# Neutrino Mixing from a Fresh Perspective

Pralay Chakraborty<EMAIL_ADDRESS>Physics Department, Gauhati
University, India  Manash Dey<EMAIL_ADDRESS>Physics Department,
Gauhati University, India  Biswajit Karmakar<EMAIL_ADDRESS>Institute of Physics, University of Silesia, Katowice, Poland  Subhankar Roy
<EMAIL_ADDRESS>Physics Department, Gauhati University, India

###### Abstract

We propose a new mixing scheme for neutrinos which eventually can be linked to
a neutrino mass matrix texture bearing a suitable correlation
($m_{22}+2\,m_{13}=0$) among its elements. The texture predicts three mass
eigenvalues in the light of normal ordering, in addition to the Majorana
phases. The texture is realized from $A_{4}\times Z_{10}\times Z_{2}$ group in
the framework of general type-I+II seesaw mechanism.

##  I Introduction

In the realm of particle physics, neutrinos are mysterious creatures that can
transform from one type to another as they traverse space. This phenomenon is
called neutrino oscillation and this idea is first introduced by Bruno
Pontecorvo in the year 1957 [1]. The neutrino oscillation arises from the fact
that the three neutrino flavour states ($\nu_{l=e,\mu,\tau}$) are admixtures
of three mass eigenstates $(\nu_{i=1,2,3})$ with corresponding mass values
($m_{i=1,2,3}$). The Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix [2] is the
mathematical representation that connects the flavour states to the mass
states, providing a framework for understanding how neutrinos transform
between different flavours as they propagate through space. The neutrino
oscillation demands the existence of a non-zero neutrino mass, contradicting
the predictions of the Standard Model (SM) [3, 4, 5]. In this regard, to
understand the origin of neutrino mass, it is necessary to explore theories
that go beyond the SM. The neutrino mass matrix ($M_{\nu}$) originates from
the Yukawa Lagrangian, which is formulated within the framework of the seesaw
mechanism [6, 7]. In the literature, the widely used seesaw mechanisms are the
type-I seesaw [8, 9, 10] and type-II seesaw mechanism [11, 12, 13, 14].

The neutrino mass matrix $M_{\nu}$ contains twelve parameters: three mass
eigenvalues ($m_{i=1,2,3}$), three mixing angles
($\theta_{12},\theta_{13},\theta_{23}$), three CP phases
($\delta,\alpha,\beta$), and three unphysical phases
($\phi_{1},\phi_{2},\phi_{3}$). In recent years, the experiments predict the
mixing angles with high precision but the precise determination of Dirac CP
phase $\delta$ and the octant of the atmospheric mixing angle $\theta_{23}$
yet to be settled [15]. In addition, the experiments are unable to provide
exact values for individual mass eigenvalues; instead, they offer predictions
for two mass-squared differences. In addition, the Majorana phases are
physical parameters which are still untouched in the experiments. In order to
get the information of the physical parameters, we need to diagonalize
$M_{\nu}$: $\text{diag}(m_{1},m_{2},m_{3})=V^{T}.M_{\nu}.V$, where, $V$ is the
PMNS matrix. The Particle Data Group (PDG) [16] has adopted a parametrization
scheme for $U$ known as standard parametrization as shown in the following,

$V=P_{\phi}.U.P_{M},$ (1)

where, $P_{\phi}=\text{diag}(e^{i\phi_{1}},e^{i\phi_{2}},e^{i\phi_{3}})$ and
$P_{M}=\text{diag}(e^{i\alpha},e^{i\beta},1)$. The matrix $U$ is represented
as shown in the following,

$\displaystyle U$ $\displaystyle=$ $\displaystyle\begin{bmatrix}1&0&0\\\
0&c_{23}&s_{23}\\\
0&-s_{23}&c_{23}\end{bmatrix}\times\begin{bmatrix}c_{13}&0&s_{13}\,e^{-i\delta}\\\
0&1&0\\\
-s_{13}e^{i\delta}&0&c_{13}\end{bmatrix}\times\begin{bmatrix}c_{12}&s_{12}&0\\\
-s_{12}&c_{12}&0\\\ 0&0&1\end{bmatrix},$ (2)

where, $s_{ij}=\sin\theta_{ij}$ and $c_{ij}=\cos\theta_{ij}$. By using the
standard parametrization, the Jarlskog invariant ($J_{CP}$) [17] can be
expressed as shown below,

$J_{CP}=s_{13}c_{13}^{2}s_{12}c_{12}s_{23}c_{23}\sin\delta.$ (3)

In literature, several mixing patterns were proposed. Among them, bimaximal
mixing (BM) [18], tri-bimaximal mixing (TBM) [19], hexagonal mixing (HM) [20]
and golden ratio (GR) [21] are notable. All of these fixed or constant mixing
schemes predict vanishing value for the reator mixing angle $\theta_{13}$. For
instance, in tri-bimaximal mixing, specific values are assigned to the mixing
parameters: $s_{12}=\frac{1}{\sqrt{3}}$, $s_{23}=\frac{1}{\sqrt{2}}$ and
$s_{13}=0$. This mixing scheme holds significance due to its inherent
connection to the $\mu-\tau$ symmetric neutrino mass matrix, which is
particularly fascinating from the model-building perspective [22, 23]. In this
regard, the tri-bimaximal mixing has been investigated within the framework of
various discrete flavour symmetries [24, 25], for a recent review on discrete
flavour symmetries and their implication in neutrino mixing, see Ref. [26] and
for specific examples, see Refs. [27, 28] and references there in. It is to be
emphasized that the recent experiments have ruled out the possibility of
$s_{13}=0$. However, the fundamental philosophy behind the mixing pattern and
its connection to specific textures of the neutrino mass matrix remains
sacred. All mixing patterns such as bimaximal, tri-bimaximal, hexagonal,
golden ratio are characterized by constant values of the neutrino mixing
angles and can be explained within the frameworks of non-Abelian dscrete
flavour symmetries. In that light, the aim of this paper is to propose a new
mixing pattern in terms of a fixed or constant mixing scheme and deviating
from $s_{13}=0$ (hence accommodating Dirac CP phase $\delta$) which is
consistent with the experiments:

$\displaystyle s_{12}=(2\sqrt{2})/5,\quad\quad\quad s_{13}=1/(3\sqrt{5}),$
$\displaystyle s_{23}=3/4,\quad\quad\quad\quad\delta=3\pi/4.$

Figure 1:  The comparison of the tri-bimaximal , proposed and best fit [15]
values of the mixing angles and Dirac CP phase $\delta$.

For better visualization, we have compared the experimental best fit values
[15], our proposed values of the mixing angles and tri-bimaximal mixing angles
as illustrated in Fig. 1. In order to derive the proposed mixing scheme from a
neutrino mass matrix, we adhere to the following texture bearing a suitable
correlation $m_{22}+\,2\,m_{13}=0$ as shown in the following,

$M_{\nu}=\begin{bmatrix}A&B&F\\\ B&-2F&G\\\ F&G&J\end{bmatrix},$ (4)

where, $A,B,F,G$ and $J$ are complex parameters.

The plan for rest of the paper is outlined as follows: In Section (II), we
show the formalism and numerical analysis to show the compatibility of the
proposed mixing scheme and the mass matrix presented in Eq. (4). Predictions
involving neutrino masses, associated Majorana phases, and effective mass
parameters appearing in the neutrinoless double beta decay. Section (III) is
devoted to the relation of the proposed mixing scheme with a specific model.
As an example, here we have showed a model for the proposed mixing scheme
based on $A_{4}$ discrete flavour symmetry and study of model parameters.
Finally, we summarize and conclude in Section (IV).

##  II Formalism and Numerical Analysis

The structure of the PMNS matrix in the light of proposed mixing pattern can
be written as,

$U_{P}=\left(\begin{array}[]{ccc}\frac{2\sqrt{187}}{15\sqrt{5}}&\frac{4\sqrt{12}}{15\sqrt{5}}&-\frac{(1+i)}{3\sqrt{10}}\\\
\frac{\sqrt{17}(1-i)}{20\sqrt{10}}-\frac{7}{5\sqrt{2}}&\frac{\sqrt{119}}{20}+\frac{(1-i)}{10\sqrt{5}}&\frac{\sqrt{11}}{2\sqrt{5}}\\\
\frac{3}{5\sqrt{2}}+\frac{\sqrt{119}(1-i)}{60\sqrt{10}}&\frac{3\sqrt{17}}{20}+\frac{\sqrt{7}(1-i)}{30\sqrt{5}}&\frac{\sqrt{77}}{6\sqrt{5}}\\\
\end{array}\right).$ (5)

We now carry out a model-independent analysis of the texture given in Eq. (4)
which can be diagonalized by $U_{P}$. To get the information of the physical
parameters, we diagonalize $M_{\nu}$ in the following way,

$\text{diag}(\tilde{m_{1}},\tilde{m_{2}},m_{3})=U_{P}^{T}.M_{\nu}.U_{P},$ (6)

Figure 2:  Shows the correlation plots between Re[$J$] and Im[$J$] vs Re[$G$].

Figure 3:  Shows the correlation plots between (a) $\Delta m^{2}_{21}$ vs $\Delta m^{2}_{31}$. (b) $m_{1}$ vs $m_{2}\,\text{and}\,m_{3}$. (c) $m_{1}$ vs $\Sigma m_{i}$. (d) $\alpha$ vs $\beta$. (e) Lightest neutrino mass ($m_{1}$) vs $m_{\beta\beta}$.  Parameters | Minimum Value | Maximum Value
---|---|---
$m_{1}/\text{eV}$ | 0.018 | 0.021
$m_{2}/\text{eV}$ | 0.020 | 0.022
$m_{3}/\text{eV}$ | 0.052 | 0.055
$\sum{m_{i}}/\text{eV}$ | 0.091 | 0.098
$\alpha/^{\circ}$ | $-13.96$ | $-3.18$
$\beta/^{\circ}$ | $-29.37$ | $-0.75$
$m_{\beta\beta}/\text{meV}$ | 17.27 | 20.22
Table 1:  Shows the maximum and minimum values of $m_{1}$, $m_{2}$, $m_{3}$,
$\Sigma m_{i}$, $\alpha$, $\beta$ and $m_{\beta\beta}$.

where, $\tilde{m_{1}}=m_{1}\,e^{-2i\alpha}$ and
$\tilde{m_{2}}=m_{2}\,e^{-2i\beta}$, where, $m_{i=1,2,3}$ are the real
positive mass eigen values and $\alpha,\beta$ are associated Majorana phases.
By applying the diagonalizing conditions, we restrict the number of free
parameters to three Re$[G]$, Re$[J]$ and Im$[J]$ and express three mass
eigenvalues and two Majorana phases in terms of the said parameters. We
generate a set of random numbers for Re$[G]$, Re$[J]$, and Im$[J]$ ensuring
that $\Sigma m_{i}\leq 0.12,eV$ [29] and that the two mass squared differences
align with experimental observations [15]. For graphical visualization, we
obtain the correlation plots. The Fig. (2) shows how the texture parameters
Re$[G]$, Re$[J]$, and Im$[J]$ vary. From the Figs. 3-3, we identify the
regions to where the observational parameters are constrained, where, the
correlation between the parameters in the $\Delta m^{2}_{21}-\Delta
m^{2}_{31}$, $m_{1}-m_{2,3}$, $m_{1}-\sum m_{i}$ and $\alpha-\beta$ are
visualized. The maximum and minimum values of the physical parameters are
listed in Table 1. In addition, the Jarlskog invariant ($J_{CP}$), which
determines the size of the leptonic CP violation, is obtained from the
proposed mixing scheme as 0.0238. The parameter $m_{\beta\beta}$ which is the
effective Majorana neutrino mass is an observational parameter and several
experiments have provided the upper bounds of $m_{\beta\beta}$:
SuperNEMO(Se82) as $67-131$ meV, GERDA(Ge76) as $104-228$ meV, EXO-200(Xe136)
as $111-477$ meV, CUORE(Te130) as $75-350$ meV and KamLAND-Zen(Xe136) as
$61-165$ meV [30, 31, 32, 33, 34, 35, 36]. In this regard, the variation of
the parameter $m_{\beta\beta}$ with respect to the lightest neutrino mass is
visualized from the proposed (see Fig. 3). We emphasize that the texture only
yields the normal hierarchy for neutrino masses when diagonalized with the
proposed mixing matrix.

##  III The Model

To realize the texture shown in the earlier section, compatible with the
proposed mixing scheme, we now consider a framework based on non-Abelian
discrete flavour symmetry. As an example, we have considered $A_{4}$ discrete
flavour symmetry, which is the smallest group having a three-dimensional
irreducible representation where three SM lepton doublets can be accommodated
and unified into the triplet representation. Here, the whole framework is
embedded within $A_{4}\times Z_{10}\times Z_{2}$ symmetry. The light neutrino
mass is generated via most general type-I+II seesaw mechanism [37]. Hence, in
this regard, we extend the field content of the SM by adding three right-
handed neutrinos $(\nu_{e_{R}},\nu_{\mu_{R}},\nu_{\tau_{R}})$ and two scalar
triplets ($\Delta_{1,2}$). To implement the flavour symmetry and appropriate
texture of the mass matrix, we also include seven scalar singlets ( $\Phi$,
$\xi,\eta,\chi$, $\psi$, $\kappa$, $\varsigma$). The transformation properties
of the field content associated with $A_{4}\times Z_{10}\times Z_{2}$ symmetry
are summarised in Table 2.

Fields |  $D_{l_{L}}$ |  $l_{R}$ |  $\nu_{l_{R}}$ |  $H$ |  $\Delta_{1}$ |  $\Delta_{2}$ |  $\Phi$ |  $\xi$ |  $\eta$ |  $\chi$ |  $\psi$ |  $\kappa$ |  $\varsigma$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$SU(2)_{L}$ |  2 |  1 |  1 |  2 |  3 |  3 |  1 |  1 |  1 |  1 |  1 |  1 |  1
$A_{4}$ |  3 |  $(1,1^{\prime\prime},1^{\prime})$ |  $(1^{\prime\prime},1,1^{{}^{\prime}})$ |  $1$ |  1 |  1 |  $3$ |  $1$ |  $1^{\prime\prime}$ |  $1$ |  3 |  3 |  1
$Z_{10}$ |  0 |  $(0,0,0)$ |  $(0,4,8)$ |  $0$ |  3 |  1 |  $0$ |  $2$ |  $5$ |  6 |  7 |  9 |  5
$Z_{2}$ |  1 |  $(1,1,1)$ |  $(1,1,1)$ |  $-1$ |  1 |  1 |  $-1$ |  $1$ |  $1$ |  1 |  1 |  1 |  1
Table 2:  The transformation properties of various fields under $A_{4}\times
Z_{10}\times Z_{2}$.

The $A_{4}\times Z_{10}\times Z_{2}$ invariant Lagrangian can be constructed
in the following manner,

$\displaystyle-\mathcal{L}_{Y}$ $\displaystyle=$
$\displaystyle\frac{y_{e}}{\Lambda}(\overline{D}_{l_{L}}\Phi)_{1}H\,e_{R_{1}}+\frac{y_{\mu}}{\Lambda}(\overline{D}_{l_{L}}\Phi)_{1^{\prime}}\,H\mu_{R_{1^{\prime\prime}}}+\,\frac{y_{\tau}}{\Lambda}\,(\overline{D}_{l_{L}}\Phi)_{1^{\prime\prime}}H\tau_{R_{1^{\prime}}}+\frac{y_{1}}{\Lambda}\,(\overline{D}_{l_{L}}\Phi)_{1^{\prime}}$
$\displaystyle\,\tilde{H}\nu_{e_{R_{1^{\prime\prime}}}}+\frac{y_{2}}{\Lambda^{2}}(\overline{D}_{l_{L}}\,\Phi)_{1}\,\tilde{H}\nu_{{\mu}_{R_{1}}}\chi+\frac{y_{3}}{\Lambda^{2}}(\overline{D}_{l_{L}}\Phi)_{1^{\prime\prime}}\,\tilde{H}\nu_{{\tau}_{R_{1^{\prime}}}}\xi+\,\frac{y_{a}}{2}\,\,(\overline{\nu^{c}}_{\mu_{R}}\,\nu_{\mu_{R}})_{1}\,\xi_{1}$
$\displaystyle+\frac{y_{b}}{2}\,[(\overline{\nu^{c}}_{e_{R}}\,\nu_{\tau_{R}})\,+(\overline{\nu^{c}}_{\tau_{R}}\,\nu_{e_{R}})]_{1}\,{\xi_{1}}+\,\frac{y_{c}}{2\Lambda}\,(\overline{\nu^{c}}_{e_{R}}\,\nu_{e_{R}})_{1}\,\eta_{1^{\prime\prime}}\,\varsigma+\,\frac{y_{t}}{\Lambda}\,(\overline{D}_{l_{L}}\,D_{l_{L}}^{c})_{3_{S}}\,\psi\Delta_{1}$
$\displaystyle+\,\frac{y_{s}}{\Lambda}\,(\overline{D}_{l_{L}}\,D_{l_{L}}^{c})_{3_{S}}\,\kappa\,\Delta_{2}\,+h.c.$

The product rules under $A_{4}$ are given in appendix A. The details for the
$A_{4}$ vacuum alignment have been discussed in Appendix B. The additional
symmetry groups $Z_{10}$ and $Z_{2}$ are considered to restrict the next to
leading order corrections. The details of $Z_{10}$ group can be found in Refs.
[38, 39, 40]. Here $\Lambda$ is the cut-off scale of the theory. The charged
lepton mass matrix under the choice of vacuum alignment
$\langle\Phi\rangle_{\circ}=v_{\Phi}\,(1,0,0)^{T}$ and $\langle
H\rangle_{\circ}=v_{H}$ can be derived from Eq.(LABEL:Yukawa_Lagrangian_M2) as
shown below,

$M_{l}=\begin{bmatrix}\frac{y_{e}}{\Lambda}v_{\Phi}\,v_{H}&0&0\\\
0&\frac{y_{\mu}}{\Lambda}v_{\Phi}\,v_{H}&0\\\
0&0&\frac{y_{\tau}}{\Lambda}v_{\Phi}\,v_{H}\\\ \end{bmatrix},$ (8)

We choose the vacuum alignments $\langle\eta\rangle_{\circ}=v_{\eta}$,
$\langle\chi\rangle_{\circ}=v_{\chi}$, $\langle\xi\rangle_{\circ}=v_{\xi}$,
$\langle\varsigma\rangle_{\circ}=v_{\varsigma}$ to derive the Dirac neutrino
mass matrix ($M_{D}$) and right handed neutrino mass matrix ($M_{R}$) as shown
in the following,

$\displaystyle M_{D}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}0&\frac{y_{2}}{\Lambda^{2}}\,v_{\Phi}\,v_{H}v_{\chi}&0\\\
\frac{y_{1}}{\Lambda}v_{\Phi}\,v_{H}&0&0\\\
0&0&\frac{y_{3}}{\Lambda^{2}}\,v_{\Phi}\,v_{H}\,v_{\xi}\\\ \end{bmatrix},$ (9)
$\displaystyle M_{R}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}\frac{y_{c}}{2\,\Lambda}\,v_{\eta}\,v_{\varsigma}&0&\frac{y_{b}}{2}v_{\xi}\\\
0&\frac{y_{a}}{2}v_{\xi}&0\\\ \frac{y_{b}}{2}v_{\xi}&0&0\\\ \end{bmatrix},$
(10)

The two matrices $M^{1}_{T_{2}}$ and $M^{2}_{T_{2}}$ from two type-II seesaw
mechanisms can be derived from Eq.(LABEL:Yukawa_Lagrangian_M2) by considering
the vacuum alignment$\langle\Delta_{1}\rangle_{\circ}=v_{\Delta_{1}}$,
$\langle\Delta_{2}\rangle_{\circ}=v_{\Delta_{2}}$,
$\langle\psi\rangle_{\circ}=v_{\psi}(0,1,-1)^{T}$ and
$\langle\kappa\rangle_{\circ}=v_{\kappa}(0,1,-1)^{T}$ as shown in the
following,

$\displaystyle M^{1}_{T_{2}}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}0&\frac{1}{3\,\Lambda}\,y_{t}\,v_{\Delta_{1}}\,v_{\psi}&-\frac{1}{3\,\Lambda}\,y_{t}\,v_{\Delta_{1}}\,v_{\psi}\\\
\frac{1}{3\,\Lambda}\,y_{t}\,v_{\Delta_{1}}\,v_{\psi}&\frac{2}{3\,\Lambda}\,y_{t}\,v_{\Delta_{1}}\,v_{\psi}&0\\\
-\frac{1}{3\,\Lambda}\,y_{t}\,v_{\Delta_{1}}\,v_{\psi}&0&-\frac{2}{3\,\Lambda}\,y_{t}\,v_{\Delta_{1}}\,v_{\psi}\\\
\end{bmatrix},$ (11) $\displaystyle M^{2}_{T_{2}}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}\frac{2}{3\,\Lambda}\,y_{s}\,v_{\kappa}\,v_{\Delta_{2}}&-\frac{1}{3\Lambda}\,y_{s}\,v_{\kappa}\,v_{\Delta_{2}}&0\\\
-\frac{1}{3\Lambda}\,y_{s}\,v_{\kappa}\,v_{\Delta_{2}}&0&-\frac{1}{3\Lambda}\,y_{s}\,v_{\kappa}\,v_{\Delta_{2}}\\\
0&-\frac{1}{3\Lambda}\,y_{s}\,v_{\kappa}\,v_{\Delta_{2}}&\frac{2}{3\,\Lambda}\,y_{s}\,v_{\kappa}\,v_{\Delta_{2}}\\\
\end{bmatrix}.$ (12)

The effective neutrino mass matrix $M_{\nu}$ is constructed after taking the
contributions from type-I ($-\,M_{D}M^{-1}_{R}M^{T}_{D}$) and two type-II
seesaw mechanisms as shown below,

$\displaystyle M_{\nu}$ $\displaystyle=$ $\displaystyle\begin{bmatrix}A&B&F\\\
B&-2F&G\\\ F&G&J\\\ \end{bmatrix},$ (13)

where, $A$, $B$, $F$, $G$, $J$ are purely complex parameters. The said texture
parameters can be expressed in terms of the model parameters as shown in the
following,

$\displaystyle A$ $\displaystyle=$
$\displaystyle\frac{2\,y_{s}\,v_{\Delta_{2}}\,v_{\kappa}}{3\,\Lambda}-\frac{2\,y_{2}^{2}\,v_{H}^{2}\,v_{\chi}^{2}\,v_{\phi}^{2}}{\Lambda^{4}\,y_{a}\,v_{\xi}},$
(14) $\displaystyle B$ $\displaystyle=$
$\displaystyle\frac{y_{t}\,v_{\Delta_{1}}\,v_{\psi}}{3\,\Lambda}-\frac{y_{s}\,v_{\Delta_{2}}\,v_{\kappa}}{3\,\Lambda},$
(15) $\displaystyle F$ $\displaystyle=$
$\displaystyle-\frac{y_{t}\,v_{\Delta_{1}}\,v_{\psi}}{3\,\Lambda},$ (16)
$\displaystyle G$ $\displaystyle=$
$\displaystyle-\frac{2\,y_{1}\,y_{3}\,v_{H}^{2}\,v_{\phi}^{2}}{\Lambda^{3}\,y_{b}}-\frac{y_{s}\,v_{\Delta_{2}}\,v_{\kappa}}{3\,\Lambda},$
(17) $\displaystyle J$ $\displaystyle=$
$\displaystyle\frac{2\,y_{3}^{2}\,y_{c}\,v_{H}^{2}\,v_{\zeta}\,v_{\eta}\,v_{\phi}^{2}}{\Lambda^{5}\,y_{b}^{2}}+\frac{2\,y_{s}\,v_{\Delta_{2}}\,v_{\kappa}}{3\,\Lambda}-\frac{2\,y_{t}\,v_{\Delta_{1}}\,v_{\psi}}{3\,\Lambda}.$
(18)

It is important to mention that it is very difficult to predict the Yukawa
couplings individually. However, the present work predicts the combinations of
the model parameters $Y_{1}=\frac{y_{t}v_{\Delta_{1}}v_{\psi}}{\Lambda}$,
$Y_{2}=\frac{y_{s}v_{\Delta_{2}}v_{\kappa}}{\Lambda}$,
$Y_{3}=\frac{\Lambda^{4}y_{a}v_{\xi}}{2y_{2}^{2}v_{H}^{2}v_{\chi}^{2}v_{\phi}^{2}}$,
$Y_{4}=\frac{\Lambda^{3}y_{b}}{2y_{1}y_{3}v_{H}^{2}v_{\phi}^{2}}$ and
$Y_{5}=\frac{y_{c}\text{$\Lambda$v}_{\zeta}v_{\eta}}{y_{1}^{2}v_{H}^{2}v_{\phi}^{2}}$.
These combinations can be expressed in terms of the texture parameters $A$,
$B$, $F$, $G$ and $J$ as shown below,

$\displaystyle Y_{1}$ $\displaystyle=$ $\displaystyle-3\,F,$ (19)
$\displaystyle Y_{2}$ $\displaystyle=$ $\displaystyle-3(B+F),$ (20)
$\displaystyle Y_{3}$ $\displaystyle=$ $\displaystyle-\frac{1}{A+2B+2F},$ (21)
$\displaystyle Y_{4}$ $\displaystyle=$ $\displaystyle\frac{2}{B+F-G},$ (22)
$\displaystyle Y_{5}$ $\displaystyle=$
$\displaystyle\frac{2(2B+F)}{(B+F-G)^{2}}.$ (23)

Figure 4:  Shows the correlation plots between (a) $|Y_{1}|$ and $|Y_{2}|$.
(b) $|Y_{4}|$ vs $|Y_{3}|$ and $|Y_{5}|$. (c) Arg $[Y_{1}]$ vs
Arg$[Y_{2,3,4,5}]$.

Model Parameter | Minimum Value | Maximum Value
---|---|---
$|Y_{1}|/\,\text{eV}$ | 0.465 | 0.054
$|Y_{2}|/\,\text{eV}$ | 0,032 | 0.039
$|Y_{3}|/\,\text{eV}^{-1}$ | 27.70 | 41.59
$|Y_{4}|/\,\text{eV}^{-1}$ | 50.97 | 64.82
$|Y_{5}|/\,\text{eV}^{-1}$ | 23.47 | 26.86
$Arg[Y_{1}]/^{\circ}$ | -177.99 | 13.85
$Arg[Y_{2}]/^{\circ}$ | -31.73 | 150.80
$Arg[Y_{3}]/^{\circ}$ | -170.04 | 26.22
$Arg[Y_{4}]/^{\circ}$ | -171.01 | 19.01
$Arg[Y_{5}]/^{\circ}$ | -66.59 | 151.65

Table 3: Allowed values of the model parameters compatible with 3$\sigma$
range of neutrino oscillation data[15]

.

For graphical representation, we generate correlation plots to visualize how
the absolute values and arguments of the combinations of the model parameters
vary. It is evident from Figs. 3-3 that the studied parameters are constrained
to certain bounds. The maximum and minimum values of the model parameters are
listed in Table 3 corresponding to the 3$\sigma$ range of neutrino oscillation
data [15].

Deriving the proposed model-independent neutrino mass matrix texture as shown
in Eq. (4) from the symmetry framework has some inherent challenges. Moreover,
it’s crucial to ensure the independence of parameters $A,B,F,G$, and $J$ from
each other. This is achieved by including contributions from both type-I and
type-II contributions, leading to an increase in the field content. This is a
necessary step to achieve the high productivity of the framework. To obtain
the texture in its exact form, we introduce the field $\varsigma$, to cut
short certain non-leading terms in the Yukawa Lagrangian. Similarly, we need
to cut short specific elements within the right-handed Majorana neutrino mass
matrix which requires two additional scalar fields $\zeta$ and $\chi$. If
$\Delta_{1}$ and $\Delta_{2}$ were chosen as $A_{4}$ triplets, the field
content could have been reduced. However, in that case, certain components
within $\Delta_{i}$ will not get vevs and consequently, there remains the
finite possibility that these components will contribute towards the neutrino
mass via radiative correction. To avoid this, we need to treat $\Delta_{1}$
and $\Delta_{2}$ as singlets under $A_{4}$ and introduce the $A_{4}$ scalar
triplet fields $\psi$ and $\kappa$ to restore the invariance of the
Lagrangian.

##  IV Summary and Discussion

In the present work, we have proposed a new mixing scheme aligning with the
philosophy of historical fixed/constant mixing schemes such as bimaximal, tri-
bimaximal, golden ratio, and hexagonal mixing schemes. Unlike its
predecessors, in the proposed mixing scheme, the Dirac CP phase $\delta$ is
fixed at $3\pi/4$. The other mixing angles are fixed at
$s_{12}=(2\sqrt{2})/5,\,s_{13}=1/(3\sqrt{5})$ and $s_{23}=3/4$. The proposed
mixing scheme is special in the sense that it can be linked to a neutrino mass
matrix texture bearing a suitable correlation ($m_{22}+2\,m_{13}=0$) among its
elements. The texture predicts three non-vanishing neutrino mass eigenvalues
which follow normal ordering ensuring that $\Sigma m_{i}\leq 0.12$ eV. In
addition, the texture constraints of the Majorana phase $\alpha$ with minimum
and maximum values $-13.96^{\circ}$ and $-3.18^{\circ}$ respectively. On the
other hand, the phase $\beta$ is constrained with minimum and maximum values
$-29.37^{\circ}$ and $-0.75^{\circ}$ respectively. The effective mass
parameter $m_{\beta\beta}$ appearing in the neutrinoless double beta decay is
also predicted in the range with the minimum value $17.28$ MeV and maximum
value $20.22$ MeV respectively which falls within the sensitivity of future
experiments such as LEGEND-1000 [41]. We demonstrate that the texture
compatible with the proposed mixing scheme can be derived from the interplay
of general type-I+II seesaw mechanisms in the framework of $A_{4}\times
Z_{10}\times Z_{2}$ group. As a future prospect, we aim to study the baryon
asymmetry of the Universe and lepton flavour violating decays in the context
of the proposed texture. The detailed analysis in this regard is left for a
future work.

##  V Acknowledgement

The research work of PC is supported by Innovation in Science Pursuit for
Inspired Research (INSPIRE), Department of Science and Technology, Government
of India, New Delhi, vide grant No. IF190651. MD acknowledges the financial
support from the Council of Scientific and Industrial Research (CSIR),
Government of India, New Delhi, through a NET Junior Research Fellowship vide
grant No. 09/0059(15346)/2022-EMR-I. The work of BK has been supported in part
by the Polish National Science Center (NCN) under grant 2020/37/B/ST2/02371,
the Freedom of Research, Mobility of Science, and the Research Excellence
Initiative of the University of Silesia in Katowice.

##  Appendix A Product Rules of $A_{4}$

The discrete group $A_{4}$ is a subgroup of $SU(3)$ which has 12 elements. The
group has four irreducible representations, out of which three are one-
dimensional and one is three-dimensional. In T-basis, the product of two
triplets gives,

$3\times 3=1+1^{\prime}+1^{\prime\prime}+3_{S}+3_{A},$ (24)

where,

$\displaystyle 1$ $\displaystyle=$
$\displaystyle(a_{1}b_{1}+a_{2}b_{3}+a_{3}b_{2})$ (25) $\displaystyle
1^{\prime}$ $\displaystyle=$ $\displaystyle(a_{3}b_{3}+a_{1}b_{2}+a_{2}b_{1})$
(26) $\displaystyle 1^{\prime\prime}$ $\displaystyle=$
$\displaystyle(a_{2}b_{2}+a_{1}b_{3}+a_{3}b_{1})$ (27) $\displaystyle(3\times
3)_{S}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\begin{bmatrix}2a_{1}b_{1}-a_{2}b_{3}-a_{3}b_{2}\\\
2a_{3}b_{3}-a_{1}b_{2}-a_{2}b_{1}\\\ 2a_{2}b_{2}-a_{1}b_{3}-a_{3}b_{1}\\\
\end{bmatrix},$ (28) $\displaystyle(3\times 3)_{A}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\begin{bmatrix}a_{2}b_{3}-a_{3}b_{2}\\\
a_{1}b_{2}-a_{2}b_{1}\\\ a_{3}b_{1}-a_{1}b_{3}\\\ \end{bmatrix}.$ (29)

The trivial singlet can be obtained from the following singlet product rules,

$1\times 1=1,\,\,1^{\prime}\times
1^{\prime\prime}=1,\,\,1^{\prime\prime}\times 1^{\prime}=1.$ (30)

##  Appendix B The Scalar Potential

The $A_{4}\times Z_{10}\times Z_{2}$ invariant scalar potential can be
constructed in the following manner,

$\displaystyle V$ $\displaystyle=$ $\displaystyle
V(\Phi)+V(H)+V(\Delta_{1})+V(\Delta_{2})+V(\eta)+V(\xi)+V(\chi)+V(\psi)+V(\kappa)$
(31) $\displaystyle+V(\zeta)+V(\Phi
H)+V(\Phi\Delta)+V(\Phi\Delta_{2})+V(\Phi\eta)+V(\Phi\xi)+V(\Phi\chi)+V(\Phi\psi)$
$\displaystyle+V(\Phi\kappa)+V(\Phi\zeta)+V(H\Delta)+V(H\Delta_{2})+V(H\eta)+V(H\xi)+V(H\chi)+$
$\displaystyle
V(H\psi)+V(H\kappa)+V(H\zeta)+V(\Delta_{1}\Delta_{2})+V(\Delta_{1}\eta)+V(\Delta_{1}\xi)+V(\Delta_{1}\chi)+V(\Delta_{1}\psi)$
$\displaystyle+V(\Delta_{1}\kappa)+V(\Delta_{1}\zeta)+V(\Delta_{2}\eta)+V(\Delta_{2}\xi)+V(\Delta_{2}\chi)+V(\Delta_{2}\psi)+V(\Delta_{2}\kappa)+V(\Delta_{2}\zeta)$
$\displaystyle+V(\eta\xi)+V(\eta\chi)+V(\eta\psi)+V(\eta\kappa)+V(\eta\zeta)+V(\xi\chi)+V(\xi\psi)+V(\xi\kappa)+$
$\displaystyle
V(\xi\zeta)+V(\chi\psi)+V(\chi\kappa)+V(\chi\zeta)++V(\psi\kappa)+V(\psi\zeta)+V(\kappa\zeta),$

where,

$\displaystyle V(\Phi)$ $\displaystyle=$
$\displaystyle\mu^{2}_{\Phi}(\Phi^{\dagger}\Phi)+\lambda^{\Phi}_{1}(\Phi^{\dagger}\Phi)(\Phi^{\dagger}\Phi)+\lambda^{\Phi}_{2}(\Phi^{\dagger}\Phi)_{1^{\prime}}(\Phi^{\dagger}\Phi)_{1^{\prime\prime}}+\lambda^{\Phi}_{3}(\Phi^{\dagger}\Phi)_{3_{s}}(\Phi^{\dagger}\Phi)_{3_{s}}+\lambda^{\Phi}_{4}(\Phi^{\dagger}\Phi)_{3_{s}}$
(32)
$\displaystyle(\Phi^{\dagger}\Phi)_{3_{a}}+\lambda^{\Phi}_{5}(\Phi^{\dagger}\Phi)_{3_{a}}(\Phi^{\dagger}\Phi)_{3_{a}},$
$\displaystyle V(H)$ $\displaystyle=$
$\displaystyle\mu^{2}_{H}(H^{\dagger}H)+\lambda^{H}(H^{\dagger}H)(H^{\dagger}H),$
(33) $\displaystyle V(\Delta_{1})$ $\displaystyle=$
$\displaystyle\mu^{2}_{\Delta_{1}}Tr(\Delta_{1}^{\dagger}\Delta_{1})+\lambda^{\Delta}_{1}\,Tr(\Delta_{1}^{\dagger}\Delta_{1})Tr(\Delta_{1}^{\dagger}\Delta_{1}),$
(34) $\displaystyle V(\Delta_{2})$ $\displaystyle=$
$\displaystyle\mu^{2}_{\Delta_{2}}Tr(\Delta_{2}^{\dagger}\Delta_{2})+\lambda^{\Delta_{2}}\,Tr(\Delta_{2}^{\dagger}\Delta_{2})Tr(\Delta_{2}^{\dagger}\Delta_{2}),$
(35) $\displaystyle V(\eta)$ $\displaystyle=$
$\displaystyle\mu^{2}_{\eta}(\eta^{\dagger}\eta)+\lambda^{\eta}(\eta^{\dagger}\eta)(\eta^{\dagger}\eta),$
(36) $\displaystyle V(\xi)$ $\displaystyle=$
$\displaystyle\mu^{2}_{\xi}(\xi^{\dagger}\xi)+\lambda^{\xi}(\xi^{\dagger}\xi)(\xi^{\dagger}\xi),$
(37) $\displaystyle V(\chi)$ $\displaystyle=$
$\displaystyle\mu^{2}_{\chi}(\chi^{\dagger}\chi)+\lambda^{\chi}(\chi^{\dagger}\chi)(\chi^{\dagger}\chi),$
(38) $\displaystyle V(\zeta)$ $\displaystyle=$
$\displaystyle\mu^{2}_{\zeta}(\zeta^{\dagger}\zeta)+\lambda^{\zeta}(\zeta^{\dagger}\zeta)(\zeta^{\dagger}\zeta),$
(39) $\displaystyle V(\psi)$ $\displaystyle=$
$\displaystyle\mu^{2}_{\psi}(\psi^{\dagger}\psi)+\lambda^{\psi}_{1}(\psi^{\dagger}\psi)(\psi^{\dagger}\psi)+\lambda^{\psi}_{2}(\psi^{\dagger}\psi)_{1^{\prime}}(\psi^{\dagger}\psi)_{1^{\prime\prime}}+\lambda^{\psi}_{3}(\psi^{\dagger}\psi)_{3_{s}}(\psi^{\dagger}\psi)_{3_{s}}+\lambda^{\psi}_{4}(\psi^{\dagger}\psi)_{3_{s}}$
(40)
$\displaystyle(\psi^{\dagger}\psi)_{3_{a}}+\lambda^{\psi}_{5}(\psi^{\dagger}\psi)_{3_{a}}(\psi^{\dagger}\psi)_{3_{a}},$
$\displaystyle V(\kappa)$ $\displaystyle=$
$\displaystyle\mu^{2}_{\kappa}(\kappa^{\dagger}\kappa)+\lambda^{\kappa}_{1}(\kappa^{\dagger}\kappa)(\kappa^{\dagger}\kappa)+\lambda^{\kappa}_{2}(\kappa^{\dagger}\kappa)_{1^{\prime}}(\kappa^{\dagger}\kappa)_{1^{\prime\prime}}+\lambda^{\kappa}_{3}(\kappa^{\dagger}\kappa)_{3_{s}}(\kappa^{\dagger}\kappa)_{3_{s}}+\lambda^{\kappa}_{4}(\kappa^{\dagger}\kappa)_{3_{s}}$
(41)
$\displaystyle(\kappa^{\dagger}\kappa)_{3_{a}}+\lambda^{\kappa}_{5}(\kappa^{\dagger}\kappa)_{3_{a}}(\kappa^{\dagger}\kappa)_{3_{a}},$
$\displaystyle V(\Phi H)$ $\displaystyle=$ $\displaystyle\lambda^{\Phi
H}_{1}(\Phi^{\dagger}\Phi)(H^{\dagger}H),$ (42) $\displaystyle
V(\Phi\Delta_{1})$ $\displaystyle=$
$\displaystyle\lambda^{\Phi\Delta_{1}}_{1}(\Phi^{\dagger}\Phi)\,Tr(\Delta_{1}^{\dagger}\Delta_{1}),$
(43) $\displaystyle V(\Phi\Delta_{2})$ $\displaystyle=$
$\displaystyle\lambda^{\Phi\Delta_{2}}_{1}(\Phi^{\dagger}\Phi)\,Tr(\Delta_{2}^{\dagger}\Delta_{2}),$
(44) $\displaystyle V(\Phi\eta)$ $\displaystyle=$
$\displaystyle\lambda^{\Phi\eta}_{1}(\Phi^{\dagger}\Phi)(\eta^{\dagger}\eta),$
(45) $\displaystyle V(\Phi\xi)$ $\displaystyle=$
$\displaystyle\lambda^{\Phi\xi}_{1}(\Phi^{\dagger}\Phi)(\xi^{\dagger}\xi),$
(46) $\displaystyle V(\Phi\chi)$ $\displaystyle=$
$\displaystyle\lambda^{\Phi\chi}_{1}(\Phi^{\dagger}\Phi)(\chi^{\dagger}\chi),$
(47) $\displaystyle V(\Phi\zeta)$ $\displaystyle=$
$\displaystyle\lambda^{\Phi\zeta}_{1}(\Phi^{\dagger}\Phi)(\zeta^{\dagger}\zeta),$
(48) $\displaystyle V(\Phi\psi)$ $\displaystyle=$
$\displaystyle\lambda^{\Phi\psi}_{1}(\Phi^{\dagger}\Phi)(\psi^{\dagger}\psi)+\lambda^{\Phi\psi}_{2}[(\Phi^{\dagger}\Phi)_{1^{\prime}}(\psi^{\dagger}\psi)_{1^{\prime\prime}}+(\Phi^{\dagger}\Phi)_{1^{\prime\prime}}(\psi^{\dagger}\psi)_{1^{\prime}}]+\lambda^{\Phi\psi}_{3}$
(49)
$\displaystyle(\Phi^{\dagger}\Phi)_{3_{s}}(\psi^{\dagger}\psi)_{3_{s}}+\lambda^{\Phi\psi}_{4}[(\Phi^{\dagger}\Phi)_{3_{s}}(\psi^{\dagger}\psi)_{3_{a}}+(\Phi^{\dagger}\Phi)_{3_{a}}(\psi^{\dagger}\psi)_{3_{s}}]+\lambda^{\Phi\psi}_{5}$
$\displaystyle(\Phi^{\dagger}\Phi)_{3_{a}}(\psi^{\dagger}\psi)_{3_{a}},$
$\displaystyle V(\Phi\kappa)$ $\displaystyle=$
$\displaystyle\lambda^{\Phi\kappa}_{1}(\Phi^{\dagger}\Phi)(\kappa^{\dagger}\kappa)+\lambda^{\Phi\kappa}_{2}[(\Phi^{\dagger}\Phi)_{1^{\prime}}(\kappa^{\dagger}\kappa)_{1^{\prime\prime}}+(\Phi^{\dagger}\Phi)_{1^{\prime\prime}}(\kappa^{\dagger}\kappa)_{1^{\prime}}]+\lambda^{\Phi\kappa}_{3}(\Phi^{\dagger}\Phi)_{3_{s}}$
(50)
$\displaystyle(\kappa^{\dagger}\kappa)_{3_{s}}+\lambda^{\Phi\kappa}_{4}[(\Phi^{\dagger}\Phi)_{3_{s}}(\kappa^{\dagger}\kappa)_{3_{a}}+(\Phi^{\dagger}\Phi)_{3_{a}}(\kappa^{\dagger}\kappa)_{3_{s}}]+\lambda^{\Phi\kappa}_{5}(\Phi^{\dagger}\Phi)_{3_{a}}(\kappa^{\dagger}\kappa)_{3_{a}},$
$\displaystyle V(H\Delta_{1})$ $\displaystyle=$
$\displaystyle\lambda^{H\Delta_{1}}_{1}(H^{\dagger}H)\,Tr(\Delta_{1}^{\dagger}\Delta_{1})+\frac{1}{\Lambda}(H^{T}i\sigma_{2}\Delta_{1}^{\dagger}H\Phi\psi),$
(51) $\displaystyle V(H\Delta_{2})$ $\displaystyle=$
$\displaystyle\lambda^{H\Delta_{2}}_{1}(H^{\dagger}H)\,Tr(\Delta_{2}^{\dagger}\Delta_{2})+\frac{1}{\Lambda}(H^{T}i\sigma_{2}\Delta_{2}^{\dagger}H\Phi\psi),$
(52)

$\displaystyle V(H\eta)$ $\displaystyle=$
$\displaystyle\lambda^{H\eta}_{1}(H^{\dagger}H)(\eta^{\dagger}\eta),$ (54)
$\displaystyle V(H\xi)$ $\displaystyle=$
$\displaystyle\lambda^{H\xi}_{1}(H^{\dagger}H)(\xi^{\dagger}\xi),$ (55)
$\displaystyle V(H\chi)$ $\displaystyle=$
$\displaystyle\lambda^{H\chi}_{1}(H^{\dagger}H)(\chi^{\dagger}\chi),$ (56)
$\displaystyle V(H\psi)$ $\displaystyle=$
$\displaystyle\lambda^{H\psi}_{1}(H^{\dagger}H)(\psi^{\dagger}\psi),$ (57)
$\displaystyle V(H\kappa)$ $\displaystyle=$
$\displaystyle\lambda^{H\kappa}_{1}(H^{\dagger}H)(\kappa^{\dagger}\kappa),$
(58) $\displaystyle V(H\zeta)$ $\displaystyle=$
$\displaystyle\lambda^{H\zeta}_{1}(H^{\dagger}H)(\zeta^{\dagger}\zeta),$ (59)
$\displaystyle V(\Delta_{1}\Delta_{2})$ $\displaystyle=$
$\displaystyle\lambda^{\Delta_{1}\Delta_{2}}_{1}(\Delta_{1}^{\dagger}\Delta_{1})\,Tr(\Delta_{2}^{\dagger}\Delta_{2}),$
(60) $\displaystyle V(\Delta_{1}\eta)$ $\displaystyle=$
$\displaystyle\lambda^{\Delta_{1}\eta}_{1}\,Tr(\Delta_{1}^{\dagger}\Delta_{1})(\eta^{\dagger}\eta),$
(61) $\displaystyle V(\Delta_{1}\xi)$ $\displaystyle=$
$\displaystyle\lambda^{\Delta_{1}\xi}_{1}\,Tr(\Delta_{1}^{\dagger}\Delta_{1})(\xi^{\dagger}\xi),$
(62) $\displaystyle V(\Delta_{1}\chi)$ $\displaystyle=$
$\displaystyle\lambda^{\Delta_{1}\chi}_{1}\,Tr(\Delta_{1}^{\dagger}\Delta_{1})(\chi^{\dagger}\chi),$
(63) $\displaystyle V(\Delta_{1}\psi)$ $\displaystyle=$
$\displaystyle\lambda^{\Delta_{1}\psi}_{1}\,Tr(\Delta_{1}^{\dagger}\Delta_{1})(\psi^{\dagger}\psi),$
(64) $\displaystyle V(\Delta_{1}\kappa)$ $\displaystyle=$
$\displaystyle\lambda^{\Delta_{1}\kappa}_{1}\,Tr(\Delta_{1}^{\dagger}\Delta_{1})(\kappa^{\dagger}\kappa),$
(65) $\displaystyle V(\Delta_{1}\zeta)$ $\displaystyle=$
$\displaystyle\lambda^{\Delta_{1}\zeta}_{1}\,Tr(\Delta_{1}^{\dagger}\Delta_{1})(\zeta^{\dagger}\zeta),$
(66) $\displaystyle V(\Delta_{2}\eta)$ $\displaystyle=$
$\displaystyle\lambda^{\Delta_{2}\eta}_{1}\,Tr(\Delta_{2}^{\dagger}\Delta_{2})(\eta^{\dagger}\eta),$
(67) $\displaystyle V(\Delta_{2}\xi)$ $\displaystyle=$
$\displaystyle\lambda^{\Delta_{2}\xi}_{1}\,Tr(\Delta_{2}^{\dagger}\Delta_{2})(\xi^{\dagger}\xi),$
(68) $\displaystyle V(\Delta_{2}\chi)$ $\displaystyle=$
$\displaystyle\lambda^{\Delta_{2}\chi}_{1}\,Tr(\Delta_{2}^{\dagger}\Delta_{2})(\chi^{\dagger}\chi),$
(69) $\displaystyle V(\Delta_{2}\psi)$ $\displaystyle=$
$\displaystyle\lambda^{\Delta_{2}\psi}_{1}\,Tr(\Delta_{2}^{\dagger}\Delta_{2})(\psi^{\dagger}\psi),$
(70) $\displaystyle V(\Delta_{2}\kappa)$ $\displaystyle=$
$\displaystyle\lambda^{\Delta_{2}\kappa}_{1}\,Tr(\Delta_{2}^{\dagger}\Delta_{2})(\kappa^{\dagger}\kappa),$
(71) $\displaystyle V(\Delta_{2}\zeta)$ $\displaystyle=$
$\displaystyle\lambda^{\Delta_{2}\zeta}_{1}\,Tr(\Delta_{2}^{\dagger}\Delta_{2})(\zeta^{\dagger}\zeta),$
(72) $\displaystyle V(\eta\xi)$ $\displaystyle=$
$\displaystyle\lambda^{\eta\xi}_{1}(\eta^{\dagger}\eta)(\xi^{\dagger}\xi),$
(73) $\displaystyle V(\eta\chi)$ $\displaystyle=$
$\displaystyle\lambda^{\eta\chi}_{1}(\eta^{\dagger}\eta)(\chi^{\dagger}\chi),$
(74) $\displaystyle V(\eta\psi)$ $\displaystyle=$
$\displaystyle\lambda^{\eta\psi}_{1}(\eta^{\dagger}\eta)(\psi^{\dagger}\psi),$
(75) $\displaystyle V(\eta\kappa)$ $\displaystyle=$
$\displaystyle\lambda^{\eta\kappa}_{1}(\eta^{\dagger}\eta)(\kappa^{\dagger}\kappa),$
(76) $\displaystyle V(\eta\zeta)$ $\displaystyle=$
$\displaystyle\lambda^{\eta\zeta}_{1}(\eta^{\dagger}\eta)(\zeta^{\dagger}\zeta),$
(77) $\displaystyle V(\xi\chi)$ $\displaystyle=$
$\displaystyle\lambda^{\xi\chi}_{1}(\xi^{\dagger}\xi)(\chi^{\dagger}\chi),$
(78) $\displaystyle V(\xi\psi)$ $\displaystyle=$
$\displaystyle\lambda^{\xi\psi}_{1}(\xi^{\dagger}\xi)(\psi^{\dagger}\psi),$
(79) $\displaystyle V(\xi\kappa)$ $\displaystyle=$
$\displaystyle\lambda^{\xi\kappa}_{1}(\xi^{\dagger}\xi)(\kappa^{\dagger}\kappa),$
(80) $\displaystyle V(\xi\zeta)$ $\displaystyle=$
$\displaystyle\lambda^{\xi\zeta}_{1}(\xi^{\dagger}\xi)(\zeta^{\dagger}\zeta),$
(81) $\displaystyle V(\chi\psi)$ $\displaystyle=$
$\displaystyle\lambda^{\chi\psi}_{1}(\chi^{\dagger}\chi)(\psi^{\dagger}\psi),$
(82) $\displaystyle V(\chi\kappa)$ $\displaystyle=$
$\displaystyle\lambda^{\chi\kappa}_{1}(\chi^{\dagger}\chi)(\kappa^{\dagger}\kappa),$
(83) $\displaystyle V(\chi\zeta)$ $\displaystyle=$
$\displaystyle\lambda^{\chi\zeta}_{1}(\chi^{\dagger}\chi)(\zeta^{\dagger}\zeta),$
(84) $\displaystyle V(\psi\kappa)$ $\displaystyle=$
$\displaystyle\lambda^{\psi\kappa}_{1}(\psi^{\dagger}\psi)(\kappa^{\dagger}\kappa)+\lambda^{\psi\kappa}_{2}[(\psi^{\dagger}\psi)_{1^{\prime}}(\kappa^{\dagger}\kappa)_{1^{\prime\prime}}+(\psi^{\dagger}\psi)_{1^{\prime\prime}}(\kappa^{\dagger}\kappa)_{1^{\prime}}]+\lambda^{\psi\kappa}_{3}$
(85)
$\displaystyle(\psi^{\dagger}\psi)_{3_{s}}(\kappa^{\dagger}\kappa)_{3_{s}}+\lambda^{\psi\kappa}_{4}[(\psi^{\dagger}\psi)_{3_{s}}(\kappa^{\dagger}\kappa)_{3_{a}}+(\psi^{\dagger}\psi)_{3_{a}}(\kappa^{\dagger}\kappa)_{3_{s}}]+\lambda^{\psi\kappa}_{5}$
$\displaystyle(\psi^{\dagger}\psi)_{3_{a}}(\kappa^{\dagger}\kappa)_{3_{a}},$
$\displaystyle V(\psi\zeta)$ $\displaystyle=$
$\displaystyle\lambda^{\psi\zeta}_{1}(\psi^{\dagger}\psi)(\zeta^{\dagger}\zeta),$
(86) $\displaystyle V(\kappa\zeta)$ $\displaystyle=$
$\displaystyle\lambda^{\kappa\zeta}_{1}(\kappa^{\dagger}\kappa)(\zeta^{\dagger}\zeta),$
(87)

The following equations hold good for the chosen vacuum expectation values
(VEVs) $\langle\Phi\rangle_{\circ}=v_{\Phi}\,(1,0,0)^{T}$, $\langle
H\rangle_{\circ}=v_{H}$, $\langle\eta\rangle_{\circ}=v_{\eta}$,
$\langle\chi\rangle_{\circ}=v_{\chi}$, $\langle\xi\rangle_{\circ}=v_{\xi}$,
$\langle\varsigma\rangle_{\circ}=v_{\varsigma}$,
$\langle\Delta_{1}\rangle_{\circ}=v_{\Delta_{1}}$,
$\langle\Delta_{2}\rangle_{\circ}=v_{\Delta_{2}}$,
$\langle\psi\rangle_{\circ}=v_{\psi}(0,1,-1)^{T}$ and
$\langle\kappa\rangle_{\circ}=v_{\kappa}(0,1,-1)^{T}$, based on the
minimization conditions of the scalar potential:

$\displaystyle\frac{dV}{d\Phi_{1}}$ $\displaystyle=$
$\displaystyle\frac{1}{9}v_{\Phi}(-\frac{24}{\Lambda}\mu_{2}v_{\Delta_{1}}v_{H}^{2}v_{\chi}^{2}\lambda_{1}^{\Phi\chi}+9v_{\Delta_{2}}^{2}\lambda_{1}^{\Phi\Delta_{2}}+9v_{\eta}^{2}\lambda_{1}^{\Phi\eta}+9v_{H}^{2}\lambda_{1}^{\Phi
H}+9v_{\kappa}^{2}\lambda_{1}^{\Phi\kappa}+4v_{\kappa}^{2}\lambda_{3}^{\Phi\kappa}$
(88)
$\displaystyle+9\mu_{\Phi}^{2}-18v_{\psi}^{2}\lambda_{1}^{\Phi\psi}+4v_{\psi}^{2}\lambda_{3}^{\Phi\psi}+2v_{\Phi}^{2}\left(9\lambda_{1}^{\Phi}+4\lambda_{3}^{\Phi}\right)+9\lambda_{1}^{\Phi\xi}v_{\xi}^{2}+9\lambda_{1}^{\phi\zeta}v_{\zeta}^{2}$
$\displaystyle+9v_{\Delta_{1}}^{2}\lambda_{1}^{\Phi\Delta_{1}})=0,$
$\displaystyle\frac{dV}{d\Phi_{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{9}v_{\Phi}(\frac{12}{\Lambda}v_{H}^{2}v_{\Delta_{1}}(\mu_{2}v_{\kappa}-\mu_{1}v_{\psi})+18v_{\kappa}^{2}\lambda_{2}^{\Phi\kappa}+2v_{\kappa}^{2}\lambda_{3}^{\Phi\kappa}-3v_{\kappa}^{2}\lambda_{4}^{\Phi\kappa}+9v_{\psi}^{2}\lambda_{2}^{\Phi\psi}$
(89)
$\displaystyle-2v_{\psi}^{2}\lambda_{3}^{\Phi\psi}+3v_{\psi}^{2}\lambda_{4}^{\Phi\psi})=0,$
$\displaystyle\frac{dV}{d\Phi_{3}}$ $\displaystyle=$
$\displaystyle\frac{1}{9}v_{\Phi}(v_{\kappa}^{2}(9\lambda_{2}^{\Phi\kappa}-2\lambda_{3}^{\Phi\kappa}-3\lambda_{4}^{\Phi\kappa})+v_{\psi}^{2}(\frac{12}{\Lambda^{2}}\mu_{1}v_{H}^{2}v_{\Delta_{1}}+9\lambda_{2}^{\Phi\psi}-2\lambda_{3}^{\Phi\psi}$
(90) $\displaystyle-3\lambda_{4}^{\Phi\psi}))=0,$
$\displaystyle\frac{dV}{d\psi_{1}}$ $\displaystyle=$
$\displaystyle-\frac{4}{3\Lambda}\mu_{1}v_{H}^{2}v_{\Delta_{1}}v_{\Phi}^{2}+\frac{1}{9}v_{\psi}(v_{\kappa}^{2}(9\lambda_{2}^{\kappa\psi}+4\lambda_{3}^{\kappa\psi})-6v_{\psi}^{2}\lambda_{4}^{\psi})=0,$
(91) $\displaystyle\frac{dV}{d\psi_{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{9}v_{\psi}(-9v_{H}^{2}\lambda_{1}^{H\psi}-9\mu_{\psi}^{2}-9\lambda_{1}^{\Delta_{1}\psi}v_{\Delta_{1}}^{2}-9v_{\Delta_{2}}^{2}\lambda_{1}^{\Delta_{2}\psi}-9\lambda_{1}^{\psi\zeta}v_{\zeta}^{2}-9\lambda_{1}^{\eta\psi}v_{\eta}^{2}-9\lambda_{1}^{\kappa\psi}v_{\kappa}^{2}$
(92)
$\displaystyle+9\lambda_{2}^{\kappa\psi}v_{\kappa}^{2}+6\lambda_{3}^{\kappa\psi}v_{\kappa}^{2}+3\lambda_{4}^{\kappa\psi}v_{\kappa}^{2}-9\lambda_{1}^{\xi\psi}v_{\xi}^{2}-9\lambda_{1}^{\Phi\psi}v_{\Phi}^{2}+2\lambda_{3}^{\Phi\psi}v_{\Phi}^{2}+3\lambda_{4}^{\Phi\psi}v_{\Phi}^{2}$
$\displaystyle-9\lambda_{1}^{\chi\psi}v_{\chi}^{2}+36\lambda_{1}^{\psi}v_{\psi}^{2}+9\lambda_{2}^{\psi}v_{\psi}^{2}+12\lambda_{3}^{\psi}v_{\psi}^{2}+3\lambda_{4}^{\psi}v_{\psi}^{2})=0,$
$\displaystyle\frac{dV}{d\psi_{3}}$ $\displaystyle=$
$\displaystyle\frac{1}{9}v_{\psi}(9v_{H}^{2}\lambda_{1}^{H\psi}+9\mu_{\psi}^{2}+9\lambda_{1}^{\Delta_{1}\psi}v_{\Delta_{1}}^{2}+9v_{\Delta_{2}}^{2}\lambda_{1}^{\Delta_{2}\psi}+9\lambda_{1}^{\psi\zeta}v_{\zeta}^{2}+9\lambda_{1}^{\eta\psi}v_{\eta}^{2}+9\lambda_{1}^{\kappa\psi}v_{\kappa}^{2}$
(93)
$\displaystyle-18\lambda_{2}^{\kappa\psi}v_{\kappa}^{2}+2\lambda_{3}^{\kappa\psi}v_{\kappa}^{2}+3\lambda_{4}^{\kappa\psi}v_{\kappa}^{2}+9\lambda_{1}^{\xi\psi}v_{\xi}^{2}-2\lambda_{3}^{\Phi\psi}v_{\Phi}^{2}+3\lambda_{4}^{\Phi\psi}v_{\Phi}^{2}+9\lambda_{1}^{\Phi\psi}v_{\psi}^{2}$
$\displaystyle+9\lambda_{1}^{\chi\psi}v_{\chi}^{2}-36\lambda_{1}^{\psi}v_{\psi}^{2}-9\lambda_{2}^{\psi}v_{\psi}^{2}-12\lambda_{3}^{\psi}v_{\psi}^{2}+3\lambda_{4}^{\psi}v_{\psi}^{2})=0,$
$\displaystyle\frac{dV}{d\kappa_{1}}$ $\displaystyle=$
$\displaystyle-\frac{4}{3\Lambda}\mu_{2}v_{H}^{2}v_{\Delta_{1}}v_{\Phi}^{2}+\lambda_{1}^{\Delta_{1}\kappa}v_{\Delta_{1}}^{2}v_{\kappa}+\frac{1}{9}v_{\kappa}(9v_{H}^{2}\lambda_{1}^{H\kappa}+9\mu_{\kappa}^{2}+9v_{\Delta_{2}}^{2}\lambda_{1}^{\Delta_{2}\kappa}+9\lambda_{1}^{\eta\kappa}v_{\eta}^{2}+9\lambda_{2}^{\kappa\psi}v_{\kappa}^{2}$
(94)
$\displaystyle+18\lambda_{1}^{\kappa}v_{\kappa}^{2}+9\lambda_{2}^{\kappa}v_{\kappa}^{2}+4\lambda_{3}^{\kappa}v_{\kappa}^{2}+3\lambda_{4}^{\kappa}v_{\kappa}^{2}+9\lambda_{1}^{\kappa\zeta}v_{\xi}^{2}-18\lambda_{1}^{\kappa\psi}v_{\psi}^{2}+2\lambda_{3}^{\kappa\psi}v_{\psi}^{2}+3\lambda_{4}^{\kappa\psi}v_{\psi}^{2}$
$\displaystyle+9\lambda_{1}^{\xi\kappa}v_{\xi}^{2}+9\lambda_{1}^{\Phi\kappa}v_{\Phi}^{2}+4\lambda_{3}^{\Phi\kappa}v_{\Phi}^{2}+9\lambda_{1}^{\chi\kappa}v_{\chi}^{2})=0,$
$\displaystyle\frac{dV}{d\kappa_{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{9}v_{\kappa}(9v_{H}^{2}\lambda_{1}^{H\kappa}+9\mu_{\kappa}^{2}+9\lambda_{1}^{\Delta_{1}\kappa}v_{\Delta_{1}}^{2}+9v_{\Delta_{2}}^{2}\lambda_{1}^{\Delta_{2}\kappa}+9\lambda_{1}^{\kappa\zeta}v_{\zeta}^{2}+9\lambda_{1}^{\eta\kappa}v_{\eta}^{2}+18\lambda_{1}^{\kappa}v_{\kappa}^{2}$
(95)
$\displaystyle+18\lambda_{2}^{\kappa}v_{\kappa}^{2}-6\lambda_{4}^{\kappa}v_{\kappa}^{2}-18\lambda_{1}^{\kappa\psi}v_{\psi}^{2}+9\lambda_{2}^{\kappa\psi}v_{\psi}^{2}-4\lambda_{3}^{\kappa\psi}v_{\psi}^{2}+9\lambda_{1}^{\xi\kappa}v_{\xi}^{2}+9\lambda_{1}^{\Phi\kappa}v_{\Phi}^{2}$
$\displaystyle-2\lambda_{3}^{\Phi\kappa}v_{\Phi}^{2}-3\lambda_{4}^{\phi\kappa}v_{\Phi}^{2}+9\lambda_{1}^{\chi\kappa}v_{\chi}^{2})=0,$
$\displaystyle\frac{dV}{d\kappa_{3}}$ $\displaystyle=$
$\displaystyle\frac{1}{9}v_{\kappa}(27\lambda_{2}^{\kappa}v_{\kappa}^{2}-12\lambda_{3}^{\kappa}v_{\kappa}^{2}+18\lambda_{2}^{\kappa\psi}v_{\psi}^{2}+2\lambda_{3}^{\kappa\psi}v_{\psi}^{2}-3(\lambda_{4}^{\kappa}v_{\kappa}^{2}+\lambda_{4}^{\kappa\psi}v_{\psi}^{2}))=0,$
(96) $\displaystyle\frac{dV}{dH}$ $\displaystyle=$
$\displaystyle\frac{1}{9}v_{H}(-\frac{8}{\Lambda}\mu_{2}v_{\Delta_{1}}v_{\kappa}v_{\Phi}^{2}+3\lambda_{1}^{H\Delta_{1}}v_{\Delta_{1}}^{2}+3(\mu_{H}^{2}+v_{\Delta_{2}}^{2}\lambda_{1}^{\Delta_{2}H}+v_{\kappa}^{2}\lambda_{1}^{H\kappa}+2\lambda^{H}v_{H}^{2}+\lambda_{1}^{H\zeta}v_{\zeta}^{2}$
(97)
$\displaystyle+\lambda_{1}^{H\eta}v_{\eta}^{2}+\lambda_{1}^{H\xi}v_{\xi}^{2}+\lambda_{1}^{H\chi}v_{\chi}^{2}-2\lambda_{1}^{H\psi}v_{\psi}^{2}+\lambda_{1}^{\Phi
H}v_{\Phi}^{2}))=0,$ $\displaystyle\frac{dV}{d\Delta_{1}}$ $\displaystyle=$
$\displaystyle
2\lambda^{\Delta_{1}}v_{\Delta_{1}}^{3}-\frac{4}{\Lambda}\mu_{2}v_{H}v_{\kappa}v_{\Phi}^{2}+v_{\Delta_{1}}(\mu_{\Delta_{1}}^{2}+v_{H}^{2}\lambda_{1}^{\Delta_{1}H}+v_{\kappa}^{2}\lambda_{1}^{\Delta_{1}\kappa}+\lambda_{1}^{\Delta_{1}\Delta_{2}}v_{\Delta_{2}}^{2}+\lambda_{1}^{\Delta_{1}\zeta}v_{\zeta}^{2}+\lambda_{1}^{\Delta_{1}\eta}v_{\eta}^{2}$
(98)
$\displaystyle+\lambda_{1}^{\Delta_{1}\xi}v_{\xi}^{2}+\lambda_{1}^{\Delta_{1}\chi}v_{\chi}^{2}-2\lambda_{1}^{\Delta_{1}\psi}v_{\psi}^{2}+\lambda_{1}^{\Phi\Delta_{1}}v_{\Phi}^{2})=0,$
$\displaystyle\frac{dV}{d\Delta_{2}}$ $\displaystyle=$ $\displaystyle
v_{\Delta_{2}}(\mu_{\Delta_{2}}^{2}+v_{H}^{2}\lambda_{1}^{\Delta_{2}H}+v^{2}{}_{\Phi}\lambda_{1}^{\Delta_{2}\phi}+v_{\zeta}^{2}\lambda_{1}^{\Delta_{2}\zeta}+v_{\eta}^{2}\lambda_{1}^{\Delta_{2}\eta}+v_{\kappa}^{2}\lambda_{1}^{\Delta_{2}\kappa}+2\lambda^{\Delta_{2}}v_{\Delta_{2}}^{2}$
(99)
$\displaystyle+v_{\xi}^{2}\lambda_{1}^{\Delta_{2}\xi}+v_{\chi}^{2}\lambda_{1}^{\Delta_{2}\chi}-2v_{\psi}^{2}\lambda_{1}^{\Delta_{2}\psi}+\lambda_{1}^{\Delta_{1}\Delta_{2}}v_{\Delta_{1}}^{2})=0,$
$\displaystyle\frac{dV}{d\eta}$ $\displaystyle=$ $\displaystyle
v_{\eta}(\mu_{\eta}^{2}+v_{H}^{2}\lambda_{1}^{\text{H$\eta$}}+\lambda_{1}^{\Delta_{1}\eta}v_{\Delta_{1}}^{2}+v_{\Delta_{2}}^{2}\lambda_{1}^{\Delta_{2}\eta}+\lambda_{1}^{\eta\zeta}v_{\zeta}^{2}+2\lambda^{\eta}v_{\eta}^{2}+\lambda_{1}^{\eta\kappa}v_{\kappa}^{2}+\lambda_{1}^{\eta\xi}v_{\xi}^{2}$
(100)
$\displaystyle+\lambda_{1}^{\eta\chi}v_{\chi}^{2}-2\lambda_{1}^{\eta\psi}v_{\psi}^{2}+\lambda_{1}^{\Phi\eta}v_{\Phi}^{2})=0,$

$\displaystyle\frac{dV}{d\xi}$ $\displaystyle=$ $\displaystyle
v_{\xi}(v_{H}^{2}\lambda_{1}^{H\xi}+\mu_{\xi}^{2}+\lambda_{1}^{\Delta_{1}\xi}v_{\Delta_{1}}^{2}+v_{\Delta_{2}}^{2}\lambda_{1}^{\Delta_{2}\xi}+\lambda_{1}^{\xi\zeta}v_{\zeta}^{2}+\lambda_{1}^{\eta\xi}v_{\eta}^{2}+\lambda_{1}^{\xi\kappa}v_{\kappa}^{2}+2\lambda^{\xi}v_{\xi}^{2}$
(102)
$\displaystyle+\lambda_{1}^{\xi\chi}v_{\chi}^{2}-2\lambda_{1}^{\xi\psi}v_{\psi}^{2}+\lambda_{1}^{\Phi\xi}v_{\Phi}^{2})=0,$
$\displaystyle\frac{dV}{d\chi}$ $\displaystyle=$ $\displaystyle
v_{\chi}(v_{H}^{2}\lambda_{1}^{H\chi}+\mu_{\chi}^{2}+\lambda_{1}^{\Delta_{1}\chi}v_{\Delta_{1}}^{2}+v_{\Delta_{2}}^{2}\lambda_{1}^{\Delta_{2}\chi}+\lambda_{1}^{\chi\zeta}v_{\zeta}^{2}+\lambda_{1}^{\eta\chi}v_{\eta}^{2}+\lambda_{1}^{\chi\kappa}v_{\kappa}^{2}+2\lambda^{\chi}v_{\chi}^{2}$
(103)
$\displaystyle+\lambda_{1}^{\xi\chi}v_{\xi}^{2}+\lambda_{1}^{\Phi\chi}v_{\Phi}^{2}-2\lambda_{1}^{\chi\psi}v_{\psi}^{2})=0,$
$\displaystyle\frac{dV}{d\zeta}$ $\displaystyle=$ $\displaystyle
v_{\zeta}(\mu_{\zeta}^{2}+v_{H}^{2}\lambda_{1}^{H\zeta}+\lambda_{1}^{\Delta_{1}\zeta}v_{\Delta_{1}}^{2}+v_{\Delta_{2}}^{2}\lambda_{1}^{\Delta_{2}\zeta}+2\lambda^{\zeta}v_{\zeta}^{2}+\lambda_{1}^{\eta\zeta}v_{\eta}^{2}+\lambda_{1}^{\kappa\zeta}v_{\kappa}^{2}+\lambda_{1}^{\xi\zeta}v_{\xi}^{2}$
(104)
$\displaystyle+\lambda_{1}^{\Phi\zeta}v_{\Phi}^{2}+\lambda_{1}^{\chi\zeta}v_{\chi}^{2}-2\lambda_{1}^{\psi\zeta}v_{\psi}^{2})=0.$

## References

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