }_{13}$$

$$M^{flavour}_{\nu L}=U^{0}M^{mass}_{\nu L}U^{0T}=\pmatrix{a&c&c\cr c&b&d\cr c&d&b}$$

$$a$$

$$b$$

$$c$$

$$d$$

$$\tan 2{\theta }_{12}^0=\frac{2\sqrt{2}c}{b+d-a}$$

$$V_{relevant}$$

$$\langle\Phi\rangle_{case1}=v\pmatrix{0&1\cr 0&0\cr 0&0}\;,\;\langle\Phi\rangle_{case2}=v\pmatrix{0&1\cr 0&e^{i\alpha}\cr 0&0}\;,\;\langle\Phi\rangle_{case3}=v\pmatrix{0&1\cr 0&1\cr 0&1}\;,\;\langle\Phi\rangle_{case4}=v\pmatrix{0&1\cr 0&\omega\cr 0&\omega^{2}}\;.$$

$$(M_{{\nu }_L}^{flavour}{)}_{ij}=\lambda \frac{v_m{v}_n}{8{\pi }^2}\sum _{\alpha ,k,l\ne (i,j)}\frac{h_{i\alpha k}{h}_{j\alpha l}}{M_{\alpha }}\left[\ln z_{\alpha }-1\right].$$

$$M_{\nu_{L}}^{flavour}=\pmatrix{\chi_{1}&\chi_{4}&\chi_{5}\cr\chi_{4}&\chi_{2}&\chi_{6}\cr\chi_{5}&\chi_{6}&\chi_{3}}$$

$${\chi }_1$$

$${\chi }_2$$

$${\chi }_3$$

$${\chi }_4$$

$${\chi }_5$$

$${\chi }_6$$

$$M_{\nu_{L}}^{flavour}=\pmatrix{\lambda_{123}(2rv^{2})&\lambda_{124}rv^{2}&\lambda_{124}rv^{2}\cr\lambda_{124}rv^{2}&\lambda_{123}(r+r_{1})v^{2}&\lambda_{124}r_{1}v^{2}\cr\lambda_{124}rv^{2}&\lambda_{124}r_{1}v^{2}&\lambda_{123}(r+r_{1})v^{2}}$$

$$a$$

$$b$$

$$c$$

$$d$$

$$M_{{\nu }_L}^{flavour}=M^0+M^{\prime }$$

$$M^{0}=\pmatrix{\lambda_{123}(2r_{2}v^{2})&\lambda_{124}r_{2}v^{2}&\lambda_{124}r_{2}v^{2}\cr\lambda_{124}r_{2}v^{2}&\lambda_{123}(r_{1}+r_{2})v^{2}&\lambda_{124}r_{1}v^{2}\cr\lambda_{124}r_{2}v^{2}&\lambda_{124}r_{1}v^{2}&\lambda_{123}(r_{1}+r_{2})v^{2}}\,{\rm and}\,M^{\prime}=\epsilon\pmatrix{x&y&0\cr y&x&0\cr 0&0&0}$$

$$a^{\prime }$$

$$b^{\prime }$$

$$c^{\prime }$$

$$d^{\prime }$$

$$\gamma \equiv (b^{\prime }-3d^{\prime }-a^{\prime })\mathrm{and}\rho \equiv \sqrt{a^{{}^{\prime }2}+b^{{}^{\prime }2}+8c^{{}^{\prime }2}+d^{{}^{\prime }2}-2a^{{}^{\prime }}{b}^{{}^{\prime }}-2a^{{}^{\prime }}{d}^{{}^{\prime }}+2b^{{}^{\prime }}{d}^{{}^{\prime }}}$$

$$|{\psi }_3⟩=\text{pmatrix}\frac{ϵ}{{\gamma }^2-{\rho }^2}\left[\gamma (x\sin 2{\theta }_{12}^0-\sqrt{2}y\cos 2{\theta }_{12}^0)+\rho \sqrt{2}y\right]\text{cr}-\frac{1}{\sqrt{2}}[1+\xi ϵ]\text{cr}\frac{1}{\sqrt{2}}[1-\xi ϵ].$$

$$\xi \equiv [\gamma x+\rho (x\cos 2{\theta }_{12}^0+\sqrt{2}y\sin 2{\theta }_{12}^0)]/({\gamma }^2-{\rho }^2).$$

$$\sin {\theta }_{13}=\frac{ϵ}{{\gamma }^2-{\rho }^2}\left[\gamma (x\sin 2{\theta }_{12}^0-\sqrt{2}y\cos 2{\theta }_{12}^0)+\rho \sqrt{2}y\right].$$

$$\tan \phi \equiv \tan ({\theta }_{23}-\pi /4)=\xi ϵ.$$

$$\tan {\theta }_{12}=\frac{\sin {\theta }_{12}^0+ϵ\beta \cos {\theta }_{12}^0}{\cos {\theta }_{12}^0-ϵ\beta \sin {\theta }_{12}^0}$$

Full text

I Introduction

Neutrino oscillation observations have clearly demonstrated the massive nature of neutrinos. The mass eigenstates are non-degenerate and distinct from the flavour eigenstates and are connected to each other by the Pontecorvo, Maki, Nakagawa, Sakata – PMNS – matrix usually parametrized as:

[TABLE]

where $c_{ij}=\cos\theta_{ij}$ and $s_{ij}=\sin\theta_{ij}$.

In 2012, the short-baseline reactor anti-neutrino experiments observed non-zero $\theta_{13}$, yet small compared to the other two mixing angles [1]. Prior to this observation, models leading to several structures like the Tribimaximal (TBM), Bimaximal (BM) and Golden Ratio (GR) mixings (which we refer henceforth as popular lepton mixings) were studied all of which were constructed with $\theta_{13}=0$ and $\theta_{23}=\pi/4$ and varying $\theta_{12}^{0}$ yielded the different alternatives.

Putting $\theta_{13}=0$ and $\theta_{23}=\pi/4$ in Eq. (4) can lead to the common structure for all popular mixings:

[TABLE]

where $\theta_{12}^{0}$ for TBM, BM and GR are listed in Table 1.

The present 3$\sigma$ global fits for the three mixing angles [2, 3]:

[TABLE]

The numbers are from NuFIT3.2 of 2018 [2].

Thus the popular mixings are in disagreement with the observed non-zero $\theta_{13}$. A plethora of activities had been taking place since this discovery to incorporate non-zero $\theta_{13}$ in these mixings. Attempts to relate the smallness of solar splitting with that of $\theta_{13}$ can be found in [4]. In [5], $\Delta m^{2}_{atmos}$ and $\theta_{23}=\pi/4$ were embedded in the dominant component of neutrino masses and mixing and the other oscillation parameters such as $\theta_{13},\theta_{12}$, the deviation of $\theta_{23}$ from $\pi/4$, and $\Delta m^{2}_{solar}$ were obtained perturbatively from a smaller see-saw [6] contribution111Some such earlier models can be found in [7].. Vanishing $\theta_{13}$ can be induced by certain symmetries and models generating non-zero $\theta_{13}$ through perturbation to such symmetric structures are also studied [8, 9].

In [10, 11], discrete flavour symmetries like $A4$, $S3$ were used to devise a two-component Lagrangian formalism at tree-level to ameliorate all the popular mixing patterns in single stroke. The dominant contribution to the Lagrangian was obtained from Type II see-saw mechanism characterized by popular mixing patterns, to which corrections were obtained from a sub-dominant Type I see-saw contribution. In [12] the same scheme was performed for the no solar mixing (NSM) case i.e., $\theta_{12}^{0}=0$ case with $A4$ symmetry222 The dominant Type II see-saw structure was kept devoid of solar splitting and degenerate perturbation theory was used to obtain large $\theta_{12}$.. The basic difference between [11, 12] and earlier works with $A4$ [13, 14, 15] is that in the earlier works Type II see-saw was used to generate the mass matrices and obtaining TBM was the prime goal. More realistic mixings can be found in recent works [16, 17].

In this paper, we intend to generate:

the popular mixing structure in Eq. (5) with $\theta_{13}=0$, $\theta_{23}=\pi/4$ and $\theta_{12}^{0}$ as listed in Table 1 2. 2.

non-zero $\theta_{13}$, deviations of $\theta_{23}$ from $\pi/4$ and small corrections to $\theta_{12}$

radiatively333For review of radiative neutrino mass models see [18]. with $A4$ flavour symmetry444For a brief discussion on $A4$ see Appendix of the paper.. Precisely, this $A4\times Z_{2}$ symmetric model will produce neutrino masses at one-loop level using three right-handed neutrinos that transform as a triplet under $A4$. To get Eq. (5) it is necessary that two of these right-handed neutrinos are degenerate. A little shift from that degeneracy will yield non-zero $\theta_{13}$, deviations of atmospheric mixing from maximality and tinker the solar mixing by a small amount in one go.

In order to accomplish this we also need to introduce a $Z_{2}$ odd $A4$ triplet scalar field $\eta$, the lightest of which could be a potential dark matter candidate.

II The Model

The neutrino mass matrix in the mass basis is given by $M^{mass}_{\nu L}=$ $diag\;(m_{1},m_{2},m_{3})$. This when expressed in flavour basis using the common structure of $U^{0}$ in Eq. (5) for the popular lepton mixings, leads to:

[TABLE]

where,

[TABLE]

Equivalently,

[TABLE]

For non-degenerate realistic neutrino masses $a,b,c$ and $d$ are non-zero.

Our objective is to obtain the structure of the matrix shown in Eq. (7) at one-loop level. For that we assign specific $A4\times Z_{2}$ charges to the scalars and fermions in our model. This model has three right-handed neutrino fields. As we will see in course of the discussion that in order to obtain the structure in Eq. (7), two of these right-handed neutrino states will require to be degenerate in masses. Once the structure in Eq. (7) is produced, we will exploit small relaxation of this degenerate feature in right-handed neutrino sector to yield the realistic neutrino mixings, namely non-zero $\theta_{13}$.

In this model, apart from the three $SU(2)_{L}$ lepton doublets we have three right-handed neutrinos, $N_{\alpha R}$, ($\alpha=1,2,3$) invariant under the standard model (SM) gauge group. Under $A4$ these three right-handed neutrinos transform as a triplet and so does the three $SU(2)_{L}$ lepton doublets. In the scalar sector we have two $A4$ symmetric triplet fields $\Phi$ and $\eta$ each of which comprises of three $SU(2)_{L}$ doublet fields $\Phi_{i}\equiv(\phi_{i}^{+},\phi_{i}^{0})^{T}$ and $\eta_{j}\equiv(\eta_{j}^{+},\eta_{j}^{0})^{T}$, $(i,j=1,2,3)$. In addition to $A4$ we have an unbroken $Z_{2}$ under which all the fields are even except the scalar field $\eta$ and the right-handed neutrinos. Thus the scalars $\eta_{j}$ do not acquire vacuum expectation value (vev) after spontaneous symmetry breaking (SSB), whereas the fields $\phi_{i}$ do. All the fields along with their quantum numbers are listed in Table 2. Here we restrict ourselves to the neutrino sector only555This model differs from [19] in terms of the particle content. Unlike [19], here we consider all the popular mixings viz. TBM, BM and GR and also generate non-zero $\theta_{13}$, deviations of $\theta_{23}$ from maximality and small corrections to $\theta_{12}$ simultaneously through small mass splitting in the right-handed neutrino sector. . We work in a basis in which the charged lepton mass matrix is diagonal and the mixing is entirely from the neutrino sector.

With these fields one can generate neutrino mass at one-loop level as shown in Fig. (1). The relevant part of the scalar potential from the four-point scalar vertex contributing to the neutrino mass matrix is given by666Note at the four-point scalar vertex, both the $\phi$ fields are annihilated and both the $\eta$ fields are created. So terms of the scalar potential of $(\eta^{\dagger}\phi)(\eta^{\dagger}\phi)$ kind will contribute to the neutrino mass matrix and are therefore relevant. :

[TABLE]

where all the quartic couplings $\lambda_{i}$ ($i=1,2,3,4$) are considered to be real.

As discussed earlier, after SSB, $\phi_{i}^{0}$ will get vevs whereas the $\eta_{j}^{0}$ will not owing to the $Z_{2}$ assignments. Let $\langle\Phi_{i}\rangle=v_{i}$ where ($i=1,2,3$). In [21, 22], it has been shown that for $A4$ symmetric three-Higgs-doublets the four vev configurations for which the scalar potential acquires the global minima777In [23], it has been shown that alignment follows as a natural consequence of the discrete symmetry $A4$ in $A4$ symmetric three-Higgs-doublet model for all the four global minima configurations mentioned in Eq. (11). are:

[TABLE]

Let $\eta_{Rj}$ and $\eta_{Ij}$ be the real and imaginary parts of $\eta_{j}^{0}$ respectively. Splitting among the masses of $\eta_{Rj}$ and $\eta_{Ij}$ is proportional to $\lambda v_{j}$ and is expected to be small. Here $\lambda$ stands for the quartic couplings $\lambda_{1}$, $\lambda_{2}$, $\lambda_{3}$ and $\lambda_{4}$ in Eq. (10). Also the mass splittings between $\eta_{j}$ ($j=1,2,3$) constituting the $A4$ triplet are neglected and $m_{0}$ is their common mass. Our model has three right-handed neutrinos, $N_{\alpha R}$ ($\alpha=1,2,3$), $M_{\alpha}$ being their masses. In the limit $M_{\alpha}^{2}>>m_{0}^{2}$, the diagram in Fig. (1) leads to neutrino mass of the kind [20]:

[TABLE]

where $z_{\alpha}\equiv\frac{M_{\alpha}^{2}}{m_{0}^{2}}$. The vevs of $\phi^{0}_{m}$ and $\phi^{0}_{n}$ are given by $v_{m}$ and $v_{n}$ respectively. The Yukawa couplings at the two vertices where the left-handed and right-handed neutrinos couple to the inert $SU(2)_{L}$ doublet fields $\eta$ are given by $h_{i\alpha k}$ and $h_{j\alpha l}$. No two of the three indices appearing in each of $h_{i\alpha k}$ and $h_{j\alpha l}$ individually can be same owing to $A4$ symmetry. Thus $h_{i\alpha k}$ and $h_{j\alpha l}$ are determined by the $A4$ invariance which in its turn governs the structure of the neutrino mass matrix. Since the logarithm is a slowly varying function and the heavy right handed neutrino masses $M_{\alpha}$ ($\alpha=1,2,3$) are expected to be close to each other, the RHS of Eq. (12) can be approximated as proportional to $\frac{1}{M_{\alpha}}$. Leaving the vevs $v_{m}$, $v_{n}$ and the quartic couplings $\lambda$, let us denote the contribution to left-handed neutrino mass matrix $(M_{\nu_{L}}^{flavour})_{ij}$ from everything else in Eq. (12) by loop contributing factors $r_{\alpha}\propto\frac{1}{M_{\alpha}}$.

For simplicity let us consider the right-handed neutrino mass matrix to be already diagonal i.e., $M_{N_{R}}\equiv diag(M_{1},M_{2},M_{3})$. In terms of the right-handed neutrino loop contributing factors $r_{\alpha}$ we have the contribution coming from right-handed neutrino sector as $diag(r_{1},r_{2},r_{3})$. Using Eqs. (12) and (10), the left-handed neutrino mass matrix that arises from Fig. (1) is given by888All the symmetries under consideration were conserved at each of the three vertices in Fig. (1).:

[TABLE]

where,

[TABLE]

In order to obtain the neutrino mass matrix of the form of Eq. (7) from Eq. (13), one will simultaneously require $\chi_{1}\neq\chi_{2}=\chi_{3}$ and $\chi_{4}=\chi_{5}$. Let us now try each of the vev configurations in Eq. (11) and find out the one suitable to obtain this feature along with the constraints put on to $r_{1},\,r_{2}$ and $r_{3}$.

Choice A: For $(v_{1},v_{2},v_{3})=v(1,0,0)$, irrespective of the choices for $r_{1},\,r_{2}$ and $r_{3}$, the off-diagonal entries in Eq. (13) will vanish and one cannot obtain mixing in the neutrino sector. 2. 2.

Choice B: For $(v_{1},v_{2},v_{3})=v(1,e^{i\alpha},0)$, two of the three off-diagonal entries in Eq. (13) will vanish for any $r_{1},\,r_{2}$ and $r_{3}$, and one cannot obtain the structure in Eq. (7). 3. 3.

Choice C: For $(v_{1},v_{2},v_{3})=v(1,\omega,\omega^{2})$, one cannot achieve $\chi_{2}=\chi_{3}$ as required to obtain the structure of the mass matrix in Eq. (7) from Eq. (13), whatever may be the choices for $r_{1},\,r_{2},\,r_{3}$. 4. 4.

Choice D: For $(v_{1},v_{2},v_{3})=v(1,1,1)$, note first that $r_{1}=r_{2}=r_{3}$ implies all the diagonal terms to be equal to each other and the off-diagonal entries are equal among themselves. This leads to two left-handed degenerate states and only TBM is admissible. We will not consider that choice. However the form in Eq. (7) starting from Eq. (13) is achieved for $r_{1}\neq r_{2}=r_{3}=r$ when $(v_{1},v_{2},v_{3})=v(1,1,1)$, which we refer to as choice D from now onwards. This choice allows all three mixings viz. TBM, BM, GR and all three left-handed neutrinos to be non-degenerate. Hence we will consider this case for further analysis. Such choice of $r_{2}=r_{3}$ is achieved when the right-handed neutrinos $N_{2R}$ and $N_{3R}$ are degenerate in masses.

Putting choice D ie., $v_{1}=v_{2}=v_{3}=v$ and $r_{1}\neq r_{2}=r_{3}=r$ in Eq. (13) one gets the following form of the left-handed neutrino mass matrix in the flavour basis:

[TABLE]

where, $\lambda_{123}=\lambda_{1}+\lambda_{2}+2\lambda_{3}$ and $\lambda_{124}=\lambda_{4}+2\lambda_{1}-\lambda_{2}$. Thus the neutrino mass matrix generated at one-loop level as shown in Fig. (1) can produce the form of $M_{\nu_{L}}^{flavour}$ as in Eq. (7) that corresponds to $\theta_{13}=0$, $\theta_{23}=\pi/4$ and $\theta_{12}^{0}$ of the popular mixing alternatives, with the vevs and right-handed neutrino masses as specified in choice D. This follows from the identifications:

[TABLE]

Having achieved this, next we concentrate on generation of realistic neutrino mixing i.e., non-zero $\theta_{13}$, deviations of $\theta_{23}$ from maximality and small corrections in the solar mixing $\theta_{12}$. For that one has to deviate from the $r_{\alpha}$ ($\alpha=1,2,3$) of choice D. Let us now split the degeneracy in the right handed neutrino sector by a small amount $\epsilon$ i.e., consider $r_{3}=r_{2}+\epsilon$ and $r_{1}\neq r_{2}\neq r_{3}\neq r_{1}$, keeping the vevs still to be $v_{1}=v_{2}=v_{3}=v$. With such a choice one is expected to get a dominant contribution of the form of $M_{\nu_{L}}^{flavour}$ as was achieved in Eq. (15), say $M^{0}$, together with small shift from it, $M^{\prime}$, proportional to $\epsilon$. Thus,

[TABLE]

where,

[TABLE]

where $x=\lambda_{123}v^{2}$ and $y=\lambda_{124}v^{2}$. Here $M^{0}$ is the form of the $M_{\nu_{L}}^{flavour}$ required for $\theta_{13}=0$, $\theta_{23}=\pi/4$ and $\theta_{12}^{0}$ of the popular mixings. Thus in analogy to Eq. (16), one can identify the followings999To distinguish from $r_{2}=r_{3}$ case, let us use a primed notation.:

[TABLE]

It is straightforward to incorporate the corrections offered by $M^{\prime}$ to $M^{0}$ using the non-degenerate perturbation theory. Columns of $U^{0}$ in Eq. (5) is the unperturbed flavour basis. From Eq. (19) one can define:

[TABLE]

The first order corrected third ket is then given by:

[TABLE]

where,

[TABLE]

Thus one can write,101010Here we restrict ourselves to no CP-violation.:

[TABLE]

Using Eqs. (19), (20) and (23), one can easily read off non-zero $\theta_{13}$ in terms of the model parameters, namely, $\epsilon$, the quartic couplings and the vevs. Throughout our discussion we have assumed $r_{\alpha}$ ($\alpha=1,2,3$) are real and restricted ourselves to a CP-conserving scenario. In principle, the right-handed neutrino masses can have Majorana phases causing these $r_{\alpha}$ to be complex. Then one can have a complex $\epsilon$, from which one can generate CP-violation in the lepton sector.

From Eq.(21) the deviation of atmospheric mixing from maximality is given by:

[TABLE]

Similarly, one can obtain small corrections to $\theta_{12}$ from the corrections of the first and second kets. The solar mixing angle after receiving first order corrections is given by:

[TABLE]

where,

[TABLE]

The corrections to the solar mixing and deviations of atmospheric mixing from $\pi/4$ in Eq. (25) and (24) respectively can be expressed in terms of the model parameters using Eqs. (19), (20), (22) and (26).

Summing up, a scotogenic $A4\times Z_{2}$ symmetric model of radiatively obtaining realistic neutrino mixing is proposed. Among others, the model comprises of three gauge singlet right-handed neutrino fields $N_{\alpha R}$, ($\alpha=1,2,3$). If $N_{2R}$ and $N_{3R}$ are degenerate in masses, one can obtain the common structure of the left-handed neutrino mass matrix required by $\theta_{13}=0$, $\theta_{23}=\pi/4$ and $\theta_{12}^{0}$ of the particular choices leading to popular lepton mixing scenarios viz. TBM, BM, GR at one-loop level. A slight shift from this degeneracy of right-handed neutrino masses could generate realistic mixing viz. non-zero $\theta_{13}$, deviations of $\theta_{23}$ from $\pi/4$ and also tweak $\theta_{12}$ by a small amount. The model has three inert $SU(2)_{L}$ doublet scalars $\eta$, odd under the unbroken $Z_{2}$, the lightest of which can be a dark matter candidate.

Acknowledgements: I thank Prof. Amitava Raychaudhuri for discussions at different stages of this work.

A Appendix: The discrete group $A4$

$A4$ being the group of even permutations of four objects has 12 elements. The group $A4$ has two generators $S$ and $T$. These generators satisfy $S^{2}=T^{3}=(ST)^{3}=\mathbb{I}$. The inequivalent irreducible representations for $A4$ are four in number out of which three are 1-dimensional viz. $1,1^{\prime}$ and $1^{\prime\prime}$ and one is 3-dimensional. The 1-dimensional representations transform as 1, $\omega$, and $\omega^{2}$ under111111Here $\omega$ is a cube root of 1. $T$ but are invariant under $S$. Thus, $1^{\prime}\times 1^{\prime\prime}=1$. The generators are represented by,

[TABLE]

Below is the combination rule for two $A4$ triplets:

[TABLE]

Let us have two $A4$ triplet fields, $3_{a}\equiv{a_{i}}$ and $3_{b}\equiv{b_{i}}$, where $i=1,2,3$, and combine them according to Eq. (A.2). The triplets that we get can be written as $3_{c}\equiv{c_{i}}$ and $3_{d}\equiv{d_{i}}$ where,

[TABLE]

The $1$, $1^{\prime}$ and $1^{\prime\prime}$ in this case are:

[TABLE]

The group was studied in context of neutrino mass and mixings in the pioneering works [13, 14].