# Octant of $\theta_{23}$, MH, $0\nu\beta\beta$ decay and vacuum alignment of
$A_{4}$ flavour symmetry in an inverse seesaw model

Maibam Ricky Devi<EMAIL_ADDRESS>Kalpana Bora<EMAIL_ADDRESS>Department of Physics, Gauhati University, Guwahati-781014, Assam, India

###### Abstract

Measurements of disappearance channel of long baseline accelerator based
experiments (like NO$\nu$A) are inflicted with the problem of octant
degeneracy. In these experiments, the mass hierarchy (MH) sensitivity depends
upon the value of CP-violating phase $\delta_{CP}$. Moreover, MH of light
neutrino masses is still not fixed. Also, the flavour structure of fermions is
yet not fully understood. We discuss all these issues, in a highly predictive,
low-scale inverse seesaw (ISS) model within the framework of $A_{4}$ flavour
symmetry. Recent global analysis has shown a preference for normal hierarchy
and higher octant of $\theta_{23}$, and hence we discuss our results with
reference to these, and find that the vacuum alignment of $A_{4}$ triplet
flavon (1,-1,-1) favours these results. Finally, we check if our very precise
prediction on $m_{ee}$ and the lightest neutrino mass falls within the range
of sensitivities of the neutrinoless double beta decay ($0\nu\beta\beta$)
experiments. We note that when octant of $\theta_{23}$ and MH is fixed by more
precise measurements of future experiments, then through our results, it would
be possible to precisely identify the favourable vacuum alignment
corresponding to the $A_{4}$ triplet field as predicted in our model.

Neutrino oscillations; inverse seesaw model; $A_{4}$ flavour symmetry.

##  I Introduction

Experimental evidences in support of neutrino oscillations during last few
decades have led theoretical physicists to construct new models. Recent
results from experiments worldwide involving neutrinos have not only
ascertained that the neutrinos are massive, with the mixing of flavours while
they travel, but also provided precise values of their mixing parameters. Yet
these experiments failed to explain the nature of the neutrinos, i.e., if they
are Dirac or Majorana, the hierarchy of masses in the three families of
neutrino and their absolute values, octant of the atmospheric mixing angle
$\theta_{23}$ and the leptonic CP-violating (CPV) phases. It is well known
that measurements of disappearance channel of long baseline (LBL) accelerator
based experiments (like NO$\nu$A) are inflicted with the problem of octant
degeneracy, where the solutions in both octants of $\theta_{23}$ are obtained
for both the mass hierarchies. Moreover, MH of neutrino masses is still not
fixed and its sensitivity depends upon the value of CP-violating phase
$\delta_{CP}$ (see for example discussions in Cao:2020ans  , deSalas:2020pgw
). So, in a way, it can be stated that measurements of octant of
$\theta_{23}$, MH and CPV phase are dependent on each other (or we can say
that they are entangled).

It is well known that the models based on type-I and type-II seesaw mechanism
Penedo:2017knr  are very unlikely to get tested in experiments due to the
presence of very heavy right-handed neutrinos whose mass scale can go up to
($\sim 10^{10}-10^{14}$) GeV. To circumvent this problem, one can construct
low-scale seesaw models with Majorana neutrinos of a relatively lower mass
scale of few TeV. Such models are much more accessible to get tested in
ongoing neutrino experiments to find new physics, and this is the motivation
behind considering low-scale seesaw models. Also, flavour structure of
fundamental particles still remains unexplained. Hence in this work, we choose
to consider an ISS model with $A_{4}$ flavour symmetry Ishimori:2012zz  .Many
flavour symmetries have been used by researchers for the purpose, however, no
information is available so far about the scale of flavour symmetry breaking
or the direction in which flavon fields acquire VEV.

To obtain some insight about VEV alignment of flavon field of $A_{4}$ flavour
symmetry, we use the information available on neutrino oscillation parameters
from various experiments. To be specific, we use the information on octant-MH
degeneracy present in the measurements of disappearance channels of NO$\nu$A
Cao:2020ans  , T2K Abe:2011sj  etc, and the fact that MH among neutrino masses
is yet not fixed. Though disappearance channel suffers from this degeneracy,
appearance channel does not. We would like to mention here that this work is
an extension of our previous work Devi:2021ujp  . The Lagrangian is
constructed up to dimension 6 with mass term $\mu_{s}$, as we need to make
sure that $\mu_{s}$ has a very low energy scale compared to the other mass
terms. Some extra symmetries such as $Z_{4}$, $Z_{5}$ and $U(1)_{X}$ are
integrated along with the $A_{4}$ symmetry to make the model feasible where
$U(1)_{X}$ is a global symmetry. We obtain the light neutrino mass matrix from
the Lagrangian, which when compared to the phenomenological one, gives us a
set of equations for the triplet and singlet flavon fields. These equations
are then solved by using the known values of mixing angles and the neutrino
mass differences as inputs, to find the unknown parameters such as the
$m_{lightest}$ and CP-violating phases $\delta_{CP}$(Dirac phase), $\alpha$
and $\beta$ (Majorana phases). This makes our model highly testable with a
heavy Majorana neutrino of $\approx$1 TeV. We constrain the values of the
known parameters within the 3$\sigma$ ranges of their latest global best fit
data which are summarised in Table (1). The new values of unknown oscillation
parameters predicted here can be used in pinpointing the favoured vacuum
alignment of triplet flavon field of $A_{4}$ symmetry. Next, these are used to
study implication on effective neutrino mass $m_{ee}$ measured in
$0\nu\beta\beta$ experiments, and correlations between lightest neutrino mass
$m_{lightest}$ and $m_{ee}$ are obtained. All these interesting results along
with low scale of seesaw mechanism are testable in future experiments.

Some earlier works on $A_{4}$ flavour symmetry based neutrino models can be
found in Dinh:2016tu ; Chen:2012st ; Kalita:2015jaa ; Sarma:2018bgf  . $A_{4}$
based inverse and linear seesaw models at low scale can be found in
Hirsch:2009mx ; Sruthilaya:2017mzt  , where in Hirsch:2009mx  a linear as well
as an inverse seesaw models incorporating $SU(2)_{L}$, $Z_{3}$ and $A_{4}$
symmetries was discussed. Similarly, in Sruthilaya:2017mzt  , a discussion on
linear seesaw model with $A_{4}$ flavour symmetry, $Z_{4}$, $Z_{3}$ discrete
symmetries and a $U(1)_{X}$ global symmetry is presented. In Borah:2017dm  ,
$A_{4}$ symmetry based type I as well as inverse seesaw model was considered
and in Sahu:2020tqe  an $A_{4}$ LRSM was discussed. To study other flavour
symmetry based neutrino models one can also refer to the works in
Verma:2018lro ; Girardi:2015rwa ; Petcov:2018snn ; Petcov:2018mvm ;
Borah:2018nvu ; Sethi:2019bfu ; Cai:2018upp  . Since we are interested to link
resolving of octant-MH degeneracy with VEV alignment of flavon field, it is
relevant to understand the importance of the former issue, for the sake of
pedagogy, and for that purpose, we take a note of some earlier works to
resolve this. Several papers have discussed the current status of light
neutrino oscillation parameters, and methods have been suggested earlier to
resolve octant-MH degeneracy Cao:2020ans ; Rahaman:2021cgc ; Rout:2020cxi ;
Yasuda:2020cff ; Ghosh:2019sfi ; Haba:2018klh ; Verma:2018gwi ; Bharti:2018eyj
; Fogli:1996pv ; Barger:2001yr ; Smirnov:2018jzl ; Bora:2014zwa  . A recent
discussion on 2$\sigma$ tension in T2K and NO$\nu$A can be found in
deSalas:2020pgw ; Denton:2020uda ; Kelly:2020fkv ; Esteban:2020itz ;
alex_himmel_2020_3959581 ; Nizam:2018got  .

Table 1: Updated values of neutrino oscillation parameters in 3$\sigma$ values taken from deSalas:2020pgw . Neutrino Oscillation Parameters | for normal hierarchy (NH) | for inverted hierarchy (IH)
---|---|---
$\Delta m_{21}^{2}/(10^{-5}eV^{2})$ | $6.94-8.14$ | $6.94-8.14$
$|\Delta m_{31}^{2}|/(10^{-3}eV^{2})$ | $2.47-2.63$ | $2.37-2.53$
$\theta_{12}/^{\circ}$ | $31.4-37.4$ | $31.4-37.4$
$\theta_{23}/^{\circ}$ | $41.20-51.33$ | $41.16-51.25$
$\theta_{13}/^{\circ}$ | $8.13-8.92$ | $8.17-8.96$
$\delta_{CP}/^{\circ}$ | $128-359$ | $200-353$

It is worth mentioning however, that in a very recent global analysis
deSalas:2020pgw , they have presented some preferences for higher octant of
atmospheric mixing angle and normal mass hierarchy, and we have discussed our
results in context of this analysis too and found that the VEV
(1,-1,-1)/(-1,1,1) of the $A_{4}$ triplet flavon field is preferred. Such
information on pinpointing VEV of flavour symmetry has not been presented
earlier. This novel information is then used to predict very precise values of
$m_{lightest}$ and $m_{ee}$ (of $0\nu\beta\beta$ decay), which can be tested
in future. We have done our computation up to an accuracy of $10^{-5}$, which
indicates the precision level of the computation.

The paper has been organised as follows. A brief discussion on octant-MH
degeneracy and the current status of values of neutrino oscillation parameter
values is presented in section II. In section III, we present the implication
of neutrino oscillation parameters on neutrinoless double beta decay. We
discuss our inverse seesaw model with $A_{4}$ symmetry in section IV and in
section V, the numerical analysis and results are presented. A discussion on
our results is summarised in section VI and then we conclude in section VII.

##  II Octant-MH degeneracy and current status of neutrino oscillation
parameters

The Super-Kamiokande experiment (SK) Super-Kamiokande:1998kpq  and Sudbury
Neutrino Observatory (SNO) SNO:2001kpb ; SNO:2002tuh  after discovering
neutrino oscillation phenomena, provided us strong evidence that neutrinos are
massive particles whose mass can not be explained within Standard Model, and
that there is a mixing of flavours while they travel. The probability in the
disappearance channel of the long baseline experiments can be expressed as
Bora:2014zwa  ,

$P_{\mu\mu}=1-sin^{2}2\theta_{23}sin^{2}\dfrac{1.27\Delta
m^{2}_{31}L}{E}+4sin^{2}\theta_{13}sin^{2}\theta_{23}cos{2\theta_{23}}sin^{2}\dfrac{1.27\Delta
m^{2}_{31}L}{E}$ (1)

where $\Delta m^{2}_{ij}=m_{i}^{2}-m_{j}^{2}$, $\Delta_{ij}=\dfrac{\Delta
m^{2}_{ij}L}{4E}$ and $\dfrac{\Delta^{2}_{21}}{\Delta^{2}_{31}}$ is assumed to
be small. The measurements of disappearance channel of long baseline
experiments cannot differentiate the octant of the atmospheric mixing angle,
as it is clear from Eq. (1) that the probability is sensitive to
$sin^{2}2\theta_{23}$ and hence $P(\theta_{23})=P(\pi/2-\theta_{23})$. This is
called octant degeneracy as one cannot differentiate if $\theta_{23}>\pi/4$ or
$\theta_{23}<\pi/4$ Fogli:1996pv ; Barger:2001yr  (or in other words it cannot
differentiate if $sin^{2}\theta_{23}>0.5$ or $sin^{2}\theta_{23}<0.5$)
deSalas:2020pgw  from experimental measurements. The LBL experiments are also
quite sensitive to measurement of CP-violating phase $\delta_{CP}$. Recent
measurements have shown that $\delta_{CP}\approx 0.8\pi$ from NO$\nu$A
analysis while it disfavours the region $\delta_{CP}\approx 1.5\pi$ which
coincides with the T2K best fit values deSalas:2020pgw  . Hence, there is
ambiguity in CPV phase measurements at T2K and NO$\nu$A. The atmospheric
neutrino results from Super-Kamiokande Super-Kamiokande:2017yvm  and Deep Core
(Ice Cube) experiments IceCube:2017lak ; IceCube:2019dqi  when are combined
with the data of the long baseline experiments Denton:2020uda ; Kelly:2020fkv
; Esteban:2020itz  , then substantially more preference for normal hierarchy
is observed. However, as the measurements of $\delta_{CP}$ in T2K and NO$\nu$A
are inflicted with discrepancy as discussed above, no clearcut preference for
a particular MH is seen in their data. In a recent work Cao:2020ans  , it was
shown that the data of T2K, NO$\nu$A and JUNO can be combined to enhance the
measurement of $\delta_{CP}$ and resolve MH and octant degeneracy problem too.

Thus it is seen that the neutrino oscillation experiments face the problem of
parameter degeneracies and measurement of one parameter depends on that of
another, and it is still a challenge before theorists as well as
experimentalists alike. Many proposals have been given to resolve these
ambiguities Bharti:2018eyj ; Bora:2014zwa  . Suppose in future, these
parameter degeneracies are resolved, then, we propose in this work, that we
can identify the vacuum alignment of the triplet flavon field. It may be noted
that at present, no information about flavour symmetries is available, and
hence findings of this work command importance.

##  III Implication of neutrino oscillation parameters on neutrinoless double
beta decay

The search for $0\nu\beta\beta$ decay is very intriguing as its detection will
confirm the Majorana nature of neutrinos. In this process, neutrinos are not
emitted due to the immediate reabsorption of the emitted neutrinos. These
neutrinos thus have an effective mass that can be expressed as
Rodejohann:2011mu ; Bilenky:2014uka  ,

$m^{\nu}_{ee}=|cos\theta^{2}_{12}cos\theta^{2}_{13}m_{1}+sin\theta^{2}_{12}cos\theta^{2}_{13}m_{2}e^{2i\alpha}+sin\theta^{2}_{13}m_{3}e^{2i\beta}|\textrm{.}$
(2)

Thus $m^{\nu}_{ee}$ depends exclusively on the neutrino oscillation parameters
that can be obtained from our model as well as from the 3$\sigma$ values of
global neutrino oscillation data. The half-life of the isotope
$T^{0\nu}_{1/2}(\mathcal{N})$ which is involved in the $0\nu\beta\beta$ decay
is constrained by its decay amplitude deSalas:2020pgw ; DellOro:2016tmg  .
This lifetime of isotopes can constrain the bounds on the $m^{\nu}_{ee}$ from
the $0\nu\beta\beta$ events as deSalas:2020pgw

$T^{0\nu}_{1/2}(\mathcal{N})=\dfrac{m^{2}_{e}}{G^{\mathcal{N}}_{0\nu}|\mathcal{M}^{\mathcal{N}}_{0\nu}|^{2}m^{2}_{\beta\beta}}$
(3)

where the electron mass is denoted by $m_{e}$, $G^{\mathcal{N}}_{0\nu}$ and
$\mathcal{M}^{\mathcal{N}}_{0\nu}$ represents the factor involving phase space
and the nuclear matrix element that depends on the isotope used in the
experiment respectively. The strongest bounds on half-life
$T^{0\nu}_{1/2}(\mathcal{N})$ set by GERDA GERDA:2019ivs  , CUORE
CUORE:2019yfd  and KamLAND-Zen KamLAND-Zen:2016pfg  are
$T^{0\nu}_{1/2}>9\times 10^{25}$ years for 76Ge, $T^{0\nu}_{1/2}>3.2\times
10^{25}$ years for 130Te and $T^{0\nu}_{1/2}>1.07\times 10^{26}$ years for
136Xe respectively at $90\%$ confidence level. Different $0\nu\beta\beta$
experiments use different experimental techniques such as external trackers,
bolometers, semiconductor detectors to probe this rare decay.
The current upper limits of $m_{\beta\beta}$ are $m_{0\nu\beta\beta}<$ 104-228
meV by GERDA GERDA:2019ivs  , $m_{0\nu\beta\beta}<$ 75-350 meV by CUORE
CUORE:2019yfd  experiment, and $m_{0\nu\beta\beta}<$ 61-165 meV by KamLAND-Zen
experiment KamLAND-Zen:2016pfg  . Some currently operating experiments are
CANDLES III, COBRA, MAJORANA Demonstrator, CUORE, NEXT WHITE and KamLand - Zen
800 Calibbi:2017uvl  . For these ongoing as well experiments under
construction such as SuperNEMO, LEGEND 200, Amore I and Amore II, NEXT etc.,
the future sensitivity of $m_{\beta\beta}$ is expected to reach up to $5\times
10^{-3}$ eV. In later sections, we compute $m_{0\nu\beta\beta}$ allowed in our
model and find that it agrees well with its current upper limits of some of
these experiments.

##  IV  An $A_{4}\times U(1)_{X}\times Z_{5}\times Z_{4}$ inverse seesaw model

To develop our model based on seesaw mechanism PhysRevLett.56.561 ;
PhysRevD.34.1642  , we use cyclic groups $Z_{4}$ and $Z_{5}$ along with
$A_{4}$ group and $U(1)_{X}$ global symmetry. Our model includes an extra
family of three singlet sterile neutrinos, $\mathcal{S}$ apart from the SU(2)
singlet neutrinos, $\mathcal{N}$. Thus we can write the neutrino mass matrix
obtained from the Lagrangian of the seesaw model in the basis
($\nu^{c}_{L},\mathcal{N},\mathcal{S}$) as Wyler:1982dd  :

$\left(M_{\nu}\right)_{iss}=\begin{bmatrix}0&m_{D}&0\\\ m^{T}_{D}&0&M\\\
0&M^{T}&\mu_{s}\end{bmatrix}\textrm{.}$ (4)

To implement the inverse seesaw mechanism, the necessary condition
$m_{D},M\gg\mu_{s}$ should be satisfied. This condition is required to
generate the light neutrino mass matrix in $\mathcal{O}(eV)$ which is doubly
suppressed by M as,

$m_{\nu}=m_{D}(M^{T})^{-1}\mu_{s}M^{-1}m^{T}_{D}\textrm{ .}$ (5)

Here the $\mu_{s}$ term of the low-scale seesaw mechanism breaks down the
lepton number conservationtHooft:1980xss  . We present the particle content of
the various fields considered in the model in Table (2) where $\Phi_{s}$,
$\Phi_{t}$ are triplet scalar fields and $\eta$, $\xi$, $\rho$, $\kappa$,
$\kappa^{\prime}$, $\kappa^{\prime\prime}$ are singlet scalar fields under
$A_{4}$ group transformation. $\mathcal{L}$ is the SM LH leptonic doublet and
$\mathcal{H}$ is the SM Higgs doublet.

Table 2: Transformation of the fields in our model under $A_{4}\times
Z_{4}\times Z_{5}\times U(1)_{X}$ symmetry for neutrino mass model realising
inverse seesaw mechanism.

$\mathcal{L}$ $\mathcal{H}$ $\Phi_{t}$ $\Phi_{s}$ $\mathcal{N}$ $\mathcal{S}$
$\mathcal{e}_{r}$ $\mathcal{\mu}_{r}$ $\mathcal{\tau}_{r}$ $\eta$ $\xi$ $\rho$
$\kappa$ $\kappa^{\prime}$ $\kappa^{\prime\prime}$ $A_{4}$ 3 1 3 3 3 3 1
$1^{\prime\prime}$ $1^{\prime}$ 1 $1^{\prime}$ $1^{\prime\prime}$ 1 1 1
$U(1)_{X}$ -1 0 0 -1 -1 1 -1 -1 -1 -1 -1 -1 0 -4 -3 $Z_{5}$ 1 1 $\omega$ 1
$\omega^{2}$ 1 $\omega$ $\omega$ $\omega$ 1 1 1 $\omega^{2}$ 1 1 $Z_{4}$ 1 1 i
-i i 1 i i i -i -i -i i i 1

If we now apply the $A_{4}$ product rules on above fields, we obtain the
Lagrangian for the charged leptons as:

$\mathcal{L}_{c.l.}\supset\dfrac{y_{1}}{\Lambda}(\bar{\mathcal{L}}\Phi^{\dagger}_{t})\mathcal{H}\mathcal{e}_{r}+\dfrac{y_{2}}{\Lambda}(\bar{\mathcal{L}}\Phi^{\dagger}_{t})^{\prime}\mathcal{H}\mathcal{\mu}_{r}+\dfrac{y_{3}}{\Lambda}(\bar{\mathcal{L}}\Phi^{\dagger}_{t})^{\prime\prime}\mathcal{H}\mathcal{\tau}_{r}\textrm{
.}$ (6)

Eq. (6) can be expressed in matrix form as:

$M_{c.l.}=\dfrac{\upsilon^{\dagger}_{t}\upsilon_{h}}{\Lambda}\begin{bmatrix}y_{1}&0&0\\\
0&y_{2}&0\\\ 0&0&y_{3}\end{bmatrix}\textrm{,}$ (7)

Here we have taken the VEVs of the standard model Higgs as $\langle
h\rangle=\upsilon_{h}$ and $\langle\Phi_{t}\rangle=\upsilon_{t}$ respectively,
$y_{1}$, $y_{2}$ and $y_{3}$ represent the coupling constants and $\Lambda$
denotes the usual cut-off scale of the theory. We can proceed to write the
Lagrangian of the neutrino sector after the application of $A_{4}$ product
rules: $1^{\prime}\times 1^{\prime}=1^{\prime\prime}$, $1^{\prime}\times
1^{\prime\prime}=1,1^{\prime\prime}\times 1^{\prime\prime}=1^{\prime}$ and
$3\times 3=1+1^{\prime}+1^{\prime\prime}+3_{A}+3_{S}$ Altarelli:2010gt  . This
can be written as

$\displaystyle\mathcal{L}_{\rm y}\supset
y_{d}\frac{\bar{\mathcal{L}}\tilde{\mathcal{H}}\mathcal{N}\kappa^{\dagger}}{\Lambda}+y_{m}\mathcal{N}\mathcal{S}\kappa^{\dagger}+y_{\mu}\mathcal{S}\mathcal{S}[\frac{\kappa^{\prime}\kappa^{\prime\prime^{\dagger}}(\Phi_{s}+\eta+\xi+\rho)}{\Lambda^{2}}]+h.c.\textrm{,}$
(8)

where $y_{d}$, $y_{m}$, $y_{\mu}$ are the coupling constants corresponding to
the mass terms $m_{D}$, M and $\mu_{s}$ in Eq. (8) respectively. The VEVs
obtained by the scalar fields apart from $\langle h\rangle$ and
$\langle\Phi_{t}\rangle$ after spontaneous symmetry breaking are
$\langle\Phi_{s}\rangle=\upsilon_{s}(\Phi_{a},\Phi_{b},\Phi_{c})$,
$\langle\eta\rangle=\upsilon_{\eta}$, $\langle\xi\rangle=\upsilon_{\xi}$,
$\langle\rho\rangle=\upsilon_{\rho}$,
$\langle\kappa\rangle=\upsilon_{\kappa}$,
$\langle\kappa^{\prime}\rangle=\upsilon_{\kappa^{\prime}}$ and
$\langle\kappa^{\prime\prime}\rangle=\upsilon_{\kappa^{\prime\prime}}$. The
block matrices M, $m_{D}$ and $\mu_{s}$ can be expressed in matrix form as:

$m_{D}=\dfrac{y_{d}\upsilon_{h}\upsilon^{\dagger}_{\kappa}}{\Lambda}\begin{bmatrix}1&0&0\\\
0&0&1\\\ 0&1&0\end{bmatrix}\textrm{,}$ (9)
$M=y_{m}\upsilon^{\dagger}_{\kappa}\begin{bmatrix}1&0&0\\\ 0&0&1\\\
0&1&0\end{bmatrix}\textrm{,}$ (10)

and,

$\mu_{s}=\dfrac{y_{\mu}\upsilon_{\kappa^{\prime}}\upsilon^{\dagger}_{\kappa^{\prime\prime}}}{\Lambda^{2}}\left(\begin{array}[]{ccc}\upsilon_{\eta}+2\upsilon_{s}\phi_{a}&\upsilon_{\xi}-\upsilon_{s}\phi_{c}&\upsilon_{\rho}-\upsilon_{s}\phi_{b}\\\
\upsilon_{\xi}-\upsilon_{s}\phi_{c}&\upsilon_{\rho}+2\upsilon_{s}\phi_{b}&\upsilon_{\eta}-\upsilon_{s}\phi_{a}\\\
\upsilon_{\rho}-\upsilon_{s}\phi_{b}&\upsilon_{\eta}-\upsilon_{s}\phi_{a}&\upsilon_{\xi}+2\upsilon_{s}\phi_{c}\\\
\end{array}\right)\textrm{.}$ (11)

Substituting these matrices in the Eq. (5) we obtain,

$\Rightarrow
m_{\nu}=F\left(\begin{array}[]{ccc}\upsilon_{\eta}+2\upsilon_{s}\phi_{a}&\upsilon_{\xi}-\upsilon_{s}\phi_{c}&\upsilon_{\rho}-\upsilon_{s}\phi_{b}\\\
\upsilon_{\xi}-\upsilon_{s}\phi_{c}&\upsilon_{\rho}+2\upsilon_{s}\phi_{b}&\upsilon_{\eta}-\upsilon_{s}\phi_{a}\\\
\upsilon_{\rho}-\upsilon_{s}\phi_{b}&\upsilon_{\eta}-\upsilon_{s}\phi_{a}&\upsilon_{\xi}+2\upsilon_{s}\phi_{c}\\\
\end{array}\right)\textrm{,}$ (12)

where,
$F=\dfrac{y^{2}_{d}y_{\mu}}{y^{2}_{m}}(\dfrac{\upsilon^{2}_{h}\upsilon_{\kappa^{\prime}}\upsilon^{\dagger}_{\kappa^{\prime\prime}}}{\Lambda^{4}})$
is a constant of the neutrino mass matrix that depends on the scale of the
three mass terms of the Lagrangian and various coupling constants of the
theory, as given in Eq. (8).

##  V Numerical method and results

To obtain the undetermined neutrino oscillation parameters such as the
lightest mass of the neutrino, CP-violating phases (both Dirac ($\delta_{CP}$)
and Majorana ($\alpha$, $\beta$) phases), we first compare the matrix in Eq.
(12) with the matrix of light neutrino mass obtained after parametrisation by
the PMNS mixing matrix, $U_{P}$ i.e.,

$m_{\nu}=U_{P}.m_{\nu_{diag}}.U^{T}_{P}\textrm{ .}$ (13)

The $U_{P}$ matrix can be parametrised as:

$U_{P}=\left(\begin{array}[]{ccc}1&0&0\\\ 0&c_{23}&s_{23}\\\
0&-s_{23}&c_{23}\\\
\end{array}\right).\left(\begin{array}[]{ccc}c_{13}&0&e^{-i\delta_{CP}}s_{13}\\\
0&1&0\\\ -e^{i\delta_{CP}}s_{13}&0&c_{13}\\\
\end{array}\right).\left(\begin{array}[]{ccc}c_{12}&s_{12}&0\\\
-s_{12}&c_{12}&0\\\ 0&0&1\\\
\end{array}\right).\left(\begin{array}[]{ccc}1&0&0\\\ 0&e^{i\alpha}&0\\\
0&0&e^{i(\beta+\delta_{CP})}\\\ \end{array}\right)\textrm{ ,}$ (14)

The mixing angles in Eq. (14) can be denoted as $c_{ij}$=$\cos{\theta_{ij}}$,
$s_{ij}$=$\sin{\theta_{ij}}$ and $m_{\nu_{diag}}$$=$$diag(m_{1},m_{2},m_{3})$
which can be reduced in the form
$m^{diag}_{\nu}=diag(m_{1},\sqrt{m^{2}_{1}+\Delta
m_{21}^{2}},\sqrt{m_{1}^{2}+\Delta m_{31}^{2}})$ and
$m^{diag}_{\nu}=diag(\sqrt{m_{3}^{2}+\Delta m_{23}^{2}-\Delta
m_{21}^{2}},\sqrt{m_{3}^{2}+\Delta m_{23}^{2}},m_{3})$ for normal and inverted
mass hierarchy respectively.

On comparison of these matrices, a set of six flavon equations are obtained
that depend purely on the neutrino oscillation parameters (both known and
unknown). Keeping the mixing angles and mass-squared differences as known
neutrino oscillation parameters, we solve these equations simultaneously to
obtain the undetermined parameters as mentioned previously in this section. We
use data for $\theta_{12}$, $\theta_{13}$, $\theta_{23}$ and $\Delta
m_{21}^{2}$, $\Delta m_{31}^{2}$ and $\Delta m_{23}^{2}$ from the recent
global fit data obtained from various neutrino oscillation experiments
deSalas:2020pgw  . We solve these equations for different vacuum alignments of
$\upsilon_{s}(\Phi_{a},\Phi_{b},\Phi_{c})$, and have followed numerical
analysis as in earlier works Chen:2012st ; Kalita:2015jaa  and Sarma:2018bgf
. Our results have been summarised in Table (4) and in Figs (1-4), we present
the correlation between the determined and undetermined neutrino parameters
using scattered plots. Latest global fit results from Ref.deSalas:2020pgw
have been summarised in Table (5), for comparison with our results. An earlier
summary of $0\nu\beta\beta$ decay can also be found in Ref.DellOro:2014ysa  .

Table 3: Summary of allowed and disallowed vacuum alignments of the flavon $\Phi_{s}$ for neutrino oscillation parameters constrained within their latest $3\sigma$ range. Allowed VEV in NH | Allowed VEV in IH
---|---
(0,1,1), (1,-1,-1), (-1,1,1), (1,-1,-1) | (0,-1,1), (0,1,-1)
Disallowed VEV in NH | Disallowed VEV in IH
(1,0,0) , (0,1,0), (0,0,1), (1,1,0), | (1,0,0) , (0,1,0), (0,0,1), (1,1,0),
(1,0,1), (1,-1,0), (1,0,-1), (-1,0,1), | (1,0,1), (0,1,1), (1,-1,0), (1,0,-1),
(-1,1,0), (0,-1,1), (0,1,-1), (-1,1,-1), | (-1,0,1), (-1,1,0), (1,-1,-1), (-1,1,-1),
(-1,-1,1), (1,1,-1), (1,-1,1), (-1,-1,0), | (-1,-1,1), (1,1,-1), (1,-1,1), (-1,1,1),
(-1,0,-1), (-1,0,0), (0,-1,0), (0,0,-1), | (-1,-1,0), (-1,0,-1), (0,-1,-1), (-1,0,0),
(1,1,1), (-1,-1,-1) | (0,-1,0), (0,0,-1), (1,1,1), (-1,-1,-1)

Figure 1: The first three plots shows the correlation between the lightest
neutrino mass, $m_{1}$ and the sinusoidal of CPV phases $\delta_{CP}$,
$\alpha$ and $\beta$ respectively and the fourth plot shows the correlation
between the atmospheric mixing angle, $\theta_{23}$ and $\delta_{CP}$ phase.
All of them are plotted for VEV alignment $(1,-1,-1)$ or $(-1,1,1)$ with
normal hierarchy.
Figure 2:  The first three plots shows the correlation between the lightest
neutrino mass, $m_{1}$ and the sinusoidal of CPV phases $\delta_{CP}$,
$\alpha$ and $\beta$ respectively and the fourth plot shows the correlation
between the atmospheric mixing angle, $\theta_{23}$ and $\delta_{CP}$ phase.
All of them are plotted for VEV alignment $(0,-1,-1)$ or $(0,1,1)$ with normal
hierarchy.
Figure 3: The first three plots shows the correlation between the lightest
neutrino mass, $m_{3}$ and the sinusoidal of CPV phases $\delta_{CP}$,
$\alpha$ and $\beta$ respectively and the fourth plot shows the correlation
between the atmospheric mixing angle, $\theta_{23}$ and $\delta_{CP}$ phase.
All of them are plotted for VEV alignment $(0,-1,1)$ or $(0,1,-1)$ with
inverted hierarchy.  Figure 4:  Plot showing the correlation between
$m_{lightest}$ and $m_{ee}$ of $0\nu\beta\beta$ decay for all the allowed
vacuum alignments of $\Phi_{s}$. Table 4: Allowed $3\sigma$ range of neutrino
oscillation parameters obtained from our ISS model

1\. VEV: (0,-1,-1) or (0, 1,1) with normal mass hierarchy $m_{1}$ (eV)
$m_{ee}$ $Sin\alpha$ $Sin\beta$ $Sin\delta_{CP}$ $\theta_{23}$ 0.01527
$\rightarrow$ 0.01762 0.01599 $\rightarrow$ 0.01760 -0.0023 $\rightarrow$
0.0024 -1. $\rightarrow$ 0.2449 -0.2323 $\rightarrow$ 1. 43.17 $\rightarrow$
44.07 (lower octant) 2\. VEV: (1,-1,-1) or (-1, 1,1) with normal mass
hierarchy $m_{1}$ (eV) $m_{ee}$ $Sin\alpha$ $Sin\beta$ $Sin\delta_{CP}$
$\theta_{23}$ 0.00243 $\rightarrow$ 0.00366 $8.24651\times 10^{-6}$
$\rightarrow$ 0.00094 -0.3191 $\rightarrow$ 1. -0.4782 $\rightarrow$ 1.
-0.4860 $\rightarrow$ 0.4980 50.06 $\rightarrow$ 50.70 (higher octant) 3\.
VEV: (0,-1,1) or (0, 1,-1) with inverted mass hierarchy $m_{3}$ (eV) $m_{ee}$
$Sin\alpha$ $Sin\beta$ $Sin\delta_{CP}$ $\theta_{23}$ 0.02092 $\rightarrow$
0.02198 0.02215 $\rightarrow$ 0.02337 0.9996 $\rightarrow$ 1. -1.
$\rightarrow$ 1. -0.6903 $\rightarrow$ 0.7008 43.09 $\rightarrow$ 43.30 (lower
octant)

##  VI Discussion on Results

From the recent data of T2K patrick_dunne_2020_3959558  and NO$\nu$A
alex_himmel_2020_3959581  experiments, it is seen that both the experiments
prefer normal hierarchy (NH), without any new physics (for standard
oscillation picture). However, a slight tension is seen at the $2\sigma$
level, where T2K prefers $\delta_{CP}\sim 3\pi/2$ for NH which is excluded by
NO$\nu$A at 90$\%$ confidence level. NO$\nu$A, in general, does not have any
strong preference for any particular value of the CPV phase and has its best
fit value around $\delta_{CP}\sim\pi$ for NH. This discrepancy is perhaps due
to the configuration of baselines and the effect of matter density as
neutrinos in NO$\nu$A experience a much stronger matter effect, and hope that
this discrepancy can be alleviated from the robust data obtained by
forthcoming neutrino experiments. A slight preference of IH over NH (without
new physics) is seen when the recent data of T2K and NO$\nu$A are combined
Denton:2020uda  , though data from Super Kamiokande still prefer NH over IH
Kelly:2020fkv  . However, it is seen that in the combined experimental data of
T2K, NOvA, Super-K Takeuchi:2020slv  , the NH is preferred over the IH (please
see Denton:2020uda ; Kelly:2020fkv ; Esteban:2020itz ), and this feature is
also observed in our analysis. From Table (3), we can pinpoint the six allowed
VEV alignments of $\Phi_{s}$ as (0,1,1), (0,-1,-1), (1,-1,-1) and (-1,1,1)
which favours NH while only two of them, i.e., (0,-1,1) and (0,1,-1) favours
IH. The rest of the cases out of the 26 possible cases in Table (3) are
rejected as we do not obtain any event point within the $3\sigma$ range of the
parameters from current experimental data fitting. The number of allowed cases
are very few as we are taking only those high-precision solutions whose
accuracy is $<10^{-5}$ with $10^{6}$ random points being generated (for each
case).

Results in the correlation plots in Figs. (1\- 3) show that our model can
predict the values of the unknown neutrino parameters, the lightest neutrino
mass ($m_{1}$ or $m_{3}$, satisfying the limits on sum of the mass of three
light neutrinos from cosmological and tritium beta decay experiments) and
Majorana phases (which can vary between $0-2\pi$), which can be tested when
measured in future. In these plots, we have chosen only those points for which
the known parameters like $\theta_{12}$, $\theta_{13}$, $\theta_{23}$, $\Delta
m^{2}_{12}$, $\Delta m^{2}_{23}$ and $\delta_{CP}$ lie in their $3\sigma$
range of current global best fit values (Please see Table 1). Next, we discuss
these results with reference to Octant-MH degeneracy Cao:2020ans ;
Bora:2014zwa  . In the future neutrino experiments, if the mass hierarchy is
fixed, say if it is normal hierarchy, then our model will be able to predict
precisely the triplet flavon VEV, i.e., the VEV alignment of $\Phi_{s}$ is
either (0,1,1) with $\theta_{23}<45^{o}$ or (-1,1,1) with
$\theta_{23}>45^{o}$. Similarly, if the mass hierarchy comes out to be
inverted in the future experiments, then our model would be able to identify
the corresponding VEV alignment of $\Phi_{s}$ as (0,-1,1) with $\theta_{23}$
in the lower octant and vice-versa. Thus, the preferred direction along which
flavon field vacuum stabilises can be pinpointed once this entanglement among
parameters is fixed, and vice-versa (if the preferred VEV of flavon field is
known, it can help us resolve the parameter entanglement). The global fit data
of neutrino oscillation deSalas:2020pgw  used in our analysis is summarised in
Table (5). In this work, they have considered data from T2K, NO$\nu$A as LBLs,
and have presented global analysis on $0\nu\beta\beta$ decay too. The results
from nEXO nEXO:2021ujk  have also been included for the results of
$0\nu\beta\beta$ decay in Table (5).

Table 5: Summary of updated values of oscillation parameters and the
preference for octant of $\theta_{23}$, $\delta_{CP}$ and mass hierarchy taken
from various global experiments deSalas:2020pgw ; nEXO:2021ujk  .

Preference for octant | Preferred value of | Favoured mass | Results
---|---|---|---
of atmospheric angle | CPV phase ( $\delta_{CP}$ ) | ordering MO | from $0\nu\beta\beta$
($\theta_{23}$) |  | (sign of $\Delta m^{2}_{31}$) | Experiments
LBLs- two | NO$\nu$A - preference | Independent analysis of |  $m_{\beta\beta}\leq 104-228$ meV
degenerate solutions | for $\delta_{CP}=0.8\pi$, | both T2K and NO$\nu$A |
for both the | disfavouring a region | does not show any | by GERDA
Octants (LO and HO) | around best fit of T2K $\delta_{CP}=1.5\pi$ | preference for Mass ordering |
Combination | Combination | All LBL data |  $m_{\beta\beta}\leq 75-350$ meV
of all | of LBL+reactor- | favour IO |
Acc. LBL + Reactor | CP conserving value | -as a | by CUORE
|  $\delta_{CP}=0$ is disfavoured, | consequence of |
\- shifts the best fit HO | while other CP conserving values | tension in T2K |
|  $\delta_{CP}=\pi$ is still allowed | and NO$\nu$A data |
Combination of all | - | Combination of all |  $m_{\beta\beta}\leq 61-165$ meV
Acc. LBL +atm data (SK) |  | LBL + reactor data favour NO | by KamLand Zen
|  | Atmospheric SK data favour NO, |
shifts preference to HO |  | whole combination of |
|  | LBL+SK favour NO |
Best fit - | Best fit - | Best fit is for NO . | nEXO
$Sin^{2}\theta_{23}=0.574(0.578)$ | $\delta_{CP}=1.08\pi(1.58\pi)$ | A small tension in IO | 1\. Sensitivity at 90$\%$ CL
for NO(IO) | for NO(IO) |  |  $m_{\beta\beta}\leq 4.7-26$ meV for NH.
|  |  |  $m_{\beta\beta}<15$ meV for IH.
|  |  | 2\. Discovery potential at
|  |  | ($3\sigma$): $m_{\beta\beta}\leq 5.0-26$ meV for NH.
|  |  |  $m_{\beta\beta}<15$ meV for IH.

Figure 5:  Our predictions on $m_{ee}$ are compared with $3\sigma$ discovery
potential limits of nEXO experiment for both NH and IH (see Table 5), allowed
region for NH is below two purple lines, while that for IH is below red line.
It can be seen that our results for all the three cases - (0,1,1)NH, (1,-1,-1)
NH and (0,1,-1) IH lie in the $3\sigma$ allowed region of nEXO. We note that
the limits of other experiments are more in value than that of nEXO, hence we
can say that all the three cases results lie in allowed region of all the
experiments listed in Table 5. Also, since the allowed limits of other
experiments are much larger than that of nEXO, so they all could not be
presented in this same plot due to variation in scale, hence we showed our
results with only nEXO limits.

From a careful comparison of our results in Fig(1-3) with Table (5), we find
that their preferences for NH and HO (higher octant) indicates that the VEV
(1,-1,-1)/(-1,1,1) of the $A_{4}$ triplet flavon field is favoured (case B in
Table (4)). Next, we use this novel information to predict very precise values
of $m_{lightest}$ and $m_{ee}$ in our model, which corresponds to upper right
panel (NH, HO) in Fig. (4) - i.e., $m_{lightest}=(0.00243-0.00366)$ eV, and
$m_{ee}=(8.24\times 10^{-6}-0.00094)$ eV, which falls in the allowed region as
shown in Table 5. Moreover, it is interesting to note that the preferred mass
ordering as well as the range of lightest neutrino mass and that of $m_{ee}$
obtained for the allowed cases in our model lie within the range allowed from
experiments such as GERDA GERDA:2019ivs  , CUORE CUORE:2019yfd  and KamLAND-
Zen KamLAND-Zen:2016pfg  for 76Ge, for 130Te and for 136Xe respectively at
$90\%$ confidence level. In Fig. 5, we have compared our results of Fig. 4
with latest $3\sigma$ discovery potential limits from nEXO experiment. Here,
allowed region for NH is below two purple lines, while that for IH is below
red line (please see Table 5). It can be seen that our results for all the
three cases - (0,1,1)NH, (1,-1,-1) NH and (0,1,-1) IH lie in the $3\sigma$
allowed region of nEXO. We note that the limits of other experiments are more
in value than that of nEXO, hence we can say that all the three cases results
lie in allowed region of all the experiments listed in Table 5. Hence, our
results of Fig (1-4) show coherence with reference to that of recent
experimentally allowed regions, and so our model is testable with new
predictions.

##  VII Conclusion

To conclude, in this work, we presented new ideas on how the resolution of
Octant-MH degeneracy present in the measurements of disappearance channel of
LBL experiments can be used to get novel information on the VEV alignment of
the triplet flavon field of $A_{4}$ flavour symmetry. We analysed the
viability of our ISS model with $A_{4}\times U(1)_{X}\times Z_{5}\times Z_{4}$
symmetry by determining both the Dirac ($\delta_{CP}$) as well as the Majorana
($\alpha$, $\beta$) phases and the corresponding lightest mass eigenvalue of
active neutrinos, keeping the mixing angles and mass squared differences
constrained within allowed $3\sigma$ range. Since computation was done for the
accuracy $<10^{-5}$, we obtained solution points for a narrow range of
neutrino oscillation parameters which depicts a very precise solution. The
correlation among $m_{lightest}$ and $m_{ee}$ of $0\nu\beta\beta$ decay agrees
very well with their current experimental $3\sigma$ bounds and sensitivity
limits for corresponding MH. By carefully comparing our results with a very
recent global analysis (which seems to favour NH and HO), we pinpointed the
favoured VEV alignment of $A_{4}$ triple flavon to be (1,-1,-1)/(-1,1,1), as
well as very precise value of $m_{lightest}$ and $m_{ee}$, as shown in Fig.
(4) (to be compared with values in Table 5.). Thus the ideas presented here
can answer some of the open problems in the neutrino sector and future
experiments would be able to confirm or refute the results presented here.

## Acknowledgements

Authors acknowledge support from FIST and RUSA grants (Govt. of India) in
upgrading the computer laboratory of the department where this work was done.

##  Appendix A Equations for $A_{4}$ Flavons present in $A_{4}\times
U(1)_{X}\times Z_{5}\times Z_{4}$ inverse seesaw model

$\displaystyle\Phi_{a}=\dfrac{J}{3}(-e^{2i\alpha}m_{2}(-\sin(\theta_{23})\cos(\theta_{12})$
(15)
$\displaystyle-e^{i\delta}\sin(\theta_{12})\sin(\theta_{13})\cos(\theta_{23}))(\cos(\theta_{12})\cos(\theta_{23})$
$\displaystyle-e^{i\delta}\sin(\theta_{12})\sin(\theta_{13})\sin(\theta_{23}))+e^{2i\alpha}m_{2}\sin^{2}(\theta_{12})\cos^{2}(\theta_{13})$
$\displaystyle+m_{3}e^{2i(\beta+\delta)-2i\delta}\sin^{2}(\theta_{13})-m_{3}e^{2i(\beta+\delta)}\sin(\theta_{23})\cos(\theta_{23})\cos^{2}(\theta_{13})$
$\displaystyle-
m_{1}(\sin(\theta_{12})\sin(\theta_{23})-e^{i\delta}\sin(\theta_{13})\cos(\theta_{12})\cos(\theta_{23}))(\sin(\theta_{12})(-\cos(\theta_{23}))$
$\displaystyle-e^{i\delta}\sin(\theta_{13})\sin(\theta_{23})\cos(\theta_{12}))+m_{1}\cos^{2}(\theta_{12})\cos^{2}(\theta_{13}))$
$\displaystyle\Phi_{b}=\frac{J}{3}(e^{2i\alpha}m_{2}(\cos(\theta_{12})\cos(\theta_{23})-e^{i\delta}\sin(\theta_{12})\sin(\theta_{13})\sin(\theta_{23})){}^{2}$
(16)
$\displaystyle-e^{2i\alpha}m_{2}\sin(\theta_{12})\cos(\theta_{13})(-\sin(\theta_{23})\cos(\theta_{12})-e^{i\delta}\sin(\theta_{12})\sin(\theta_{13})\cos(\theta_{23}))$
$\displaystyle+m_{3}e^{2i(\beta+\delta)}\sin^{2}(\theta_{23})\cos^{2}(\theta_{13})-m_{3}e^{2i(\beta+\delta)-i\delta}\sin(\theta_{13})\cos(\theta_{13})\cos(\theta_{23})$
$\displaystyle+m_{1}(\sin(\theta_{12})(-\cos(\theta_{23}))-e^{i\delta}\sin(\theta_{13})\sin(\theta_{23})\cos(\theta_{12})){}^{2}$
$\displaystyle-
m_{1}\cos(\theta_{12})\cos(\theta_{13})(\sin(\theta_{12})\sin(\theta_{23})-e^{i\delta}\sin(\theta_{13})\cos(\theta_{12})\cos(\theta_{23})))$

$\displaystyle\Phi_{c}=\frac{J}{3}(e^{2i\alpha}m_{2}(-\sin(\theta_{23})\cos(\theta_{12})-e^{i\delta}\sin(\theta_{12})\sin(\theta_{13})\cos(\theta_{23})){}^{2}$
(17)
$\displaystyle-e^{2i\alpha}m_{2}\sin(\theta_{12})\cos(\theta_{13})(\cos(\theta_{12})\cos(\theta_{23})-e^{i\delta}\sin(\theta_{12})\sin(\theta_{13})\sin(\theta_{23}))$
$\displaystyle+m_{3}e^{2i(\beta+\delta)}\cos^{2}(\theta_{13})\cos^{2}(\theta_{23})-m_{3}e^{2i(\beta+\delta)-i\delta}\sin(\theta_{13})\sin(\theta_{23})\cos(\theta_{13})$
$\displaystyle+m_{1}(\sin(\theta_{12})\sin(\theta_{23})-e^{i\delta}\sin(\theta_{13})\cos(\theta_{12})\cos(\theta_{23})){}^{2}$
$\displaystyle-
m_{1}\cos(\theta_{12})\cos(\theta_{13})(\sin(\theta_{12})(-\cos(\theta_{23}))-e^{i\delta}\sin(\theta_{13})\sin(\theta_{23})\cos(\theta_{12})))$

$\displaystyle\eta=\frac{J}{3}(2(e^{2i\alpha}m_{2}(-\sin(\theta_{23})\cos(\theta_{12})$
(18)
$\displaystyle-e^{i\delta}\sin(\theta_{12})\sin(\theta_{13})\cos(\theta_{23}))(\cos(\theta_{12})\cos(\theta_{23})$
$\displaystyle-e^{i\delta}\sin(\theta_{12})\sin(\theta_{13})\sin(\theta_{23}))+m_{3}e^{2i(\beta+\delta)}\sin(\theta_{23})\cos(\theta_{23})\cos^{2}(\theta_{13})$
$\displaystyle+m_{1}(\sin(\theta_{12})\sin(\theta_{23})-e^{i\delta}\sin(\theta_{13})\cos(\theta_{12})\cos(\theta_{23}))(\sin(\theta_{12})(-\cos(\theta_{23}))$
$\displaystyle-e^{i\delta}\sin(\theta_{13})\sin(\theta_{23})\cos(\theta_{12})))+e^{2i\alpha}m_{2}\sin^{2}(\theta_{12})\cos^{2}(\theta_{13})$
$\displaystyle+m_{3}e^{2i(\beta+\delta)-2i\delta}\sin^{2}(\theta_{13})+m_{1}\cos^{2}(\theta_{12})\cos^{2}(\theta_{13}))$

$\displaystyle\rho=\frac{J}{3}(2(e^{2i\alpha}m_{2}\sin(\theta_{12})\cos(\theta_{13})(-\sin(\theta_{23})\cos(\theta_{12})$
(19)
$\displaystyle-e^{i\delta}\sin(\theta_{12})\sin(\theta_{13})\cos(\theta_{23}))+m_{3}e^{2i(\beta+\delta)-i\delta}\sin(\theta_{13})\cos(\theta_{13})\cos(\theta_{23})$
$\displaystyle+m_{1}\cos(\theta_{12})\cos(\theta_{13})(\sin(\theta_{12})\sin(\theta_{23})-e^{i\delta}\sin(\theta_{13})\cos(\theta_{12})\cos(\theta_{23})))$
$\displaystyle+e^{2i\alpha}m_{2}(\cos(\theta_{12})\cos(\theta_{23})-e^{i\delta}\sin(\theta_{12})\sin(\theta_{13})\sin(\theta_{23})){}^{2}$
$\displaystyle+m_{3}e^{2i(\beta+\delta)}\sin^{2}(\theta_{23})\cos^{2}(\theta_{13})+m_{1}(\sin(\theta_{12})(-\cos(\theta_{23}))$
$\displaystyle-e^{i\delta}\sin(\theta_{13})\sin(\theta_{23})\cos(\theta_{12})){}^{2})$
$\displaystyle\xi=\frac{J}{3}(2(e^{2i\alpha}m_{2}\sin(\theta_{12})\cos(\theta_{13})(\cos(\theta_{12})\cos(\theta_{23})$
(20)
$\displaystyle-e^{i\delta}\sin(\theta_{12})\sin(\theta_{13})\sin(\theta_{23}))+m_{3}e^{2i(\beta+\delta)-i\delta}\sin(\theta_{13})\sin(\theta_{23})\cos(\theta_{13})$
$\displaystyle+m_{1}\cos(\theta_{12})\cos(\theta_{13})(\sin(\theta_{12})(-\cos(\theta_{23}))-e^{i\delta}\sin(\theta_{13})\sin(\theta_{23})\cos(\theta_{12}))$
$\displaystyle+e^{2i\alpha}m_{2}(-\sin(\theta_{23})\cos(\theta_{12})-e^{i\delta}\sin(\theta_{12})\sin(\theta_{13})\cos(\theta_{23})){}^{2}$
$\displaystyle+m_{3}e^{2i(\beta+\delta)}\cos^{2}(\theta_{13})\cos^{2}(\theta_{23})+m_{1}(\sin(\theta_{12})\sin(\theta_{23})$
$\displaystyle-e^{i\delta}\sin(\theta_{13})\cos(\theta_{12})\cos(\theta_{23})){}^{2})$

where,
$J=\dfrac{y^{2}_{m}}{y^{2}_{d}y_{\mu}}[\dfrac{\Lambda^{4}}{v^{2}_{h}v_{\kappa^{\prime}}v^{\dagger}_{\kappa^{\prime\prime}}}]$.
We have absorbed $v_{s}$ in $\Phi_{a},\Phi_{b},\Phi_{c}$.

##  Appendix B Scalar potential minimisation of the model and possible
solutions of triplet scalar field, $\Phi_{s}$

We present here the minimisation of the scalar potential considered in our
model and later include the all possible solutions of the triplet scalar
flavon, $\Phi_{s}$ after its minimisation:

$\displaystyle
V=V(h)+V(\Phi_{t})+V(\Phi_{s})+V(\eta)+V(\xi)+V(\rho)+V(\kappa)+V(\kappa^{\prime})+V(\kappa^{\prime\prime})$
$\displaystyle+V(h,\Phi_{t},\Phi_{s},\eta,\xi,\rho,\kappa,\kappa^{\prime},\kappa^{\prime\prime})+V(\Phi_{t},\Phi_{s},\eta,\xi,\rho,\kappa,\kappa^{\prime},\kappa^{\prime\prime})+V_{ex}(h,\Phi_{t},\Phi_{s},\eta,\xi,\rho,\kappa,\kappa^{\prime},\kappa^{\prime\prime})$

where,

$V(h)=\mu^{2}_{h}h^{\dagger}h+\lambda_{h}(h^{\dagger}h)(h^{\dagger}h)$ (21)

$\displaystyle
V(\Phi_{s})=-\mu^{2}_{s}[\Phi^{\dagger}_{a}\Phi_{a}+\Phi^{\dagger}_{b}\Phi_{c}+\Phi^{\dagger}_{c}\Phi_{b}]+\lambda_{s}[(\Phi^{\dagger}_{a}\Phi_{a}+\Phi^{\dagger}_{b}\Phi_{c}+\Phi^{\dagger}_{c}\Phi_{b})^{2}+(\Phi^{\dagger}_{b}\Phi_{b}+\Phi^{\dagger}_{a}\Phi_{c}+\Phi^{\dagger}_{c}\Phi_{a})(\Phi^{\dagger}_{c}\Phi_{c}$
(22)
$\displaystyle+\Phi^{\dagger}_{a}\Phi_{b}+\Phi^{\dagger}_{b}\Phi_{a})+(2\Phi^{\dagger}_{a}\Phi_{a}-\Phi^{\dagger}_{b}\Phi_{c}+\Phi^{\dagger}_{c}\Phi_{b})^{2}+2(2\Phi^{\dagger}_{c}\Phi_{c}-\Phi^{\dagger}_{a}\Phi_{b}-\Phi^{\dagger}_{b}\Phi_{a})(2\Phi^{\dagger}_{b}\Phi_{b}-\Phi^{\dagger}_{a}\Phi_{c}-\Phi^{\dagger}_{c}\Phi_{a})$

1\. $(\Phi_{a},\Phi_{b},\Phi_{c})\to(0,0,0)$
2\.
$(\Phi_{a},\Phi_{b},\Phi_{c})\to(0,1,-1)*(-\frac{0.242536i\mu_{s}}{\sqrt{\lambda_{s}}})$
3.$(\Phi_{a},\Phi_{b},\Phi_{c})\to(0,-1,1)*(-\frac{0.242536i\mu_{s}}{\sqrt{\lambda_{s}}})$
4\.
$(\Phi_{a},\Phi_{b},\Phi_{c})\to(0,-\frac{(0.210042\,+0.121268i)\mu_{s}}{\sqrt{\lambda_{s}}},-\frac{(0.210042\,-0.121268i)\mu_{s}}{\sqrt{\lambda_{s}}})$
5\.
$(\Phi_{a},\Phi_{b},\Phi_{c})\to(0,\frac{(0.210042\,+0.121268i)\mu_{s}}{\sqrt{\lambda_{s}}},\frac{(0.210042\,-0.121268i)\mu_{s}}{\sqrt{\lambda_{s}}})$
6\.
$(\Phi_{a},\Phi_{b},\Phi_{c})\to(0,\frac{(0.210042\,-0.121268i)\mu_{s}}{\sqrt{\lambda_{s}}},\frac{(0.210042\,+0.121268i)\mu_{s}}{\sqrt{\lambda_{s}}})$
7\.
$(\Phi_{a},\Phi_{b},\Phi_{c})\to(0,-\frac{(0.210042\,-0.121268i)\mu_{s}}{\sqrt{\lambda_{s}}},-\frac{(0.210042\,+0.121268i)\mu_{s}}{\sqrt{\lambda_{s}}})$
8\.
$(\Phi_{a},\Phi_{b},\Phi_{c})\to(1,1,1)*-\frac{0.288675\mu_{s}}{\sqrt{\lambda_{s}}}$
9\.
$(\Phi_{a},\Phi_{b},\Phi_{c})\to(-\frac{0.288675\mu_{s}}{\sqrt{\lambda_{s}}},\frac{(0.144338\,+0.25i)\mu_{s}}{\sqrt{\lambda_{s}}},\frac{(0.144338\,-0.25i)\mu_{s}}{\sqrt{\lambda_{s}}})$
10\.
$(\Phi_{a},\Phi_{b},\Phi_{c})\to(-\frac{0.288675\mu_{s}}{\sqrt{\lambda_{s}}},\frac{(0.144338\,-0.25i)\mu_{s}}{\sqrt{\lambda_{s}}},\frac{(0.144338\,+0.25i)\mu_{s}}{\sqrt{\lambda_{s}}})$
11\.
$(\Phi_{a},\Phi_{b},\Phi_{c})\to(1,1,1)*\frac{0.288675\mu_{s}}{\sqrt{\lambda_{s}}}$
12\.
$(\Phi_{a},\Phi_{b},\Phi_{c})\to(\frac{0.288675\mu_{s}}{\sqrt{\lambda_{s}}},-\frac{(0.144338\,+0.25i)\mu_{s}}{\sqrt{\lambda_{s}}},-\frac{(0.144338\,-0.25i)\mu_{s}}{\sqrt{\lambda_{s}}})$
13\.
$(\phi_{a},\phi_{b},\phi_{c})\to(\frac{0.288675\mu_{s}}{\sqrt{\lambda_{s}}},-\frac{(0.144338\,-0.25i)\mu_{s}}{\sqrt{\lambda_{s}}},-\frac{(0.144338\,+0.25i)\mu_{s}}{\sqrt{\lambda_{s}}})$
14\.
$(\phi_{a},\phi_{b},\phi_{c})\to(1,0,0)*-\frac{0.316228\mu_{s}}{\sqrt{\lambda_{s}}}$
15\.
$(\phi_{a},\phi_{b},\phi_{c})\to(-1,2,2)*\frac{0.105409\mu_{s}}{\sqrt{\lambda_{s}}}$
16\.
$(\phi_{a},\phi_{b},\phi_{c})\to(-\frac{0.105409\mu_{s}}{\sqrt{\lambda_{s}}},-\frac{(0.105409\,-0.182574i)\mu_{s}}{\sqrt{\lambda_{s}}},-\frac{(0.105409\,+0.182574i)\mu_{s}}{\sqrt{\lambda_{s}}})$
17\.
$(\phi_{a},\phi_{b},\phi_{c})\to(-\frac{0.105409\mu_{s}}{\sqrt{\lambda_{s}}},-\frac{(0.105409\,+0.182574i)\mu_{s}}{\sqrt{\lambda_{s}}},-\frac{(0.105409\,-0.182574i)\mu_{s}}{\sqrt{\lambda_{s}}})$
18\.
$(\phi_{a},\phi_{b},\phi_{c})\to(1,-2,-2)*\frac{0.105409\mu_{s}}{\sqrt{\lambda_{s}}}$
19\.
$(\phi_{a},\phi_{b},\phi_{c})\to(\frac{0.105409\mu_{s}}{\sqrt{\lambda_{s}}},\frac{(0.105409\,-0.182574i)\mu_{s}}{\sqrt{\lambda_{s}}},\frac{(0.105409\,+0.182574i)\mu_{s}}{\sqrt{\lambda_{s}}})$
20\.
$(\phi_{a},\phi_{b},\phi_{c})\to\frac{0.105409\mu_{s}}{\sqrt{\lambda_{s}}},\frac{(0.105409\,+0.182574i)\mu_{s}}{\sqrt{\lambda_{s}}},\frac{(0.105409\,-0.182574i)\mu_{s}}{\sqrt{\lambda_{s}}})$
21\.
$(\phi_{a},\phi_{b},\phi_{c})\to(1,0,0)*\frac{0.316228\mu_{s}}{\sqrt{\lambda_{s}}}$
22\.
$(\phi_{a},\phi_{b},\phi_{c})\to(-2,1,1)*\frac{0.140028\mu_{s}}{\sqrt{\lambda_{s}}}$
23\.
$(\phi_{a},\phi_{b},\phi_{c})\to(-\frac{0.280056\mu_{s}}{\sqrt{\lambda_{s}}},-\frac{(0.070014\,+0.121268i)\mu_{s}}{\sqrt{\lambda_{s}}},-\frac{(0.070014\,-0.121268i)\mu_{s}}{\sqrt{\lambda_{s}}})$
24\.
$(\phi_{a},\phi_{b},\phi_{c})\to(-\frac{0.280056\mu_{s}}{\sqrt{\lambda_{s}}},-\frac{(0.070014\,-0.121268i)\mu_{s}}{\sqrt{\lambda_{s}}},-\frac{(0.070014\,+0.121268i)\mu_{s}}{\sqrt{\lambda_{s}}})$
25\.
$(\phi_{a},\phi_{b},\phi_{c})\to(2,-1,-1)*\frac{0.140028\mu_{s}}{\sqrt{\lambda_{s}}}$
26\.
$(\phi_{a},\phi_{b},\phi_{c})\to(\frac{0.280056\mu_{s}}{\sqrt{\lambda_{s}}},\frac{(0.070014\,+0.121268i)\mu_{s}}{\sqrt{\lambda_{s}}},\frac{(0.070014\,-0.121268i)\mu_{s}}{\sqrt{\lambda_{s}}})$
27\.
$(\phi_{a},\phi_{b},\phi_{c})\to(\frac{0.280056\mu_{s}}{\sqrt{\lambda_{s}}},\frac{(0.070014\,-0.121268i)\mu_{s}}{\sqrt{\lambda_{s}}},\frac{(0.070014\,+0.121268i)\mu_{s}}{\sqrt{\lambda_{s}}})$.

## References

  * (1) S. Cao, A. Nath, T. V. Ngoc, P. T. Quyen, N. T. Hong Van and N. K. Francis, Phys. Rev. D 103 (2021) no.11, 112010 doi:10.1103/PhysRevD.103.112010 [arXiv:2009.08585 [hep-ph]].
  * (2) P. F. de Salas, D. V. Forero, S. Gariazzo, P. Martínez-Miravé, O. Mena, C. A. Ternes, M. Tórtola and J. W. F. Valle, JHEP 02 (2021), 071 doi:10.1007/JHEP02(2021)071 [arXiv:2006.11237 [hep-ph]].
  * (3) J. T. Penedo, S. T. Petcov and T. Yanagida, Nucl. Phys. B 929 (2018), 377-396 doi:10.1016/j.nuclphysb.2018.02.018 [arXiv:1712.09922 [hep-ph]].
  * (4) H. Ishimori, T. Kobayashi, H. Ohki, H. Okada, Y. Shimizu and M. Tanimoto, Lect. Notes Phys. 858 (2012), 1-227 doi:10.1007/978-3-642-30805-5
  * (5) K. Abe et al. [T2K], Phys. Rev. Lett. 107 (2011), 041801 doi:10.1103/PhysRevLett.107.041801 [arXiv:1106.2822 [hep-ex]].
  * (6) M. R. Devi and K. Bora, [arXiv:2103.10065 [hep-ph]].
  * (7) D. N. Dinh, N. Anh Ky, P. Q. Văn and N. T. H. Vân, [arXiv:1602.07437 [hep-ph]].
  * (8) M. C. Chen, J. Huang, J. M. O’Bryan, A. M. Wijangco and F. Yu, JHEP 02 (2013), 021 doi:10.1007/JHEP02(2013)021 [arXiv:1210.6982 [hep-ph]].
  * (9) R. Kalita, D.Borah, Phys. Rev. D 92 (2015) no.5, 055012 doi:10.1103/ PhysRevD.92.055012 [arXiv:1508.05466 [hep-ph]].
  * (10) N. Sarma, K. Bora and D. Borah, Eur. Phys. J. C 79 (2019) no.2, 129 doi:10.1140/epjc/s10052-019-6584-z [arXiv:1810.05826 [hep-ph]].
  * (11) M. Hirsch, S. Morisi and J. W. F. Valle, Phys. Lett. B 679 (2009), 454-459 doi:10.1016/j.physletb.2009.08.003 [arXiv:0905.3056 [hep-ph]].
  * (12) M. Sruthilaya, R. Mohanta and S. Patra, Eur. Phys. J. C 78 (2018) no.9, 719 doi:10.1140/epjc/s10052-018-6181-6 [arXiv:1709.01737 [hep-ph]].
  * (13) D. Borah and B. Karmakar, Phys. Lett. B 780 (2018), 461-470 doi:10.1016/j.physletb.2018.03.047 [arXiv:1712.06407 [hep-ph]].
  * (14) P. Sahu, S. Patra and P. Pritimita, [arXiv:2002.06846 [hep-ph]].
  * (15) S. Verma, M. Kashav and S. Bhardwaj, Nucl. Phys. B 946 (2019), 114704 doi:10.1016/j.nuclphysb.2019.114704 [arXiv:1811.06249 [hep-ph]].
  * (16) I. Girardi, S. T. Petcov, A. J. Stuart and A. V. Titov, Nucl. Phys. B 902 (2016), 1-57 doi:10.1016/j.nuclphysb.2015.10.020 [arXiv:1509.02502 [hep-ph]].
  * (17) S. T. Petcov and A. V. Titov, Phys. Rev. D 97 (2018) no.11, 115045 doi:10.1103/PhysRevD.97.115045 [arXiv:1804.00182 [hep-ph]].
  * (18) S. T. Petcov and A. V. Titov, Int. J. Mod. Phys. A 33 (2018) no.31, 1844024 doi:10.1142/S0217751X18440244
  * (19) D. Borah and B. Karmakar, Phys. Lett. B 789 (2019), 59-70 doi:10.1016/j.physletb.2018.12.006 [arXiv:1806.10685 [hep-ph]].
  * (20) I. Sethi and S. Patra, J. Phys. G 48 (2021) no.10, 105003 doi:10.1088/1361-6471/ac1d99 [arXiv:1909.01560 [hep-ph]].
  * (21) H. Cai, T. Nomura and H. Okada, Nucl. Phys. B 949 (2019), 114802 doi:10.1016/j.nuclphysb.2019.114802 [arXiv:1812.01240 [hep-ph]].
  * (22) U. Rahaman and S. Razzaque, [arXiv:2108.11783 [hep-ph]].
  * (23) J. Rout, S. Roy, M. Masud, M. Bishai and P. Mehta, Phys. Rev. D 102 (2020), 116018 doi:10.1103/PhysRevD.102.116018 [arXiv:2009.05061 [hep-ph]].
  * (24) O. Yasuda, PTEP 2020 (2020) no.6, 063B03 doi:10.1093/ptep/ptaa033 [arXiv:2002.01616 [hep-ph]].
  * (25) M. Ghosh and T. Ohlsson, Mod. Phys. Lett. A 35 (2020) no.05, 2050058 doi:10.1142/S0217732320500583 [arXiv:1906.05779 [hep-ph]].
  * (26) N. Haba, Y. Mimura and T. Yamada, Phys. Rev. D 101 (2020) no.7, 075034 doi:10.1103/PhysRevD.101.075034 [arXiv:1812.10940 [hep-ph]].
  * (27) S. Verma and S. Bhardwaj, Adv. High Energy Phys. 2019 (2019), 8464535 doi:10.1155/2019/8464535 [arXiv:1808.04263 [hep-ph]].
  * (28) S. Bharti, S. Prakash, U. Rahaman and S. Uma Sankar, JHEP 09 (2018), 036 doi:10.1007/JHEP09(2018)036 [arXiv:1805.10182 [hep-ph]].
  * (29) G. L. Fogli and E. Lisi, Phys. Rev. D 54 (1996), 3667-3670 doi:10.1103/PhysRevD.54.3667 [arXiv:hep-ph/9604415 [hep-ph]].
  * (30) V. Barger, D. Marfatia and K. Whisnant, Phys. Rev. D 65 (2002), 073023 doi:10.1103/PhysRevD.65.073023 [arXiv:hep-ph/0112119 [hep-ph]].
  * (31) M. V. Smirnov, Z. Hu, S. Li and J. Ling, Chin. Phys. C 43 (2019) no.3, 033001 doi:10.1088/1674-1137/43/3/033001 [arXiv:1808.03795 [hep-ph]].
  * (32) K. Bora, D. Dutta and P. Ghoshal, Mod. Phys. Lett. A 30 (2015) no.14, 1550066 doi:10.1142/S0217732315500662 [arXiv:1405.7482 [hep-ph]].
  * (33) P. B. Denton, J. Gehrlein and R. Pestes, Phys. Rev. Lett. 126 (2021) no.5, 051801 doi:10.1103/PhysRevLett.126.051801 [arXiv:2008.01110 [hep-ph]].
  * (34) K. J. Kelly, P. A. N. Machado, S. J. Parke, Y. F. Perez-Gonzalez and R. Z. Funchal, Phys. Rev. D 103 (2021) no.1, 013004 doi:10.1103/PhysRevD.103.013004 [arXiv:2007.08526 [hep-ph]].
  * (35) I. Esteban, M. C. Gonzalez-Garcia and M. Maltoni, [arXiv:2004.04745 [hep-ph]].
  * (36) A. Himmel, New Oscillation Results from the NOvA Experiment, doi:10.5281/zenodo. 3959581, https://doi.org/10.5281/zenodo.3959581.
  * (37) M. Nizam, S. Bharti, S. Prakash, U. Rahaman and S. Uma Sankar, Mod. Phys. Lett. A 35 (2019) no.06, 06 doi:10.1142/S0217732320500212 [arXiv:1811.01210 [hep-ph]].
  * (38) Y. Fukuda et al. [Super-Kamiokande], Phys. Rev. Lett. 81 (1998), 1562-1567 doi:10.1103/PhysRevLett.81.1562 [arXiv:hep-ex/9807003 [hep-ex]].
  * (39) Q. R. Ahmad et al. [SNO], Phys. Rev. Lett. 87 (2001), 071301 doi:10.1103/PhysRev Lett.87.071301 [arXiv:nucl-ex/0106015 [nucl-ex]].
  * (40) Q. R. Ahmad et al. [SNO], Phys. Rev. Lett. 89 (2002), 011301 doi:10.1103/PhysRev Lett.89.011301 [arXiv:nucl-ex/0204008 [nucl-ex]].
  * (41) K. Abe et al. [Super-Kamiokande], Phys. Rev. D 97 (2018) no.7, 072001 doi:10.1103/PhysRevD.97.072001 [arXiv:1710.09126 [hep-ex]].
  * (42) M. G. Aartsen et al. [IceCube], Phys. Rev. Lett. 120 (2018) no.7, 071801 doi:10.1103/PhysRevLett.120.071801 [arXiv:1707.07081 [hep-ex]].
  * (43) M. G. Aartsen et al. [IceCube], Phys. Rev. D 99 (2019) no.3, 032007 doi:10.1103/PhysRevD.99.032007 [arXiv:1901.05366 [hep-ex]].
  * (44) W. Rodejohann, Int. J. Mod. Phys. E 20 (2011), 1833-1930 doi:10.1142/S0218301311020186 [arXiv:1106.1334 [hep-ph]].
  * (45) S. M. Bilenky and C. Giunti, Int. J. Mod. Phys. A 30 (2015) no.04n05, 1530001 doi:10.1142/S0217751X1530001X [arXiv:1411.4791 [hep-ph]].
  * (46) S. Dell’Oro, S. Marcocci, M. Viel and F. Vissani, Adv. High Energy Phys. 2016 (2016), 2162659 doi:10.1155/2016/2162659 [arXiv:1601.07512 [hep-ph]].
  * (47) M. Agostini et al. [GERDA], Science 365 (2019), 1445 doi:10.1126/science.aav8613 [arXiv:1909.02726 [hep-ex]].
  * (48) D. Q. Adams et al. [CUORE], Phys. Rev. Lett. 124 (2020) no.12, 122501 doi:10.1103/PhysRevLett.124.122501 [arXiv:1912.10966 [nucl-ex]].
  * (49) A. Gando et al. [KamLAND-Zen], Phys. Rev. Lett. 117 (2016) no.8, 082503 doi:10.1103/PhysRevLett.117.082503 [arXiv:1605.02889 [hep-ex]].
  * (50) L. Calibbi and G. Signorelli, Riv. Nuovo Cim. 41 (2018) no.2, 71-174 doi:10.1393/ncr/i2018-10144-0 [arXiv:1709.00294 [hep-ph]].
  * (51) R. N. Mohapatra, Phys. Rev. Lett. 56 (1986), 561-563 doi:10.1103/PhysRevLett.56.561
  * (52) R. N. Mohapatra and J. W. F. Valle, Phys. Rev. D 34 (1986), 1642 doi:10.1103/PhysRevD.34.1642
  * (53) D. Wyler and L. Wolfenstein, Nucl. Phys. B 218 (1983), 205-214 doi:10.1016/0550-3213(83)90482-0
  * (54) G. ’t Hooft, C. Itzykson, A. Jaffe, H. Lehmann, P. K. Mitter, I. M. Singer and R. Stora, NATO Sci. Ser. B 59 (1980), pp.1-438 doi:10.1007/978-1-4684-7571-5
  * (55) G. Altarelli and F. Feruglio, Rev. Mod. Phys. 82 (2010), 2701-2729 doi:10.1103/RevMod Phys.82.2701 [arXiv:1002.0211 [hep-ph]].
  * (56) S. Dell’Oro, S. Marcocci and F. Vissani, Phys. Rev. D 90 (2014) no.3, 033005 doi:10.1103/PhysRevD.90.033005 [arXiv:1404.2616 [hep-ph]].
  * (57) P. Dunne, Latest Neutrino Oscillation Results from T2K, doi: 10.5281/zenodo.3959558, https://doi.org/10.5281/zenodo.3959558
  * (58) Y. Takeuchi [Super-Kamiokande], Nucl. Instrum. Meth. A 952 (2020), 161634 doi:10.1016/j.nima.2018.11.093
  * (59) G. Adhikari et al. [nEXO], [arXiv:2106.16243 [nucl-ex]].