#  Absolute neutrino mass scale and dark matter stability from flavour
symmetry

Salvador Centelles Chuliá<EMAIL_ADDRESS>Max-Planck-Institut für
Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany  Ricardo Cepedello
<EMAIL_ADDRESS>Institut für Theoretische Physik und
Astrophysik, University of Würzburg,
Campus Hubland Nord, D-97074 Würzburg, Germany  Omar Medina
<EMAIL_ADDRESS>AHEP Group, Institut de Física Corpuscular –
C.S.I.C./Universitat de València, Parc Científic de Paterna.
C/ Catedrático José Beltrán, 2 E-46980 Paterna (Valencia), Spain

###### Abstract

We explore a simple but extremely predictive extension of the scotogenic
model. We promote the scotogenic symmetry $\mathbb{Z}_{2}$ to the flavour non-
Abelian symmetry $\Sigma(81)$, which can also automatically protect dark
matter stability. In addition, $\Sigma(81)$ leads to striking predictions in
the lepton sector: only Inverted Ordering is realised, the absolute neutrino
mass scale is predicted to be $m_{\text{lightest}}\approx 7.5\times 10^{-4}$
eV and the Majorana phases are correlated in such a way that $|m_{ee}|\approx
0.018$ eV. The model also leads to a strong correlation between the solar
mixing angle $\theta_{12}$ and $\delta_{CP}$, which may be falsified by the
next generation of neutrino oscillation experiments. The setup is minimal in
the sense that no additional symmetries or flavons are required.

##  I Introduction

Motivated by two fundamental problems of particle and astroparticle physics,
namely the origin of neutrino masses Minkowski:1977sc ; Yanagida:1979as ;
GellMann:1980vs ; mohapatra:1980ia ; Schechter:1980gr ; Schechter:1982cv ;
ma:1998dn ; CentellesChulia:2018gwr ; CentellesChulia:2020dfh  and the nature
of dark matter Aghanim:2018eyx , there’s been a great effort to relate them
within a single, predictive framework. Unarguably, they both point towards the
presence of physics beyond the Standard Model (SM), presumably with the
addition of new particles and symmetries that account for a mass mechanism for
neutrinos, a viable dark matter candidate and its stability.

An _economical_ approach to combine all these appealing properties is to
consider radiative neutrino mass models Bonnet:2012kz ; Farzan:2012ev ;
Sierra:2014rxa ; Cepedello:2017eqf ; Yao:2017vtm ; Cepedello:2018rfh ;
Klein:2019iws ; CentellesChulia:2019xky  (for a review see Cai:2017jrq ). In
this kind of models, fields running in a loop generate neutrino masses, giving
rise to two clearly distinguishable particle sectors, one of which can be
regarded as a dark sector by means of a symmetry. The stability of the dark
matter candidate, i.e. the lightest of the particles belonging to the dark
sector, is determined by the transformation properties of the SM fields and
the dark sector under symmetries Bonilla:2018ynb ; CentellesChulia:2019gic ;
Srivastava:2019xhh . In the most simple scenarios, the SM fields transform
only under an invariant subgroup of the symmetry, while any particle beyond
those of the SM not belonging to this subgroup will not be able to decay
solely to the SM, i.e. it will be part of the dark sector. A popular
implementation of this principle is the scotogenic model Ma:2006km  and its
many variants (see for instance Rojas:2018wym ; Kang:2019sab ; Leite:2019grf ;
Barreiros:2020gxu ; Borah:2021rbx ; Escribano:2021ymx ; Mandal:2021yph ;
Sarma:2022bhl ; Ma:2022bfa ; Sarazin:2021nwo ).

While a large number of models built following the described approach are
consistent with experimental data from neutrino oscillations and bounds from
dark matter searches, there are further unknowns about fundamental particles
that are also important to address. The SM lacks a suitable theoretical
explanation for the masses and the mixing pattern of fermions. Furthermore,
the majority of input parameters of the SM are directly related to this
_flavour puzzle_. The lepton mixing angles, being large and with a completely
different structure in comparison to their analogues in the quark sector,
manifest the lack of a first principle explanation of the flavour
phenomenology Kajita:2016cak ; McDonald:2016ixn ; ParticleDataGroup:2020ssz .
Here is where _flavour symmetries_ can play a major role in explaining such
mixing patterns and mass hierarchies.111While there is a vast bibliography on
this topic, we direct the interested readers to the reviews King:2015aea ;
Feruglio:2019ybq . By means of imposing a flavour symmetry between the three
generations it’s possible to predict strong correlations between different
observables. This is essential for a flavour symmetry model to be verifiable.

In this paper we build a model for radiative neutrino masses with a flavour
symmetry $\Sigma(81)$. We focus on such a discrete group due to an interesting
feature: $\Sigma(81)$ contains a non-trivial subgroup formed by the singlets
and one of the triplet representations. This ensures, as we will show in
section IV, that for a reasonable choice of the transformation properties of
the field content under the flavour symmetry, one can straightforwardly obtain
a stable dark matter candidate. Thus, providing a _natural_ framework to
account for dark matter stability along with light radiative Majorana neutrino
masses through a scotogenic-like mechanism. Other works with _flavoured
stability_ are, for example, Hirsch:2010ru ; Lavoura:2012cv ; Boucenna:2012qb
; Ma:2019iwj ; deAnda:2021jzc . Other works with a flavour group $\Sigma(81)$
are for example BenTov:2012xp ; Hagedorn:2008bc .

A more conventional role played by $\Sigma(81)$ symmetry is to strongly
constrain the structure of the mass matrices of fermions, leading to strong
predictions that can be tested in the following years by the next generation
of neutrino oscillation abe2014long ; Hyper-Kamiokande:2016srs ; DUNE:2020jqi
; JUNO:2015zny ; Cao:2014bea , and neutrinoless double beta decay experiments
KamLAND-Zen:2016pfg ; GERDA:2018zzh ; Agostini:2017iyd ; Alduino:2017ehq ;
Arnold:2016qyg ; Albert:2014awa . While the idea of imposing a flavour
symmetry is certainly not new, we will show that our setup has a series of
attractive and unique features, namely explaining the lepton mixing pattern,
as well as predicting the absolute mass scale of neutrinos, their ordering and
the Majorana phases, and therefore leading to a definite prediction for
neutrinoless double beta decay ($0\nu ee$). Moreover, this is obtained without
the need of extra flavons, i.e. extra scalars that further break the flavour
symmetry. In our setup the breaking of the flavour symmetry is done by
extending the number of Higgs doublets, as a variant of a 3HDM and giving them
non-trivial charges under $\Sigma(81)$.

The paper is structured as follows: in section II we present the model setup,
i.e. the field content, the charges under the SM gauge group and flavour
symmetry and discuss some of its most important attributes. In section III we
delve into its most important phenomenological predictions: absolute neutrino
mass scale and ordering, strong correlations between oscillation observables
and the implications for neutrinoless double beta decay. In section IV we
explicitly flesh out the non-Abelian stability mechanism provided by
$\Sigma(81)$. The paper then closes with a short summary and conclusions.
Details about the symmetry group $\Sigma(81)$ are relegated to Appendix A.

##  II The model setup

We extend the Standard Model gauge symmetry $\mathrm{SU(3)_{C}\times
SU(2)_{L}\times U(1)_{Y}}$ by a global, discrete flavour group $\Sigma(81)$.
This group is of the type $\Sigma(3N^{3})$ and contains 9 singlets and 4
complex triplets, denoted as $\mathbf{1}_{(i,j)}$ with $i,j=0,1,2$ and
$\mathbf{3}_{X}(\mathbf{\bar{3}}_{X})$ with $X=A,B,C,D$ (see Appendix A for
details). The irreducible representations $\mathbf{3}_{D}$
($\mathbf{\bar{3}}_{D}$), together with the singlets, form a closed set under
tensor products, implying that if every Standard Model field transforms as
$\mathbf{3}_{D}$, $\mathbf{\bar{3}}_{D}$ or as one of the singlets, then any
field transforming as $\mathbf{3}_{A,B,C}$ and their conjugates, will belong
to the dark sector. The lightest among them will then be a dark matter
candidate. This relation between $\Sigma(81)$ and dark matter will be further
discussed in section IV.

|  Fields |  $\mathrm{SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}}$ |  $\Sigma(81)$
---|---|---|---
Visible | $L$ | ($\mathbf{1},\mathbf{2},{-1/2}$) | $\mathbf{3}_{D}$
$\ell_{R}$ | ($\mathbf{1},\mathbf{1},{-1}$) | $\mathbf{\bar{3}}_{D}$
$H$ | ($\mathbf{1},\mathbf{2},{1/2}$) | $\mathbf{\bar{3}}_{D}$
Dark | $N_{L,R}$ | ($\mathbf{1},\mathbf{1},{0}$) | $\mathbf{3}_{A}$
$\eta$ | ($\mathbf{1},\mathbf{2},{1/2}$) | $\mathbf{3}_{A}$
$\phi$ | ($\mathbf{1},\mathbf{2},{1/2}$) | $\mathbf{\bar{3}}_{A}$
Table 1: Particle content and symmetry transformation properties under the SM
gauge group and the flavour symmetry $\Sigma(81)$. Note that the fields of the
visible sector transform as $\mathbf{3}_{D}$, $\mathbf{\bar{3}}_{D}$ or
$\mathbf{1}_{(i,j)}$, while the dark sector transforms as $\mathbf{3}_{A}$ or
$\mathbf{\bar{3}}_{A}$. The lightest particle of the dark sector will be
automatically stable. See text for details.

The field content of the SM is extended by adding a vector-like singlet $N$
and two Higgs-like scalars, transforming non-trivially under $\Sigma(81)$. All
the fields and charges are given in table 1. Comparing to the original
scotogenic model Ma:2006km , new fields were also required to generate
neutrino masses at one-loop with $\Sigma(81)$. While for the simple
$\mathbb{Z}_{2}$ symmetry of the scotogenic model, any product of an odd field
under $\mathbb{Z}_{2}$ times itself transforms as a singlet under
$\mathbb{Z}_{2}$, this is not the case for any of the triplet representations
of $\Sigma(81)$. For this reason, one needs to promote the right-handed
neutrino to a vector-like fermion and, on a similar footing, two copies of the
inert doublet Higgs are required, $\eta$ and $\phi$. For simplicity, we split
the most relevant parts of the Lagrangian as,

$\mathcal{L}\;\supset\;\mathcal{L}^{V}_{\text{Y}}+\mathcal{L}^{D}_{\text{Y}}-\mathcal{V}_{\text{s}}\,,$
(1)

where the scalar potential is further divided into parts, for convenience, as
$\mathcal{V}_{s}=\mathcal{V}_{\nu}+\mathcal{V}_{\text{soft}}+...$ . The first
part of the potential contains the scalar interactions that enter in the
neutrino mass, the second of soft breaking terms of mass dimension 2 and
“$...$” denotes the rest of the usual four-scalar interactions, that are not
interesting for the purpose of our discussion. The terms in
$\mathcal{V}_{\text{soft}}$ are of the form,

$\mathcal{V}_{\text{soft}}=\mu_{ij}^{2}\,H_{i}^{\dagger}H_{j}\,.$ (2)

These terms are necessary in order to satisfy phenomenological bounds. In the
limit $\mu_{ij}\rightarrow 0$, which we will call the “symmetric limit”, the
allowed VEV alignments will be highly restricted by the $\Sigma(81)$ symmetry.
A preliminary analysis of the scalar potential, solving the tadpole equations,
always yields highly symmetrical VEV alignments in this limit, for example
$(v_{1},v_{2},v_{3})=v\,(1,0,0)$ or $v\,(1,1,1)$. However, from the mass
matrices shown in sections II.1 and II.2, it’s evident that realistic lepton
mixing and masses cannot be realized from such symmetric alignments. Including
the $\mu_{ij}$ terms will add nine new parameters to the tadpole equations,
allowing enough deviations from the symmetric limit to get a realistic lepton
mixing pattern. In particular, the alignment that has been obtained from the
phenomenological analysis is a perturbation from the tadpole equation
solution,

$v(1,0,0)\rightarrow
v(1,\epsilon_{1},\epsilon_{2})\quad\text{with}\quad\epsilon_{1},\epsilon_{2}\sim\mathcal{O}(10^{-2}).$
(3)

It is worth mentioning that such alignment approximately preserves a residual
$\mathbb{Z}_{3}$ symmetry, which originates due to the invariance of the
tadpole equation solution $v(1,0,0)$ under the $a^{\prime}$ generator of
$\Sigma(81)$ in the $\mathbf{\bar{3}}_{D}$ representation, as can be seen
directly from equation (36) in Appendix A. On the other hand, the symmetric
limit faces other phenomenological challenges. Strong FCNCs are expected in
this type of models, as well as deviations from the Standard Model fermion-
Higgs interactions. Following Georgi:1978ri , after we have chosen a
particular VEV alignment, we can rotate to the Higgs basis
$H=\sum_{i}\frac{v_{i}}{v}H_{i}$. In this basis, only one doublet $H$ has a
non-zero VEV, while the other two orthogonal combinations remain VEV-less.
Then, the diagonal $\mu$ terms for these two doublets can be taken to be
arbitrarily large, thus effectively decoupling them from the rest of the model
without affecting the VEV structure. The doublet $H$ will be SM-like. Finally,
for a more detailed analysis of the 3HDM scalar potential, see for example
deMedeirosVarzielas:2021zqs ; Ivanov:2012ry ; Keus:2013hya ; Ivanov:2014doa ;
Maniatis:2014oza ; Vergeest:2022mqm ; Hernandez:2021iss ;
deMedeirosVarzielas:2022kbj .

###  II.1 Charged lepton masses

In this section, we derive the mass matrix for the charged leptons given the
particle content of table 1. The relevant piece of the Lagrangian in equation
(1) is the first term, which contains the Yukawa interaction terms among
fields of the visible sector. To make the derivation clearer, all terms have
been expanded in $\Sigma(81)$ components, for example,
$L=(L_{1},L_{2},L_{3})^{\text{T}}$, and similarly for the other triplets,
following the tensor products given in the second edition of the book “An
Introduction to Non-Abelian Discrete Symmetries for Particle Physicists”
Kobayashi:2022moq  (see also Appendix A for more details). In this way, it’s
made explicit in the Lagrangian itself that several contractions may lead to a
singlet under $\Sigma(81)$. For instance, three triplets $\mathbf{3}_{D}$ have
three different contractions to an invariant singlet $\mathbf{1}_{(0,0)}$.

The Yukawa interactions among the visible sector are given by,

$\displaystyle\mathcal{L}^{V}_{\text{Y}}\;$ $\displaystyle=$
$\displaystyle\;Y^{e}_{1}\sum_{i=1}^{3}\bar{L}_{i}\ell_{R_{i}}H_{i}$
$\displaystyle+$ $\displaystyle
Y^{e}_{2}\left(\bar{L}_{1}\ell_{R_{3}}H_{2}+\bar{L}_{2}\ell_{R_{1}}H_{3}+\bar{L}_{3}\ell_{R_{2}}H_{1}\right)$
$\displaystyle+$ $\displaystyle
Y^{e}_{3}\left(\bar{L}_{1}\ell_{R_{2}}H_{3}+\bar{L}_{2}\ell_{R_{3}}H_{1}+\bar{L}_{3}\ell_{R_{1}}H_{2}\right)$
$\displaystyle+$ $\displaystyle\text{h.c.}\,,$

where $SU(2)_{L}$ indices and contractions have been omitted for simplicity.
After electroweak symmetry breaking (EWSB) the Lagrangian
$\mathcal{L}^{V}_{\text{Y}}$ in last equation gives rise to the mass matrix
for charged leptons,

$M_{e}=\frac{1}{\sqrt{2}}\,\left(\begin{array}[]{ccc}Y_{1}^{e}\,v1&Y_{3}^{e}\,v3&Y_{2}^{e}\,v2\\\
Y_{2}^{e}\,v3&Y_{1}^{e}\,v2&Y_{3}^{e}\,v1\\\
Y_{3}^{e}\,v2&Y_{2}^{e}\,v1&Y_{1}^{e}\,v3\end{array}\right),$ (5)

in the basis
$\\{(L_{1},L_{2},L_{3}),\,(\ell_{R_{1}},\ell_{R_{2}},\ell_{R_{3}})\\}$, with
the vacuum expectation values of the Higgs defined as, $\left\langle
H_{i}\right\rangle=v_{i}/\sqrt{2}$ and $\sum_{i}v_{i}^{2}=v_{\text{SM}}^{2}$
the Standard Model VEV. The mass matrix $M_{e}$ is then diagonalised by the
unitary rotations $U_{\ell}$ and $V_{\ell}$ as,

$\displaystyle\hat{M}_{e}=U_{\ell}^{\dagger}M_{e}V_{\ell},\quad\text{with}\quad
L\rightarrow U_{\ell}\,L,\hskip 14.22636pt\ell_{R}\rightarrow
V_{\ell}\,\ell_{R}\,,\quad\hat{M}_{e}=\text{diag}(m_{e},m_{\mu},m_{\tau})\,.$
(6)

###  II.2 Neutrino masses

The term $\mathcal{L}^{D}_{\text{Y}}$ in the RHS of equation (1), describes
the interactions with the fields of the dark sector. This piece, together with
the scalar potential, will give rise to the one-loop neutrino mass diagram
depicted in figure 1 and its corresponding mass matrix. This term in the
Lagrangian is given by,

$\displaystyle\mathcal{L}^{D}_{\text{Y}}\;$ $\displaystyle=$
$\displaystyle\;M_{N}\left({\color[rgb]{.5,0,.5}\bar{N}_{L_{1}}N_{R_{1}}}+{\color[rgb]{0,0,1}\bar{N}_{L_{2}}N_{R_{2}}}+{\color[rgb]{0,0,0}\bar{N}_{L_{3}}N_{R_{3}}}\right)$
$\displaystyle+$ $\displaystyle
Y^{N}_{1}\left({\color[rgb]{0,0,1}L_{1}\bar{N}_{R_{2}}\eta_{1}}+{\color[rgb]{0,0,0}L_{2}\bar{N}_{R_{3}}\eta_{2}}+{\color[rgb]{.5,0,.5}L_{3}\bar{N}_{R_{1}}\eta_{3}}\right)$
$\displaystyle+$ $\displaystyle
Y^{N}_{2}\left({\color[rgb]{.5,0,.5}L_{1}N_{L_{1}}\phi_{2}}+{\color[rgb]{0,0,1}L_{2}N_{L_{2}}\phi_{3}}+{\color[rgb]{0,0,0}L_{3}N_{L_{3}}\phi_{1}}\right)$
$\displaystyle+$ $\displaystyle\text{h.c.}\,.$

Figure 1: Diagram generating neutrino masses at one-loop in our model,
analogous to the original scotogenic model. Fields running in the loop are
charged as $\mathbf{3}_{A}$ or $\mathbf{\bar{3}}_{A}$ under $\Sigma(81)$,
while fields in the external legs (SM) transform as $\mathbf{3}_{D}$ or
$\mathbf{\bar{3}}_{D}$.

The relevant scalar couplings, analogous to the $\lambda_{5}$ interaction from
the original scotogenic model Ma:2006km , are

$\displaystyle\mathcal{V}_{\nu}\;$ $\displaystyle\supset$
$\displaystyle\lambda_{5}^{(1)}\left[{\color[rgb]{0,0,0}(H_{1}\eta_{2}^{\dagger})(H_{1}\phi_{1}^{\dagger})}+{\color[rgb]{.5,0,.5}(H_{2}\eta_{3}^{\dagger})(H_{2}\phi_{2}^{\dagger})}+{\color[rgb]{0,0,1}(H_{3}\eta_{1}^{\dagger})(H_{3}\phi_{3}^{\dagger})}\right]$
(8) $\displaystyle+$
$\displaystyle\lambda_{5}^{(2)}\left[{\color[rgb]{0,0,1}(H_{1}\eta_{1}^{\dagger})(H_{2}\phi_{3}^{\dagger})}+{\color[rgb]{.5,0,.5}(H_{1}\eta_{3}^{\dagger})(H_{3}\phi_{2}^{\dagger})}+{\color[rgb]{0,0,0}(H_{2}\eta_{2}^{\dagger})(H_{3}\phi_{1}^{\dagger})}\right]$
$\displaystyle+$ $\displaystyle\text{h.c.}\,.$

The expansion in components of $\Sigma(81)$ makes explicit that not every
entry of the neutrino mass matrix will be generated. In fact, there are only
six possible diagrams with different $\Sigma(81)$ components outside the loop
and running in it. After EWSB, the resultant neutrino mass matrix is of the
form,

$M_{\nu}\sim\frac{1}{2}\,\left(\begin{array}[]{ccc}0&{\color[rgb]{0,0,1}C_{1}v_{3}^{2}+C_{2}v_{1}v_{2}}&{\color[rgb]{.5,0,.5}C_{1}v_{2}^{2}+C_{2}v_{1}v_{3}}\\\
{\color[rgb]{0,0,1}C_{1}v_{3}^{2}+C_{2}v_{1}v_{2}}&0&{\color[rgb]{0,0,0}C_{1}v_{1}^{2}+C_{2}v_{2}v_{3}}\\\
{\color[rgb]{.5,0,.5}C_{1}v_{2}^{2}+C_{2}v_{1}v_{3}}&{\color[rgb]{0,0,0}C_{1}v_{1}^{2}+C_{2}v_{2}v_{3}}&0\end{array}\right)\,.$
(9)

For the sake of clarity, we have assigned colours to each entry of the matrix
and to its corresponding terms in the Lagrangian $\mathcal{L}^{D}_{\text{Y}}$
and the scalar potential $\mathcal{V}_{\nu}$ in equations (II.2) and (8)
respectively. The coefficients $C_{a}$ are obtained by computing the different
diagrams of the type of figure 1 that contribute,

$\displaystyle
C_{a}\sim\frac{1}{16\pi^{2}}\,\frac{\lambda_{5}^{(a)}\,(Y^{N}_{1})\,(Y^{N}_{2})}{M_{N}}\,.$
(10)

A very remarkable feature of the UV-realisation with $\Sigma(81)$ that we
present here, is the fact that the neutrino matrix is exactly traceless with
vanishing diagonal entries. This feature is protected by the symmetry and
yields several strong predictions in the neutrino sector, as we will discuss
in the next section. The matrix in equation (9) coefficients $C_{a}$
correspond to the dominant contribution. The neutrino mass matrix is, in
general, given by,

$\displaystyle(M_{\nu})_{\alpha\beta}=\frac{1}{16\pi^{2}}\,(Y^{N}_{1})_{\beta
ij}\,(Y^{N}_{2})_{\alpha
ij}\,M_{N}\sum_{X=R,I}\sigma_{X}\,(U_{X}^{\alpha})_{1i}\,(U_{X}^{\alpha})_{i2}\,B_{0}(0,M_{N},m_{X_{i}}^{2})\,,$
(11)

where $\sigma_{R,I}=\pm 1$. The expression for the neutrino mass matrix (11)
is very similar to that of the original scotogenic model, where after
electroweak symmetry breaking the neutral part of the scalar doublet in the
loop splits into its $CP$-even and $CP$-odd components (denoted as $R$ and
$I$, respectively) due to the quartic coupling $\lambda_{5}$. The result is
the sum of two $B_{0}$ Passarino-Veltman loop functions Passarino:1978jh  with
a relative minus sign. Also, similar to the generalised scotogenic models with
several scalars Escribano:2020iqq , the mixing among the different scalar
doublets in the loop need to be considered. The main subtlety is that, given
the flavour symmetry $\Sigma(81)$, not every coupling is allowed. The only
non-zero Yukawa couplings are
$(Y_{1}^{N})_{112}=(Y_{1}^{N})_{223}=(Y_{1}^{N})_{331}=Y_{1}^{N}$ and
$(Y_{2}^{N})_{121}=(Y_{2}^{N})_{232}=(Y_{2}^{N})_{313}=Y_{2}^{N}$. While the
mass matrices mixing the neutral components of the scalars can be trivially
obtained from (8), with diagonalising matrices $U^{\alpha}_{R}$ and
$U^{\alpha}_{I}$, for the $CP$-even and odd components respectively, in the
basis $(\eta_{\alpha},\phi_{k})$. Note that again $\Sigma(81)$ only allows the
mixing among specific pairs of $\eta$ and $\phi$ (see the scalar potential
(8)).

It is worth noting that while lepton flavour is violated in the neutrino
sector, the usual dominant one-loop contribution to cLFV, mediated directly by
$Y_{i}^{N}$, is absent. The Yukawa structure (II.2) leads to a diagonal
contribution proportional to $(|Y_{1}^{N}|^{2}+|Y_{2}^{N}|^{2})$ and the
charged lepton masses. Consequently, any cLFV process, like $\mu\rightarrow
e\gamma$, is suppressed.

##  III Predictions

The neutrino mass matrix in equation (9) is diagonalised as,

$\displaystyle
U_{\nu}^{T}\,M_{\nu}\,U_{\nu}\,=\,\text{diag}(m_{1},m_{2},m_{3})\,,$ (12)

where $U_{\nu}$ is the neutrino unitary mixing matrix and $m_{i}$ are the
neutrino masses. In the Normal Ordering case $m_{3}>m_{2}>m_{1}$, while in the
Inverted Ordering case $m_{2}>m_{1}>m_{3}$. Considering both equations (6) and
(12) we obtain the lepton mixing matrix,

$U_{\text{lep}}=U_{\ell}^{\dagger}\,U_{\nu}\,.$ (13)

$U_{\text{lep}}$ is constrained by neutrino oscillation experiments. We choose
the so-called symmetric parametrisation of a general unitary matrix
Schechter:1980gr ; Rodejohann:2011vc ,

$U_{\text{lep}}=P(\delta_{1},\delta_{2},\delta_{3})\,U_{23}(\theta_{23},\phi_{23})\,U_{13}(\theta_{13},\phi_{13})\,U_{12}(\theta_{12},\phi_{12})\,,$
(14)

where $P(\delta_{1},\delta_{2},\delta_{3})$ is a diagonal matrix of unphysical
phases and the $U_{ij}$ are complex rotations in the $ij$ plane, as for
example,

$U_{23}(\theta_{23},\phi_{23})=\left(\begin{matrix}1&0&0\\\
0&\cos\theta_{23}&\sin\theta_{23}\,e^{-i\phi_{23}}\\\
0&-\sin\theta_{23}\,e^{i\phi_{23}}&\cos\theta_{23}\end{matrix}\right)\,.$ (15)

The phases $\phi_{12}$ and $\phi_{13}$ are relevant for neutrinoless double
beta decay while the combination $\delta_{CP}=\phi_{13}-\phi_{12}-\phi_{23}$
is the usual Dirac $CP$ phase measured in neutrino oscillations.

Before going into the numerical results, let us note an interesting analytical
property of the matrix (9). The shape of this mass matrix, due to the
$\Sigma(81)$ flavour symmetry, implies that the neutrino masses satisfy the
relation,

$\frac{1}{2}\,\sum\,m_{i}\,=\,m_{\text{heaviest}}\,,$ (16)

where $m_{\text{heaviest}}$ is the heaviest neutrino mass. Equation (16) is
actually a general prediction for a complex, symmetric, diagonal-less neutrino
mass matrix. If we call such a mass matrix $A$ and define it in general as,

$A=\left(\begin{matrix}0&a&b\\\ a&0&c\\\ b&c&0\end{matrix}\right),\hskip
14.22636pt\text{with }a,\,b,\,c\in\mathbb{C}\,,$ (17)

diagonalised as usual by

$\displaystyle U^{T}A\,U$ $\displaystyle=$ $\displaystyle
m_{d}=\text{diagonal}(m_{1},m_{2},m_{3})\,,$ (18) $\displaystyle
U^{\dagger}A^{\dagger}A\,U$ $\displaystyle=$ $\displaystyle m_{d}^{2}\,,$ (19)

where $m_{d}$ is real, diagonal and positive. With this definition the traces
of the matrices $A^{\dagger}A$ and $(A^{\dagger}A)^{2}$ can be computed
straightforwardly,

$\displaystyle
Tr(A^{\dagger}A)=2(|a|^{2}+|b|^{2}+|c|^{2})=m_{1}^{2}+m_{2}^{2}+m_{3}^{2}\,,$
(20) $\displaystyle
Tr\left[(A^{\dagger}A)^{2}\right]=2(|a|^{2}+|b|^{2}+|c|^{2})^{2}=m_{1}^{4}+m_{2}^{4}+m_{3}^{4}\,.$
(21)

These traces fulfill the general relation,

$\frac{1}{2}\left[Tr(A^{\dagger}A)\right]^{2}=Tr\left[(A^{\dagger}A)^{2}\right]\,,$
(22)

which translated to the mass eigenvalues reads,

$m_{3}^{2}=(m_{1}\pm m_{2})^{2}\,.$ (23)

Since $m_{i}$ are real and positive, only one solution survives after
specifying the ordering. In particular,

$\displaystyle m_{3}^{\text{NO}}=m_{1}^{\text{NO}}+m_{2}^{\text{NO}}\,,$ (24)
$\displaystyle m_{2}^{\text{IO}}=m_{1}^{\text{IO}}+m_{3}^{\text{IO}}\,,$ (25)

or in general, irrespective of the ordering, the sum rule (16).

Neutrino oscillations measure the mass squared differences of neutrino masses
IceCube:2019dqi ; Super-Kamiokande:2017yvm ; T2K:2021xwb ;
alex_himmel_2020_3959581 ; DayaBay:2018yms ; KamLAND:2010fvi , which in
combination with the mass sum rules (24) and (25) lead to the prediction of
the absolute scale of the neutrino masses:

$\displaystyle m_{\text{lightest}}^{\text{NO}}\,$ $\displaystyle\approx$
$\displaystyle\,2.8\times 10^{-2}\,\text{eV}\,,$ (26) $\displaystyle
m_{\text{lightest}}^{\text{IO}}\,$ $\displaystyle\approx$
$\displaystyle\,7.5\times 10^{-4}\,\text{eV}\,.$ (27)

Both values are well below cosmological bounds Planck:2018vyg  and direct
measurements of neutrino mass KATRIN:2021uub ; KATRIN:2022ayy . However, they
may be probed with astrophysical sources Kyutoku:2017wnb .

Note, however, that the neutrino mass matrix in equation (9) is more
restricted than the matrix in equation (17). In particular, the strong
hierarchy in the masses of the charged leptons implies a strong hierarchy
between the VEVs of the Higgs doublets, further restricting the neutrino mass
matrix. We have performed a numerical scan and found the following results and
predictions for both orderings.

###  III.1 Inverted ordering

In the Inverted Ordering case, a strong correlation appears between
$\theta_{12}$ and $\delta_{CP}$ when the charged lepton masses, neutrino
masses, $\theta_{13}$ and $\theta_{23}$ are fitted to their experimental
values. The model can accommodate all oscillation observables inside their
$3\sigma$ ranges with a slight tension in the $\theta_{12}$ vs $\delta_{CP}$
plane, see figure 2. However, the best fit value of $\delta_{CP}$ is very
sensible to new data sets. An update from the Nova collaboration
alex_himmel_2020_3959581  may change the picture in 2022.

Moreover, we are using the global fit deSalas:2020pgw  to produce the plots,
although the other two global fits Esteban:2020cvm ; Marrone:2015nip  yield
slightly lower values for $\delta_{CP}$, thus reducing the tension of the
model. Taking $\theta_{12}$ alone, we can see that the model can accommodate
$\theta_{12}\approx\theta_{12}^{\text{best fit}}$ if $\delta_{CP}\approx\pi$.
In other words, this model prediction may be tested in the following data
releases of neutrino oscillation experiments. Furthermore, the Majorana phases
$\phi_{12}$ and $\phi_{13}$, relevant for neutrinoless beta decay experiments,
also obtain a strong correlation, as seen in figure 3. The striking
similarities between these correlations and the ones in Chen:2018lsv ;
Chen:2019fgb  may indicate that our setup leads to the partial conservation of
some of the TBM symmetries of the neutrino mass matrix.

Figure 2: Correlation between $\theta_{12}$ and $\delta_{CP}$ in the IO case
when the other observables are fitted inside their $3\sigma$ experimental
ranges: charged lepton masses, neutrino squared mass differences,
$\theta_{13}$ and $\theta_{23}$. The $3\sigma$ tension in the combined
$\theta_{12}-\delta_{CP}$ plane may be relieved if $\delta_{CP}$ is measured
to be around $CP$ conserving values. In that case, $\theta_{12}$ would lie in
its $1\sigma$ experimental region. $\chi^{2}$ profiles extracted from the
global fit deSalas:2020pgw .

Figure 3: Correlations between physical phases in the IO case. Left up:
Correlation between $\phi_{12}$ and $\delta_{CP}$. Right up: Correlation
between $\phi_{12}$ and $\delta_{CP}$. Down: Correlation between the Majorana
phases $\phi_{13}$ and $\phi_{13}$. In all the plots blue dots arise when the
other observables are fitted inside their $3\sigma$ experimental ranges except
for $\theta_{12}$ and $\delta_{CP}$, which are free. In addition, purple dots
fit $\theta_{12}$ and red dots also fit $\delta_{CP}$ at the $3\sigma$ level.
By imposing all the experimental constraints, the model predicts
$\delta_{CP}\approx 1.2\pi$, $\phi_{12}\approx 0.45\pi$ and $\phi_{13}\approx
0.12\pi$ plus a small variance.

For neutrinoless double beta decay, if the Majorana neutrino mass mechanism is
the dominant contribution to $0\nu\beta\beta$, its rate will be proportional
to the quantity $|m_{ee}|$, given by,

$|m_{ee}|=\left|\sum_{i}U_{ei}^{2}m_{i}\right|=|c_{12}^{2}c_{13}^{2}m_{1}+s_{12}^{2}c_{13}^{2}e^{2i\phi_{12}}m_{2}+s_{13}^{2}e^{2i\phi_{13}}m_{3}|\,.$
(28)

In our model the Majorana phases are approximately fixed as $\phi_{12}\approx
0.45\pi$ and $\phi_{13}\approx 0.12\pi$, while the neutrino masses are also
predicted to be around $m_{3}\approx 7.51\times 10^{-4}$ eV, $m_{1}\approx
4.95\times 10^{-2}$ eV, $m_{2}\approx 5.02\times 10^{-2}$ eV. Small deviations
from these values are possible due to the experimental uncertainty on $\Delta
m^{2}_{ij}$ and the variance in $\phi_{ij}$. This automatically leads to a
definite prediction of $|m_{ee}|$ in our model:

$|m_{ee}^{\text{model}}|\approx 0.018\text{ eV}\,,$ (29)

Note that the term with $\phi_{13}$ in (28) interferes constructively to
$|m_{ee}|$ but is strongly suppressed by $s_{13}^{2}m_{3}$, while the term
with $\phi_{12}$ interferes destructively. This is why the allowed points in
the model are in the lower region of $|m_{ee}|$ as seen in figure 4. The nEXO
experiment is expected to test this model prediction in the future
nEXO:2017nam ; nEXO:2021ujk .

Figure 4: $|m_{ee}|$ is restricted to a small region in this model. The reason
is that $m_{light}$ and the Majorana phases are predicted, as well as the
ordering. The deviation from a single point comes from the experimental
uncertainties in $\Delta m^{2}_{ij}$ and a small variance in $\phi_{ij}$. The
current experimental constraints are given by KamLAND KamLAND-Zen:2016pfg  and
Planck Planck:2018vyg . In the future, nEXO is expected to have enough
sensitivity to completely rule out the inverted ordering region nEXO:2021ujk .

###  III.2 Normal ordering

In the Normal Ordering case, after imposing the correct charged lepton and
neutrino masses at $3\sigma$, a strong correlation appears between the mixing
angles $\theta_{23}$ and $\theta_{13}$ in the neutrino sector. This
correlation is not compatible with experimental constraints by more than
$7\sigma$, as can be seen in figure 5. Therefore, Normal Ordering of neutrino
masses cannot be realised in this model.

Figure 5: Model prediction in the NO case. The correlation between
$\theta_{13}$ and $\theta_{23}$ is incompatible with current experimental
constraints at more than $7\sigma$. Therefore the model predicts IO.
$\chi^{2}$ profiles extracted from the global fit deSalas:2020pgw .

##  IV Dark matter sector

The $\Sigma(81)$ flavour symmetry has the additional property of stabilizing
the lightest of the dark sector fields. In order to see how this mechanism
works, we must first note that the singlets $\mathbf{1}_{(i,j)}$ and the
$\mathbf{3}_{D}$, $\mathbf{\bar{3}}_{D}$ triplets form a closed subset under
the tensor products, i.e.

$\displaystyle\mathbf{1}_{(i,j)}\times\mathbf{1}_{(k,l)}=\mathbf{1}_{(i+k,\,j+l)},$
$\displaystyle\mathbf{1}_{(i,j)}\times\mathbf{3(\bar{3})}_{D}=\mathbf{3(\bar{3})}_{D},$
(30)
$\displaystyle\mathbf{3(\bar{3})}_{D}\times\mathbf{3(\bar{3})}_{D}=\mathbf{\bar{3}(3)}_{D},$
$\displaystyle\mathbf{3}_{D}\times\mathbf{\bar{3}}_{D}=\mathbf{1}_{(i,j)}\,.$
(31)

We start by imposing the condition that all of the visible sector fields, i.e.
the Standard Model fermions and Higgs, transform as either
$\mathbf{1}_{(i,j)}$, $\mathbf{3}_{D}$ or $\mathbf{\bar{3}}_{D}$. This
automatically implies that any effective operator formed by any arbitrary
combination of SM particles, $\mathcal{O}_{\text{visible}}$, will still
transform under the same subgroup, i.e. as $\mathbf{1}_{(i,j)}$,
$\mathbf{3}_{D}$ or $\mathbf{\bar{3}}_{D}$.

Consider now a field $\eta$ belonging to the dark sector and transforming as,
for example, $\mathbf{3}_{A}$. It’s clear that the effective operator
$\eta\,.\,\mathcal{O}_{\text{visible}}$ cannot be invariant under
$\Sigma(81)$, because no operator of the type $\mathcal{O}_{\text{visible}}$
transforms as $\mathbf{\bar{3}}_{A}$.

In conclusion, any symmetry invariant decay operator of a particle belonging
to the dark sector must involve, at least, one dark sector particle in the
final state and, thus, the lightest of them will necessarily be stable (see
figure 6). In our model the dark matter could be either the lightest neutral
mass eigenstate of the scalars $\eta$, $\phi$ or the vector-like fermion $N$,
if lighter than the scalars.

Note that this is a generalised, non-Abelian version of the original
scotogenic mechanism of Ma:2006km , where the stability of the dark matter
candidate is enforced by a $\mathbb{Z}_{2}$ symmetry. This mechanism was
extended to Abelian symmetries in Bonilla:2018ynb ; CentellesChulia:2019gic .
Moreover, in Lavoura:2012cv  the authors present a similar mechanism for some
discrete subgroups of $SU(2)$.

Figure 6: Stability diagrams of the dark matter sector in the model. The
lightest particle charged as $3(\bar{3})_{A,B,C}$ cannot decay into only
Standard Model fields. Left: allowed decay channels for a dark sector particle
will necessarily include at least one dark sector particle in the final state
(if kinematically allowed). Right: the lightest dark sector particle cannot
decay into Standard Model particles due to the flavour symmetry, thus ensuring
its stability.

While similar to the original scotogenic model, $\Sigma(81)$ makes a clear
distinction as already explained, even at the neutrino mass level: there are
three independent scotogenic-like diagrams. Each set of unconnected fields and
couplings were denoted with colours on Eqs. (II.2)-(9) for clarity. Regarding
the dark matter, this means that the model contains a three-component DM,
where the candidates are the lightest of each triad
$\\{\eta_{1},\phi_{3},N_{2}\\}$, $\\{\eta_{2},\phi_{1},N_{3}\\}$ and
$\\{\eta_{3},\phi_{2},N_{1}\\}$. Of course, after EWSB the neutral part of
each scalar doublet will split into its CP-odd and CP-even components due to
the usual $\lambda_{5}$ term in equation (8). For more details on the scalar
sector, we refer to Appendix B.

It is worth noting that while in most of the Scotogenic scenarios the scalar
DM is normally preferred Avila:2021mwg , due to the problems of
underproduction versus large cLFV for the fermion singlet candidate
Vicente:2014wga , in this case the issue is largely mitigated. The fact that
we have a three-component DM and no large contributions to cLFV, makes the
singlet fermion DM case again phenomenologically interesting. Nevertheless, a
complete study of the dark matter phenomenology is beyond the scope of this
paper.

##  V Conclusions

We have presented a simple but extremely predictive variant of the scotogenic
model. We promoted the scotogenic $\mathbb{Z}_{2}$ symmetry of the original
work to a non-Abelian $\Sigma(81)$ symmetry, which will satisfy the same role
of stabilizing the dark matter candidates running in the neutrino mass loop.
We considered that leptons, as well as the Higgs doublet $H$, transform as
triplets under the flavour symmetry, thus, resembling a 3HDM. These three
scalar doublets are responsible for all the spontaneous symmetry breaking,
which implies that the model does not need extra flavons in order to fit the
experimental data. We found that such a model can, not only satisfy the
current experimental constraints, but also lead to very strong and testable
predictions in the close future. Fitting the charged lepton masses,
$\theta_{13}$ and $\theta_{23}$ inside their $3\sigma$ allowed ranges, we
automatically obtained the following predictions:

  * •

Neutrino mass sum rule: $\frac{1}{2}\sum m_{i}=m_{\text{heaviest}}$.

  * •

Only Inverted Ordering is realised.

  * •

These two conditions together lead to $m_{\text{lightest}}\approx 7.5\times
10^{-4}$ eV, with some small deviations due to experimental uncertainty in
$\Delta m^{2}_{ij}$.

  * •

Strong correlation between $\theta_{12}$ and $\delta_{CP}$ as shown in figure
2, testable in the near future Smith:2022ezh ; NOvA:2016kwd .

  * •

Majorana phases predicted to be around $\phi_{12}\approx 0.45\pi$ and
$\phi_{13}\approx 0.12\pi$, when all the other observables are in their
experimental allowed ranges (see figure 3).

  * •

The prediction of the Majorana phases and $m_{\text{lightest}}$ lead to
$|m_{ee}|\approx 0.018\text{ eV}$, testable in future neutrinoless double beta
decay experiments nEXO:2021ujk  (see figure 4).

  * •

The flavour symmetry $\Sigma(81)$ ensures the stability of the dark matter
candidate, which could be either fermionic or scalar. No other symmetries are
required apart from the Standard Model gauge symmetries and the spontaneous
symmetry breaking comes solely from the three Higgs gauge doublets arranged
into a flavour triplet.

##  Appendix A $\Sigma(81)$ Group

The group $\Sigma(81)$ is a discrete, non-Abelian subgroup of $U(3)$
Jurciukonis:2017mjp  and belongs to the family of groups $\Sigma(3N^{3})$. It
has four generators denoted by $a$, $a^{\prime}$, $a^{\prime\prime}$, and $b$,
which fulfill the relations,

$a^{3}={a^{\prime}}^{3}={a^{\prime\prime}}^{3}=1,\qquad
aa^{\prime}=a^{\prime}a,\qquad aa^{\prime\prime}=a^{\prime\prime}a,\qquad
a^{\prime}a^{\prime\prime}=a^{\prime\prime}a^{\prime},$ (32) $b^{3}=1,\qquad
b^{2}ab=a^{\prime\prime},\qquad b^{2}a^{\prime\prime}b=a^{\prime},\qquad
b^{2}a^{\prime}b=a.$ (33)

All the elements of $\Sigma(81)$ can be written in terms of the four
generators as,

$\forall g\in\Sigma(81),\qquad
g=b^{k}a^{n}{a^{\prime}}^{n}{a^{\prime\prime}}^{l},\qquad\text{with }\quad
k,n,n,l=0,1,2.$ (34)

The representations of $\Sigma(81)$ used for the fields multiplets in this
model are $\mathbf{3}_{A}$, $\mathbf{\bar{3}}_{A}$, $\mathbf{3}_{D}$, and
$\mathbf{\bar{3}}_{D}$. We choose the following basis for these
representations: in the $\mathbf{\bar{3}}_{A}$,

$b=\begin{pmatrix}0&1&0\\\ 0&0&1\\\ 1&0&0\end{pmatrix},\quad
a=\begin{pmatrix}\omega&0&0\\\ 0&1&0\\\
0&0&1\end{pmatrix},\quad{a}^{\prime}=\begin{pmatrix}1&0&0\\\ 0&1&0\\\
0&0&\omega\end{pmatrix},\quad{a}^{\prime\prime}=\begin{pmatrix}1&0&0\\\
0&\omega&0\\\ 0&0&1\end{pmatrix},$ (35)

where $\omega=e^{i\nicefrac{{2\pi}}{{3}}}$. In the $\mathbf{\bar{3}}_{D}$
representation,

$b=\begin{pmatrix}0&1&0\\\ 0&0&1\\\ 1&0&0\end{pmatrix},\quad
a=\begin{pmatrix}\omega^{2}&0&0\\\ 0&1&0\\\
0&0&\omega\end{pmatrix},\quad{a}^{\prime}=\begin{pmatrix}1&0&0\\\
0&\omega&0\\\
0&0&\omega^{2}\end{pmatrix},\quad{a}^{\prime\prime}=\begin{pmatrix}\omega&0&0\\\
0&\omega^{2}&0\\\ 0&0&1\end{pmatrix}.$ (36)

Notice that the generators in the $\mathbf{3}_{A}$ representation are the
complex conjugate of the generators in the $\mathbf{\bar{3}}_{A}$, and
similarly between the $\mathbf{3}_{D}$, and $\mathbf{\bar{3}}_{D}$.

It’s worth showing explicitly one of the key properties of $\Sigma(81)$ that
give rise to dark matter stability in the model presented, i.e. the singlet
irreps together with $\bm{3}_{D}$ and $\bm{\bar{3}}_{D}$ form a close
subgroup. This can be seen by looking at the products (38)-(42).

$\mathbf{1}_{(k,l)}\times\mathbf{3}_{D}(\mathbf{\bar{3}}_{D})=\mathbf{3}_{D}(\mathbf{\bar{3}}_{D})\,,\quad\mathbf{3}_{D}\times\mathbf{3}_{D}=\mathbf{\bar{3}}_{D}+\mathbf{\bar{3}}_{D}+\mathbf{\bar{3}}_{D}\,,\quad\mathbf{3}_{D}\times\mathbf{\bar{3}}_{D}=\mathbf{1}_{(k,l)}\,,$
(37)

with $k,l=0,1,2$.

Expanding in components in the basis defined by (35) and (36), we have the
tensor products,

$\displaystyle\begin{pmatrix}x_{1}\\\ x_{2}\\\
x_{3}\end{pmatrix}_{\bm{3}_{D}}\otimes\begin{pmatrix}y_{1}\\\ y_{2}\\\
y_{3}\end{pmatrix}_{\bm{\bar{3}}_{D}}=\sum_{k=0,1,2}[(x_{1}y_{1}+\omega^{2k}x_{2}y_{2}+\omega^{k}x_{3}y_{3})_{\mathbf{1}_{(k,0)}}$
$\displaystyle\oplus(x_{2}y_{3}+\omega^{2k}x_{3}y_{1}+\omega^{k}x_{1}y_{2})_{\mathbf{1}_{(k,2)}}$
$\displaystyle\oplus(x_{3}y_{2}+\omega^{2k}x_{1}y_{3}+\omega^{k}x_{2}y_{1})_{\mathbf{1}_{(k,1)}}].$
(38) $\begin{pmatrix}x_{1}\\\ x_{2}\\\
x_{3}\end{pmatrix}_{\bm{3}_{D}}\otimes\begin{pmatrix}y_{1}\\\ y_{2}\\\
y_{3}\end{pmatrix}_{\bm{3}_{D}}=\begin{pmatrix}x_{1}y_{1}\\\ x_{2}y_{2}\\\
x_{3}y_{3}\end{pmatrix}_{\bm{\bar{3}}_{D}}\oplus\begin{pmatrix}x_{2}y_{3}\\\
x_{3}y_{1}\\\
x_{1}y_{2}\end{pmatrix}_{\bm{\bar{3}}_{D}}\oplus\begin{pmatrix}x_{3}y_{2}\\\
x_{1}y_{3}\\\ x_{2}y_{1}\end{pmatrix}_{\bm{\bar{3}}_{D}},$ (39)
$(x)_{\mathbf{1}_{(k,0)}}\otimes\begin{pmatrix}y_{1}\\\ y_{2}\\\
y_{3}\end{pmatrix}_{\bm{3}(\bm{\bar{3}})_{D}}=\begin{pmatrix}xy_{1}\\\
\omega^{k}xy_{2}\\\ \omega^{2k}xy_{3}\end{pmatrix}_{\bm{3}(\bm{\bar{3}})_{D}}$
(40) $(x)_{\mathbf{1}_{(k,1)}}\otimes\begin{pmatrix}y_{1}\\\ y_{2}\\\
y_{3}\end{pmatrix}_{\bm{3}(\bm{\bar{3}})_{D}}=\begin{pmatrix}xy_{3}\\\
\omega^{k}xy_{1}\\\ \omega^{2k}xy_{2}\end{pmatrix}_{\bm{3}(\bm{\bar{3}})_{D}}$
(41) $(x)_{\mathbf{1}_{(k,2)}}\otimes\begin{pmatrix}y_{1}\\\ y_{2}\\\
y_{3}\end{pmatrix}_{\bm{3}(\bm{\bar{3}})_{D}}=\begin{pmatrix}xy_{2}\\\
\omega^{k}xy_{3}\\\ \omega^{2k}xy_{1}\end{pmatrix}_{\bm{3}(\bm{\bar{3}})_{D}}$
(42)

The label $\mathbf{1}_{(k,l)}$, with $k,l=0,1,2$, represent the nine different
one dimensional irreps of $\Sigma(81)$, being $\mathbf{1}_{(0,0)}$ the
invariant singlet.

For further details of the properties of the $\Sigma(81)$ group and the
explicit expressions of the tensor products of dark sector fields we refer the
reader to the second edition of the book “An Introduction to Non-Abelian
Discrete Symmetries for Particle Physicists” Kobayashi:2022moq , since the
first edition had inconsistencies in the representations used in the tensor
products.

##  Appendix B Scalar sector

We now explicitly write down the relevant Lagrangian terms that lead to the
masses of the scalars $\eta$ and $\phi$. These are given by equations (8) and

$\displaystyle\mathcal{V}_{\nu}\;$ $\displaystyle\supset$ $\displaystyle
m_{\eta}^{2}\,(\eta^{\dagger}\eta)+m_{\phi}^{2}\,(\phi^{\dagger}\phi)$
$\displaystyle+$ $\displaystyle\sum_{i=1}^{3}\lambda_{H\eta
i}\left[(H^{\dagger}H)(\eta^{\dagger}\eta)\right]_{i}+\lambda_{H\phi
i}\left[(H^{\dagger}H)(\phi^{\dagger}\phi)\right]_{i}$ $\displaystyle+$
$\displaystyle\sum_{i=4}^{6}\lambda_{H\eta
i}\left[(H^{\dagger}\eta)(\eta^{\dagger}H)\right]_{i}+\lambda_{H\phi
i}\left[(H^{\dagger}\phi)(\phi^{\dagger}H)\right]_{i}\,.$

where the sub-index outside a bracket $\left[X\right]_{i}$ represent the
different contractions under $\Sigma(81)$ to the trivial singlet. Explicitly,

$\displaystyle m_{X}^{2}(X^{\dagger}X)$ $\displaystyle=$ $\displaystyle
m_{X}^{2}(X_{1}^{\dagger}X_{1}+X_{2}^{\dagger}X_{2}+X_{3}^{\dagger}X_{3})\,,$
(44) $\displaystyle\lambda_{HX1}\left[(H^{\dagger}H)(X^{\dagger}X)\right]_{1}$
$\displaystyle=$
$\displaystyle\lambda_{HX1}\left((H_{1}^{\dagger}H_{1})(X_{1}^{\dagger}X_{1})+(H_{2}^{\dagger}H_{2})(X_{2}^{\dagger}X_{2})+(H_{3}^{\dagger}H_{3})(X_{3}^{\dagger}X_{3})\right)\,,$
(45) $\displaystyle\lambda_{HX2}\left[(H^{\dagger}H)(X^{\dagger}X)\right]_{2}$
$\displaystyle=$
$\displaystyle\lambda_{HX2}\left((H_{1}^{\dagger}H_{1})(X_{2}^{\dagger}X_{2})+(H_{2}^{\dagger}H_{2})(X_{3}^{\dagger}X_{3})+(H_{3}^{\dagger}H_{3})(X_{1}^{\dagger}X_{1})\right)\,,$
(46) $\displaystyle\lambda_{HX3}\left[(H^{\dagger}H)(X^{\dagger}X)\right]_{3}$
$\displaystyle=$
$\displaystyle\lambda_{HX3}\left((H_{1}^{\dagger}H_{1})(X_{3}^{\dagger}X_{3})+(H_{2}^{\dagger}H_{2})(X_{1}^{\dagger}X_{1})+(H_{3}^{\dagger}H_{3})(X_{2}^{\dagger}X_{2})\right)\,,$
(47) $\displaystyle\lambda_{HX4}\left[(H^{\dagger}X)(X^{\dagger}H)\right]_{4}$
$\displaystyle=$
$\displaystyle\lambda_{HX4}\left((H_{1}^{\dagger}X_{1})(X_{1}^{\dagger}H_{1})+(H_{2}^{\dagger}X_{2})(X_{2}^{\dagger}H_{2})+(H_{3}^{\dagger}X_{3})(X_{3}^{\dagger}H_{3})\right)\,,$
(48) $\displaystyle\lambda_{HX5}\left[(H^{\dagger}X)(X^{\dagger}H)\right]_{5}$
$\displaystyle=$
$\displaystyle\lambda_{HX5}\left((H_{1}^{\dagger}X_{2})(X_{2}^{\dagger}H_{1})+(H_{2}^{\dagger}X_{3})(X_{3}^{\dagger}H_{2})+(H_{3}^{\dagger}X_{1})(X_{1}^{\dagger}H_{3})\right)\,,$
(49) $\displaystyle\lambda_{HX6}\left[(H^{\dagger}X)(X^{\dagger}H)\right]_{6}$
$\displaystyle=$
$\displaystyle\lambda_{HX6}\left((H_{1}^{\dagger}X_{3})(X_{3}^{\dagger}H_{1})+(H_{2}^{\dagger}X_{1})(X_{1}^{\dagger}H_{2})+(H_{3}^{\dagger}X_{2})(X_{2}^{\dagger}H_{3})\right)\,,$
(50)

with $X\in\\{\eta,\phi\\}$.

As usual, the scalars $\eta_{i}$ and $\phi_{i}$ can be written down into their
$SU(2)_{L}$ components as,

$\displaystyle\eta_{i}$ $\displaystyle=$
$\displaystyle\left(\eta_{i}^{+},\,\,\frac{1}{\sqrt{2}}(\eta_{Ri}+i\,\eta_{Ii})\right)^{T}\
,$ (51) $\displaystyle\phi_{i}$ $\displaystyle=$
$\displaystyle\left(\phi_{i}^{+},\,\,\frac{1}{\sqrt{2}}(\phi_{Ri}+i\,\phi_{Ii})\right)^{T}\,.$
(52)

After EWSB, the charged components of each $\eta_{i}$ and $\phi_{i}$ remain as
mass eigenstates, with masses

$\displaystyle m_{\eta_{i}^{+}}$ $\displaystyle\approx$ $\displaystyle
m_{\eta}^{2}+v_{1}^{2}\lambda_{H\eta i},$ (53) $\displaystyle
m_{\phi_{i}^{+}}$ $\displaystyle\approx$ $\displaystyle
m_{\phi}^{2}+v_{1}^{2}\lambda_{H\phi i},$ (54)

in the limit $v_{1}\gg v_{2},v_{3}$ (see (3)).

On the other hand, the neutral components mix through the $\lambda_{5}^{(k)}$
terms given in equation (8). Assuming a CP conserving scalar potential, which
implies $\lambda_{5}^{(1)},\lambda_{5}^{(2)}\in\text{Reals}$, the mixing only
happens among CP-odd and CP-even scalars and in pairs
$\\{\eta_{1},\phi_{3}\\}$, $\\{\eta_{2},\phi_{1}\\}$ and
$\\{\eta_{3},\phi_{2}\\}$. Respectively, the mass matrices for each pair is
given by

$\displaystyle
m_{R,I1}\approx\left(\begin{matrix}m_{\eta}^{2}+v_{1}^{2}(\lambda_{H\eta
1}+\lambda_{H\eta 4})&&\pm v_{1}v_{2}\lambda_{5}^{(2)}\\\ \pm
v_{1}v_{2}\lambda_{5}^{(2)}&&m_{\phi}^{2}+v_{1}^{2}(\lambda_{H\phi
3}+\lambda_{H\phi 6})\end{matrix}\right)\,,$ (55) $\displaystyle
m_{R,I2}\approx\left(\begin{matrix}m_{\eta}^{2}+v_{1}^{2}(\lambda_{H\eta
2}+\lambda_{H\eta 5})&&\pm v_{1}^{2}\lambda_{5}^{(1)}\\\ \pm
v_{1}^{2}\lambda_{5}^{(1)}&&m_{\phi}^{2}+v_{1}^{2}(\lambda_{H\phi
1}+\lambda_{H\phi 4})\end{matrix}\right)\,,$ (56) $\displaystyle
m_{R,I3}\approx\left(\begin{matrix}m_{\eta}^{2}+v_{1}^{2}(\lambda_{H\eta
3}+\lambda_{H\eta 6})&&\pm v_{1}v_{3}\lambda_{5}^{(2)}\\\ \pm
v_{1}v_{3}\lambda_{5}^{(2)}&&m_{\phi}^{2}+v_{1}^{2}(\lambda_{H\phi
2}+\lambda_{H\phi 5})\end{matrix}\right)\,.$ (57)

By choosing appropriate signs for the $\lambda$ couplings one can make sure
that a neutral component is the lightest field of each set.

##  Appendix C Acknowledgements

The authors want to thank Andreas Trautner for double covering us with wisdom
and Rahul Srivastava for helpful comments. We also thank professor Luís
Lavoura and professor Martin K. Hirsch for helpful insight. O.M. is supported
by the Spanish grant PID2020-113775GB-I00 (AEI/10.13039/501100011033) and
Programa Santiago Grisolía (No. GRISOLIA/2020/025). R.C. is supported by the
Alexander von Humboldt Foundation Fellowship.

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