# Reviving keV sterile Neutrino Dark Matter

Carlos Jaramillo111email<EMAIL_ADDRESS>Max-Planck-Institut
für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany

###### Abstract

We propose a new production mechanism for keV sterile neutrino dark matter
which relies neither on the oscillations between sterile and active neutrinos
nor on the decay of additional heavier particles. The dark matter neutrinos
are instead produced by thermal freeze-out, much like a typical WIMP. The
challenge consists in balancing a large Yukawa coupling so that the sterile
neutrinos thermalize in the early universe on the one hand, and a small enough
Yukawa coupling such that they are stable on cosmological scales on the other.
We solve this problem by implementing varying Yukawa couplings. We achieve
this by using a three-sterile neutrino seesaw extension to the SM and
embedding it in a Froggatt-Nielsen model with a single flavon. Because the vev
of the flavon changes during the electroweak phase transition, the effective
Yukawa couplings of the fermions have different values before and after the
phase transition, thus allowing for successful dark matter genesis.
Additionally, the hierarchy in the flavour structure is alleviated and the
origin of the light neutrino masses is explained by the interplay of the
seesaw and Froggatt-Nielsen mechanisms.

https://doi.org/10.1088/1475-7516/2022/10/093 Carlos Jaramillo JCAP10(2022)093

##  1 Introduction

The phenomenon of neutrino flavour oscillations, which has been observed [1,
2, 3] and confirmed by many experiments [4, 5] and for which the 2015 Nobel
Prize was awarded [6], implies definitively that active neutrinos have tiny
non-vanishing masses. Also, there is an overwhelming amount of evidence for
the existence of a non-baryonic substance, which makes up almost one quarter
of the energy density of the Universe today and is generally referred to as
Dark Matter (DM) [7, 8, 9, 10, 11]. It is widely accepted that the origin of
the light neutrino masses and the fundamental nature of the Dark Matter of the
Universe are evidence of the incompleteness of the Standard Model (SM) of
particle physics as a theory of nature. Perhaps the simplest and most elegant
solution to the problem of neutrino masses is delivered by the well known
seesaw mechanism of type I [12, 13, 14, 15, 16]. Therein, at least two right-
handed (RH) SM singlet fermions are introduced, which have a Majorana mass
term and couple to the SM neutrinos through a Yukawa coupling, thus repairing
the left-right asymmetry in the matter content of the theory; these are the
right-handed (also called sterile) neutrinos. Just like in the case of all
other fermions, the neutrinos get a Dirac mass term through the Higgs
mechanism. The full neutrino-mass-matrix contains the Majorana mass matrix and
the Dirac mass matrix. After diagonalization, and assuming that the elements
of the Majorana matrix are much larger than those of the Dirac matrix, the
active-neutrino mass matrix arises and is suppressed by the large entries in
the Majorana matrix, thus elegantly explaining the smallness of the masses of
the active neutrinos. This minimal extension of the SM is also highly
appealing because one (or many) of the sterile neutrinos could play the role
of the DM of the Universe (see for instance $\nu$MSM [17]). There are two
basic requirements for this to be the case: firstly, there has to be a viable
mechanism to produce the right density of sterile neutrinos in the early
universe, i.e. the observed relic abundance. Secondly, the sterile neutrinos
must be stable on cosmological timescales, i.e. they must be compatible with
the indirect (non-) detection observations. This last issue is non-trivial
because the sterile neutrinos, being massive and coupled to the active
neutrinos by the Yukawa term, may decay into SM particles. Given the results
of the indirect detection searches, it has turned notoriously difficult to
reconcile these two requirements with each other: the most straightforward way
to produce sterile neutrino DM in the early Universe is through the
oscillations with active neutrinos, the so-called ”Dodelson-Widrow mechanism”
[18]. However, this simplest scenario has already been ruled out by indirect
search observations [19, 20] because the Dodelson-Widrow mechanism requires a
sizeable mixing between active and sterile neutrinos; however, if one allows
for large enough mixing angles, the decay rate of the DM neutrinos increases,
thus conflicting with indirect detection results. This constrains can be
somewhat alleviated if the oscillations that produce the DM occur resonantly
(the so-called Shy-Fuller mechanism [21]) but this requires the presence of a
lepton asymmetry at temperatures well below the Electroweak (EW) scale, which
is hard to motivate. Another possibility is that sterile neutrinos are
produced by the decay of additional dark-sector heavy particles, see for
example [22, 23, 24, 25, 26, 27, 28], or that the production is aided by the
presence of non-standard neutrino interactions [29, 30, 31, 32, 33, 34].
However, such models depart from the minimal seesaw extension to the SM and in
them the active-sterile coupling plays little to no role at all, thus
rendering indirect searches less relevant.

Therefore, an alternative mechanism for sterile neutrino DM genesis on the
basis of the Yukawa coupling is desirable. We obtained our inspiration from
another very popular candidate as a DM particle, a generic Weakly Interacting
Massive Particle (WIMP). WIMPs are coupled to the SM by some portal with an
interaction strength similar or smaller than the weak interactions of the SM.
Thanks to such interactions WIMPs are generally thought to be in thermal
equilibrium with the primordial plasma until their interaction rate falls
below the expansion rate of the Universe as it expands and cools, at which
point the comoving density of the WIMPs gets frozen-out and remains constant
thereafter. This production mechanism is usually referred to as “freeze-out”,
and until now it has not been considered for sterile neutrinos because the
Yukawa coupling necessary to thermalize them would also make them highly
unstable - they would not survive long.

Here we propose a scenario where this problem is solved by implementing
varying Yukawa couplings. The main idea is simple: if the neutrino Yukawa
coupling is allowed to be large enough in the early universe for the sterile
neutrinos to thermalize, but later drops to much smaller values such that
their interaction rate turns smaller that the expansion rate of the Universe,
then the relic abundance of sterile neutrinos will be fixed by a process
similar to the WIMP freeze-out. Varying Yukawa couplings have been studied in
the past, particularly in the context of EW Baryogenesis [35, 36, 37]. There
are many ways to implement varying Yukawa couplings and, in principle, any of
them could work to thermalize sterile neutrinos in the early Universe and then
induce their freeze-out to generate the observed DM relic abundance. For
concreteness, however, we will examine the implementation of varying Yukawa
couplings involving the Froggatt-Nielsen mechanism. The Froggatt-Nielsen (FN)
mechanism is a popular way to explain the flavour structure in the SM (i.e.
the mass hierarchy in the fermion sector). It employs a new scalar field,
called the flavon, and an additional $U(1)$ flavour symmetry under which the
flavon and the fermions are charged. The bare Yukawa couplings, which within
this framework were all initially of order 1, become effectively suppressed by
the vev of the flavon in a hierarchical pattern determined by the different
flavour charges of the fermions. In this work we consider the possibility that
the flavon vev changes during the Electroweak Phase Transition (EWPT), varying
from a value similar to the flavour scale before the EWPT to a somewhat
smaller value afterwards. This is a reasonable consideration because the
scalar potential responsible for the EWPT will contain both the Higgs and the
flavon fields and therefore it will find its minimum in the Higgs-flavon field
space. Thus, during the EWPT, as the potential relaxes to its true minimum,
one should not expect the coordinate of the new minimum along the flavon axis
to remain constant. On the contrary, it is sensible to expect that the flavon-
coordinate of the minimum will also change as the Higgs-coordinate changes. As
a consequence, the Yukawa couplings will have different effective values
before and after the phase transition. As we will see, something similar to
this will also happen to the Majorana masses of the sterile neutrinos.

Here we will formulate a concrete low energy FN model which explains neutrino
masses and oscillations via the seesaw mechanism as well as the flavour
hierarchy in the lepton sector via the FN mechanism. Moreover, we will show
that the variation in the flavon vev leads to effectively varying Yukawa
couplings that allow for the thermalization of the sterile neutrinos before
the EWPT and their freeze-out afterwards, such that the lightest sterile
neutrino with keV mass plays the role of the DM and its relic abundance is set
by thermal freeze-out.

The outline of this paper is as follows. We begin in Section 2 by reviewing
the seesaw mechanism and briefly discussing the keV neutrino as a DM
candidate. Then, in Section 3 we introduce the DM production mechanism and
compute the DM relic abundance. In Section 4 we formulate a low energy FN
model to implement the varying Yukawa couplings for DM production, the light
neutrino masses and the lepton flavour hierarchy, before stating some
concluding remarks in Section 5.

##  2 The keV sterile neutrino as the Dark Matter particle

We start by reviewing the general Type I seesaw mechanism. The SM is extended
by three222The low energy neutrino data can be explained within the type I
seesaw framework with two neutrinos or more. However, anticipating Section 4
we discuss the case of three sterile neutrinos, which is also the case in the
well known $\nu$MSM. neutral Majorana fermions that are also singlets with
respect to the SM symmetry, $(\nu_{R})_{i}$. With the presence of these
sterile neutrinos the Lagrangian gets the following additional terms:

$\displaystyle-{\cal L}\supset
i\bar{\nu}_{R}\not{\partial}\nu_{R}+\bar{L}\,Y_{\nu}\,\tilde{\phi}\nu_{R}+\frac{1}{2}\overline{\nu^{c}_{R}}\,M_{R}\,\nu_{R}+\mathrm{h.c.},$
(1)

where $\tilde{\phi}$ is the dual Higgs field, $L$ stands for the $SU(2)$
lepton doublets, $Y_{\nu}$ is the neutrino Yukawa matrix and $M_{R}$ is the
Majorana mass matrix. After the spontaneous breaking of the EW symmetry by the
vev of the Higgs $v$, a Dirac mass matrix for the neutrinos is generated with
$m_{D}=Y_{\nu}\,v/\sqrt{2}$. The mass term for all six neutrinos can then be
formulated as

$\displaystyle-{\cal
L}_{\mathrm{\nu,mass}}=\frac{1}{2}\,\overline{\nu^{c}_{\cal M}}\,{\cal
M}_{\nu}\,\nu_{\cal M}+\mathrm{h.c.}=\frac{1}{2}\,\overline{\nu^{c}_{\cal
M}}\begin{pmatrix}0&m_{D}\\\ m_{D}^{T}&M_{R}\end{pmatrix}\nu_{\cal
M}+\mathrm{h.c.},$ (2)

where the basis is defined by $\nu_{\cal
M}=(\nu_{L,e},\nu_{L,\mu},\nu_{L,\tau},\nu_{R,1}^{c},\nu_{R,2}^{c},\nu_{R,3}^{c})^{T}$
and ${\cal M}_{\nu}$ is a $6\times 6$ matrix. The full neutrino mass matrix
can first be block diagonalized to

$\displaystyle{\cal M}_{\nu}^{\prime}=\mathrm{diag}(m_{\nu},\,M_{N})$ (3)

and, assuming that the seesaw condition holds, namely that $m_{D}\ll M$, we
obtain

$\displaystyle m_{\nu}\approx m_{D}\,M^{-1}\,m_{D}^{T},\qquad M_{N}\approx
M_{R},$ (4)

whereby these matrices can be further diagonalized by the appropriate matrices
$V$ and $U$

$\displaystyle
m_{\nu}^{d}=V^{T}\,m_{\nu}\,V=\mathrm{diag}(m_{1},m_{2},m_{3}),\qquad
M_{N}^{d}=U^{T}\,M_{N}\,U=\mathrm{diag}(M_{1},M_{2},M_{3}),$ (5)

resulting in three very light neutrino mass eigenstates $\nu_{i}$, mostly
composed of the active flavours, and three heavy states $N_{i}$, mostly
composed of the sterile neutrinos. The active-sterile mixing is then of the
order $\theta\sim m_{D}\,M_{R}^{-1}$.

It is the lightest sterile neutrino $N_{1}$ which we want to consider as the
DM particle. The so-called Tremaine-Gunn Bound [38] states that a fermionic DM
particle candidate must be heavier than $\sim 1\,\mathrm{keV}$. This model
independent constrain is based on the fact that the densest phase space
distributions for fermionic DM particles may not be larger than that allowed
by the Pauli exclusion principle. Thus, sterile neutrinos with masses in the
keV scale are the lightest viable fermionic DM candidates. They are also
appealing because they would be warm dark matter (WDM), meaning that they tend
to suppress the power of structure formation at small scales, offering a
possible solution to the satellite problem and the core-cusp problem [39, 40].

Regardless of how they are produced, sterile neutrinos with a Yukawa coupling
to the SM as in Eq. 1 are never completely stable. The interaction
strength/probability of $N_{1}$ with the SM will be characterised by the sum
of the square of its mixing angle with all active neutrinos [41], i.e.

$\displaystyle\theta_{1}^{2}=\sum_{\alpha=e,\mu,\tau}\theta_{\alpha,1}^{2}=\sum_{\alpha=e,\mu,\tau}\frac{v^{2}\,|(Y_{\nu})_{\alpha
1}|^{2}}{M_{1}^{2}}=\sum_{\alpha=e,\mu,\tau}\frac{|(m_{D})_{\alpha
1}|^{2}}{M_{1}^{2}}.$ (6)

Through this mixing, sterile neutrinos with keV masses can decay at tree level
to three active neutrinos, $N_{1}\rightarrow\nu_{i}\,\bar{\nu}_{j}\,\nu_{j}$.
The fact that DM must still exist today means that the lifetime of unstable DM
should be comparable to the age of the universe, delivering a constraining
relationship between $M_{1}$ and $\theta_{1}$. However, an even more
constraining relation is obtained by the sub-dominant one-loop decay
$N_{1}\rightarrow\nu_{i}\,\gamma$, where the produced photon is almost
monochromatic with energy $E_{\gamma}=M_{1}/2$. These photons, which for
$M_{1}\sim\mathrm{keV}$ will lie in the $X$-Ray part of the spectrum, can be
searched for in space with $X$-Ray telescopes [42, 43, 44]. The non-
observation of this smoking-gun signal has placed a limit on the mixing angle
of sterile neutrino DM [45],

$\displaystyle\sin^{2}(2\theta_{1})\lesssim 3\times
10^{-5}\left(\frac{\mathrm{keV}}{M_{1}}\right)^{5},$ (7)

assuming that $N_{1}$ makes up 100% of the DM abundance.

Since the year 2014 there has been a lot of debate concerning a suspicious
$X$-Ray line with an energy of $E_{\gamma}=3.55\,\mathrm{keV}$. The signal has
been detected from a multitude of sources, such as galaxy clusters, the
Andromeda galaxy and our own galactic center, and a multitude of instruments
e.g. XMM-Newton, Chandra, EPIC, Suzaku, ACIS and Fermi, just to name a few.
The reader may find a review in [46]. This signal can be interpreted as the
smoking-gun photon from sterile neutrino DM decay, implying a DM mass of
$M_{1}=7.1\,\mathrm{keV}$. While this is a very attractive hypothesis and
there have been many arguments made in its favour [45], it remains
controversial. The line could also originate from astrophysical processes [47]
or appear as a consequence of unaccounted systematics [48]. Particularly, the
line has not been observed from dwarf spheroidal galaxies, which were expected
to provide the cleanest DM decay signal because of their high mass-to-light
ratio and correspondingly low $X$-Ray background [49]. The debate for the
correct interpretation of this signal is still ongoing and one can only hope
that more data and further analysis will bring clarity.

In any case, if we insist on the DM interpretation of the signal, an
alternative to the Dodelson-Widrow mechanism for DM production is needed,
because the mixing angle necessary to explain the relic abundance by the
Dodelson-Widrow mechanism is incompatible with the value that would correspond
to the 3.55 keV line. Although resonant production is still allowed, it would
require an additional mechanism to generate a large lepton asymmetry well
after the EWPT, which is not well motivated. To address this situation, we
propose in the following section an alternative production mechanism that
provides an explanation to the 3.55 keV signal that is compatible with the DM
hypothesis.

##  3 Sterile neutrino Dark Matter genesis by freeze-out

To simultaneously explain the phenomena of active neutrino oscillations and to
offer a viable DM candidate, three sterile neutrinos are needed: one to play
the role of the DM and the other two of them to generate the masses of the
light neutrinos. Thus, we consider an extension to the SM by three sterile
neutrinos. In this work we will insist that the relic abundance of the
lightest sterile neutrino $N_{1}$, which shall play the role of the DM
particle, is produced in the early Universe by thermal freeze-out, similar to
a typical WIMP, and not by oscillations from active neutrinos or by the decay
of additional heavier particles. For this to work, $N_{1}$ must have a large
enough Yukawa coupling so that it comes into thermal equilibrium with the SM
bath. On the other hand, a large Yukawa coupling translates into a large
active-sterile mixing angle $\theta_{1}$, which would lead to a rapid decay of
the DM particles and contribute a sharp line at $E=M_{1}/2$ to the $X$-Ray
background and from sources with a large DM density or mass to light ratio.
These problems are solved if, by some mechanism, the Yukawa coupling varies
during the early universe, going from large values at early times, to smaller
values at later times. Even though there are many ways to implement varying
Yukawa couplings [35], we here chose to do so by embedding the seesaw
extension in a Froggatt-Nielsen model. This has the added benefit that it also
helps to explains the flavour hierarchy.

Field | $L_{i}$ | $E_{R_{j}}$ | $N_{k}$ | $\Theta$ | $\phi$
---|---|---|---|---|---
$U(1)_{\mathrm{FN}}$ Charge | $q_{L_{i}}$ | $q_{R_{j}}$ | $q_{N_{k}}$ | $-1$ | 0
Table 1:  FN charges for the lepton doublets $L_{i}$, charged singlets
$E_{R_{j}}$, sterile neutrinos $N_{k}$, the flavon $\Theta$ and the Higgs
$\phi$, respectively.

The Froggatt-Nielsen (FN) mechanism [50] attempts to resolve the flavour
hierarchy puzzle by introducing a scalar field, called the flavon333The flavon
encodes the Froggatt-Nielsen messengers, which are heavier particles that are
integrated out below the flavour scale $\Lambda_{\mathrm{FN}}$. $\Theta$, and
a global flavour symmetry $U(1)_{\mathrm{FN}}$, under which the flavon and all
fermions are charged. The flavon is set to have a $U(1)_{\mathrm{FN}}$ charge
of $-1$, while the fermions $f_{i}$ may have any integer as their charge
$q_{f_{i}}$ and the Higgs field $\phi$ is not charged under the flavour
symmetry, see Table 1. Consequently, the Yukawa terms in the Lagrangian arise
from non-renormalizable operators that contain powers of the flavon such that
the terms are invariant under the $U(1)_{\mathrm{FN}}$ symmetry. Thus, below
the flavour scale $\Lambda_{\mathrm{FN}}$ the usual Yukawa terms in the SM
Lagrangian must be replaced by

$\displaystyle Y_{ij}\,\overline{f_{L}}_{i}\,\phi\,f_{R\,j}\longrightarrow
Y_{ij}\,\overline{f_{L}}_{i}\,\phi\,f_{R\,j}\,\left(\frac{\langle\Theta\rangle}{\Lambda_{\mathrm{FN}}}\right)^{q_{L,i}\,+\,q_{R,j}}.$
(8)

The key idea is that below the flavour scale $\Lambda_{\mathrm{FN}}$ the
flavon has a non-vanishing vev, and therefore each entry in the Yukawa and
Majorana matrices gets rescaled by powers of the FN factor, defined as
$\lambda:=\langle\Theta\rangle/\Lambda_{\mathrm{FN}}$. This resolves the
flavour hierarchy puzzle because, if $\lambda<1$, then each entry is
effectively suppressed by powers of $\lambda$ according to the sum of FN
charges of the fermions involved. Thus, the observed flavour hierarchy is the
result of the different FN charges of the fermions and not due to a mysterious
hierarchy in the Yukawa couplings, which in this framework are all allowed to
be of order unity. This mechanism is usually employed in the quark sector,
where the FN suppression factor is set to $\lambda\approx 0.22$, in
resemblance of the Cabibbo angle. However, in this work we will apply these
tools to the lepton sector to study the posibility of sterile neutrino DM. For
the lepton sector, the relevant terms in the effective Lagrangian are

$\displaystyle-{\cal L_{\mathrm{eff}}}$
$\displaystyle\supset\bar{L_{i}}\,Y^{E}_{ij}\,{\phi}\,E_{R\,j}\,\left(\frac{\langle\Theta\rangle}{\Lambda_{\mathrm{FN}}}\right)^{q_{\bar{L}i}+q_{Rj}}+\,\bar{L_{i}}\,Y^{\nu}_{ij}\,\tilde{\phi}\,\nu_{Rj}\,\left(\frac{\langle\Theta\rangle}{\Lambda_{\mathrm{FN}}}\right)^{q_{\bar{L}i}+q_{Nj}}$
(9)
$\displaystyle+\,\frac{1}{2}\overline{\nu_{Ri}^{c}}\,(M_{R})_{ij}\,\nu_{Rj}\,\left(\frac{\langle\Theta\rangle}{\Lambda_{\mathrm{FN}}}\right)^{q_{Ni}+q_{Nj}}\,+\,\mathrm{h.c.},$

where $q_{L,R,N}$ are the $U(1)_{\mathrm{FN}}$ charges of the $SU(2)$ charged
doublets, singlets and sterile neutrinos respectively, as given in Table 1,
and all fermions should be understood as containing all three generations.

Figure 1: Location of the minimum of the scalar potential in field space and
shift in the value of the flavon vev caused by the electroweak symmetry
breaking (EWSB).

To achieve the thermal freeze-out of sterile neutrinos, we propose to
implement varying Yukawa couplings in the following way, resembling [37, 51].
Since the scalar potential of the theory will include both the Higgs $\phi$
and the flavon $\Theta$, the true minimum of the scalar potential will be
located at a point in the $(\phi,\Theta)$-field space. Below the flavour scale
but prior to the EWPT (i.e. at a temperature $T$ with
$\Lambda_{\mathrm{EW}}<T<\Lambda_{\mathrm{FN}}$), the vev of the Higgs is
$\langle\phi\rangle=v=0$ while the vev of the flavon is
$\langle\Theta\rangle=w_{e}\neq 0$ and has already broken the flavour symmetry
(the subscript $e$ stands for early, as in $w_{e}$ is the vev of the flavon at
times earlier than the EWPT). If we set $w_{e}=\Lambda_{\mathrm{FN}}$, then
the FN factor would be $\lambda=1$ at early times and the Yukawa couplings of
order unity would not be suppressed. Crucially, this means that the sterile
neutrinos would be in thermal equilibrium because their Yukawa couplings would
be as strong as that of the top quark. Then, as the Universe cools down, the
EWPT kicks in and plays a key role: as the Higgs gets its vev and the scalar
potential relaxes to its true minimum, it is reasonable to expect that this
minimum lies at a completely new position in the $(\phi,\Theta)$-field space,
or in other words, during the EWPT the vev of the flavon gets dragged along by
the vev of the Higgs. This is illustrated in Fig. 1. If after the EWPT the vev
of the flavon ends up at a value
$\langle\Theta\rangle=w_{l}=\epsilon\,\Lambda_{\mathrm{FN}}$ with
$\epsilon<1$, then the Yukawa couplings get suppressed by powers of $\epsilon$
(here, the subscript $l$ stands for late, as in $w_{l}$ is the vev of the
flavon at times later than the EWPT). This suppression could be drastic enough
to force the sterile neutrinos to decouple from thermal equilibrium and could
even make the lightest sterile neutrino stable on cosmological time scales and
thus evade the $X$-Ray bounds. In this context, the FN suppression factor
$\lambda$ can be understood as describing a path in field space with
boundaries given by

$\displaystyle\lambda(\langle\phi\rangle)=\begin{cases}1,\quad&\mathrm{for}\quad\langle\phi\rangle=0\\\
\epsilon,\quad&\mathrm{for}\quad\langle\phi\rangle=v.\end{cases}$ (10)

The precise trajectory in field space during the EWPT is not relevant. What
matters is that the flavon vev has different values in the different phases,
as sketched in Fig. 1. We will comment on the scalar potential necessary to
achieve this behaviour at the end of this section.

Note from Eq. 9 that this mechanism not only suppresses the Yukawa matrices
but also the Majorana mass matrix is affected. Just like the Yukawa matrices,
the Majorana mass matrix can be understood as effectively having different
values before and after the EWPT. As we will discuss below, this issue is
highly significant for DM genesis. In terms of the $\lambda$ parameter, and
recalling Eq. 4, we can write the $(j,k)$ elements of the effective sterile
neutrino mass matrix $M_{\mathrm{eff}}$ as

$\displaystyle(M_{\mathrm{eff}})_{jk}=(M_{N})_{jk}\,[\lambda(\langle\phi\rangle)]^{q_{Nj}\,+\,q_{Nk}}=\begin{cases}(\tilde{M})_{jk}=(M_{N})_{jk},&\mathrm{for}\quad
T>T_{EW}\\\
(M)_{jk}=(M_{N})_{jk}\,\epsilon^{q_{Nj}\,+\,q_{Nk}},&\mathrm{for}\quad
T<T_{EW}\end{cases},$ (11)

where $\tilde{M}$ stands for the early Majorana mass matrix, i.e. before it is
suppressed by the shift in the flavon vev at the EWPT and $M$ is the late
Majorana mass matrix, i.e. the suppressed Majorana mass matrix after the EWPT.
Diagonalising the $\tilde{M}$ and $M$ gives us the masses of the sterile
neutrinos before and after the EWPT respectively. We will refer to the early
(i.e. unsuppressed) and late (i.e. suppressed) mass of the lightest sterile
neutrino, i.e. the DM neutrino, as $\tilde{M}_{1}$ and $M_{1}$ respectively.

We can convince ourselves that the sterile neutrinos were indeed in thermal
equilibrium before being forced to decouple by the EWPT by computing the ratio
of the sterile neutrino interaction rate $\Gamma_{\nu}$ to the expansion rate
of the Universe $H(T)$. Prior the EWPT, the main interactions keeping the
sterile neutrinos in thermal equilibrium are the decays and inverse decays
governed by the Yukawa couplings, which are all $\mathcal{O}(1)$ since they
have not been suppressed yet. The corresponding interaction rate and the
Hubble rate during radiation domination are given by

$\displaystyle\Gamma_{\nu}=\frac{(Y^{\nu})^{2}\tilde{M}_{1}}{16\,\pi},\qquad
H(T)\approx\frac{\sqrt{g_{\rho}(T)}}{M_{\mathrm{P}}}\,T^{2},$ (12)

where $g_{\rho}(T)$ stands for the effective number of energetic d.o.f. in the
plasma at temperature $T$, $M_{\mathrm{P}}$ is the reduced Planck mass and we
have ignored the fact that the Yukawa coupling is actually a matrix, just for
simplicity. Inserting the values for $T=\Lambda_{\mathrm{EW}}$,
$\tilde{M}_{1}=10^{3}\,$GeV and taking $Y^{\nu}\sim\mathcal{O}(1)$ we arrive
at

$\displaystyle\left.\frac{\Gamma_{\nu}}{H}\right|_{T\gtrsim
T_{\mathrm{EW}}}\sim 10^{14}\gg 1,$ (13)

thus confirming that prior to the suppression induced by the EWPT, the sterile
neutrinos are very much in thermal equilibrium. However, after the EWPT the
Yukawa couplings will be suppressed by a factor of
$\epsilon^{q_{\bar{L}}+q_{N}}$. Assuming that $\epsilon=0.1$, as we will do
throughout this paper, this implies that the sum of the FN charges of the
$SU(2)$ leptons and sterile neutrinos should be at least 7 in order for the
suppression to force the sterile neutrinos to decouple, i.e.
$q_{\bar{L}}+q_{N}>7$. In fact, the concrete example which will be studied in
the next section, will be such that $q_{\bar{L}}+q_{N}=11$, leading to

$\displaystyle\left.\frac{\Gamma_{\nu}}{H}\right|_{T\lesssim
T_{\mathrm{EW}}}\sim 10^{-8}\ll 1,$ (14)

which is more than enough of a suppression to decouple the sterile neutrinos
(the choice of $q_{\bar{L}}+q_{N}=11$ is made in order to also avoid the
stringent $X$-ray bounds, as will be discussed in the next section).
Furthermore, the fact that the EWPT also causes a suppression of the Majorana
mass matrix will lead to a further reduction of the interaction rate.
Particularly, if the suppressed mass is lighter than the Higgs mass, i.e.
$M_{1}<m_{\phi}$, as would be the case for
$M_{1}\sim\mathcal{O(\mathrm{keV})}$, then the direct decay of the sterile
neutrinos becomes kinematically forbidden. Then, the only interactions the
sterile neutrinos can have with the SM are through active-sterile
oscillations, which will be negligible because of the tiny mixing angle, which
is proportional to the Yukawa coupling, $\theta_{\alpha i}=Y^{\nu}_{\alpha
i}\,v/M_{i}$. Indeed, although further production (or depletion) of sterile
neutrinos through the Dodelson-Widrow mechanism (active-sterile oscillations)
is in principle still possible, the drastically suppressed mixing angle
renders this effect negligible [52].

The knowledge that the suppression of the Yukawa couplings is enough to force
the sterile neutrinos to drop out of thermal equilibrium (i.e. to freeze-out)
is now enough to compute the relic abundance of the lightest sterile neutrino
as described in the following. The contribution to the total energy density of
the Universe from the frozen-out DM neutrinos, which are non-relativistic
today, is given by [53]

$\displaystyle\Omega_{\mathrm{DM}}\,h^{2}=\frac{s_{0}\,y_{\mathrm{fo}}\,M_{1}}{\rho_{\mathrm{crit}}/h^{2}},$
(15)

where $s_{0}$ is the entropy density today, $y_{\mathrm{fo}}$ is the frozen-
out comoving number density of the sterile neutrinos, $\rho_{\mathrm{crit}}$
is the critical density of the Universe and the Hubble parameter today is
given by $H_{0}=100\,h\;\mathrm{km\,s}^{-1}\mathrm{Mpc}^{-1}$. The early
masses of the sterile neutrinos will be of the same order as the Majorana
scale, which is expected to be higher than $\Lambda_{\mathrm{EW}}$. Thus,
shortly before the EWPT the sterile neutrinos with early mass $\tilde{M}_{1}$
will be non-relativistic and their comoving density in equilibrium will be
given by

$\displaystyle
y_{\mathrm{eq}}(T,\tilde{M}_{1})=\frac{45}{(2\pi^{5})^{3/2}g_{s}}\left(\frac{\tilde{M}_{1}}{T}\right)^{3/2}\,e^{-\tilde{M}_{1}/T},$
(16)

where $g_{s}$ is the effective entropic number of degrees of freedom in the
plasma. Then, when the EWPT occurs and shifts the vev of the flavon, the
drastically suppressed Yukawa couplings force the sterile neutrinos to drop
out of thermal equilibrium and their comoving density is frozen at
$y_{\mathrm{fo}}=y_{\mathrm{eq}}(T_{EW},\tilde{M}_{1})$. Notice that while it
is the early mass $\tilde{M}_{1}$ that comes into $y_{\mathrm{fo}}$ in Eq. 16,
the relic abundance today is proportional to the late mass $M_{1}$ in Eq. 15,
because the energy density of non-relativistic DM today is given by number
density of DM particles $n_{\mathrm{DM}}=s_{0}\cdot y_{\mathrm{fo}}$ times
their mass today, i.e. the late mass $M_{1}$. Thus, the relic abundance of our
DM sterile neutrinos in this framework as given in Eq. 15 can be understood as
a function of three parameters, namely the freeze-out temperature, which here
is equal to the temperature of the EWPT, $T_{\mathrm{fo}}=T_{\mathrm{EW}}$ and
the early and late mass of the DM neutrinos $\tilde{M}_{1}$ and $M_{1}$.

Figure 2: Left panel: lines for which the combinations of early and late
masses result in a 100% contribution from the sterile neutrinos to the DM
abundance of the Universe for different freeze-out temperatures
$T_{\mathrm{fo}}=T_{EW}$. For example, with a late mass of $M_{1}=7.1\,$keV
and at a decoupling temperature of $T_{\mathrm{fo}}=150\,$GeV, one can see
that if the early mass is $\tilde{M}_{1}\approx 1\,$TeV, then the sterile
neutrino can account for 100% of the DM density. Right panel: At a fixed
freeze-out temperature of $T_{\mathrm{fo}}=T_{EW}=150\,$GeV, the lines show
the combinations of early and late masses for which the sterile neutrinos
contribute 100%, 10% and 1% to the DM abundance. Above the uppermost line, the
contribution can be considered negligible.

Using the values from reference [54] for $\Omega_{\mathrm{DM}}\,h^{2}$,
$s_{0}$ and $\rho_{\mathrm{crit}}$, we can plot Eq. 15 in the
$(\tilde{M}_{i},M_{i})$ plane. The results can be seen in the left panel of
Fig. 2. Each line represents a combination of suppressed and unsuppressed
masses such that Eq. 15 returns the full DM abundance with the freeze-out
temperature $T_{\mathrm{fo}}=T_{EW}$ as an additional parameter. For example,
at $T_{EW}=150\,$GeV and $M_{1}=7.1\,$keV, the early DM mass should be
$\tilde{M}_{1}\approx 1\,$TeV for $N_{1}$ to account for all of the DM of the
Universe. Similarly, the right panel of Fig. 2 depicts the relic abundance of
sterile neutrinos for a fixed freeze-out temperature. For instance, with
$T_{EW}=150\,$GeV, if the two heavier sterile neutrinos have a mass of
$M_{2}=10\,$keV and $M_{3}=50\,$keV after the EWPT and a mass of
$\tilde{M}_{2}=2\,$TeV and $\tilde{M}_{3}=3\,$TeV before, then their
contribution to the DM abundance will be negligible. It is easy to understand
why this is the case: when the induced freeze-out at $T_{EW}$ generates the DM
abundance from the $N_{1}$ population, the population of $N_{2,3}$ is already
depleted by the Boltzmann factor $\exp(-M_{2,3}/T_{EW})$ because of their
larger masses $M_{2,3}>M_{1}$.

It is thus clear that successful DM production by the mechanism proposed in
this work depends only on three parameters: the temperature of the EWPT and
the early and late masses of the sterile neutrinos.

As has been discussed in [51, 37] 444In [51], the PT during which the flavon
vev changes is not necessarily the EWPT, but the PT of some generic scalar
field $\Sigma$ which undergoes SSB. To be applied in case in question here,
one simply has to replace $\Sigma$ by the Higgs field $\phi$. The potential
discussed in [51] was taken and adapted from [37]., a tree-level, zero
temperature scalar potential capable of exhibiting the desired behaviour, i.e.
a simultaneous change in the vev’s of both the Higgs and the flavon, should
feature negligible Higgs-flavon mixing and could be of the form

$\displaystyle
V(\phi,\,\Theta)=\mu_{\phi}^{2}\,\phi^{\dagger}\phi+\lambda_{\phi}\,(\phi^{\dagger}\phi)^{2}+\mu_{\Theta}^{2}\,\Theta^{\dagger}\Theta+\lambda_{\Theta}\,(\Theta^{\dagger}\Theta)^{2}+\lambda_{\phi\,\Theta}(\Theta^{\dagger}\Theta)(\phi^{\dagger}\phi),$
(17)

where $\lambda_{\phi},\;\lambda_{\Theta},\;\lambda_{\phi\,\Theta}$ are the
self couplings of the Higgs, the flavon and the Higgs-flavon coupling,
respectively. These couplings should not be confused with the FN suppression
factor $\lambda=\langle\Theta\rangle/\Lambda_{\mathrm{FN}}$ from (10). The
temperature and quantum correction to such a potential with the desired
behaviour during the phase transition are discussed in [55]. We point out that
the simultaneous variation in the Higgs and flavon vev’s is actually not
really crucial for the DM production mechanism proposed here: the shift in the
flavon vev could also occur before or after the EWPT; whenever it occurs, that
will be the moment in which the sterile neutrinos are forced to decouple. Such
two-stepped phase transitions, in which the two scalar fields develop their
vev’s one after the other, have been considered e.g. in [56]. The assumption
that the shift in the flavon vev is realised during the EWPT simply allows us
to fix the freeze-out temperature to the temperature of the EWPT, i.e.
$T_{\mathrm{fo}}=T_{\mathrm{EW}}$. A more detailed study concerning the
phenomenology of the scalar potential and the phase transition itself,
however, lies beyond the scope of this work.

##  4 Towards a model with a keV neutrino as Dark Matter, active neutrino
masses and lepton flavour hierarchy

Now we attempt to construct a low energy FN model that accomplishes the
following goals:

  1. (i)

generate the light masses and mixing of the active neutrinos in accordance
with experimental data through the type I seesaw mechanism.

  2. (ii)

alleviate the flavour hierarchy in the lepton sector by exploiting the synergy
of the seesaw and FN mechanisms.

  3. (iii)

provide a DM candidate by implementing varying Yukawa couplings in the early
universe to allow sterile neutrinos to be in thermal equilibrium and then
freeze-out into long-lived stability, as described in the previous section.

In other words, the task is to specify the FN charges of all leptons so that,
with ${\cal O}(1)$ elements in all Yukawa matrices, the above conditions are
satisfied. The relevant part of the Lagrangian is given in Eq. 9. We postulate
that the FN suppression factor $\lambda$ has the following values before and
after the EWPT:

$\displaystyle\lambda=\frac{\langle\Theta\rangle}{\Lambda_{\mathrm{FN}}}=\begin{cases}1&:\quad\text{before
EWPT}\\\ \epsilon=0.1&:\quad\text{after EWPT}\end{cases}.$ (18)

Our starting point is the mass Lagrangian for the lepton sector in the broken
EW phase:

$\displaystyle-\mathcal{L}_{m}=\overline{E_{L\alpha}}\,(m_{E})_{\alpha\beta}\,E_{R\,\beta}+\overline{\nu_{L\alpha}}\,(m_{D})_{\alpha
i}\,\nu_{Ri}+\frac{1}{2}\,\overline{\nu_{Ri}^{c}}\,(M)_{ij}\,\nu_{Rj},$ (19)
$\displaystyle(m_{E})_{\alpha\beta}=v\,\epsilon^{q_{\bar{L}_{\alpha}}}\,Y^{E}_{\alpha\beta}\,\epsilon^{q_{E_{\beta}}},\quad(m_{D})_{\alpha
i}=v\,\epsilon^{q_{\bar{L}_{\alpha}}}\,Y^{\nu}_{\alpha
i}\,\epsilon^{q_{N_{i}}},\quad(M)_{ij}=\epsilon^{q_{N_{i}}}\,(M_{N})_{ij}\,\epsilon^{q_{N_{i}}},$
(20)

recalling that $q_{f}$ stands for the FN charge of the fermion $f$ (cf. Table
1). Note that we can write the matrices in Eq. 20 with the help of diagonal
matrices containing the corresponding FN suppression factors, i.e.

$\displaystyle
Q_{\bar{L}}=\mathrm{diag}(\epsilon^{q_{\bar{L}_{e}}},\epsilon^{q_{\bar{L}_{\mu}}},\epsilon^{q_{\bar{L}_{\tau}}}),\qquad
Q_{N}=\mathrm{diag}(\epsilon^{q_{N_{1}}},\epsilon^{q_{N_{2}}},\epsilon^{q_{N_{3}}}),$
(21)

and analogously for $Q_{E}$. Also, we can extract the Majorana scale
$\Lambda_{M}$ from the bare Majorana matrix such that we have
$M_{N}=\Lambda_{M}\,Y^{N},$ with a coefficient matrix $Y^{N}$ whose entries
are all close to $\mathcal{O}(1)$. Then the mass matrices can be written as

$\displaystyle m_{E}=v\,{Q_{L}}\,Y^{E}\,{Q_{E}},\quad
m_{D}=v\,{Q_{L}}\,Y^{\nu}\,{Q_{N}},\quad
M=\Lambda_{M}\,{Q_{N}}\,Y^{N}\,{Q_{N}}.$ (22)

Contrary to the situation in the $\nu$MSM [57, 17], where one can usually
choose a flavour basis in which $m_{E}$ and $M$ are both diagonal, we cannot
do that here because each entry in the matrices above gets an individual FN
suppression factor.

We begin by considering the Majorana matrix. We are aiming for DM sterile
neutrinos with late-time mass in the keV range. From Eq. 11 and Fig. 2 we see
that the early mass eigenvalues should be in the TeV range, i.e.

$\displaystyle\tilde{M}^{d}=\tilde{U}^{T}\,M_{N}\,\tilde{U}=\tilde{U}^{T}\,\Lambda_{M}\,Y^{N}\,\tilde{U}\sim
10^{3}\,\mathrm{GeV}$ (23)

with an orthogonal matrix $\tilde{U}\sim\mathcal{O}(0.1)$ and
$Y^{N}\sim\mathcal{O}(1)$ it follows that we should set

$\displaystyle\Lambda_{M}=10^{4}\,\mathrm{GeV}.$ (24)

We can now estimate the FN charges of the sterile neutrinos necessary to get
suppressed masses in the keV range. The late mass eigenvalues are obtained
with another orthogonal matrix $U$ as

$\displaystyle M^{d}$
$\displaystyle=U^{T}\,Q_{N}(\Lambda_{M}\,Y^{N})\,Q_{N}\,U$ (25)

Demanding that $M^{d}\sim 10^{-6}\,\mathrm{GeV}$ and with
$U\sim\mathcal{O}(0.1)$, $Y^{N}\sim\mathcal{O}(1)$ and
$\Lambda_{M}=10^{4}\,\mathrm{GeV}$ we arrive at

$\displaystyle M^{d}\sim 10^{-6}\,\mathrm{GeV}\quad\text{implying
that}\quad\epsilon^{q_{N_{i}}+q_{N_{j}}}\sim 10^{-9},$ (26)

which means that a good guess would be $q_{N_{i}}\sim 4$. Here we will discuss
two particular choices:

$\displaystyle Q_{N}=\begin{pmatrix}\epsilon^{4}&&\\\ &\epsilon^{4}&\\\
&&\epsilon^{4}\end{pmatrix}\quad\text{as option 1, and}\quad
Q_{N}=\begin{pmatrix}\epsilon^{5}&&\\\ &\epsilon^{4}&\\\
&&\epsilon^{4}\end{pmatrix}\quad\text{as option 2}.$ (27)

In the case of option 1, because all sterile neutrinos have the same FN
charge, meaning that $Q_{N}$ is proportional to the identity matrix, both the
early and late mass matrices $M$ and $\tilde{M}$ are diagonalised by the same
orthogonal matrices, or in other words, $U=\tilde{U}$. This is not the case
for option 2. Up to this point we have been fixed the Majorana scale
$\Lambda_{M}$ and have two possible choices for the FN charges of the sterile
neutrinos such that we the early and late masses of the sterile neutrinos lie
in the required range for successful DM production (see Eqs. 15 and 16 and
Fig. 2).

Next we continue and consider the Dirac matrix $m_{D}$ from Eq. 22. Our goal
is to choose $q_{L_{i}}$ and the entries of $Y^{\nu}$ such that the
requirements from the beginning of this section are satisfied. We start by
considering the Casas-Ibarra parametrization [58], which assumes the type I
seesaw mechanism and parametrizes the Dirac mass matrix with the mass
eigenvalues of the active and sterile neutrinos as input. However, the
standard form of the Casas-Ibarra parametrization assumes that the charged
lepton mass matrix $m_{E}$ and the heavy neutrino mass matrix $M$ can be
chosen to be diagonal - Casas and Ibarra did not have a FN embedding in mind
[58]. In contrary, since we are in a FN model, this is not the case. Therefore
we rederive the Casas-Ibarra parametrization using our non-diagonal matrices
Eq. 22 and obtain

$\displaystyle
m_{D}=i\,V^{\star}\,\sqrt{m_{\nu}^{d}}\,R\,\tilde{U}\sqrt{\tilde{M}^{d}}\,\tilde{U}^{T}\,Q_{N},$
(28)

or equivalently

$\displaystyle
Q_{\bar{L}}\,Y^{\nu}=\frac{i}{v}\,V^{\star}\,\sqrt{m_{\nu}^{d}}\,R\,\tilde{U}\sqrt{\tilde{M}^{d}}\,\tilde{U}^{T}.$
(29)

where $R$ is any arbitrary orthogonal matrix. Here, the matrix $V$ is the
matrix that diagonalises the light neutrino mass matrix $m_{\nu}$, i.e.
$m_{\nu}^{d}=V^{\dagger}m_{\nu}V$. The freedom one has to choose $R$ reflects
the fact that there are multiple possible choices for the entries of $Y^{\nu}$
that lead to the eigenvalues in $m_{\nu}^{d}$. Also, recall from Eq. 6 that to
compute $\theta_{1}^{2}$ we need $m_{D}$,

$\displaystyle\theta_{1}^{2}\propto\sum_{\alpha}|(m_{D})_{\alpha
1}|^{2}=|m_{D}^{\dagger}\,m_{D}|_{11}=(Q_{N}\,\tilde{U}\,\sqrt{\tilde{M}^{d}}\,\tilde{U}^{T}\,R^{\dagger}\,m_{\nu}^{d}\,R\,\tilde{U}\,\sqrt{\tilde{M}^{d}}\,\tilde{U}^{T}\,Q_{N})_{11}.$
(30)

Concerning these last three equations the following remarks are in place:

  * •

The matrix $m_{\nu}^{d}$ contains the masses of the light neutrinos. Although
the absolute scale remains to be determined, we know from oscillation data
that [59]

$\displaystyle\Delta m_{21}^{2}=m_{2}^{2}-m_{1}^{2}=7.5\times
10^{-5}\,\mathrm{eV}^{2},\qquad|\Delta
m_{31}^{2}|=|m_{3}^{2}-m_{1}^{2}|=2.5\times 10^{-3}\,\mathrm{eV}^{2}.$ (31)

Thus, assuming normal ordering and taking the lightest neutrino to be massless
as a benchmark point, we can set

$\displaystyle m_{1}=0\,\mathrm{eV},\qquad m_{2}=8.7\times
10^{-3}\,\mathrm{eV},\qquad m_{3}=5\times 10^{-2}\,\mathrm{eV}.$ (32)

With this, we can now consider $m_{\nu}^{d}$ as known. At the end of this
section we will also investigate the case of non-vanishing $m_{1}$.

  * •

Another ingredient is $V$, which is the matrix that diagonalises $m_{\nu}$. In
the SM, where one can assume the charged lepton mass matrix to be diagonal,
$V$ is just the PMNS matrix. However, within a FN model $m_{E}$ is in general
non-diagonal, but can be diagonalised by two unitary matrices, $W_{L}$ and
$W_{R}$,

$\displaystyle m_{E}^{d}=W_{L}^{\dagger}\,m_{E}\,W_{R}.$ (33)

The PMNS matrix, whose parameters have been measured in oscillation
experiments with a few percent precision [59], is given by
$V_{\mathrm{PMNS}}=W_{L}^{\dagger}\,V$. Then, for a given $W_{L}$, we can
compute $V$ by

$\displaystyle V=W_{L}\,V_{\mathrm{PMNS}}.$ (34)

Thus, by analysing the diagonalization pattern in the charged lepton sector we
can easily obtain the matrix $V$.

  * •

Notice that Eq. 29 is independent of $Q_{N}$, i.e. it is independent of the FN
charges of the sterile neutrinos. We can use Eq. 29 to estimate the FN charges
of the $SU(2)$ lepton doublets. By simply taking $\tilde{M}^{d}\sim
10^{4}\,$GeV, $Y^{\nu}\sim\mathcal{O}(1)$ and
$R,V,\tilde{U}\sim\mathcal{O}(0.1)$ we conclude that $q_{\bar{L}_{i}}=7$, or
in other words

$\displaystyle
Q_{\bar{L}}=\mathrm{diag}(\epsilon^{7},\epsilon^{7},\epsilon^{7}).$ (35)

  * •

From Eq. 30 we see that $\theta_{1}^{2}$ turns out to be independent of $V$.
Also, with regards to $Q_{N}$ it actually only depends on
$(Q_{N})_{11}=\epsilon^{q_{N_{1}}}$, i.e. it is independent of $q_{N_{2,3}}$.
We can again use Eqs. 6 and 30 to get a naive estimate of $\theta_{1}^{2}$ by
taking $\tilde{M}^{d}\sim 10^{4}\,$GeV and $R,\tilde{U}\sim\mathcal{O}(0.1)$.
For a DM mass of $M_{1}=7.1\,$keV we find $\theta_{1}^{2}\sim 10^{-7}$ which
is not small enough to evade the $X$-Ray bound Eq. 7 which demands
$\theta_{1}^{2}\lesssim 4\cdot 10^{-10}$. Therefore, we have to investigate
Eq. 30 further to see if the freedom to choose any orthogonal matrix $R$
allows us to make $\theta_{1}^{2}$ small enough.

By now we have already specified

$\displaystyle\Lambda_{M}=10^{4}\,\mathrm{GeV},\qquad
Q_{L}=\mathrm{diag}(\epsilon^{7},\epsilon^{7},\epsilon^{7}),\qquad
Q_{N}=\begin{cases}\mathrm{diag}(\epsilon^{4},\epsilon^{4},\epsilon^{4}):\quad\text{option
1},\\\ \mathrm{diag}(\epsilon^{5},\epsilon^{4},\epsilon^{4}):\quad\text{option
2}.\end{cases}$ (36)

To continue and be able to properly compute $m_{D}$ and $\theta_{1}^{2}$ from
Eqs. 30 and 28, we must make a choice to specify $V$, $\tilde{U}$ and
$\tilde{M}^{d}$. To determine $V$ we begin by considering the charged leptons,
for which the mass eigenstates are known to be
$m_{E}^{d}=\mathrm{diag}(m_{e},m_{\mu},m_{\tau})=(0.511\times
10^{-3},0.105,1.776)\,\mathrm{GeV}.$ There are infinitely many non-diagonal
matrices $m_{E}$ that have these eigenvalues. We pick a particular choice by
randomly generating two unitary matrices $W_{L}$ and $W_{R}$ and inverting Eq.
33. From the resulting $m_{E}$ we see that a good choice is

$\displaystyle Q_{E}=\mathrm{diag}(\epsilon^{-3},\epsilon^{-4},\epsilon^{-4})$
(37)

in order for Yukawa matrix $Y^{E}$ to have $\mathcal{O}(1)$ entries, or in
other words, we find that $q_{E}=\\{-3,-4,-4\\}$ is the right choice of FN
charges for the charged $SU(2)$ singlets. In this framework, this is the
origin of the hierarchy in the charged lepton masses (see the Appendix for an
example). Furthermore, having selected a specific $W_{L}$ we can then compute
the matrix $V$ by plugging $W_{L}$ into Eq. 34 and using the values from [59]
for the PMNS matrix. After doing so $Y^{E}$, $q_{E}$ and the matrix $V$ can be
considered known.

To determine $\tilde{U}$ and $\tilde{M}^{d}$ we now turn to the Majorana
sector, whereby we first consider option 1 from Eq. 27. Notice first that
because in option 1 all sterile neutrinos have the same FN charge, $Q_{N}$
will be proportional to the unit matrix, i.e.
$Q_{N}=\epsilon^{4}\,\mathbb{1}$, and as a consequence, the late and early
Majorana matrices will be related to each other simply by
$M=\epsilon^{8}\,\tilde{M}$. They will also be diagonalized by the same
orthogonal matrix, i.e. $U=\tilde{U}$ and their diagonal versions will also be
proportional to each other, $M^{d}=\epsilon^{8}\,\tilde{M}^{d}$. This means
that, if we make a choice for the late masses of the sterile neutrinos
$M_{i}$, the early masses $\tilde{M}_{i}$ are also immediately fixed. For the
DM neutrino we set $M_{1}=7.1\,$keV and for the other two sterile neutrinos we
arbitrarily choose $M_{2}=20\,$keV and $M_{3}=30\,$keV; other choices are also
possible and lead to qualitatively similar results as long as $N_{2}$ and
$N_{3}$ do not contribute to the DM density (see comments below and also the
right panel of Fig. 2). Thus, from our choice of

$\displaystyle M^{d}=\mathrm{diag}(7.1,\,20,\,30)\,\mathrm{keV}\qquad\text{it
follows
that}\qquad\tilde{M}^{d}=\mathrm{diag}(710,\,2000,\,3000)\,\mathrm{GeV}.$ (38)

We can see from the left panel of Fig. 2 that the lightest sterile neutrino
can indeed account for 100% of the DM if the EWPT occurs at $T_{EW}\sim
100\,$GeV. Furthermore, when the DM neutrino freezes-out at $T_{EW}$ and sets
the DM relic abundance, the other two heavier neutrinos, which were also in
thermal equilibrium with $\tilde{M}_{2,3}\gtrsim 2000\,$GeV, have already been
depleted by their Boltzmann distributions and do not contribute to
$\Omega_{\mathrm{DM}}h^{2}$, as can also be seen from the right panel of Fig.
2. Again, there are many non-diagonal Majorana matrices that have the
eigenvalues specified in Eq. 38. We can pick a specific one by randomly
generating an orthogonal matrix $\tilde{U}$ and computing
$\tilde{M}=\tilde{U}\,\tilde{M}^{d}\,\tilde{U}^{T}$, and because
$\tilde{M}=\Lambda_{M}\,Y^{N}$ and since we have already fixed
$\Lambda_{M}=10^{4}\,$GeV, we expect to obtain the coefficient matrix
$Y^{N}\sim\mathcal{O}(1)$ close to order unity.

Now we finally have all ingredients to compute $\theta_{1}^{2}$, which, from
Eqs. 6 and 30, is given by

$\displaystyle\theta_{1}^{2}=\frac{|m_{D}^{\dagger}\,m_{D}|_{11}}{(M_{1})^{2}}=\frac{1}{(M_{1})^{2}}\,(Q_{N}\,\tilde{U}\,\sqrt{\tilde{M}^{d}}\,\tilde{U}^{T}\,R^{\dagger}\,m_{\nu}^{d}\,R\,\tilde{U}\,\sqrt{\tilde{M}^{d}}\,\tilde{U}^{T}\,Q_{N})_{11},$
(39)

where the orthogonal matrix $R$ is arbitrary. We are interested in the choices
of $R$ that lead to a very small mixing angle $\theta_{1}^{2}$. As any
orthogonal matrix, $R$ has three independent parameters and one possible
parametrization of it consists of the product of three $3\times 3$ rotation
matrices with angles $\alpha$, $\beta$ and $\gamma$ around three orthogonal
directions respectively. Ideally, we would insert this general parametrization
of $R$ in Eq. 39 and simply compute its global minimum as a function of the
three rotation angles $\alpha$, $\beta$ and $\gamma$. However, the resulting
system of equations is highly nonlinear, which makes the optimazation task
difficult and inefficient. It is more feasible to numerically search for a
local minimum, but for that we need to identify a region in the parameter
space $(\alpha,\beta,\gamma)$ in the vicinity of which the minimum is expected
to be found. To graphically find such regions, we sample values of
$\theta_{1}^{2}$ in a $(\alpha,\beta,\gamma)$ cube. The result, which can be
seen in Fig. 3, indicates the regions of the parameter space
$(\alpha,\beta,\gamma)$ for which $\theta_{1}^{2}$ becomes interestingly small
in dark blue color. These regions appear to form three bands around
$\alpha=-\pi,0,\pi$ and $\beta=-\pi,0,\pi$. The parameter $\gamma$ seems to
have less of an effect.

Figure 3: The mixing angle squared for the lightest sterile neutrino with all
active neutrinos, $\theta_{1}^{2}$, as a function of the free parameters of
the arbitrary orthogonal matrix $R$ from Eq. 39, i.e. the angles
$(\alpha,\beta,\gamma)$ that parametrize $R$ as three consecutive orthogonal
rotations. The smallest values are obtained in the dark blue regions of the
parameter space, which seem to form three bands around $\alpha=-\pi,0,\pi$ and
$\beta=-\pi,0,\pi$.

To have a closer look we plot the plane at $\gamma=0$, which is representative
for the whole cube.

Figure 4: Left panel: contour lines for the mixing angle of the lightest
sterile neutrino $N_{1}$ with all active neutrinos, as defined in Eq. 39.
There are infinitely many neutrino Yukawa matrices $Y^{\nu}$ that could lead
to the eigenvalues defined in Eq. 32. The Casas-Ibarra parametrization
captures this freedom in the three free parameters contained in the arbitrary
orthogonal matrix $R$, which can be parametrised as a general rotation in
three dimensions. Thus, $\theta_{1}^{2}$ in Eq. 39 can be understood as a
function of the three rotation angles $(\alpha,\beta,\gamma)$ contained in the
matrix $R$. In order to be compatible with the $X$-Ray bounds, we want to find
the matrices $R$ that make $\theta_{1}^{2}$ the smallest. In Fig. 3 we sampled
values of $\theta_{1}^{2}$ in the parameter space $(\alpha,\beta,\gamma)$.
This plot is a cut of the cube in Fig. 3 at $\gamma=0$. We clearly find four
different regions in the $(\alpha,\beta)$ plane where $\theta_{1}^{2}$ gets
very small, potentially realising the longevity of the sterile neutrino DM
(the regions of interest in the edges should not be counted twice because of
the periodicity of this parametrization of $R$). As can be seen in Fig. 3,
this cut of the cube at $\gamma=0$ is representative of the whole parameter
space, and cuts at other values of $\gamma$ are qualitatively equivalent to
the one presented here and have always four regions of local minima.
Right panel: We zoom into the local minimum of the upper right corner in the
left panel of the figure and confirm that $\theta_{1}^{2}$ indeed turns small
enough for the DM sterile neutrinos to be safe from the $X$-Ray constrains Eq.
7, which, for $M_{1}=7.1\,$keV demand that $\theta_{1}^{2}\lesssim 4.2\times
10^{-10}$.

The result is $\theta_{1}^{2}$ as a scalar field over the $(\alpha,\beta)$
plane and can be found in Fig. 4, where in the left panel we can see that
there are four regions of interest where $\theta_{1}^{2}$ gets very small (the
regions on the edges should not be counted twice because of periodicity). As
an example, we zoom into the region of interest at $(\alpha,\beta)=(\pi,\pi)$
in the right panel of Fig. 4. For our DM sterile neutrino with mass
$M_{1}=7.1\,$keV, the $X$-Ray bound of Eq. 7 demands $\theta_{1}^{2}\lesssim
4.2\times 10^{-10}$. This bound is respected with $R(\alpha,\beta)$ for any
$(\alpha,\beta)$ point in the right panel of Fig. 4. In fact, the numerically
determined minimum is found at $(\alpha,\beta)=(3.26997,\,3.18663)$ and has a
value of $(\theta_{1}^{2})_{\mathrm{min}}=1.59\times 10^{-20}$. We emphasise
that this is not the only region where we find orthogonal matrices $R$ that
make our sterile neutrino DM compatible with the $X$-Ray bound from Eq. 7:
alone in the left panel of Fig. 4 we have three other such regions, and keep
in mind that the angle $\gamma$, which is also a free parameter of $R$, was
set to $\gamma=0$ in the right panel of Fig. 4, but any other value of
$\gamma$ would lead to qualitatively equivalent results, as we checked. Thus,
there are infinitely many regions in the parameter space of $R$ which allow
our DM neutrino to evade the $X$-Ray bounds. Furthermore, although we arrived
at the result of Fig. 4 after using one randomly generated orthogonal
matrix555We also randomly generated $W_{L}$ to calculate $V$, but that plays
no role in the computation of $\theta_{1}^{2}$; it only comes into the
computation of $m_{D}$, see Eqs. 28 and 30., namely $\tilde{U}$, we checked
that the results of this analysis are qualitatively unaltered if we use a
different $\tilde{U}$ in Eq. 39. In that case Fig. 3 might then look slightly
different, but that is irrelevant; what matters is only whether or not it is
possible to find matrices $R$ such that the $X$-Ray bound is respected, and as
it turns out, the answer yes in all cases. Thus, we conclude that it is always
possible to find many matrices $R$ such that our lightest sterile neutrino
$N_{1}$ is a good DM candidate that satisfies the $X$-Ray constrain Eq. 7.
Also, for many of these viable choices of $R$, the neutrino Yukawa couplings
turn out to be all close to order 1, as are the Yukawa matrix for the charged
leptons $Y^{E}$ and the coefficient matrix for the Majorana mass matrix
$Y^{N}$. The hierarchy in the lepton masses is thus explained by interplay
between the seesaw mechanism and the FN charges of the leptons. And, since we
started our computations with the Casas-Ibarra parametrization, the masses of
the active neutrinos also come out correctly.

We now make a few remarks about option 2 for the choice of FN charges of the
sterile neutrinos (see Eq. 27), where, instead of all sterile neutrinos having
the same charge of 4, we choose $q_{N}=\\{5,\,4,\,4\\}$ which is implies
$Q_{N}=\mathrm{diag}(\epsilon^{5},\,\epsilon^{4},\,\epsilon^{4})$. In this
case, there is an additional complication, namely that the eigenbasis of the
early and late Majorana mass matrices are not the same. Recall that their
definition is

$\displaystyle\tilde{M}=\Lambda_{M}\,Y^{N},\qquad
M=\Lambda_{M}\,{Q_{N}}\,Y^{N}\,{Q_{N}}.$ (40)

If the sterile neutrinos have different FN charges, then these two matrices
are not proportional to each other, which means that they will be diagonalized
by two different matrices $\tilde{U}\neq U$ and their diagonal versions also
will not be proportional to each other. We can still choose the late Majorana
mass eigenvalues as $M^{d}=\mathrm{diag}(7.1,\,20,\,30)\,\mathrm{keV}$ and use
a randomly generated orthogonal matrix $U$ to find $M=U\,M^{d}\,U^{T}$. Then
to calculate $\theta_{1}^{2}$ and check for the longevity of the DM neutrino,
we need to first compute the early mass matrix

$\displaystyle\tilde{M}=Q_{N}^{-1}\,M\,Q_{N}^{-1}$ (41)

and then compute its eigenvalues, which are the elements of $\tilde{M}^{d}$,
and its normalized eigenvectors, which are the columns of $\tilde{U}$.
However, recall from Eq. 15 and Fig. 2 that successful DM production with
$M_{1}=7.1\,$keV and at reasonable freeze-out temperatures for the EWPT, say
$T_{EW}\in[100,\,300]\,$GeV, requires that $\tilde{M}_{1}\in[700,\,2000]\,$GeV
and $M_{2,3}\gtrsim 3000\,$GeV. Thus, we must make sure that the early mass
eigenvalues of the sterile neutrinos satisfy these conditions666When the FN
charges of all sterile neutrinos are the same $q_{N_{i}}=q_{N}$, this is not
necessary, because the late and early Majorana mass eigenvalues are
proportional to each other as $M^{d}=\epsilon^{2\,q_{N}}\,\tilde{M}^{d}$ and
we can therefore choose the late eigenvalues such that the early eigenvalues
lie in the right range for successful DM genesis, which depend on the randomly
generated matrix $U$. We checked that DM genesis is successful for $\sim 70\%$
of the randomly generated $U$ matrices, which shows that, even with this
choice of FN charges for the sterile neutrinos, it is not rare to have early
and late mass eigenvalues in the right range for FM production. As a
representative example, consider a case in which the early Majorana mass
eigenvalues were
$\tilde{M}^{d}=\mathrm{diag}(1.19,\,2.86,\,124.94)\,\mathrm{TeV}$. The
lightest sterile neutrino contributes 100% of the DM density for
$T_{EW}\approx 165\,$GeV and the other two sterile neutrinos do not contribute
to it at all, as can be seen in Fig. 5.

Figure 5: Contribution to the DM abundance as a function of the early and late
sterile neutrino mass for freeze-out at $T_{EW}=165\,$GeV and FN charges of
the sterile neutrinos given by $q_{N}=\\{5,\,4,\,4\\}$. The dashed line marks
the position of our DM neutrino at $M_{1}=7.1\,$keV. One clearly sees that
with a corresponding early mass eigenvalue of $\tilde{M}_{1}\approx 1.2\,$TeV
the lightest sterile neutrino $N_{1}$ makes up 100% of the DM relic abundance.
The other two sterile neutrinos with late masses $M_{2}=20\,$keV and
$M_{3}=30\,$keV and early masses $\tilde{M}_{2,3}\gtrsim 2.5\,$TeV contribute
practically nothing to the DM density as their equilibrium density has already
been completely depleted by their Boltzmann suppression at the moment of
freeze-out.

Now that we know that DM production is successful with these mass eigenvalues
and freeze-out temperature $T_{EW}\approx 165\,$GeV, we can then plug
$\tilde{M}^{d}$ and $\tilde{U}$ into Eq. 39 and repeat the previous analysis
using the same parametrization of $R$ as a general rotation matrix in three
dimensions to see if the DM neutrinos are sufficiently long lived. The result
looks qualitatively very similar to Fig. 4; there are four local minima in the
vicinity of $(\alpha,\,\beta)=(0,0),\,(0,\pi),\,(\pi,0),\,(\pi,\pi)$, the
value of the minima is $\theta_{1}^{2}\sim 10^{-21}$, which means that the
$X$-Ray bound Eq. 7 are satisfied within infinitely many regions of the
parameter space of $R$ around the local minima. This shows that for this new
DM production mechanism to work it is not necessary that all sterile neutrinos
have the same FN charges.

Up to this point we have focused on a DM neutrino with mass $M_{1}=7.1\,$keV,
which is particularly interesting because a suspicious $X$-Ray signal has been
detected [60]. Now we turn to the possibility of DM neutrinos with masses
different than $M_{1}=7.1\,$keV. For this it is best to return to the previous
choice of FN charges for the sterile neutrinos, i.e. option 1 from Eq. 27,
where all three sterile neutrinos have the same FN charge
$q_{N}=\\{4,\,4,\,4\\}$. We will soon see why this is convenient. Our starting
point is the same choice of late sterile neutrino mass eigenvalues with which
we have been working all along, namely
$M^{d}=\mathrm{diag}(7.1,\,20,\,30)\,$keV. The corresponding early mass
eigenvalues are $\tilde{M}^{d}=\mathrm{diag}(0.71,\,2,\,3)\,$TeV. By
multiplying the late mass eigenvalues with a dimensionless parameter $s$ and
letting $s$ vary, we can homogeneously vary all late and early mass
eigenvalues, because with the current choice of FN charges $M^{d}$ and
$\tilde{M}^{d}$ are proportional to each other. Of course, for $s=1$ we
recover the case that we have been studying so far. Next we determine how
$\theta_{1}^{2}$ changes when the sterile neutrino mass eigenvalues vary as
determined by the dimensionless parameter $s$. From Eq. 39 we see that
$\theta_{1}^{2}$ is proportional to two powers of $\sqrt{\tilde{M}^{d}}$ while
being inversely proportional to $(M_{1})^{2}$. Consequently, the dependence is
$\theta_{1}^{2}\propto s^{-1}$. This allows us to extrapolate from the point
$(M_{1}=7.1\,\mathrm{keV},\,\theta_{1}^{2}\approx 10^{-21})$ to span a wide
area of the $(M_{1},\,\theta_{1}^{2})$ plane, the results of which can be seen
in Fig. 6. We checked that the extrapolation delivers correct results by
explicitly computing $\theta_{1}^{2}$ for different combinations of late and
early mass eigenvalues. At this point we also relax the fix-point assumption
that one of the active neutrinos is massless and compute one point in the
parameter space to extrapolate from for non-vanishing values of $m_{1}$. We
find that the ability of this DM candidate to evade the $X$-Ray bounds relies
on the lightest active neutrino being massless or at least much lighter than
the other two active neutrinos. In Fig. 6 each solid black stands for a
different mass for the lightest active neutrino, with the line at the bottom
representing the case of one massless active neutrino. The region above each
line is the portion of the parameter space where the FN model and DM
production mechanism proposed here delivers a viable DM candidate, the
lightest sterile neutrino, which is stable on cosmological timescales and
evades the $X$-Ray constrains. Furthermore, for $s=7.05$ the late DM mass
reaches $M_{1}=50\,$keV while the corresponding early mass reaches slightly
over $\tilde{M}_{1}=5\,$TeV. For this combination of masses to achieve
successful DM production according to Eq. 15, the freeze-out temperature would
need to be $T_{EW}\approx 550\,$GeV, which is not unreasonable.

Figure 6: The parameter space for sterile neutrino DM. The red region in the
upper part of the plot is excluded by the $X$-Ray survey NuSTAR GC (2016)
[19]. In the framework presented in this work, how far down the mixing angle
can be pushed depends on the absolute mass scale of the active neutrinos. Each
solid line gives, for the denoted mass of the lightest active neutrino, the
smallest mixing angle that could be reached with the given DM mass, while
still recovering the known squared mass differences and oscillation data of
the active neutrinos. Thus, the area above each line is the region of the
parameter space where the framework proposed here gives us a viable DM
candidate, the lightest sterile neutrino produced by the thermal freeze-out
induced during the EWPT. If the lightest active neutrino is massless, then
$\sin^{2}(2\theta_{1})$ can be as small as $10^{-21}$ and safely evade all
$X$-Ray bounds.

##  5 Conclusions

In this work, we have proposed a new mechanism for keV sterile neutrino DM
genesis in the early Universe that relies neither on the oscillations between
active and sterile neutrinos nor on heavier parent particles which produce the
DM sterile neutrinos in their decay. Instead, the sterile neutrinos are
produced by freeze-out from thermal equilibrium, much like a typical WIMP,
without introducing any additional coupling or interactions for the sterile
neutrinos beyond their Yukawa term, by which they generate the masses of the
light neutrinos through the type I seesaw mechanism, and the FN interactions
encoded by the flavon and visible only in the UV-complete theory. This
possibility has not been considered in the past because for the sterile
neutrinos to be in thermal equilibrium their Yukawa couplings would need to be
sizeable, which would compromise their stability. Indeed, the decay of sterile
neutrinos with a non-vanishing Yukawa coupling is unavoidable and has been
searched for with $X$-Ray surveys, placing strict upper bounds on the mixing
angle between sterile and active neutrinos, and equivalently, on the Yukawa
coupling. As we pointed out previously in [51], this can be solved by varying
Yukawa couplings: if, by some mechanism, the Yukawa couplings were large at
early times but became very small at later times, then the sterile neutrinos
could have been in thermal equilibrium in the early Universe, then decouple
and stay stable on cosmological timescales thereafter.

Here we have presented an implementation of varying Yukawa couplings by
formulating a FN model for the leptonic sector expanded by three sterile
Majorana neutrinos. In this framework the Yukawa couplings and the Majorana
mass matrix of the sterile neutrinos are rescaled powers of the FN suppression
factor, which is proportional to the vev of the flavon. We argue that during
the Electroweak phase transition, when the vev of the Higgs changes from 0 to
$v$ as the scalar potential finds its true minimum, it is reasonable to expect
that the vev of the flavon will change too. We have shown that, assuming for
concreteness that the FN factor was close to 1 before the Electroweak phase
transition and 0.1 afterwards, it is possible for the sterile neutrinos to be
produced by thermal freeze-out and later evade the current $X$-Ray constrains.
Because of their unsuppressed Yukawa couplings, the sterile neutrinos, like
all other fermions, are in thermal equilibrium before the Electroweak phase
transition. When the phase transitions occurs and the vev’s of the Higgs and
the flavon get their late-time values, the Yukawa couplings get suppressed,
and in the case of the sterile neutrinos this suppression is so drastic that
they are forced to decouple from the thermal bath and are able to stay
compatible with the $X$-Ray constrains. In this sense it is the phase
transition itself which induces the decoupling of the sterile neutrinos.
Furthermore, the change in the FN factor during the phase transition also
provokes a change in the masses of the sterile neutrinos. A consequence of
this is the fact that reproducing the observed relic abundance by this induced
freeze-out mechanism relies on the interplay between the early- and late
masses of the sterile neutrinos as well as on the temperature of the phase
transition, which is also the temperature of freeze-out. For DM masses in the
keV range, the Majorana scale and the masses of the sterile neutrinos in the
early Universe should be on the TeV scale and the temperature of the phase
transition should lie between $100-600\,$GeV.

In the specific FN realization proposed here, we show that successful DM
genesis is possible while also obtaining Yukawa couplings and entries in the
Majorana coefficient matrix that are all of order unity or close to it, thus
somewhat alleviating the flavour puzzle in the lepton sector by the different
FN charges of the leptons, and avoiding a severe hierarchy in the Yukawa
couplings. The smallness of the active neutrino masses and oscillation
phenomena are explained by the synergy of the FN model and the type I seesaw
mechanism. We empirically find that the smallest value of the mixing angle of
the DM neutrino with all active neutrinos that can be achieved in this
framework depends of the absolute scale of the active neutrino masses, and is
minimal in the case of one massless active neutrino.

At this point, one remark concerning the spectrum of the sterile neutrino dark
matter produced by this mechanism is in place. Right until the moment of
induced freeze-out by the phase transition, the sterile neutrinos have a non-
relativistic thermal spectrum, i.e. it would be cold DM. The question of
whether this changes after the phase transition due to the effective
suppression of the Majorana masses is non-trivial and will be connected to the
dynamics of the phase transition. In Ref. [55] it has been discussed that
larger Yukawa couplings could lead to the EWPT becoming a first order phase
transition, for which the friction between the bubble wall and the plasma
plays an important role in the wall dynamics, among other things [61]. In
essence, the question at hand boils down to how the coupled particles exchange
energy and momentum during the phase transition across the bubble walls. A
detailed analysis of this issue lies beyond the scope of this study and we
leave it for future work.

In summary, with only three additional sterile neutrinos and one flavon (which
encodes the FN messengers that are integrated out at low energies - of course)
the framework proposed here allows us to alleviate the flavour problem, the
origin of the masses of active neutrinos and their oscillation phenomena and
produces a viable and appealing DM candidate. Future investigations should
look into the possibility of also addressing the baryogenesis/leptogenesis
problem and explore other possible implementations of varying Yukawa couplings
to achieve the same goals.

Acknowledgements

I gratefully thank Dr. Salvador Centelles Chulia, Dr. Sudip Jana and Dr.
Andreas Trautner for many invaluable discussions and feedback as well as Prof.
Dr. Manfred Lindner for his support.

##  Appendix A One specific benchmark point in the FN model

We start with the following mass eigenvalues for the SM leptons:

$\displaystyle m_{E}^{d}$ $\displaystyle=\mathrm{diag}(0.511\times
10^{-3},\,0.105,\,1.776)\,\mathrm{GeV},$ (42) $\displaystyle m_{\nu}^{d}$
$\displaystyle=\mathrm{diag}(0,\,8.7\times 10^{-3},\,5\times
10^{-2})\,\mathrm{eV},$ (43)

and for the mass of the sterile neutrinos before the EWPT we choose

$\displaystyle\tilde{M}^{d}$
$\displaystyle=\mathrm{diag}(710,\,2000,\,3000)\,\mathrm{GeV}.$ (44)

The FN charges for the charged singlets, the doublets, and the sterile
neutrinos are taken to be

$\displaystyle q_{E}=\\{-3,\,-4,\,-4\\},\qquad q_{L}=\\{7,\,7,\,7\\},\qquad
q_{N}=\\{4,\,4,\,4\\}.$ (45)

respectively. The bare Majorana mass matrix is given by
$\tilde{M}=\Lambda_{M}\,Y^{N}$, where we set $\Lambda_{M}=10^{4}\,$GeV, and is
related to its diagonal version by the orthogonal matrix $\tilde{U}$, which we
set to

$\displaystyle\tilde{U}=\begin{pmatrix}-0.200&-0.696&-0.690\\\
0.868&0.209&-0.460\\\ -0.465&0.687&-0.559\end{pmatrix},\quad\text{leading
to}\quad Y^{N}=\begin{pmatrix}0.242&0.054&0.027\\\ 0.054&0.125&0.077\\\
0.027&0.077&0.203\end{pmatrix}$ (46)

After the EWPT, the suppressed mass eigenvalues of the sterile neutrinos are

$\displaystyle M^{d}$
$\displaystyle=\mathrm{diag}(7.1,\,20,\,30)\,\mathrm{keV}.$ (47)

Since in this case all sterile neutrinos have the same FN charge, the
eigenbasis of the matrices $\tilde{M}$ and $M$ are the same. The randomly
generated unitary matrix $W_{L}$, used to diagonalize the charged lepton mass
matrix by $m_{E}^{d}=W_{L}^{\dagger}\,m_{E}\,W_{R}$ and compute the matrix
$V=W_{L}\,V_{\mathrm{PMNS}}$ used in Eq. 28 is given by

$\displaystyle
W_{L}=\begin{pmatrix}-0.598+i\,0.289&-0.508-i\,0.023&-0.548+i\,0.009\\\
-0.377+i\,0.531&0.558+i\,0.123&0.161-i\,0.474\\\
-0.364-i\,0.057&0.485-i\,0.424&-0.054+i\,0.668\end{pmatrix}.$ (48)

Finally, the Yukawa matrices for the neutrinos and charged singlets are

$\displaystyle|Y^{\nu}|=\begin{pmatrix}0.030&7.381&1.031\\\
0.029&4.681&4.068\\\
0.025&3.308&4.142\end{pmatrix},\qquad|Y^{E}|=\begin{pmatrix}3.14&2.59&2.98\\\
1.89&2.21&2.86\\\ 5.57&3.28&3.53\end{pmatrix}.$ (49)

Although there still is some hierarchy in the columns of $Y^{\nu}$, it is
remarkably less severe. The mixing angle for the DM neutrino with all active
neutrinos is $\theta_{1}^{2}=2.86\times 10^{-10}$ which is below the bound of
$\theta_{1\,\mathrm{bound}}^{2}=4.16\times 10^{-10}$ at a DM mass
$M_{1}=7.1\,$keV.

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