#  Revamping Kaluza-Klein dark matter in an orbifold theory of flavor

Francisco J. de Anda<EMAIL_ADDRESS>Tepatitlán’s Institute for Theoretical
Studies, C.P. 47600, Jalisco, México  Dual CP Institute of High Energy
Physics, C.P. 28045, Colima, México  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  José W. F.
Valle<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  Carlos A.
Vaquera-Araujo<EMAIL_ADDRESS>Consejo Nacional de Ciencia y
Tecnología, Av. Insurgentes Sur 1582. Colonia Crédito Constructor, Del. Benito
Juárez, C.P. 03940, Ciudad de México, México  Departamento de Física, DCI,
Campus León, Universidad de Guanajuato, Loma del Bosque 103, Lomas del
Campestre C.P. 37150, León, Guanajuato, México  Dual CP Institute of High
Energy Physics, C.P. 28045, Colima, México

###### Abstract

We suggest a common origin for dark matter, neutrino mass and family symmetry
within the orbifold theory proposed in [1, 2]. Flavor physics is described by
an $A_{4}$ family symmetry that results naturally from compactification. WIMP
Dark matter emerges from the first Kaluza-Klein excitation of the same scalar
that drives family symmetry breaking and neutrino masses through the inverse
seesaw mechanism. In addition to the “golden” quark-lepton mass relation and
neutrino predictions for $\rm 0\nu\beta\beta$ decay, the model provides a good
global description of all flavor observables.

##  I Introduction

Understanding flavor constitutes one of the major challenges in particle
physics. Accounting for the observed pattern of fermion masses and the
structure of their mixing parameters continues to defy the particle physics
community, even more so after the discovery of neutrino oscillations [3, 4,
5]. The latter has demonstrated that leptons mix rather differently from the
way quarks do in the Cabibbo–Kobayashi–Maskawa (CKM) model. The Standard Model
lacks a symmetry-based organizing principle that one may use to describe
flavor properties. The striking oddness of the fermion pattern is unlikely to
result just from randomness. Rather, it indicates the presence of some
“family” or “flavor” symmetry, a compilation of possibilities for discrete
non-Abelian groups as family symmetry candidates is given in [6].

The existence of extra space-time dimensions may shed light on the two
complementary aspects of the flavor problem. Indeed, while mass hierarchies
may result from geometry [7], mixing angle relations may be predicted from
adequate symmetries [8, 9], thereby covering both sides of the coin.

Here we focus on six-dimensional orbifold theories proposed in [1, 2]. In
contrast to the warped flavordynamics picture suggested in [8, 9], where
family symmetries were imposed by hand, here an $A_{4}$ family symmetry
results naturally from compactification, a setup that was first introduced and
described in detail in references [10, 11, 12]. Dark matter emerges as the
first Kaluza-Klein (KK) excitation of the same scalar that drives lepton
number violation, and family symmetry breakdown, leading to neutrino masses
through an inverse seesaw mechanism. This new seed scalar field allows us to
resurrect the KK dark matter proposal. The model leads naturally to the
“golden” quark-lepton mass relation, that implies strong predictions for the
light-quark masses $m_{d}$ and $m_{s}$ and neutrinoless double beta decay. It
also provides an adequate description of neutrino oscillation parameters,
together with a very good global description of flavor observables.

This letter is structured as follows. In Sec. II we recapitulate the theory
framework of our model, while in Sec. III we describe the general features of
Kaluza-Klein dark matter, followed in Sec. IV by an analysis of the
phenomenology of $\sigma^{KK}$ as a WIMP dark matter candidate. In Sec. V we
present the “golden quark-lepton” mass formula, results for the neutrino
sector, and also discuss our global fit to flavor observables. In Sec. VI we
present a brief summary and additional discussion of our model.

##  II Theory Framework

Our model is a 6-dimensional version of the Standard Model implementing the
inverse seesaw mechanism [13, 14]. It provides is a simple extension of the
model presented in [1], where the orbifold compactification implies a discrete
$A_{4}$ family symmetry in four dimensions [10, 11, 12]. We stress that the
4-dimensional flavor symmetry is not arbitrary, but rather dictated by the
extra-dimensional symmetries of our construction. We assume a sequential setup
where 3 singlet fermions $S$ accompany the 3 “right-handed” neutrinos. The
transformation properties of the fields under the gauge and family symmetry
and their localization on the orbifold are shown in Table 1.

Field |  | $SU(3)_{C}$ | $SU(2)_{L}$ | $U(1)_{Y}$ |  | $A_{4}$ |  | $\mathbb{Z}_{2}$ | $\mathbb{Z}_{3}$ | Localization
---|---|---|---|---|---|---|---|---|---|---
$L$ |  | $\mathbf{1}$ | $\mathbf{2}$ | $-1/2$ |  | $\mathbf{3}$ |  | $1$ | $1$ | Brane
$d^{c}$ |  | $\bar{\mathbf{3}}$ | $\mathbf{1}$ | $1/3$ |  | $\mathbf{3}$ |  | $1$ | $1$ | Brane
$e^{c}$ |  | $\mathbf{1}$ | $\mathbf{1}$ | $1$ |  | $\mathbf{3}$ |  | $1$ | $1$ | Brane
$Q$ |  | $\mathbf{3}$ | $\mathbf{2}$ | $1/6$ |  | $\mathbf{3}$ |  | $1$ | $1$ | Brane
$u_{1,2,3}^{c}$ |  | $\bar{\mathbf{3}}$ | $\mathbf{1}$ | $-2/3$ |  | $\mathbf{1^{\prime\prime}},\mathbf{1^{\prime}},\mathbf{1}$ |  | $-1$ | $1$ | Bulk
$\nu^{c}$ |  | $\mathbf{1}$ | $\mathbf{1}$ | $0$ |  | $\mathbf{3}$ |  | $-1$ | $\omega$ | Brane
$S$ |  | $\mathbf{1}$ | $\mathbf{1}$ | $0$ |  | $\mathbf{3}$ |  | $-1$ | $\omega^{2}$ | Brane
$H_{u}$ |  | $\mathbf{1}$ | $\mathbf{2}$ | $1/2$ |  | $\mathbf{3}$ |  | $-1$ | $1$ | Brane
$H_{d}$ |  | $\mathbf{1}$ | $\mathbf{2}$ | $-1/2$ |  | $\mathbf{3}$ |  | $1$ | $1$ | Brane
$H_{\nu}$ |  | $\mathbf{1}$ | $\mathbf{2}$ | $1/2$ |  | $\mathbf{3}$ |  | $-1$ | $\omega^{2}$ | Brane
$\sigma$ |  | $\mathbf{1}$ | $\mathbf{1}$ | $0$ |  | $\mathbf{3}$ |  | $1$ | $\omega^{2}$ | Bulk
Table 1: Field content of the model.

The scalar sector consists of 3 Higgs doublets $H_{u}$, $H_{d}$ and $H_{\nu}$
plus an extra singlet scalar $\sigma$, all transforming as flavor triplets.
Given the auxiliary discrete symmetries $\mathbb{Z}_{2}$ and $\mathbb{Z}_{3}$
in Table 1, $H_{d}$ only couples to down type fermions (charged leptons and
down quarks), while $H_{u}$ couples only to up quarks and $H_{\nu}$ only
couples to neutrinos. The effective Yukawa terms are given by

$\begin{split}\mathcal{L}_{Y}&=y_{1}^{\nu}(LH_{\nu}\nu^{c})_{1}+y_{2}^{\nu}(LH_{\nu}\nu^{c})_{2}+\frac{y^{S}}{2}\sigma
SS+m_{\nu^{c}S}\nu^{c}S\\\
&\quad+y_{1}^{d}(Qd^{c}H_{d})_{1}+y_{2}^{d}(Qd^{c}H_{d})_{2}+y_{1}^{e}(Le^{c}H_{d})_{1}+y_{2}^{e}(Le^{c}H_{d})_{2}\\\
&\quad+y_{1}^{u}(QH_{u})_{1^{\prime}}u_{1}^{c}+y_{2}^{u}(QH_{u})_{1^{\prime\prime}}u_{2}^{c}+y_{3}^{u}(QH_{u})_{1}u_{3}^{c},\end{split}$
(1)

where $()_{1,2}$ and $()_{1,1^{\prime},1^{\prime\prime}}$ indicate possible
singlet contractions
$\mathbf{3}\times\mathbf{3}\times\mathbf{3}\to\mathbf{1}_{1,2}$ and
$\mathbf{3}\times\mathbf{3}\to\mathbf{1}_{1,1^{\prime},1^{\prime\prime}}$ in
$A_{4}$. Here we will also assume all dimensionless Yukawa couplings to be
real.

Moreover, we adopt a dynamical scenario where lepton number and family
symmetries are violated spontaneously, through the expectation value of an
extra scalar field $\sigma$, whose vacuum expectation value (VEV) yields
Majorana masses to the singlet fermion $S$. The VEV of $\sigma$ must lie in
the zero mode of the extra dimensional field decomposition [15], and must
comply with the $\mathbb{Z}_{2}$ boundary condition

$P\braket{\sigma}=\braket{\sigma}.$ (2)

Here $P$ defines an arbitrary gauge twist of the orbifold, associated to its
geometry, and is chosen to be

$P=\frac{1}{3}\left(\begin{array}[]{ccc}-1&2\omega^{2}&2\omega\\\
2\omega&-1&2\omega^{2}\\\ 2\omega^{2}&2\omega&-1\end{array}\right),$ (3)

with $\omega=e^{2\pi i/3}$, the cube root of unity. Notice that it satisfies
$P^{2}=1$. This boundary condition aligns the VEV as

$\braket{\sigma}=v_{\sigma}\left(\begin{array}[]{c}1\\\ \omega\\\
\omega^{2}\end{array}\right),$ (4)

The Higgs doublets are assumed to obtain the most general vacuum alignment,
which we parametrize as in [1], these alignments require in general $A_{4}$
soft breaking terms in the scalar potential [16], nonetheless they are a
possibility for obtaining viable neutrino mixing results in models with three
Higgs-doublets as an $A_{4}$ triplet [17],

$\begin{split}\braket{H_{u}}=v_{u}\left(\begin{array}[]{c}\epsilon_{1}^{u}e^{i\phi_{1}^{u}}\\\
\epsilon_{2}^{u}e^{i\phi_{2}^{u}}\\\ 1\end{array}\right),\ \ \ \
\braket{H_{\nu}}=v_{\nu}e^{i\phi^{\nu}}\left(\begin{array}[]{c}\epsilon_{1}^{\nu}e^{i\phi_{1}^{\nu}}\\\
\epsilon_{2}^{\nu}e^{i\phi_{2}^{\nu}}\\\ 1\end{array}\right),\ \ \
\braket{H_{d}}=v_{d}e^{i\phi^{d}}\left(\begin{array}[]{c}\epsilon_{1}^{d}e^{i\phi_{1}^{d}}\\\
\epsilon_{2}^{d}e^{i\phi_{2}^{d}}\\\ 1\end{array}\right).\end{split}$ (5)

The explicit form of the mass matrices for the quarks and charged leptons (up
to unphysical rephasings) is given as

$\displaystyle
M_{u}=v_{u}\left(\begin{array}[]{ccc}y_{1}^{u}\epsilon_{1}^{u}&y_{2}^{u}\epsilon_{1}^{u}&y_{3}^{u}\epsilon_{1}^{u}\\\
y_{1}^{u}\epsilon_{2}^{u}\omega^{2}&y_{2}^{u}\epsilon_{2}^{u}\omega&y_{3}^{u}\epsilon^{u}_{2}\\\
y_{1}^{u}\omega&y_{2}^{u}\omega^{2}&y_{3}^{u}\end{array}\right),\ \ \
M_{d}=v_{d}\left(\begin{array}[]{ccc}0&y_{1}^{d}\epsilon_{1}^{d}e^{i(\phi_{1}^{d}-\phi_{2}^{d})}&y_{2}^{d}\epsilon_{2}^{d}\\\
y_{2}^{d}\epsilon_{1}^{d}e^{i(\phi_{1}^{d}-\phi_{2}^{d})}&0&y_{1}^{d}\\\
y_{1}^{d}\epsilon_{2}^{d}&y_{2}^{d}&0\end{array}\right),$ (12) $\displaystyle
M_{e}=v_{d}\left(\begin{array}[]{ccc}0&y_{1}^{e}\epsilon_{1}^{d}e^{-i(\phi_{1}^{d}-\phi_{2}^{d})}&y_{2}^{e}\epsilon_{2}^{d}\\\
y_{2}^{e}\epsilon_{1}^{d}e^{-i(\phi^{1}_{d}-\phi_{2}^{d})}&0&y_{1}^{e}\\\
y_{1}^{e}\epsilon_{2}^{d}&y_{2}^{e}&0\end{array}\right),$ (16)

Turning now to the neutral fermion mass matrix, in the $(\nu,\nu^{c},S)$
basis, it is given as

$M_{\nu}=\left(\begin{array}[]{ccc}0&M_{D}&0\\\ M^{T}_{D}&0&M_{\nu^{c}S}\\\
0&M^{T}_{\nu^{c}S}&M_{S}\end{array}\right),$ (17)

with

$\displaystyle
M_{D}=v_{\nu}\left(\begin{array}[]{ccc}0&y_{1}^{\nu}\epsilon_{1}^{\nu}e^{i(\phi^{\nu}_{1}-\phi_{2}^{\nu})}&y_{2}^{\nu}\epsilon_{2}^{\nu}\\\
y_{2}^{\nu}\epsilon_{1}^{\nu}e^{i(\phi_{1}^{\nu}-\phi_{2}^{\nu})}&0&y_{1}^{\nu}\\\
y_{1}^{\nu}\epsilon_{2}^{\nu}&y_{2}^{\nu}&0\end{array}\right),$ (21)
$\displaystyle M_{\nu^{c}S}=m_{\nu^{c}S}~{}\left(\begin{array}[]{ccc}1&0&0\\\
0&1&0\\\ 0&0&1\end{array}\right),\ \ \
M_{S}=y^{S}v_{\sigma}\left(\begin{array}[]{ccc}0&\omega^{2}&\omega\\\
\omega^{2}&0&1\\\ \omega&1&0\end{array}\right).$ (28)

After spontaneous symmetry breaking, the light neutrinos acquire masses
through the inverse seesaw mechanism, characterized by the effective mass
matrix [13, 14]

$m_{\nu}\sim
M_{D}M^{-1}_{\nu^{c}S}M_{S}M^{T-1}_{\nu^{c}S}M_{D}^{T}=\frac{M_{D}M_{S}M_{D}^{T}}{m_{\nu^{c}S}^{2}}.$
(29)

Notice that, due to the family symmetry, the singlet mass entry $\nu^{c}S$ is
trivial and the others, $M_{D}$ and $M_{S}$, have zeros along the diagonal. We
will assume the following hierarchy of scales

$M_{D}\sim\braket{H_{\nu}}\sim\mathcal{O}(\mathrm{GeV}),\ \ \
M_{S}\sim\braket{\sigma}\raise
1.29167pt\hbox{$\;<$\kern-7.5pt\raise-4.73611pt\hbox{$\sim\;$}}\mathcal{O}(\mathrm{GeV}),\
\ m_{\nu^{c}S}\sim\mathcal{O}(10\,\mathrm{TeV}).$ (30)

This choice renders naturally light neutrino masses.

##  III Revamping Kaluza-Klein Dark matter

Besides the gauge fields, our model contains only two fields that propagate
into the bulk, namely $u^{c}_{i}$ and $\sigma$, where the latter is an
electrically neutral scalar transforming as a singlet under the symmetries of
the $\mathrm{SU(3)_{c}\otimes SU(2)_{L}\otimes U(1)_{Y}}$ gauge group. In this
section we examine the possibility of identifying the lightest Kaluza-Klein
mode of $\sigma$ as a WIMP dark matter candidate.

The extra dimensions are orbifolded by a $\mathbb{Z}_{2}$. Every field has a
different $\mathbb{Z}_{2}$ charge, depending on its $A_{4}$ and Lorentz
transformation. In principle, one could choose the charge for each field
individually, which implies that some couplings are also charged under the
$\mathbb{Z}_{2}$ orbifolding, usually called kink couplings. We instead assume
that each coupling constant in the model is trivial under orbifolding. This
leaves the $\mathbb{Z}_{2}$ orbifolding symmetry as a symmetry of the
compactified Lagrangian. This surviving $\mathbb{Z}_{2}$ (not to be confused
with the discrete auxiliary symmetry in Table 1) protects the lightest KK
mode, making it stable and a potential dark matter candidate.

In order to determine the mass scale of the lightest KK mode, we start by
analyzing the spectrum of the effective 4-dimensional theory that emerges
after the $T^{2}/\mathbb{Z}_{2}$ orbifold compactification of the extra two
dimensions present in our construction. Denoting the extra dimensions by a
single complex coordinate $z$, we can decompose any field in the bulk $\Psi$
(fermion, scalar or vector) into a tower of low-energy 4-dimensional effective
modes $\Psi_{ni}(x)$ as

$\Psi(x,z)=\sum_{n=0}^{\infty}\sum_{i=\pm}\Psi_{ni}(x)f_{ni}(z),$ (31)

where the profiles $f_{ni}(z)$ are eigenfunctions of the extra dimensional
translation and the $\mathbb{Z}_{2}$ orbifold parity, these satisfy by
definition the following conditions

$f_{n\pm}(z)=f_{n\pm}(z+1)=f_{n\pm}(z+\omega)=\pm f_{n\pm}(-z).$ (32)

Note that the profiles form an orthonormal basis

$\int dzd\bar{z}f^{*}_{ni}(z)f_{mj}(z)=\delta^{m}_{n}\delta^{i}_{j}.$ (33)

As a result of the periodicity conditions in Eq. (32) and orthonormality from
Eq. (33), the profiles must be translation eigenfunctions (which for flat
extra dimensions are exponentials) with

$\partial_{z}f_{ni}=M_{ni}f_{ni},$ (34)

with $M_{ni}$ proportional to $M_{ni}\sim n/R$ where $R$ is the
compactification scale. Hence, for example, the kinetic term for a
6-dimensional scalar field becomes a collection of mass terms for their
corresponding 4-dimensional effective degrees of freedom

$\int
dzd\bar{z}(\partial_{z}\Psi)^{\dagger}\partial_{z}\Psi=\sum_{n,m,i,j}M_{ni}M_{mj}\Psi_{ni}^{\dagger}\Psi_{mj}\int
dzd\bar{z}f_{ni}^{*}f_{mj}=\sum_{n,i}M_{ni}^{2}\Psi_{ni}^{\dagger}\Psi_{ni},$
(35)

with an analogous relation for fermions. By choosing an orthonormal basis, the
KK modes are already in the mass basis. Note that the zero-mode has an
eigenvalue $M_{0i}=0$, so that its profile $f_{0i}$ is a constant. The zero-
modes remain massless after compactification, and can be identified at low
energies with the SM fields.

Concerning the heavy KK modes, there is an important result that comes from
the preservation of the orbifolding $\mathbb{Z}_{2}$ symmetry. If one tries to
build a term involving a single heavy KK mode (with $n\geq 1$) and any number
$N$ of massless zero modes $\Psi_{0}^{(k)}$, $k=1,\dots,N$, one finds that
such term automatically vanishes, as the effective 4-dimensional coupling
comes from integrating the extra dimensional profiles

$\begin{split}\int
dzd\bar{z}\Psi_{0}^{(1)}\cdot...\cdot\Psi_{0}^{(N)}\Psi_{nj}f_{0}^{N}f_{nj}&=\Psi_{0}^{(1)}\cdot...\cdot\Psi_{0}^{(N)}\Psi_{nj}f_{0}^{N}\
\ \int dzd\bar{z}f_{0}f_{nj}\\\
&=\Psi_{0}^{(1)}\cdot...\cdot\Psi_{0}^{(N)}\Psi_{nj}f_{0}^{N-1}\delta^{n}_{0}=0.\end{split}$
(36)

The first line follows from the fact that the $\Psi$ functions do not depend
on the extra dimensions, and the zero mode profiles are just a constant. The
second line results from the orthonormality of the profiles. Such terms would
in general induce the decay of KK modes into $N$ massless modes. However, in
our construction these terms clearly do not exist. There might be
destabilizing interactions between KK modes. However, only the “right-handed”
up-type quarks $u^{c}$ and the $\sigma$ scalar have KK modes, and these do not
interact with each other. Therefore there are no destabilizing interactions
and the lightest KK mode is stable.

In the present model there are three types of fields in the bulk, the gauge
fields, the “right-handed” up-type quarks $u^{c}$ and the $\sigma$ scalar
driving lepton number and family symmetry breaking. In the following we will
assume that the lightest KK mode is electrically neutral. That leaves us with
the first KK modes of the neutral SM gauge bosons: the photon, gluon or the
$Z$ boson [18], as well as first KK mode of the $\sigma$ scalar. The latter is
charged only under $A_{4}$ and drives family symmetry breaking and neutrino
mass generation. The lightest KK excitation of this field is a potential dark
matter candidate. After orbifold compactification and spontaneous electroweak
symmetry breaking the first neutral KK modes will acquire masses of the order

$M_{(\gamma,g)^{KK}}^{2}\sim\left(\frac{1}{2R}\right)^{2},\ \ \
M_{Z^{KK}}^{2}\sim\left(\frac{1}{2R}\right)^{2}+m_{Z}^{2},\ \ \
M_{\sigma^{KK}}^{2}\sim\left(\frac{1}{2R}\right)^{2}+m_{\sigma}^{2}.$ (37)

In the simplest extra dimensional extension of the SM it is natural to assume
that the lightest KK particle (LKP) is the first mode of the photon or the
gluon, as their zero mode always remains massless.

Note that Eq.(37) only gives the order of magnitude of the first excitations,
any mass hierarchy can be assumed for the neutral KK light modes. If one
assumes that the LKP is a dark matter candidate and identifies it with the
first excitation of the photon, the gluon or the $Z$ gauge boson, it is in
general difficult to successfully reproduce the observed dark matter relic
abundance, since the all gauge couplings are completely fixed, their masses
being the only free parameter, determined by the compactifcation scale. In
particular, if the first KK mode of the photon is identified with the LKP, in
order to correctly reproduce the required relic abundance a mass of order
$M_{(\gamma,Z)^{KK}}\sim 1\,\mathrm{TeV}$ is required [19, 20]. Similarly, if
the LKP is the first KK gluon then its mass must be of order $M_{g^{KK}}\sim
5\,\mathrm{TeV}$ [21]. Note that both scenarios predict compactification
scales that lie in tension the LEP bound in Eq. (38). Furthermore, the case of
vector dark matter is currently disfavored experimentally [22].

Previous proposals [18, 21, 19, 20] with the LKP as a dark matter candidate,
assumed the LKP to be associated to a Standard Model field, a possibility no
longer viable experimentally. In our present setup, the LKP comes from a
genuinely new field, beyond those of the Standard Model. This field drives
lepton number violation plays a key role in explaining the flavor structure of
the fermions [1, 2]. This choice is not only well-motivated but also relaxes
the constraints, allowing us to build a viable and predictive model. Indeed,
we will show that if the first KK mode of $\sigma$ is identified with the LKP
it can play the role of a WIMP dark matter candidate.

Before closing this section, we comment on the key role played by the
compactification scale on the Kaluza-Klein WIMP dark matter phenomenology. It
has long been noted that the existence of a KK tower generates flavor Changing
Neutral Currents (FCNCs). These arise when the extra-dimensional fields in the
bulk are allowed to have an explicit mass term, which prevents the
simultaneous flavor diagonalization of their higher KK modes with the zeroth
level Lagrangian [23, 24]. We note, however, that our model is built in six
flat dimensions, and the bulk fermions are 6-D chiral fields, with no explicit
mass terms. As a result, it is free from FCNCs of this kind by construction.
However, an important effect arises from the additional contributions to the
Electroweak Precision observables. The Peskin-Takeuchi $S$, $T$ and $U$
parameters are modified by the existence of a tower of massive vector
$\mathrm{SU(2)_{L}}$ triplets. The current experimental bound for a setup with
2 flat non-Universal Extra Dimensions is [25, 26]

$\frac{1}{2R}\gtrsim 2.1\ {\rm TeV}.$ (38)

As shown in Eq.(37), the mass of the first neutral KK mode for $\sigma$ has
two contributions. The first one comes from the compactification scale
constrained above. The second contribution $m_{\sigma}^{2}$ is actually
negative, since the zero-mode $\sigma$ must develop a VEV $\braket{\sigma}$
after spontaneous symmetry breaking. This triggers neutrino mass generation by
the inverse seesaw mechanism [13, 14] and breaks the family symmetry as well.
Its associated scale $\braket{\sigma}$ is estimated in Eq. (30). As we will
show in the next section the model can accommodate naturally a mass of a few
TeV for the LKP, which can provide a viable WIMP dark matter particle.

##  IV $\sigma^{KK}$ as a viable WIMP dark matter candidate

Here we show how the lightest KK mode of the field $\sigma$, identified as the
LKP, can be a successful WIMP dark matter candidate, while the corresponding
scalar simultaneously provides small neutrino masses through a low-scale
inverse seesaw mechanism [13, 14], and the breakdown of a flavour symmetry
following the lines proposed in [1, 2].

In our model the complex scalar singlet $\sigma$ transforms as a triplet under
the $A_{4}$ flavor symmetry, providing the source of $A_{4}$ flavor symmetry
breaking, driving low-scale neutrino mass generation through spontaneous
violation of lepton number. We now discuss the role of the first KK mode of
$\sigma$ as a DM candidate in our framework. In contrast to the case of the
gauge bosons, its couplings are not fixed by SM interactions. As a result they
can be used to fit the observed relic abundance and direct detection
constraints [21], together with a viable compactification scale and a natural
mechanism for light neutrino masses.

After compactification, one obtains a 4-dimensional scalar potential whose
relevant terms can be written generically as

$\begin{split}\mathcal{L}_{\sigma}=&\
\mu_{\sigma}^{2}\sigma^{\dagger}\sigma+\lambda_{\sigma}(\sigma^{\dagger}\sigma)^{2}+\tilde{\lambda}_{\sigma}(\sigma^{\dagger}\sigma\sigma^{\dagger}\sigma)+\lambda_{u,d,\nu}\sigma^{\dagger}\sigma
H_{u,d,\nu}^{\dagger}H_{u,d,\nu}.\end{split}$ (39)

In the above relation, the $\lambda_{\sigma}$ term involves the $3\times 3\to
1$ contraction under the $A_{4}$ symmetry, while the
$\tilde{\lambda}_{\sigma}$ term characterizes the symmetric $3\times 3\times
3\times 3\to 1$ contraction. The coefficient $\mu^{2}_{\sigma}$ is negative,
in consistency with the spontaneous symmetry breakdown driven by the zero mode
of $\sigma$. The corresponding VEV becomes

$v_{\sigma}^{2}=\frac{-\mu_{\sigma}^{2}}{2\lambda_{\sigma}+6\tilde{\lambda}_{\sigma}},$
(40)

where we have assumed that the alignment of $\sigma$ is fixed by the extra
dimensional boundary condition, entering as an input to the potential. Note
that the scale of $v_{\sigma}$ is determined by the arbitrary parameter
$\mu^{2}_{\sigma}$ which is unrelated to the compactification scale. An
important consequence of the spontaneous symmetry breaking is that Eq.(37) is
rewritten as

$M_{\sigma^{KK}}^{2}\sim\left(\frac{1}{2R}\right)^{2}+\mu^{2}_{\sigma},$ (41)

with $\mu^{2}_{\sigma}<0$, which allows the first excited mode of $\sigma$ to
be identified as the LKP and, as we will see, implement the WIMP dark matter
picture.

We can further improve the estimate for the LKP mass by decomposing a generic
bulk field into an infinite tower of orbifold eigenstates, as shown in
Eq.(31). Each 6-dimensional field should be decomposed into a series of the
extra dimensional translation and reflection eigenstates For example the bulk
$\sigma$ field, which is a flavor triplet, should be decomposed as a linear
combination of the three eigenvectors of the boundary condition matrix in Eq.
3. Therefore, its full decomposition into orbifold eigenstates is

$\sigma(x,z)=\sum_{n=0}^{\infty}\left[\sigma_{n+}(x)\frac{1}{\sqrt{3}}\left(\begin{array}[]{c}1\\\
\omega\\\
\omega^{2}\end{array}\right)f_{n+}(z)+\sigma_{n-}(x)\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}1\\\
0\\\
-\omega^{2}\end{array}\right)f_{n-}(z)+\tilde{\sigma}_{n-}(x)\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}0\\\
1\\\ -\omega\end{array}\right)f_{n-}(z)\right],$ (42)

where the three 4-dimensional towers of fields
$\sigma_{n+},\sigma_{n-},\tilde{\sigma}_{n-}$ represent the components of the
$A_{4}$ triplet. Here the labels $\pm$ denote the eigenvectors corresponding
to the $\pm 1$ eigenvalues of the extra dimensional boundary condition in
Eq.(3). In this notation, the zero mode is associated with a constant profile
$f_{0+}$ and a vanishing $f_{0-}$. The LKP would correspond to the first mode,
which can be any of the three complex scalars

$\sigma_{1LKP}=\sigma_{1+}(x)\frac{1}{\sqrt{3}}\left(\begin{array}[]{c}1\\\
\omega\\\ \omega^{2}\end{array}\right),\ \ \
\sigma_{2LKP}=\sigma_{1-}(x)\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}1\\\
0\\\ -\omega^{2}\end{array}\right),\ \ \
\sigma_{3LKP}=\tilde{\sigma}_{1-}(x)\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}0\\\
1\\\ -\omega\end{array}\right).$ (43)

Due to the orbifolding, these obtain the same mass $\sim 1/(2R)$ after
compactification. The degeneracy is lifted by the potential and the zero mode
VEV $v_{\sigma}$. However, under the natural assumption $1/(2R)\gg
v_{\sigma}$, the mass splitting is expected to be small.

From the scalar potential in Eq.(39) we can obtain the mass contributions
arising from $v_{\sigma}$ as

$\begin{split}\mathcal{L}_{m\sigma}=\Big{(}\sigma_{1+}^{*},\sigma_{1-}^{*},\tilde{\sigma}_{1-}^{*}\Big{)}\left(\begin{array}[]{ccc}\frac{1}{4R^{2}}-(5\lambda_{\sigma}+7\tilde{\lambda}_{\sigma})v^{2}_{\sigma}&0&0\\\
0&\frac{1}{4R^{2}}-\frac{10\lambda_{\sigma}+11\tilde{\lambda}_{\sigma}}{2}v^{2}_{\sigma}&-\frac{5\omega^{2}\lambda_{\sigma}+(2-\omega-5\omega^{2})\tilde{\lambda}_{\sigma}}{2}v^{2}_{\sigma}\\\
0&-\frac{5\omega\lambda_{\sigma}+(2-\omega^{2}-5\omega)\tilde{\lambda}_{\sigma}}{2}v^{2}_{\sigma}&\frac{1}{4R^{2}}-\frac{10\lambda_{\sigma}+11\tilde{\lambda}_{\sigma}}{2}v^{2}_{\sigma}\end{array}\right)\left(\begin{array}[]{c}\sigma_{1+}\\\
\sigma_{1-}\\\ \tilde{\sigma}_{1-}\\\ \end{array}\right).\end{split}$ (44)

Barring the presence of new co-annihilation channels, the parameter space
allowed by the observed relic complex-scalar dark matter abundance and direct
detection is in general very constrained. In our model the parameter space is
widened by the fact that the KK modes are nearly degenerate in mass, leading
naturally to a multicomponent dark matter picture.

In order to demonstrate the viability of our model as a theory for dark
matter, it suffices to consider a simplified scenario in which all the non-SM
fields are heavy and decouple, except for the almost degenerate complex
scalars $\sigma_{1+}$, $\sigma_{1-}$ and $\tilde{\sigma}_{1-}$. For
illustration we will include only one Higgs doublet $H$, omitting the explicit
$A_{4}$ contractions. This way the relevant Higgs portal parameters are the
effective quartic coupling constants of the extended scalar sector
characterizing the potential

$\begin{split}V(H,\sigma_{1-},\tilde{\sigma}_{1-})\supset&\lambda_{1}H^{\dagger}H\sigma_{1+}^{*}\sigma_{1+}+\lambda_{2}H^{\dagger}H\sigma_{1-}^{*}\sigma_{1-}+\lambda_{3}H^{\dagger}H\tilde{\sigma}_{1-}^{*}\tilde{\sigma}_{1-}\\\
&+\lambda_{4}H^{\dagger}H(\sigma_{1+}^{*}\sigma_{1-}+\sigma_{1-}^{*}\sigma_{1+})+\lambda_{5}H^{\dagger}H(\sigma_{1+}^{*}\tilde{\sigma}_{1-}+\tilde{\sigma}_{1-}^{*}\sigma_{1+})+\lambda_{6}H^{\dagger}H(\tilde{\sigma}_{1-}^{*}\sigma_{1-}+\sigma_{1-}^{*}\tilde{\sigma}_{1-}).\end{split}$
(45)

After electroweak spontaneous symmetry breaking, the complex scalars
$\sigma_{1+}$, $\sigma_{1-}$ and $\tilde{\sigma}_{1-}$ mix into neary
degenerate physical complex scalars $\phi_{1}$, $\phi_{2}$ and $\phi_{2}$.

In order to ensure perturbativity we have studied the relic abundance and
direct detection constraints for this simplified scenario. We have varied
randomly the relevant parameters in the range $0<\lambda_{i}<\sqrt{4\pi}$.
Moreover, we have modeled a small mass splitting amongst the three physical
scalars identifying $\phi_{1}$ as the LKP dark matter candidate, and scanning
the mass parameters within the range
$0<m_{\phi_{1}}=m_{DM}<10^{4}\,\mathrm{GeV}$,
$0<m_{\phi_{2}},m_{\phi_{3}}<1.2\times m_{DM}$.

The results are shown in Fig.1. Each magenta point corresponds to a model
parameter combination that reproduces the correct relic abundance $\Omega
h^{2}=0.120$ [27] through Higgs portal interactions, and enhanced by re-
scattering with the other nearly degenerate complex scalars in the first KK-
mode sector. This way the model opens a rather large parameter window below
the current direct detection constraints, including the recent ones of LUX-
ZEPLIN [28]. Many of these points will be probed within upcoming dark matter
experiments. [29, 30, 31]

In conclusion we find that our LKP, the lightest complex scalar contained in
the first excited mode of $\sigma$, can provide a viable WIMP dark matter
candidate.

Figure 1:  The direct detection and relic abundance constraints versus the
dark matter mass $m_{DM}=m_{\sigma_{1-}}$. Each magenta point corresponds to a
combination of the model parameters that give the correct relic abundance
$\Omega h^{2}=0.120$. The orange shaded region is ruled out by the LUX-ZEPLIN
experiment [28]. We also show the PandaX-4T [32] and XENON1T [33] limits, and
the projected sensitivities of some upcoming direct detection experiments.[29,
30, 31]

##  V Flavor predictions and global fit

Our model is based on an $A_{4}\otimes\mathbb{Z}_{2}\otimes\mathbb{Z}_{3}$
family symmetry as summarized in Table 1, with a total of 17 independent
parameters that determine the flavour properties of both the quark and lepton
sectors. These parameters, described in equations (1)-(28), can be written
compactly as

$\left\\{\frac{(y_{1,2}^{\nu}v_{\nu})^{2}y^{S}v_{\sigma}}{m^{2}_{\nu^{c}S}},y_{1,2}^{e,d}v_{d},\
y_{1,2,3}^{u}v_{u},\ \epsilon_{1,2}^{u,\nu,d},\ \
\phi_{1}^{\nu,d}-\phi_{2}^{\nu,d}\right\\}.$ (46)

We have performed a goodness of fit analysis of the model using a reduced chi-
squared test. The reference values for the flavour related observables for the
quark and lepton sectors were taken from references [5, 34, 35], and the
masses of the SM fermions were taken at the Z-boson mass ($M_{Z}$) scale.

Our chi-square test assesses how well the model is able to describe the 19
experimentally determined low-energy flavor observables, including the SM
fermion masses, the quark mixing parameters, and the neutrino oscillation
parameters,

$\left\\{m_{u,c,t,d,s,b,e,\mu,\tau},\Delta m^{2}_{21},\Delta m^{2}_{31},\
\theta_{12,13,23}^{q},\theta^{\ell}_{12,13,23},\delta^{q},\delta^{\ell}\right\\},$
(47)

therefore the number of degrees of freedom $K=19-17=2$.

Table 2 summarizes the results from our goodness-of-fit global analysis of
flavor observables. The values for the different charged lepton masses and
mixing angles are taken from [34]. For the quark masses we used an updated
determination from reference [35]. For definiteness we assume the
experimentally preferred case of normal ordered (NO) neutrino spectrum. Our
best fit point, as described in detail by Table 2, yields
$\chi^{2}_{K}=\frac{\chi^{2}}{K}=1.05$, showing that the model is in very good
agreement with the experimental data.

###  V.1  “Golden” Quark-Lepton Mass Formula

A key prediction of our model comes from the fact that the charged leptons and
down-quarks, both transforming as triplets of $A_{4}$, obtain their masses
from the coupling with the same doublet Higgs, $H_{d}$. The flavor symmetry
structure of the mass matrices $M_{e}$, and $M_{d}$ in equation (16) implies
the following relation between their masses

$\frac{m_{\tau}}{\sqrt{m_{\mu}m_{e}}}=\frac{m_{b}}{\sqrt{m_{s}m_{d}}},$ (48)

This relation has been called “golden quark-lepton mass formula” [16, 36, 37,
38, 39, 40] and constitutes a key prediction of our model, rather robust
against renormalization group evolution.

Given the precise measurements of the masses of charged lepton $m_{e}$,
$m_{\mu}$, $m_{\tau}$ and the bottom quark $m_{b}$, equation (48) can be
interpreted as a prediction for the light quark masses $m_{d}$ and $m_{s}$.
These are determined by lattice simulations with larger uncertainties [41,
42]. As shown in Fig. 2 it is in very good agreement with the reported values
for the light quark masses at the $M_{Z}$ scale [34, 35]. One sees that not
only the prediction is very close to the reported PDG values of the light
quark masses, but also the current central value (black star) got closer to
the model’s prediction than before (black dot).

Figure 2:  In magenta we give the 3$\sigma$-predicted light quark mass $m_{d}$
versus $m_{s}$ at the $M_{Z}$ scale, Eq. (48). In orange we display the
1$\sigma$ regions in each mass from [34], and in blue the recent update in
[35]. The model’s best fit is denoted with a red diamond.

###  V.2 Neutrinoless Double Beta Decay

There are other relevant observables, such as the absolute neutrino mass
scale, whose value is only bounded from above by current experiments. Apart
from cosmological observations [27], we have constraints from tritium beta
decay endpoint measurements at Katrin [43], as well as searches for
neutrinoless double beta decay [44]. The latter is a lepton number violating
process and hence crucially depends on two additional physical phases [45, 46]
which are not included in the above discussion. These phases are present in
the lepton mixing matrix but do not manifest in neutrino oscillation
experiments [46].

The most convenient description of the lepton mixing matrix is the symmetrical
parametrization proposed in [45] and revisited in [47]. The three physical
phases are parametrized as $\phi_{12}$, $\phi_{13}$, and $\phi_{23}$, so that
the leptonic “Dirac” $CP$ phase entering oscillations is given by

$\delta^{\ell}=\phi_{13}-\phi_{12}-\phi_{23}$

while the extra two phases are crucial to describe lepton number violating
processes.

In our model neutrinos are Majorana particles, so lepton number violating
processes such as $\rm 0\nu\beta\beta$ decay are expected. The associated
amplitude is proportional to

$\left\langle
m_{\beta\beta}\right\rangle=\left\lvert\cos^{2}\theta^{\ell}_{12}\cos^{2}\theta^{\ell}_{13}m^{\nu}_{1}+\sin^{2}\theta^{\ell}_{12}\cos^{2}\theta^{\ell}_{13}m^{\nu}_{2}e^{2i\phi_{12}}+\sin^{2}\theta^{\ell}_{13}m^{\nu}_{3}e^{2i\phi_{13}}\right\rvert,$
(49)

Note that, as expected, $\left\langle m_{\beta\beta}\right\rangle$ only
depends on the two Majorana phases, but not in the “Dirac” phase
$\delta^{\ell}$ 111In contrast to the PDG phase convention, the symmetrical
parametrization provides a transparent description of $\rm 0\nu\beta\beta$
decay.. In Figure 3 we show the regions for the mass parameter $\left\langle
m_{\beta\beta}\right\rangle$ characterizing the neutrinoless double beta decay
amplitude, $\left\langle m_{\beta\beta}\right\rangle$. The blue region is the
generally allowed one, given the current experimental determination of
neutrino oscillation parameters [5], while the magenta region gives the
predicted region within our model, in which all of the 19 flavor observables
in both the quark and lepton sectors, Eq.(47), lie within 3-$\sigma$ of their
measured values. The best fit value is indicated as a red diamond, and lies
somewhat below the present limit set by Kamland-Zen $(36-156\;\mathrm{meV})$
which is shown as the upper orange horizontal band in Fig. 3. We have also
displayed the projected sensitivities for some of the next generation
experiments searching for $\rm 0\nu\beta\beta$ , such as LEGEND [48], SNO +
Phase II [49], and nEXO [50], shown as horizontal dashed lines.

Figure 3:  Neutrinoless double beta decay mass parameter $\left\langle
m_{\beta\beta}\right\rangle$ versus the mass of the lightest neutrino $m_{1}$
for the normal ordered (NO) scenario. The blue region is allowed given the
current neutrino oscillation parameters [5], while the predicted magenta
region is the one consistent with the masses and mixings of quarks and leptons
at 3$\sigma$. The best fit point is indicated by a red diamond. The current
bound on $\left\langle m_{\beta\beta}\right\rangle$ from Kamland-Zen [44] as
well as future projections are shown as the upper horizontal band, and
horizontal dashed lines respectively. The vertical band indicates the
cosmological bound [27].

Parameter |  | Value
---|---|---
$y^{e}_{1}v_{d}/\mathrm{GeV}$ |  | $1.746$
$y^{e}_{2}v_{d}/(10^{-1}\mathrm{GeV})$ |  | $-9.78$
$y^{d}_{1}v_{d}/\mathrm{GeV}$ |  | $-2.85$
$y^{d}_{2}v_{d}/(10^{-2}\mathrm{GeV})$ |  | $-7.35$
$(y_{1}^{\nu}v_{\nu})^{2}y^{S}v_{\sigma}/(m^{2}_{\nu^{c}S}$ $\mathrm{eV})$ |  | $2.58$
$(y_{2}^{\nu}v_{\nu})^{2}y^{S}v_{\sigma}/(m^{2}_{\nu^{c}S}$ $\mathrm{eV})$ |  | $-0.54$
$y^{u}_{1}v_{u}/(10^{-3}\mathrm{GeV})$ |  | $-4.765$
$y^{u}_{2}v_{u}/(10^{-1}\mathrm{GeV})$ |  | $3.582$
$y^{u}_{3}v_{u}/\mathrm{GeV}$ |  | $-7.08$
$\epsilon^{u}_{1}/10^{-2}$ |  | $-8.65$
$\epsilon^{u}_{2}$ |  | $23.7$
$\epsilon^{d}_{1}/10^{-2}$ |  | $1.80$
$\epsilon^{d}_{2}/10^{-4}$ |  | $9.12$
$\epsilon^{\nu}_{1}$ |  | $1.749$
$\epsilon^{\nu}_{2}$ |  | $-7.397$
$(\phi^{d}_{1}-\phi^{d}_{2})/\pi$ |  | $-0.076$
$(\phi^{\nu}_{1}-\phi^{\nu}_{2})/\pi$ |  | $-0.321$

Observable | Data |  | Model best fit
---|---|---|---
Central value |  1$\sigma$ range |
$\sin^{2}\theta_{12}^{\ell}$$/10^{-1}$ | 3.18 |  3.02 $\to$ 3.34 |  | $3.18$
$\sin^{2}\theta_{13}^{\ell}$$/10^{-2}$ (NO) | 2.200 |  2.138 $\to$ 2.269 |  | $2.199$
$\sin^{2}\theta_{23}^{\ell}$$/10^{-1}$ (NO) | 5.74 |  5.60 $\to$ 5.88 |  | $5.74$
$\delta^{\ell}$$/\pi$ (NO) | 1.08 |  0.96 $\to$ 1.21 |  | $1.12$
$m_{e}$$/\mathrm{MeV}$ | 0.486 |  0.486 $\to$ 0.486 |  | $0.486$
$m_{\mu}$$/\mathrm{GeV}$ | 0.102 |  0.102 $\to$ 0.102 |  | $0.102$
$m_{\tau}$$/\mathrm{GeV}$ | 1.746 |  1.746 $\to$1.746 |  | $1.746$
$\Delta m_{21}^{2}/(10^{-5}\,\mathrm{eV}^{2})$ | 7.50 |  7.30 $\to$ 7.72 |  | $7.50$
$\Delta m_{31}^{2}/(10^{-3}\,\mathrm{eV}^{2})$ | 2.55 |  2.25 $\to$ 2.75 |  | $2.55$
$m_{1}$$/\mathrm{meV}$ |  |  |  | $39.02$
$m_{2}$$/\mathrm{meV}$ |  |  |  | $39.97$
$m_{3}$$/\mathrm{meV}$ |  |  |  | $63.81$
$\phi_{12}$ |  |  |  | $0.245$
$\phi_{13}$ |  |  |  | $5.22$
$\phi_{23}$ |  |  |  | $1.46$
$\langle m_{\beta\beta}\rangle$$/\mathrm{eV}$ |  |  |  | $0.036$
$\theta_{12}^{q}$ $/^{\circ}$ | 13.04 |  12.99 $\to$ 13.09 |  | $13.03$
$\theta_{13}^{q}$ $/^{\circ}$ | 0.20 |  0.19 $\to$ 0.22 |  | $0.21$
$\theta_{23}^{q}$ $/^{\circ}$ | 2.38 |  2.32 $\to$ 2.44 |  | $2.41$
$\delta^{q}$ $/^{\circ}$ | 68.75 |  64.25 $\to$ 73.25 |  | $69.11$
$m_{u}$ $/\mathrm{MeV}$ | 1.23 |  1.08$\to$ 1.51 |  | $1.23$
$m_{c}$ $/\mathrm{GeV}$ | 0.620 |  0.603 $\to$ 0.637 |  | $0.620$
$m_{t}$ $/\mathrm{GeV}$ | 168 |  167 $\to$ 169 |  | $168$
$m_{d}$ $/\mathrm{MeV}$ | 2.67 |  2.57 $\to$ 2.94 |  | $2.57$
$m_{s}$ $/\mathrm{MeV}$ | 53.1 |  51.61 $\to$ 58.32 |  | $51.7$
$m_{b}$ $/\mathrm{GeV}$ | 2.84 |  2.76 $\to$ 2.87 |  | $2.81$
$\chi^{2}_{K}$ |  |  |  | $1.05$

Table 2:  Summary of our goodness-of-fit global analysis of flavor
observables. Charged lepton masses are taken from [34] and quark masses are
based on [35]. Neutrino oscillation parameters are taken from [5] assuming
normal-ordering.

Let us now comment on Table 2. As already noted, the best fit point gives
$\chi^{2}_{K}=\frac{\chi^{2}}{K}=1.05$ showing that indeed our model is in
excellent agreement with experiment. Note also that one of the phases in the
fit, $\phi^{d}_{1}-\phi^{d}_{2}$, practically vanishes. This would suggest
that a reasonable fit of the flavor parameters could be achieved even with
these phase parameters fixed to vanish. Without them, one would still find a
not-so-bad description, with a reduced chi-squared $\chi^{2}_{K}=5.8$. The
main tension with experiment for this case lies in the physical $CP$ phases
$\delta^{l},\delta^{q}$, of about $3\sigma$. This would be the price to pay
for having an enhanced predictivity for other flavour observables. The
vanishing of these phases could follow from the imposition of a generalized CP
symmetry, in the manner described in [1].

##  VI Discussion and outlook

In this paper we have examined a low-scale realization of the orbifold theory
of flavor proposed in [1, 2]. Below compactification an $A_{4}$ family
symmetry emerges naturally, providing a good description of flavor physics,
see Table 2. The first Kaluza-Klein excitation of the same scalar which drives
family symmetry breaking as well as neutrino mass generation through the
inverse seesaw mechanism can be identified with WIMP Dark matter, see Fig. 1.
The model predicts the “golden” quark-lepton mass relation, Eq. (48) and Fig.
2. Concerning neutrinos we have interesting $\rm 0\nu\beta\beta$ decay
predictions shown in Fig. 3. For definiteness we have assumed the favored case
of normal neutrino mass ordering. Let us also mention that our model is not
necessarily meant to be a UV-complete construction. For instance, the fact
that there are three color triplets in the bulk generates 6-dimensional gauge
anomalies. However, after compactification, at low energies, these anomalies
cancel and the theory is consistent. One may assume that there are extra
fields at high energies that cancel the anomalies at that level too [51].

All in all, our model provides an excellent global description of flavor
observables both in the quark as well as lepton sectors starting from first
principles. Compared with the work in Refs. [1, 2] the results given here
correspond to a low-scale inverse seesaw mechanism and they follow from the
use of updated flavor parameters, such as oscillation parameters as well as
recent quark mass determinations.

###### Acknowledgements.

Work supported by the Spanish grants PID2020-113775GB-I00
(AEI/10.13039/501100011033) and Prometeo CIPROM/2021/054 (Generalitat
Valenciana). OM is supported by Programa Santiago Grisolía (No.
GRISOLIA/2020/025). CAV-A is supported by the Mexican Cátedras CONACYT project
749 and SNI 58928. The relic abundance and direct detection constraints were
calculated using the micrOMEGAS package [52] at GuaCAL (Guanajuato
Computational Astroparticle Lab). We thank Ignatios Antoniadis for discussions
in the early phase of the work.

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