IFIC/21-19

Dark matter in a charged variant of the Scotogenic model

Valentina De Romeri${}^{\text{a}}$, Miguel Puerta${}^{\text{a}}$, Avelino
Vicente${}^{\text{a,b}}$

${}^{(\text{a})}$ Instituto de Física Corpuscular, CSIC-Universitat de
València, 46980 Paterna, Spain

${}^{(\text{b})}$ Departament de Física Teòrica, Universitat de València,
46100 Burjassot, Spain

<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Scotogenic models are among the most popular possibilities to link dark matter
and neutrino masses. In this work we discuss a variant of the Scotogenic model
that includes charged fermions and a doublet with hypercharge $3/2$. Neutrino
masses are induced at the one-loop level thanks to the states belonging to the
dark sector. However, in contrast to the standard Scotogenic model, only the
scalar dark matter candidate is viable in this version. After presenting the
model and explaining some particularities about neutrino mass generation, we
concentrate on its dark matter phenomenology. We show that the observed dark
matter relic density can be correctly reproduced in the usual parameter space
regions found for the standard Scotogenic model or the Inert Doublet model. In
addition, the presence of the charged fermions may open up new regions,
provided some tuning of the parameters is allowed.

##  1 Introduction

The Standard Model (SM) of particle physics fails to address two of the most
important open questions in current fundamental physics: the origin of
neutrino masses and the nature of dark matter (DM). While these two issues
might be completely independent, and the explanation to the DM puzzle might
even come from a completely different branch of physics, it is tempting to
explore extensions of the SM in which they are simultaneously addressed.

Many radiative neutrino mass models are good examples of such extensions. In
this class of models, pioneered in [1, 2, 3, 4], neutrino masses vanish at
tree-level but become non-zero once loop corrections are included. This
naturally explains the smallness of neutrino masses, which get suppressed by
the usual loop factors. In many cases, they are induced at one-loop, but there
are many well-known examples leading to neutrino masses at higher loop orders,
see [5] for a review. A symmetry is often introduced to avoid the generation
of neutrino masses at tree-level. Interestingly, the lightest state charged
under this symmetry becomes completely stable and hence can be a viable DM
candidate, provided it has the correct quantum numbers [6].

One of the most popular models with this feature is the so-called Scotogenic
model [7]. This economical setup extends the SM particle content with three
singlet fermions and one inert scalar doublet, all odd under a
$\mathbb{Z}_{2}$ parity. Neutrino masses are induced at the one-loop level and
the lightest $\mathbb{Z}_{2}$-odd particle, which might be a fermion or a
scalar state, becomes stable and can play the role of DM candidate. Both
possibilities have been studied in detail and shown to be valid options. The
fermion DM candidate can be produced in the early Universe via its Yukawa
interactions. Although the bounds from lepton flavor violating (LFV) processes
[8] strongly limit the allowed parameter space in this case, the observed DM
relic density can be achieved [9] (see also [10, 11, 12, 13]). Qualitatively
similar conclusions were recently found in a variant of the Scotogenic model
with Dirac fermion DM [14]. The scalar DM candidate does not suffer from this
limitation since it can be produced via gauge and/or scalar interactions. In
this case the DM phenomenology resembles that of the Inert Doublet model [15,
16, 17, 18, 19].

While the original Scotogenic model is a very attractive and economical setup,
it is also interesting to consider variations with a richer phenomenology.
111The Scotogenic model can be generalized in many different ways, see [20]
and references therein. Furthermore, the Scotogenic mechanism can be used to
induce a small one-loop mass for a light DM candidate [21]. For instance, a
simple extension including both singlet and triplet fermions [22] already
leads to novel phenomenological signatures [23, 24]. In this work we consider
a variant of the Scotogenic model originally introduced in [25]. In addition
to the usual inert doublet, in this case one introduces charged fermions and a
doublet with hypercharge $3/2$. As a consequence, the particle spectrum
contains many new charged states, including a doubly-charged scalar. The
presence of these states naturally leads to a much richer collider
phenomenology [25]. The aim of our work is to explore the DM phenomenology of
the model. In contrast to the minimal Scotogenic model, only the scalar dark
matter candidate is viable in this version, since the rest of the
$\mathbb{Z}_{2}$-odd states are electrically charged. We will show that the
observed DM relic density can be correctly reproduced in this model and
identify the regions of the parameter space where this is achieved.

The rest of the manuscript is organized as follows. The model is introduced in
Sec. 2, where detailed discussions on the scalar sector, the neutrino mass
generation mechanism and the DM candidate can be found. Sec. 3 describes the
approach followed in our numerical analysis. This Section also discusses the
most relevant experimental bounds in our setup. Our results are given in Sec.
4, whereas a summary with the main conclusions derived from our work is given
in Sec. 5. Finally, additional details are given in Appendices A and B.

##  2 The model

| $q_{L}$ | $u_{R}$ | $d_{R}$ | $\ell_{L}$ | $e_{R}$ | $\psi_{L}$ | $\psi_{R}$ | $H$ | $\eta$ | $\Phi$
---|---|---|---|---|---|---|---|---|---|---
$\rm SU(3)_{C}$ | $\mathbf{3}$ | $\mathbf{\bar{3}}$ | $\mathbf{\bar{3}}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$
$\rm SU(2)_{L}$ | $\mathbf{2}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{2}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{2}$ | $\mathbf{2}$ | $\mathbf{2}$
$\rm U(1)_{Y}$ | $\frac{1}{6}$ | $\frac{2}{3}$ | $-\frac{1}{3}$ | $-\frac{1}{2}$ | $-1$ | $-1$ | $-1$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $\frac{3}{2}$
$\mathbf{Z}_{2}$ | $+$ | $+$ | $+$ | $+$ | $+$ | $-$ | $-$ | $+$ | $-$ | $-$
Generations | 3 | 3 | 3 | 3 | 3 | 2 | 2 | 1 | 1 | 1
Table 1: Particle content of the model. $q_{L}$, $\ell_{L}$, $u_{R}$, $d_{R}$,
$e_{R}$ and $H$ are the usual SM fields.

We consider the variant of the original Scotogenic model introduced in [25].
The SM particle content is extended by adding two $\rm SU(2)_{L}$ doublets,
$\eta$ and $\Phi$, with hypercharge $1/2$ and $3/2$, respectively, and two
pairs of singlet vector-like fermions $\psi_{L,R}^{a}$ ($a=1,2$) with
hypercharge $-1$. The scalar doublets of the model can be decomposed into $\rm
SU(2)_{L}$ components as

$H=\begin{pmatrix}H^{+}\\\
H^{0}\end{pmatrix}\,,\quad\eta=\begin{pmatrix}\eta^{+}\\\
\eta^{0}\end{pmatrix}\,,\quad\Phi=\begin{pmatrix}\Phi^{++}\\\
\Phi^{+}\end{pmatrix}\,.$ (1)

Here $H$ is the usual SM Higgs doublet. As in the Scotogenic model, we impose
an exact $\mathbb{Z}_{2}$ parity. All the new particles are odd under this
symmetry, while the SM particles are assumed to be even. The particle content
of the model is summarized in Table 1. We note that $H$ and $\eta$ are only
distinguished by their $\mathbb{Z}_{2}$ charges. In fact, the scalar doublet
$\eta$ has the same quantum numbers as the usual inert doublet present in the
Inert Scalar Doublet model [15] and the Scotogenic model [7].

The most general Yukawa Lagrangian involving the new particles can be written
as

$\mathcal{L}_{Y}=M_{\psi}\,\overline{\psi}_{L}\,\psi_{R}+Y^{L}\,\overline{\ell_{L}^{c}}\,\Phi\,\psi_{L}+Y^{R}\,\overline{\ell_{L}}\,\eta\,\psi_{R}+\text{h.c.}\,,$
(2)

where $M_{\psi}$ is a $2\times 2$ vector-like (Dirac) mass matrix, which we
take to be diagonal without loss of generality, while $Y^{L}$ and $Y^{R}$ are
dimensionless $3\times 2$ complex matrices. The most general scalar potential
is given by

$\displaystyle\mathcal{V}=\,$ $\displaystyle\mu^{2}_{1}\,\lvert
H\rvert^{2}+\mu^{2}_{2}\,\lvert\eta\rvert^{2}+\mu_{\Phi}^{2}\,\lvert\Phi\rvert^{2}+\frac{1}{2}\,\lambda_{1}\,\lvert
H\rvert^{4}+\frac{1}{2}\,\lambda_{2}\,\lvert\eta\rvert^{4}+\frac{1}{2}\,\lambda_{\Phi}\,\lvert\Phi\rvert^{4}$
$\displaystyle+$ $\displaystyle\lambda_{3}\,\lvert
H\rvert^{2}\,\lvert\eta\rvert^{2}+\lambda_{4}\,\lvert
H^{\dagger}\eta\rvert^{2}+\rho_{1}\,\lvert
H\rvert^{2}\,\lvert\Phi\rvert^{2}+\rho_{2}\,\lvert\eta\rvert^{2}\,\lvert\Phi\rvert^{2}+\sigma_{1}\,\lvert
H^{\dagger}\Phi\rvert^{2}+\sigma_{2}\,\lvert\eta^{\dagger}\Phi\rvert^{2}$ (3)
$\displaystyle+$
$\displaystyle\frac{1}{2}\left[\lambda_{5}\,(H^{\dagger}\eta)^{2}+\text{h.c.}\right]+\left[\kappa\,(\Phi^{\dagger}H)(\eta
H)+\text{h.c.}\right]\,,$

where $\mu_{1}$, $\mu_{2}$ and $\mu_{\Phi}$ are parameters with dimension of
mass and $\lambda_{j}$ ($j=1,2,3,4,5$), $\lambda_{\Phi}$, $\rho_{1}$,
$\rho_{2}$, $\sigma_{1}$, $\sigma_{2}$ and $\kappa$ are dimensionless. We note
that in the presence of a non-zero $\kappa$ term it is not possible to define
a conserved lepton number. As shown below, this plays a crucial role for the
generation of neutrino masses.

###  2.1 Scalar sector

Let us now discuss the resulting scalar particle content of the model. We will
assume a minimum of the potential characterized by the vacuum expectation
values (VEVs)

$\langle H\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}0\\\
v\end{pmatrix}\,,\quad\langle\eta\rangle=0\,,\quad\langle\Phi\rangle=0\,,$ (4)

with $v\simeq 246$ GeV the usual SM VEV. This vacuum preserves the
$\mathbb{Z}_{2}$ parity, which remains unbroken after electroweak symmetry
breaking. This forbids the mixing between $H$, even under $\mathbb{Z}_{2}$,
and the $\eta$ and $\Phi$ doublets, odd under $\mathbb{Z}_{2}$. The neutral
$H^{0}$ and $\eta^{0}$ states can be split into their CP-even and CP-odd
components as

$\displaystyle H^{0}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(h+i\,A+v\right)\,,$ (5)
$\displaystyle\eta^{0}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\eta_{R}+i\,\eta_{I}\right)\,.$ (6)

The CP-even state $h$ can be identified with the SM Higgs boson, with
$m_{h}\approx 125$ GeV, while $A$ is the Goldstone boson that becomes the
longitudinal component of the $Z$ boson. Assuming that CP is conserved in the
scalar sector, $\eta_{R}$ and $\eta_{I}$ do not mix. Their masses are given by

$m_{\eta_{R,I}}^{2}=\mu_{2}^{2}+\frac{1}{2}\left(\lambda_{3}+\lambda_{4}\pm\lambda_{5}\right)v^{2}\,.$
(7)

The charged component of $H$ becomes the longitudinal component of the $W$
boson. The charged component of $\eta$ mixes with the singly-charged component
of $\Phi$. Their Lagrangian mass term can be written as

$\mathcal{L}_{m}=\begin{pmatrix}\eta^{-}&\Phi^{-}\end{pmatrix}\,\begin{pmatrix}m^{2}_{\eta^{\pm}}&m^{2}_{12}\\\
m^{2}_{12}&m^{2}_{\Phi^{\pm}}\end{pmatrix}\,\begin{pmatrix}\eta^{+}\\\
\Phi^{+}\end{pmatrix}+\text{h.c.}\,,$ (8)

with

$\displaystyle m^{2}_{\eta^{\pm}}$
$\displaystyle=\mu_{2}^{2}+\frac{1}{2}\,\lambda_{3}\,v^{2}\,,$ (9)
$\displaystyle m^{2}_{\Phi^{\pm}}$
$\displaystyle=\mu_{\Phi}^{2}+\frac{1}{2}\,\left(\rho_{1}+\sigma_{1}\right)\,v^{2}\,,$
(10) $\displaystyle m^{2}_{12}$ $\displaystyle=\frac{1}{2}\,\kappa\,v^{2}\,.$
(11)

The gauge eigenstates $\eta^{+}$ and $\Phi^{+}$ are related to the mass
eigenstates $H_{1}^{+}$ and $H_{2}^{+}$ as

$\begin{pmatrix}H_{1}^{+}\\\
H_{2}^{+}\end{pmatrix}=\begin{pmatrix}c_{\chi}&s_{\chi}\\\
-s_{\chi}&c_{\chi}\end{pmatrix}\,\begin{pmatrix}\eta^{+}\\\
\Phi^{+}\end{pmatrix}\,,$ (12)

where $c_{\chi}=\cos\chi$, $s_{\chi}=\sin\chi$ and $\chi$ is a mixing angle.
The Lagrangian mass term can be written in terms of the mass eigenstates
$H_{1}^{+}$ and $H_{2}^{+}$ as

$\mathcal{L}_{m}=\begin{pmatrix}H_{1}^{-}&H_{2}^{-}\end{pmatrix}\,\begin{pmatrix}m_{H_{1}^{\pm}}&0\\\
0&m_{H_{2}^{\pm}}\end{pmatrix}\,\begin{pmatrix}H_{1}^{+}\\\
H_{2}^{+}\end{pmatrix}+\text{h.c.}\,.$ (13)

This allows us to obtain the mixing angle $\chi$. Combining Eqs. (8) and (13)
with Eq. (12) one finds

$c_{\chi}\,s_{\chi}=\frac{m^{2}_{H^{\pm}_{1}}-m^{2}_{H^{\pm}_{2}}}{\left[\frac{(m^{2}_{\Phi^{\pm}}-m^{2}_{\eta^{\pm}})^{2}}{m^{2}_{12}}+4\,m^{2}_{12}\right]}\approx\frac{m^{2}_{12}}{m^{2}_{H_{1}^{\pm}}-m^{2}_{H_{2}^{\pm}}}=\frac{\kappa\,v^{2}}{2\,(m^{2}_{H_{1}^{\pm}}-m^{2}_{H_{2}^{\pm}})}\,,$
(14)

where in the second step we have assumed $\chi$ to be a small mixing angle or,
equivalently, $\kappa\ll 1$. Finally, the mass of the doubly-charged
$\Phi^{++}$ is given by

$m_{\Phi^{++}}^{2}=\mu_{\Phi}^{2}+\frac{1}{2}\,\rho_{1}\,v^{2}\,.$ (15)

###  2.2 Neutrino mass generation

Figure 1: One-loop diagram which generates neutrino masses in the variant of
the Scotogenic model under consideration. Due to electric charge conservation,
in each vertex the relevant components of $\eta$ and $\Phi$ are the singly-
charged ones.

In this model, neutrino masses are induced at the one-loop level by the
diagram in Fig. 1. 222All Feynman diagrams in our paper were made with
JaxoDraw [26, 27]. The $\mathbb{Z}_{2}$-odd states in the loop are
electrically charged, in contrast to the original Scotogenic model, which has
electrically neutral states running in the loop. The resulting neutrino mass
matrix can be easily computed with the general expressions in Appendix A. In
this model, $n_{S}=n_{F}=2$, with $S\equiv\\{H_{1}^{\pm},H_{2}^{\pm}\\}$ and
$F\equiv\\{\psi_{1},\psi_{2}\\}$. The masses of the scalars in the loop are
$m^{2}_{H^{\pm}_{1}}$ and $m^{2}_{H^{\pm}_{2}}$, defined in Eq. (13), whereas
the masses of the fermions are $m_{\psi^{1}}$ and $m_{\psi^{2}}$, with
$m_{\psi^{b}}$ ($b=1,2$) the diagonal components of the matrix $M_{\psi}$,
introduced in Eq. (2). One can also read the couplings of $H^{\pm}_{1}$ and
$H^{\pm}_{2}$ from Eqs. (2) and (12), obtaining

$\displaystyle Y_{H^{\pm}_{1}}^{L}$ $\displaystyle\equiv
Y^{L}\,s_{\chi}\,,\quad Y_{H^{\pm}_{1}}^{R}\equiv Y^{L}\,c_{\chi}\,,$ (16)
$\displaystyle Y_{H^{\pm}_{2}}^{L}$ $\displaystyle\equiv
Y^{R}\,c_{\chi}\,,\quad Y_{H^{\pm}_{2}}^{R}\equiv-Y^{R}\,s_{\chi}\,.$ (17)

Therefore, the contribution to $(m_{\nu})_{\alpha\beta}$ of the diagram
displayed in Fig. 1 is proportional to $Y^{L}_{\alpha b}\,Y^{R}_{\beta b}$,
with $b$ the fermion index. It is easy to realize that there is a mirror
contribution proportional to $Y^{L}_{\beta b}\,Y^{R}_{\alpha b}$, obtained
after exchanging the scalars. In summary, the neutrino mass matrix can be
written as

$(m_{\nu})_{\alpha\beta}=\sum_{b=1}^{2}\left[(m_{\nu})^{\\{H_{1}^{\pm},\psi^{b}\\}}_{\alpha\beta}+(m_{\nu})^{\\{H_{2}^{\pm},\psi^{b}\\}}_{\alpha\beta}\right]\,,$
(18)

with

$\displaystyle(m_{\nu})^{\\{H_{1}^{\pm},\psi^{b}\\}}_{\alpha\beta}=\frac{Y^{L}_{\alpha
b}\,Y^{R}_{\beta b}+Y^{R}_{\alpha b}\,Y^{L}_{\beta
b}}{16\pi^{2}}\,s_{\chi}\,c_{\chi}\,m_{\psi^{b}}\,\left[\Delta_{\epsilon}+1-\frac{m^{2}_{\psi^{b}}\,\log
m^{2}_{\psi^{b}}-m^{2}_{H_{1}^{\pm}}\log
m^{2}_{H_{1}^{\pm}}}{m^{2}_{\psi^{b}}-m^{2}_{H_{1}^{\pm}}}\right]\,,$ (19)
$\displaystyle(m_{\nu})^{\\{H_{2}^{\pm},\psi^{b}\\}}_{\alpha\beta}=-\frac{Y^{L}_{\alpha
b}\,Y^{R}_{\beta b}+Y^{R}_{\alpha b}\,Y^{L}_{\beta
b}}{16\pi^{2}}\,s_{\chi}\,c_{\chi}\,m_{\psi^{b}}\,\left[\Delta_{\epsilon}+1-\frac{m^{2}_{\psi^{b}}\log
m^{2}_{\psi^{b}}-m^{2}_{H_{2}^{\pm}}\log
m^{2}_{H_{2}^{\pm}}}{m^{2}_{\psi^{b}}-m^{2}_{H_{2}^{\pm}}}\right]\,.$ (20)

The addition of both contributions cancels out the divergence
($\Delta_{\epsilon}$), as expected. We can finally use Eq. (14) to replace the
mixing angle $\chi$ in Eqs. (19) and (20). By doing this, and also by assuming
the hierarchy $m^{2}_{H_{1}^{\pm}},m^{2}_{H_{2}^{\pm}}\ll m^{2}_{\psi^{b}}$,
one finds

$\displaystyle(m_{\nu})_{\alpha\beta}=\sum_{b=1}^{2}\frac{Y^{L}_{\alpha
b}\,Y^{R}_{\beta b}+Y^{R}_{\alpha b}\,Y^{L}_{\beta
b}}{32\,\pi^{2}\,m_{\psi^{a}}}\frac{\kappa
v^{2}}{m^{2}_{H^{\pm}_{2}}-m^{2}_{H^{\pm}_{1}}}\left(m^{2}_{H_{2}^{\pm}}\log\frac{m^{2}_{\psi^{b}}}{m^{2}_{H_{2}^{\pm}}}-m^{2}_{H_{1}^{\pm}}\log\frac{m^{2}_{\psi^{b}}}{m^{2}_{H_{1}^{\pm}}}\right)\,.$
(21)

This expression matches the result in [25]. Neutrino masses are proportional
to the $\kappa$ parameter, as expected from symmetry arguments. In the limit
$\kappa\to 0$ lepton number is restored, and this explains that Majorana
neutrino masses can only be induced with $\kappa\neq 0$. Furthermore,
$\kappa\ll 1$ is natural, in the sense of ’t Hooft [28]. As in the Scotogenic
model, small neutrino masses can be naturally generated in this model. One can
generate $m_{\nu}\sim 1$ eV with $Y^{R,L}\sim 1$, $m_{H_{1}^{\pm}}\sim 300$
GeV, $m_{H^{\pm}_{2}}\sim 400$ GeV and $m_{\psi^{a}}\sim 1$ TeV if $\kappa\sim
10^{-12}$.

###  2.3 Dark matter candidate

As a consequence of the conservation of the $\mathbb{Z}_{2}$ parity, the
lightest $\mathbb{Z}_{2}$-odd state is stable and cannot decay. In contrast to
the Scotogenic model, the only potentially viable candidate is one of the
neutral $\eta$ scalars, either $\eta_{R}$ or $\eta_{I}$, since the other
$\mathbb{Z}_{2}$-odd states are electrically charged. Their mass difference is
controlled by the $\lambda_{5}$ quartic coupling, as shown in Eq. (7),

$m_{\eta_{R}}^{2}-m_{\eta_{I}}^{2}=\lambda_{5}\,v^{2}\,.$ (22)

Another difference with respect to the Scotogenic model is that this mass
difference can in principle be large. This is because the $\lambda_{5}$
coupling does not enter the neutrino mass formula, see Eq. (21), and can be
$\sim\mathcal{O}(1)$. Also, we note that the sign of $\lambda_{5}$ determines
the DM candidate. $\eta_{I}$ is the DM candidate if $\lambda_{5}>0$, while
$\lambda_{5}<0$ selects $\eta_{R}$ as the DM candidate.

The $\eta_{R}$ and $\eta_{I}$ fields couple to the electroweak gauge bosons,
since they originate from an $\rm SU(2)_{L}$ doublet with hypercharge $1/2$.
Their production in the early Universe is thus expected to be generally
dominated by gauge interactions. The exception to this rule will be found when
the mass of the DM candidate is close to $\sim m_{h}/2$, when the so-called
Higgs portal will be the most important channel. In general, the DM
phenomenology of the model is expected to be similar to that of the Inert
Scalar Doublet model [15], a popular and economical model for DM [16, 17].
Finally, we point out that the DM candidates, $\eta_{R}$ and $\eta_{I}$, carry
the quantum numbers of a supersymmetric sneutrino, except for lepton number.

##  3 Analysis and experimental bounds

We now proceed to discuss the approach followed in our numerical analysis and
the experimental bounds considered. The first step has been the implementation
of the model in SARAH (version 4.11.0) [29], a Mathematica package for the
analytical evaluation of the model. 333See [30] for a pedagogical introduction
to the use of SARAH. With the help of this tool, we have created a SPheno
(version 4.0.2) [31, 32] module with the required numerical routines to obtain
the particle spectrum and compute several observables of interest in our
model. This includes the calculation of flavor violation observables with
FlavorKit [33]. Finally, we have used micrOmegas (version 5.0.9) [34] for the
evaluation of DM observables, such as the DM relic density and direct and
indirect detection predictions. We performed a numerical scan with $\sim
10000$ points. Our choice of parameters is summarized in Tab. 2. In
particular, we choose the negative sign for $\lambda_{5}$ so that throughout
our analysis $\eta_{R}$ plays the role of the dark matter. Let us recall that
this is a choice just for definiteness, equivalent results would be obtained
by assuming $\eta_{I}$ as the lightest $\mathbb{Z}_{2}$-odd state.
Furthermore, we take $\mu_{2}^{2}=\mu^{2}_{\Phi}$ in most of the parameter
space to guarantee that $\eta_{R}$ (and not one of the charged states from
$\Phi$) is the lightest $\mathbb{Z}_{2}$-odd particle. An analogous reason
lies behind the range chosen for $\mu^{2}_{\Phi}$.

$\lambda_{1}=0.26$ |  $m_{\psi^{1}}=2.1$ TeV
---|---
$\lambda_{2}=0.5$ |  $m_{\psi^{2}}=2.3$ TeV
$\lambda_{3}=10^{-2}$ | $\rho_{1}=0.5$
$\lambda_{4}\in\left[-0.5,-10^{-4}\right]$ | $\rho_{2}=0.7$
$\lambda_{5}\in\left[-0.32,-0.003\right]$ | $\sigma_{1}\in\left[10^{-5},0.16\right]$
$\lambda_{\Phi}=3\times 10^{-3}$ | $\sigma_{2}=10^{-2}$
$\mu^{2}_{\Phi}\in\left[100,4.4\times 10^{6}\right]$ GeV2 | $\kappa=10^{-8}$
$\mu_{2}^{2}=\mu^{2}_{\Phi}$ except if $m_{\eta_{R}}\in[50,100]$ GeV |
Table 2: Values of the input parameters considered in the numerical scan.

The new particles introduced in this variant of the Scotogenic model may lead
to different experimental signatures and affect the SM prediction of several
observables. For this reason and throughout our analysis we have considered a
list of experimental constraints.

#### Neutrino oscillation data

The generation and smallness of neutrino masses is one of the main motivations
behind the idea of Scotogenic models. In our analysis, we demand compatibility
of the neutrino oscillation parameters with the most recent neutrino
oscillation global fit [35]. This is achieved by adjusting the entries of the
Yukawa matrices $Y^{L}$ and $Y^{R}$ by means of the master parametrization
[36, 37], which allows one to write

$\displaystyle\left(Y^{L}\right)^{T}$
$\displaystyle=\frac{1}{\sqrt{2\,f}}\,\Sigma^{-1/2}\,W\,A\,\bar{D}_{\sqrt{m}}\,U^{\dagger}\,,$
(23) $\displaystyle\left(Y^{R}\right)^{T}$
$\displaystyle=\frac{1}{\sqrt{2\,f}}\,\Sigma^{-1/2}\,W^{\ast}\,B\,\bar{D}_{\sqrt{m}}\,U^{\dagger}\,,$
(24)

where $\Sigma$ is a diagonal $2\times 2$ matrix containing the positive
singular values of the matrix $M$, with $M_{ij}=\omega_{i}\,\delta_{ij}$ and

$\omega_{i}=\frac{1}{m_{\psi^{i}}}\left(m^{2}_{H_{2}^{\pm}}\,\log\frac{m^{2}_{\psi^{i}}}{m^{2}_{H_{2}^{\pm}}}-m^{2}_{H_{1}^{\pm}}\,\log\frac{m^{2}_{\psi^{i}}}{m^{2}_{H_{1}^{\pm}}}\right)\,.$
(25)

Moreover,

$f=\frac{\kappa\,v^{2}}{32\,\pi^{2}\,(m^{2}_{H^{\pm}_{2}}-m^{2}_{H^{\pm}_{1}})}$
(26)

is a global factor, $U$ is the leptonic mixing matrix, a unitary $3\times 3$
matrix that brings the neutrino mass matrix to a diagonal form as

$U^{T}\,m_{\nu}\,U=\text{diag}(m_{1},m_{2},m_{3})\,,$ (27)

while the matrix $\bar{D}_{\sqrt{m}}$ is defined as

$\bar{D}_{\sqrt{m}}=\left\\{\begin{array}[]{ll}\text{diag}\left(\sqrt{m_{1}},\sqrt{m_{2}},\sqrt{m_{3}}\right)&\text{if}\
m_{1}\neq 0\,,\\\
P\cdot\text{diag}\left(\sqrt{v},\sqrt{m_{2}},\sqrt{m_{3}}\right)\cdot
P&\text{if}\ m_{1}=0\,.\end{array}\right.$ (28)

Here $v$ can actually be replaced by any arbitrary scale and $P$ depends on
the neutrino mass hierarchy. In case of normal ordering, $P$ is just the
identity matrix, while for inverted ordering $P$ is a permutation matrix that
exchanges the first and third elements. Finally, the matrices $W$, $A$ and $B$
are defined in Appendix B, where explicit analytical results for the elements
of the $Y^{L}$ and $Y^{R}$ matrices are also given. In our numerical analysis
we assume a normal mass ordering for light active neutrino masses and take
vanishing CP violating phases for simplicity. We also assume that the lightest
neutrino is massless.

#### Lepton flavor violation

While this model could in principle produce signals at facilities looking for
LFV processes, these have not been observed yet. Hence we can use current null
searches for LFV to constrain the parameters of the model, in particular
$\kappa$ which determines the magnitude of the Yukawa matrices $Y^{L}$ and
$Y^{R}$. We consider the following most stringent bounds on rare LFV
processes: BR$(\mu\to e\gamma)<4.2\times 10^{-13}$ [38], BR$(\mu\to
eee)<1.\times 10^{-12}$ [39] and CR$(\mu^{-},{\rm Au}\to e^{-},\rm Au)<7\times
10^{-13}$ [40].

#### Electroweak precision observables

The experimental accuracy of electroweak observables can be sensitive to the
presence of new particles, like those introduced in this model. The main
contribution to the higher-order calculation of electroweak precision
observables is parameterized via the $\delta\rho$ parameter. We require an
adequately small deviation of the $\rho$ parameter from one through the
following prescription: $-0.00022\lesssim\delta\rho\lesssim 0.00098$ [41]
($3\sigma$ range).

#### Dark matter searches

We assume our DM candidate $\eta_{R}$ to be in thermal equilibrium with the SM
particles in the early Universe. We also assume a standard cosmological
scenario. If no other dark matter candidates are present, then the relic
abundance of $\eta_{R}$ must be in agreement with the latest observations by
the Planck satellite [42], which set a limit on the cosmological content of
cold dark matter: $0.1164\leq\Omega_{\eta_{R}}h^{2}\leq 0.1236$ (3$\sigma$
range). The relic abundance of $\eta_{R}$ can also be subdominant, i.e.
$\Omega_{\eta_{R}}h^{2}<0.1164$, however in this case another DM candidate is
required to explain the totality of the cosmological dark matter. Moreover,
$\eta_{R}$ can be probed at dark matter experiments like direct detection
facilities. We apply the most stringent bound on the WIMP-nucleon spin-
independent (SI) elastic scatter cross-section from the XENON1T experiment
[43]. We compute the direct detection cross section at tree-level. Since a
more exhaustive study of this observable is out of the scope of this paper, we
have considered the constraint

$\frac{1}{2}\lvert\lambda_{3}+\lambda_{4}+\lambda_{5}\rvert\geq 10^{-3}\,,$
(29)

obtained for the inert Higgs doublet [44] in order to avoid sizable loop
corrections. If $\eta_{R}$ annihilates into SM particles with a cross section
typical of WIMPs, it may also be detected indirectly. We consider both
$\gamma$ rays and antiprotons as annihilation products and compare with the
respective current bounds on the WIMP annihilation cross section set by the
Fermi Large Area Telescope (LAT) satellite [45], the ground-based arrays of
Cherenkov telescopes H.E.S.S. [46] and the Alpha Magnetic Spectrometer
(AMS-02) [47, 48, 49] onboard the International Space Station.

#### LHC searches

The new charged particles in the model can be copiously produced and detected
at the LHC, see [25] for a discussion. It is however beyond the scope of this
work to perform a detailed collider study of the model. In order to guarantee
compatibility with the current bounds, we have chosen $m_{\psi^{1,2}}>1$ TeV
in our numerical scan. The singly-charged scalars $H_{1,2}^{+}$ have masses in
a wide range of values, always in the hundreds of GeV. Finally, the doubly-
charged scalar $\Phi^{++}$ is chosen to be always heavier than $\sim 200$ GeV
by imposing $\mu_{2}^{2}\neq\mu^{2}_{\Phi}$ in the region
$m_{\eta_{R}}\in\left[50\text{ GeV},100\text{ GeV}\right]$. This may seem as
too little restrictive, since the currently most stringent bounds on a doubly-
charged scalar range from $220$ GeV [50] (for a $\Phi^{++}$ that decays into
$W^{+}W^{+}$) to $846$ GeV [51] (for a $\Phi^{++}$ that decays into
$\mu^{+}\mu^{+}$). However, notice that in our model $\Phi^{++}$ is
$\mathbb{Z}_{2}$-odd and its decays always include DM particles. In fact, we
have found that in many parameter points $\Phi^{++}$ decays as
$\Phi^{++}\rightarrow H_{1}^{+}\,W^{+}$, followed by
$H_{1}^{+}\to\eta_{R}\,\ell^{+}\,\nu$, thus leading to large amounts of
missing energy in the final state.

#### Invisible decay width of the Higgs boson

If the new neutral scalars $\eta_{R}$ and $\eta_{I}$ are light enough, new
invisible decay channels of the Higgs boson will be kinematically accessible.
We require that their contribution to the invisible decay width of the Higgs
boson is not larger than the currently most stringent experimental bound:
BR$(h\to\rm inv)\lesssim 19\%$ [52]. Our computation of BR$(h\to\rm inv)$
includes the decay BR$(h\to\eta_{R}\,\eta_{R})$, when kinematically possible,
and also BR$(h\to\eta_{I}\,\eta_{I})$ when $m_{\eta_{I}}-m_{\eta_{R}}<10$ GeV.
We estimate that for larger mass differences between $\eta_{I}$ and
$\eta_{R}$, $\eta_{I}$ would decay visibly inside the detector. Moreover, the
presence of charged scalars may also modify the Higgs photon coupling to two
photons. For this reason we further impose that $0.84\lesssim$
BR$(h\to\gamma\gamma)/$BR$(h\to\gamma\gamma)_{\rm SM}\lesssim 1.41$, with
BR$(h\rightarrow\gamma\gamma)_{\rm SM}=(0.227\pm 0.011)\times 10^{-2}$ [41].

####  $\boldsymbol{\eta_{I}}$ lifetime

Searches for long-lived particles at the LHC usually look for displaced
vertices. In order to ensure compatibility with the non-observation of such
signatures, we require $\tau(\eta_{I})\ll 1$ s. This translates into the
constraint $\lvert\lambda_{5}\rvert\geq 5\times 10^{-4}$, which guarantees
that $\eta_{I}$ decays fast enough into $\eta_{R}$.

#### Theoretical considerations

We require that the expansion of the scalar potential (see Eq. 3) around its
minimum must be perturbatively valid. At this scope, we take all scalar
quartic couplings to be $\lesssim 1$. Similarly, the elements of the $Y^{L}$
and $Y^{R}$ Yukawa matrices are also bounded to be $\lesssim 1$.

We summarize in table 3 the list of experimental constraints applied in the
numerical scan.

BR$(\mu\rightarrow e\gamma)$ |  $<4.2\times 10^{-13}$ [38]
---|---
BR$(\mu\rightarrow eee)$ |  $<1.\times 10^{-12}$ [39]
CR$(\mu^{-},{\rm Au}\to e^{-},\rm Au)$ |  $<7\times 10^{-13}$ [40]
$\frac{\text{BR}(h\rightarrow\gamma\gamma)}{\text{BR}(h\rightarrow\gamma\gamma)_{\rm SM}}$ |  $[0.84,1.41]$ [41]
BR$(h\rightarrow\gamma\gamma)_{\rm SM}$ |  $(0.227\pm 0.011)\times 10^{-2}$ [41]
BR$(h\rightarrow\rm inv)$ |  $0.19$ [52]
$\delta\rho$ |  $[-0.00022,0.00098]$ [41]
$\Omega_{\eta_{R}}h^{2}$ |  $[0.1164,0.1236]$ [41]
Table 3: List of experimental constraints applied to the numerical scan.

##  4 Numerical results

In this section we summarize our results of the analysis of $\eta_{R}$ as a
dark matter candidate in the model. As we previously commented, this is the
only potential DM candidate in this model, given the choice of negative sign
of $\lambda_{5}$ (see Sec. 2.3).

Fig. 2 shows the expected $\eta_{R}$ relic abundance as a function of the mass
of $\eta_{R}$. Points colored in magenta denote solutions which can reproduce
the observed cold dark matter relic density, as they fall within the 3$\sigma$
range derived by the Planck satellite data [42],
$\Omega_{\eta_{R}}h^{2}=0.120\pm 0.0036$ (green band). From this analysis we
can identify that the preferred DM mass range lies around $500~{}{\rm
GeV}\lesssim m_{\eta_{R}}\lesssim 800$ GeV. Blue points also refer to allowed
solutions but where $\eta_{R}$ would be a subdominant component of dark
matter, and another candidate would be required. Finally, gray points denote
solutions excluded by any of the constraints previously discussed in Sec. 3.
In particular, the Planck constraint itself rules out most of the solutions
leading to excessively large relic density. Another large set of solutions
with $70~{}{\rm GeV}\lesssim m_{\eta_{R}}\lesssim 700$ GeV is excluded by the
current bound on WIMP-nucleon SI elastic scattering cross section set by the
experiment XENON1T [43].

Looking at the figure from left to right, the first dip refers to the
$Z$-pole, where $m_{\eta_{R}}\sim M_{Z}/2$ and annihilations via s-channel $Z$
exchange are relevant. A similar feature appears at $m_{\eta_{R}}\sim 60$ GeV,
where a narrower dip indicates efficient annihilations via s-channel Higgs
exchange. However, most solutions with $m_{\eta_{R}}\lesssim 60$ GeV appear to
be in conflict with current collider limits on BR$(h\to\rm inv)$ and with LHC
searches for doubly-charged scalars and are therefore excluded. Notice that
the s-channel Higgs annihilations are in general more efficient than the
$Z$-mediated ones, not being momentum suppressed, and in principle could lead
to solutions with $\Omega_{\eta_{R}}h^{2}<10^{-5}$. Only very few solutions
eventually survive in the Higgs-pole region at $m_{\eta_{R}}\lesssim 100$ GeV,
due to a variety of constraints, among which BR$(h\to\gamma\gamma)$ and
BR$(h\to\rm inv)$ are the most important ones. These parameter points have
very small relic density. After the Higgs pole, $\eta_{R}$ quartic
interactions with gauge bosons and heavy quarks become effective, when
kinematically allowed. In particular, $\eta_{R}$ annihilations into
$W^{+}W^{-}$ via quartic couplings are efficient at $m_{\eta_{R}}\gtrsim 80$
GeV, translating into a third drop in the relic abundance. As soon as
kinematically allowed, $\eta_{R}$ can then annihilate also into two Higgs
bosons. In general, for $m_{\eta_{R}}\gtrsim 100$ GeV, the $\eta_{R}$ relic
density increases as its annihilation cross section drops proportionally as
$\sim\frac{1}{m_{\eta_{R}}^{2}}$.

Figure 2: Relic abundance of $\eta_{R}$ as a function of $m_{\eta_{R}}$.
Magenta points depict solutions in agreement with the cold dark matter
measurement obtained from Planck data [42] (green band, 3$\sigma$ interval).
Blue points denote allowed solutions but leading to underabundant dark matter.
Gray points are excluded by any of the constraints listed in Sec. 3.  Figure
3: Dominant diagram contributing to the co-annihilation channel
$H_{1}^{-}H_{2}^{+}\rightarrow\nu\nu$. Notice that both Yukawa matrices,
$Y^{L}$ and $Y^{R}$, must have sizable entries to make this process efficient.

A very interesting feature appears at $m_{\eta_{R}}\sim 1.3$ TeV. The relic
density suddenly drops to very small values due to a very efficient co-
annihilation channel $H_{1}^{-}H_{2}^{+}\rightarrow\nu\nu$, mediated by the
vector-like fermions $\psi_{1,2}$ in t-channel, as shown in Fig. 3. This
contribution is made very efficient by the magnitude of the relevant entries
of the Yukawa matrices entering the diagram. The same Yukawas also play a
relevant role in the rare LFV process $\mu\rightarrow e\gamma$, which turns
out to exceed the current limit set by MEG [38] and thus excludes all
solutions falling in this very narrow pole. Nonetheless, given the large
freedom due to the vast array of parameters that can be varied in the
parameterization of the Yukawa matrices (see Appendix B), one can always find
a fine-tuned combination which allows to keep the co-annihilation channel
$H_{1}^{-}H_{2}^{+}\rightarrow\nu\nu$ efficient while at the same time not
leading to a BR($\mu\rightarrow e\gamma$) in conflict with current
observations. This can be seen in Fig 4, where we show the dependence of
BR($\mu\rightarrow e\gamma$) and $\Omega_{\eta_{R}}h^{2}$ as a function of the
parameters $K_{12}$ (left panel) and $T_{12}$ (right panel). To obtain these
figures we extracted one solution at $m_{\eta_{R}}\sim 1.3$ TeV from our
general scan (see Fig. 2) — which we recall was made fixing $K_{12}=0$ and
$T_{12}=0$ — and we scanned around those central values. Eventually, few
allowed solutions (both with a relic abundance matching current observations
and under-abundant) will appear at $m_{\eta_{R}}\sim 1.3$ TeV. Let us notice
that similar features with fine-tuned solutions may be present also in other
parts of the parameter space.

Figure 4: BR($\mu\rightarrow e\gamma$) (left vertical axis, blue points) and
relic abundance of $\eta_{R}$ (right vertical axis, green points) as a
function of $K_{12}$ (left panel) and $T_{12}$ (right panel). The horizontal
line represents the current upper bound on BR($\mu\rightarrow e\gamma$) [38].
Therefore, all blue points above this line are excluded.

Next we proceed to discuss the results for $\eta_{R}$ direct detection.
In full generality, the tree-level SI $\eta_{R}$-nucleon interaction cross
section receives two contributions, from scattering through the Higgs or $Z$
bosons. However, since the $\lambda_{5}$ quartic coupling induces a mass
splitting between $\eta_{R}$ and $\eta_{I}$, eventually it turns out that the
interaction through the $Z$ boson is kinematically forbidden, or leads to
inelastic scattering, in a large part of the parameter space. Should this not
be the case, the $\eta_{R}$ \- nucleon interaction through the $Z$ boson would
dominate (since the $\eta$ doublet has hypercharge different from zero) and
very likely exceed the current bounds from DM direct detection experiments. We
show in Fig. 5 the SI $\eta_{R}$-nucleon elastic scattering cross section,
weighted by $\xi=\frac{\Omega_{\eta_{R}}}{\Omega_{\rm CDM,Planck}}$ versus the
$\eta_{R}$ mass. We use the same color code as in Fig. 2. The plain green line
and dashed area indicate the current most stringent limit from XENON1T [43].
Even if not shown here, other stringent constraints on the SI
$\eta_{R}$-nucleon elastic scattering cross section apply from the liquid
xenon experiments LUX [53, 54] and PandaX-II [55]. Liquid argon experiments
like DarkSide-50 [56] and DEAP-3600 [57] are instead presently limited and
hence less constraining, due to lower exposures and the currently low
acceptance in DEAP-3600. We also illustrate for comparison the expected
discovery limit corresponding to the “$\nu$-floor” from coherent elastic
neutrino-nucleus scattering (CE$\nu$NS) for a Ge target [58] (dashed orange
line), as well as the sensitivity projection (90% CL) for the future
experiment LUX-ZEPLIN (LZ, red dashed) [59]. Other future experiments like
XENONnT [60], DarkSide-20k [61], ARGO [61] and DARWIN [62, 63] (see [64] for a
recent review), not shown here not to overcrowd the figure, will in principle
be able to explore the parameter space of this model. Given the current
experimental constraints, most of the solutions with a relic abundance falling
within the $3\sigma$ C.L. cold dark matter range obtained by the Planck
collaboration [42] lie in a narrow region around $m_{\eta_{R}}\sim 500-700$
GeV. Many more solutions leading to under-abundant dark matter lie at $\sim
150~{}\text{GeV}\lesssim m_{\eta_{R}}\sim 500$ GeV and could be tested at
future DD experiments. On the whole, the phenomenology of the scalar DM
candidate in this model, while richer, shares similar properties to that of
the analogous scalar DM candidates in the simple Scotogenic Model [7], in
Inert Scalar Doublet models, and in other Scotogenic variants like the
Singlet-Triplet Scotogenic model [24].

Figure 5: Spin-independent $\eta_{R}$-nucleon elastic scattering cross section
– weighted by the relative abundance – versus $m_{\eta_{R}}$. Same color code
as in Fig. 2. The green dashed area is excluded by the most recent upper bound
from the XENON1T experiment [43]. The dashed orange curve indicates the
expected discovery limit corresponding to the “$\nu$-floor” from CE$\nu$NS of
solar and atmospheric neutrinos for a Ge target [58]. The dashed red curve
depicts the projected sensitivity expected at LUX-ZEPLIN (LZ) [59].

Another $\eta_{R}$ search strategy is via its indirect detection. If it
annihilates into SM particles or messengers (with sizable annihilation cross
section, close to the thermal value) it can contribute to the flux of cosmic
particles that reaches the Earth. Photons, and more specifically $\gamma$ rays
are among the most suitable messengers to probe WIMP dark matter indirectly.
They are not deflected during propagation, so they carry information about
their source, and they are relatively easy to detect. When $\eta_{R}$ is
lighter than $M_{W}$, only annihilations into fermions lighter than
$m_{\eta_{R}}$ are allowed at tree level, with the heaviest kinematically
allowed fermion final states dominating (i.e. $\eta_{R}\eta_{R}\rightarrow
b\bar{b}$ and $\eta_{R}\eta_{R}\rightarrow\tau^{+}\tau^{-}$). At higher
masses, when kinematically allowed through the corresponding quartic
couplings, also the following channels open: $\eta_{R}\eta_{R}\rightarrow
W^{+}W^{-},hh,Z^{0}Z^{0}$. The hadronization of the gauge bosons, Higgs boson
or quarks produces neutral pions, which in turn decay into photons thus giving
rise to a $\gamma$-ray flux with a continuum spectrum which may be detected at
indirect detection experiments. Besides this featureless $\gamma$-ray
spectrum, the model also predicts a spectral feature from the internal
bremsstrahlung process $\eta_{R}\eta_{R}\rightarrow W^{+}W^{-}\gamma$,
similarly to the Inert Doublet model [65, 66]. We include this contribution in
our analysis.

While more challenging due to uncertainties in the treatment of their
propagation, charged particles can also be used to probe for $\eta_{R}$
annihilations. PAMELA [67, 68] and, more recently, the AMS-02 [69, 70]
positrons data allow to place constraints on annihilating WIMPs, which are
particularly stringent if they annihilate mainly to the first two generations
of charged leptons. In our case, at $m_{\eta_{R}}<m_{W}$, $\eta_{R}$
annihilates predominantly into $\tau$s (and $b\bar{b}$). Hence, bounds from
cosmic positrons are less stringent than those from $\gamma$ rays. On the
other hand, AMS-02 data on the antiproton flux and the Boron to Carbon (B/C)
ratio can be used to constrain the $\eta_{R}$ annihilation cross section [47,
48, 49]. Provided that astrophysical uncertainties on the $\bar{p}$
production, propagation and on Solar modulation are reliably taken into
account (see e.g. [71, 72, 73]), these bounds result to be stronger than
$\gamma$-ray limits from dwarf spheroidal satellite galaxies (dSphs) over a
wide mass range.

We focus on the main $\eta_{R}$ annihilation channels into $b\bar{b}$,
$\tau^{+}\tau^{-}$ and $W^{+}W^{-}$ to compare with current limits set by
$\gamma$-ray observations of Milky Way dSphs with Fermi-LAT data [45], of the
Galactic Center (GC) with the H.E.S.S. array of ground-based Cherenkov
telescopes [46] and a combination of $\bar{p}$ and B/C data of AMS-02 [47,
48]. Fig. 6 shows the results of our numerical scan of the $\eta_{R}$
annihilation cross section (weighted by $\xi^{2}$ and by the correspondent
branching ratio) versus its mass, for the main annihilation channels:
$\eta_{R}\eta_{R}\rightarrow b\bar{b}$ (green points),
$\eta_{R}\eta_{R}\rightarrow\tau^{+}\tau^{-}$ (blue points) and
$\eta_{R}\eta_{R}\rightarrow W^{+}W^{-}$ (dark red points). As in previous
plots, gray points denote solutions excluded by any of the constraints listed
in Sec. 3. In the same figure we depict the 95% C.L. upper limits currently
set by the Fermi-LAT with $\gamma$-ray observations of Milky Way dSphs (6
years, Pass 8 event-level analysis) [45] (plain curves and shaded areas,
assuming annihilation into $b\bar{b}$ (green), $\tau^{+}\tau^{-}$ (blue) and
$W^{+}W^{-}$ (dark red)). We also show the current upper limit obtained with
data accumulated over 10 years by H.E.S.S. observations of the GC [46],
assuming a $W^{+}W^{-}$ annihilation channel and an Einasto dark matter
density profile (dark red dot-dashed curve and shaded area). Current bounds
from a combination of $\bar{p}$ and B/C data of AMS-02 [47, 48] are instead
shown as dashed curves (green for $b\bar{b}$ and red for $W^{+}W^{-}$
channels, respectively). As for comparison we further illustrate projected
sensitivities for Fermi-LAT from a stacked analysis of 60 dSphs and 15 years
of data, in the $b\bar{b}$ channel [74] (dot-dashed green line) and for the
Cherenkov Telescope Array (CTA), for the Milky Way galactic halo target,
assuming the $W^{+}W^{-}$ annihilation channel and an Einasto dark matter
density profile [75]. Our model predictions, not excluded by other
constraints, lie at least a factor of few below current limits set by
$\gamma$-ray observations with Fermi-LAT and H.E.S.S.. In particular, allowed
solutions in the “low” (and narrow) mass window around $m_{\eta_{R}}\sim 60$
GeV ($b\bar{b}$ and $\tau^{+}\tau^{-}$ annihilation channels) are hardly
found, so only a couple of them are shown for illustration. Predictions for
these channels lie well below current bounds. Antiprotons and B/C data of
AMS-02 instead already allow to exclude some solutions with
$200~{}\text{GeV}\lesssim m_{\eta_{R}}\lesssim 800$ GeV ($W^{+}W^{-}$
annihilation channel). We remark that these bounds are obtained under
significant astrophysical uncertainties. Future telescopes like CTA as well as
additional LAT data will allow to further explore these regions of the
parameter space, thus allowing for multi-messenger indirect probes of this
model. Finally let us comment that the annihilation cross section of non-
relativistic $\eta_{R}$ at the current epoch can be affected by a non-
perturbative correction, the Sommerfeld enhancement [76, 77, 78, 79, 80] which
would consequently affect also its indirect detection signatures. However, we
did not include it in our calculation as a thorough treatment of this effect
lies beyond the scope of this paper.

Figure 6: $\eta_{R}$ annihilation cross section for annihilations into
$b\bar{b}$ (green), $\tau^{+}\tau^{-}$ (blue) and $W^{+}W^{-}$ (red) final
states. The green, blue and red plain lines refer to the corresponding 95%
C.L. upper limits currently set by Fermi-LAT $\gamma$-ray data from dSphs
[45]. The dark red dot-dashed curve is the current 95% C.L. upper limit
obtained by H.E.S.S. using GC data [46]. The green and red dashed lines denote
current 95% C.L. constraints derived from the antiproton and B/C data of
AMS-02 [48]. For comparison, we also show sensitivity projections for Fermi-
LAT ($b\bar{b}$ channel, assuming 60 dSphs and 15 years of data) [74] and for
CTA (looking at the galactic halo, $W^{+}W^{-}$ channel) [75].

##  5 Conclusions

We have considered a variant of the Scotogenic model with an extended scalar
content, in which one of the new doublets carries hypercharge $Y=3/2$. As a
consequence, the model particle spectrum contains new states, including a
doubly-charged scalar, singly-charged scalars and new charged fermions leading
to a rich collider phenomenology. Neutrino masses are generated radiatively as
in other Scotogenic scenarios. However, contrarily to the simple Scotogenic
model, in this setup only the lightest $\mathbb{Z}_{2}$-odd neutral scalar
$\eta$ is a viable dark matter candidate. We have analyzed the phenomenology
of the dark matter particle $\eta_{R}$ and shown that the observed relic
density can be obtained for $500~{}{\rm GeV}\lesssim m_{\eta_{R}}\lesssim 800$
GeV. This result is equivalent to that obtained in the Inert Doublet model. In
addition, we have found that the correct relic density can also be achieved at
larger $\eta_{R}$ masses if the coannihilation channel
$H_{1}^{-}H_{2}^{+}\rightarrow\nu\nu$ becomes efficient. However, this
requires some fine-tuning to avoid the stringent constraints from lepton
flavor violation. Moreover, we have presented a full numerical analysis of the
signatures expected at dark matter detection experiments, both via direct and
indirect probes. We have found that most recent direct detection data from
XENON1T already rule out a region of the parameter space in the mass range
$m_{\eta_{R}}\sim 100$ GeV. Finally we have commented about indirect detection
signatures with $\gamma$-rays and antiprotons data. While bounds derived from
$\gamma$-ray data from Fermi-LAT and H.E.S.S. currently lie above the model
predictions, AMS-02 antiproton data instead already allows to constrain some
solutions in the $200~{}{\rm GeV}\lesssim m_{\eta_{R}}\lesssim 800$ GeV mass
range.

## Acknowledgements

Work supported by the Spanish grants FPA2017-85216-P (MINECO/AEI/FEDER, UE),
SEJI/2018/033, SEJI/2020/016 (Generalitat Valenciana) and FPA2017-90566-REDC
(Red Consolider MultiDark). AV acknowledges financial support from MINECO
through the Ramón y Cajal contract RYC2018-025795-I. VDR acknowledges
financial support by the Universitat de València through the sub-programme
“ATRACCIÓ DE TALENT 2019”.

##  Appendix A Neutrino mass matrix in Scotogenic models

Figure 7: Generic one-loop contribution to the neutrino mass matrix. $F$ and
$S$ represent fermion and scalar mass eigenstates contributing to the neutrino
mass matrix.

In this Appendix we find a general analytical expression for one-loop
contributions to the neutrino mass matrix in Scotogenic models. Let us
consider a generic model with $n_{S}$ scalars $S_{a}$ with masses $m_{Sa}$
($a=1,\dots,n_{S}$) and $n_{F}$ fermions $F_{b}$ with masses $m_{Fb}$
($b=1,\dots,n_{F}$) that couple to light neutrinos with interaction terms of
the form

$\mathcal{L}=\left(Y_{S}^{F}\right)_{\alpha
ab}\,\overline{\nu_{L}}_{\alpha}\,S_{a}\,F_{b}+\text{h.c.}\,,$ (30)

where $Y^{F}_{S}$ is a $3\times n_{S}\times n_{F}$ complex object. This
interaction term induces neutrino masses at the one-loop level via the diagram
in Fig. 7. The resulting neutrino mass matrix can be expressed as

$i\,(m_{\nu})_{\alpha\beta}=i\,\sum_{a,b}(m_{\nu})^{\\{S_{a},F_{b}\\}}_{\alpha\beta}\,.$
(31)

where the sum extends over all mass eigenstates running in the loop. Each
$i\,(m_{\nu})^{\\{S_{a},F_{b}\\}}_{ij}$ represents the individual contribution
by the scalar $S_{a}$ and the fermion $F_{b}$ and is given by

$i\,(m_{\nu})^{\\{S_{a},F_{b}\\}}_{\alpha\beta}=(Y_{S}^{F})_{\alpha
ab}\,(Y_{S}^{F})_{\beta
ab}\,\int\frac{d^{D}k}{(2\pi)^{D}}\frac{(\not{k}+m_{Fb})}{(k^{2}-m^{2}_{Sa})(k^{2}-m^{2}_{Fb})}\,,$
(32)

where $D=4-\varepsilon$ is the number of space-time dimensions, $k$ is the
momentum running in the loop and the external neutrinos have been taken at
rest. The $\not{k}$ piece does not contribute since it is an odd function in
$k$. The remaining term is logarithmically divergent. Introducing the usual
Feynman’s parameters, we can rewrite our expression as

$i\,(m_{\nu})^{\\{S_{a},F_{b}\\}}_{\alpha\beta}=(Y^{F}_{S})_{\alpha
ab}\,(Y^{F}_{S})_{\beta
ab}\,\int^{1}_{0}dx\int\frac{d^{D}k}{(2\pi)^{D}}\frac{m_{Fb}}{(k^{2}-C_{ab}^{2})^{2}}\,,$
(33)

where $C_{ab}^{2}=m_{Sa}^{2}+m^{2}_{Fb}(1-x)$. One can now integrate in $k$
and obtain

$i\,(m_{\nu})^{\\{S_{a},F_{b}\\}}_{\alpha\beta}=\frac{i\,(Y_{S}^{F})_{\alpha
ab}\,(Y_{S}^{F})_{\beta
ab}}{16\pi^{2}}\,m_{Fb}\,\int_{0}^{1}dx\left[\Delta_{\epsilon}-\log
C_{ab}+\mathcal{O}(\epsilon)\right]\,,$ (34)

where $\Delta_{\epsilon}=\frac{2}{\epsilon}-\gamma+\log 4\pi$ and $\gamma$ is
Euler’s constant. Finally, we integrate in $x$ and take the limit $\epsilon\to
0$ (which corresponds to $D\to 4$). This leads us to

$(m_{\nu})^{\\{S_{a},F_{b}\\}}_{\alpha\beta}=\frac{(Y^{F}_{S})_{\alpha
ab}\,(Y_{S}^{F})_{\beta
ab}}{16\pi^{2}}\,m_{Fb}\,\left[\Delta_{\epsilon}+1-\frac{m^{2}_{Fb}\log(m^{2}_{Fb})-m^{2}_{Sa}\log(m^{2}_{Sa})}{m^{2}_{Fb}-m^{2}_{Sa}}\right]\,.$
(35)

Eq. (35) is our master expression for the one-loop contributions to the
neutrino mass matrix. In order to compute the neutrino mass matrix in
different versions of the Scotogenic model, one just needs to identify the
scalar and fermion mass eigenstates running in the loop and replace the
$Y^{F}_{S}$ couplings by their expressions in terms of the model parameters.
Then, summing over all individual contributions one finds the total one-loop
neutrino mass matrix. We note that individual contributions to $m_{\nu}$ are
divergent due to the $\Delta_{\epsilon}$ term. However, the one-loop neutrino
mass matrix is finite. Therefore, these divergences cancel out in the sum in
Eq. (31).

As an example, we show this explicitly in the original Scotogenic model [7].
In this case one introduces the inert scalar doublet $\eta$, with the same
definition as in the model in Sec. 2, as well as three generations of fermions
$N$, singlets under the SM gauge group. The new $N$ and $\eta$ fields couple
to the SM lepton doublet with the Yukawa term
$Y^{N}\,\overline{\ell_{L}}\,\widetilde{\eta}\,N$, with
$\widetilde{\eta}=i\tau_{2}\eta^{*}$ and $\tau_{2}$ the second Pauli matrix.
Moreover, the neutral component of the $\eta$ doublet can been split into its
CP-even and CP-odd components as

$\eta^{0}=\frac{1}{\sqrt{2}}\left(\eta_{R}+i\,\eta_{I}\right)\,.$ (36)

Assuming that CP is conserved in the scalar sector, $\eta_{R}$ and $\eta_{I}$
do not mix and constitute mass eigenstates. Therefore, in the Scotogenic model
one has $n_{S}=2$ and $n_{F}=3$, with $S\equiv\\{\eta_{R},\eta_{I}\\}$ and
$F\equiv\\{N_{1},N_{2},N_{3}\\}$. Their couplings are given by

$Y^{\eta_{R}}\equiv\frac{Y^{N}}{\sqrt{2}}\,,\quad Y^{\eta_{I}}\equiv
i\,\frac{Y^{N}}{\sqrt{2}}\,.$ (37)

Finally, applying Eq. (35) and summing over all the states in the loop we are
able to obtain a finite expression (the divergences of the diagrams with
$\eta_{R}$ and $\eta_{I}$ cancel each other) for the neutrino mass matrix

$(m_{\nu})_{\alpha\beta}=\sum_{b=1}^{3}\frac{Y^{N}_{\alpha b}\,Y^{N}_{\beta
b}}{32\pi^{2}}\,m_{Nb}\left[\frac{m^{2}_{\eta_{R}}}{m^{2}_{Nb}-m^{2}_{\eta_{R}}}\log\frac{m^{2}_{\eta_{R}}}{m^{2}_{Nb}}-\frac{m^{2}_{\eta_{I}}}{m^{2}_{Nb}-m^{2}_{\eta_{I}}}\log\frac{m^{2}_{\eta_{I}}}{m^{2}_{Nb}}\right]\,,$
(38)

where $m_{\eta_{R}}$ and $m_{\eta_{I}}$ are the corresponding masses for each
component. This expression differs from the expression in [7] by a factor of
$1/2$, an error that was first identified in v1 of [81].

##  Appendix B General parametrization for the Yukawa matrices

The master parametrization [36, 37] allows one to write the $3\times 2$ Yukawa
matrices $Y^{L}$ and $Y^{R}$ in terms of neutrino oscillation parameters as in
Eqs. (23) and (24). In these expressions, $W$ is a $2\times 2$ unitary matrix,
given in terms of the complex angle $\beta$ as

$W=\begin{pmatrix}\cos\beta&\sin\beta\\\
-\sin\beta^{*}&\cos\beta^{*}\end{pmatrix}\,,$ (39)

$A=T\,C_{1}$, with $T$ a $2\times 2$ upper-triangular matrix with positives
real entries in the diagonal, and $B=(T^{T})^{-1}(C_{1}C_{2}+KC_{1})$, with
$K$ a $2\times 2$ antisymmetric matrix, hence containing only one degree of
freedom. The matrices $C_{1}$ and $C_{2}$ are given by

$C_{1}=\begin{pmatrix}z_{1}&0&0\\\ z_{2}&0&0\end{pmatrix}\,,\quad
C_{2}=\begin{pmatrix}-1&0&0\\\ 0&1&0\\\ 0&0&1\end{pmatrix}\,,$ (40)

with $z_{1}$ and $z_{2}$ two complex numbers such that
$z_{1}^{2}+z_{2}^{2}=0$. With all these ingredients one can obtain explicit
expressions for the elements of $Y^{L}$ and $Y^{R}$, which will be determined
from neutrino oscillation data as well as from a set of free parameters. We
point out that these are the most general expressions for the elements of the
$Y^{L}$ and $Y^{R}$ Yukawa matrices:

$Y^{L}_{11}=\frac{c_{12}c_{13}\sqrt{m_{1}}(c_{\beta}(z_{1}T_{11}+z_{2}T_{12})+z_{2}T_{22}s_{\beta})+c_{13}e^{-i\eta_{2}}\sqrt{m_{2}}s_{12}T_{11}c_{\beta}}{\sqrt{2f}\sqrt{\omega_{1}}}\\\
+\frac{\sqrt{m_{3}}s_{13}e^{i(\delta-\eta_{3})}(T_{22}s_{\beta}+T_{12}c_{\beta})}{\sqrt{2f}\sqrt{\omega_{1}}}\,,$
(41)
$Y^{L}_{21}=-\frac{\sqrt{m_{1}}(c_{\beta}(z_{1}T_{11}+z_{2}T_{12})+z_{2}T_{22}s_{\beta})\left(c_{23}s_{12}+c_{12}e^{-i\delta}s_{13}s_{23}\right)}{\sqrt{2f}\sqrt{\omega_{1}}}\\\
-e^{-i(\eta_{2}+\eta_{3})}\frac{e^{i\eta_{3}}\sqrt{m_{2}}\
T_{11}c_{\beta}\left(c_{12}c_{23}-e^{-i\delta}s_{12}s_{13}s_{23}\right)+c_{13}e^{i\eta_{2}}\sqrt{m_{3}}s_{23}(T_{22}s_{\beta}+T_{12}c_{\beta})}{\sqrt{2f}\sqrt{\omega_{1}}}\,,$
(42)
$Y^{L}_{31}=e^{-i(\eta_{2}+\eta_{3})}\frac{e^{i\eta_{2}}c_{13}c_{23}\sqrt{m_{3}}\
(T_{22}s_{\beta}+T_{12}c_{\beta})}{\sqrt{2f}\sqrt{\omega_{1}}}\\\
+\frac{\sqrt{m_{1}}(z_{1}T_{11}c_{\beta}+z_{2}T_{22}s_{\beta}+z_{2}T_{12}c_{\beta})\left(s_{12}s_{23}-c_{12}c_{23}e^{-i\delta}s_{13}\right)}{\sqrt{2f}\sqrt{\omega_{1}}}\\\
+e^{-i(\eta_{2}+\eta_{3})}\frac{-e^{i\eta_{3}}\sqrt{m_{2}}\
T_{11}c_{\beta}\left(c_{23}e^{-i\delta}s_{12}s_{13}+c_{12}s_{23}\right)}{\sqrt{2f}\sqrt{\omega_{1}}}\,,$
(43)
$Y^{L}_{12}=e^{-i(\eta_{2}+\eta_{3})}\frac{e^{i\eta_{2}}T_{22}c_{\beta}^{*}\left(c_{12}c_{13}e^{i\eta_{3}}\sqrt{m_{1}}z_{2}+e^{i\delta}\sqrt{m_{3}}s_{13}\right)}{\sqrt{2f}\sqrt{\omega_{2}}}\\\
-e^{-i(\eta_{2}+\eta_{3})}\frac{s_{\beta}^{*}\left(c_{13}e^{i\eta_{3}}\left(\sqrt{m_{2}}s_{12}T_{11}+c_{12}e^{i\eta_{2}}\sqrt{m_{1}}(z_{1}T_{11}+z_{2}T_{12})\right)+\sqrt{m_{3}}s_{13}T_{12}e^{i(\delta+\eta_{2})}\right)}{\sqrt{2f}\sqrt{\omega_{2}}}\,,$
(44)
$Y^{L}_{22}=\frac{\sqrt{m_{1}}\left(c_{23}s_{12}+c_{12}e^{-i\delta}s_{13}s_{23}\right)\left(s_{\beta}^{*}(z_{1}T_{11}+z_{2}T_{12})-z_{2}T_{22}c_{\beta}^{*}\right)}{\sqrt{2f}\sqrt{\omega_{2}}}\\\
-e^{-i(\eta_{2}+\eta_{3})}\frac{e^{i\eta_{3}}\sqrt{m_{2}}\
T_{11}s_{\beta}^{*}\left(c_{12}c_{23}-e^{-i\delta}s_{12}s_{13}s_{23}\right)-c_{13}e^{i\eta_{2}}\sqrt{m_{3}}s_{23}\left(T_{12}s_{\beta}^{*}-T_{22}c_{\beta}^{*}\right)}{\sqrt{2f}\sqrt{\omega_{2}}}\,,$
(45)
$Y^{L}_{32}=\frac{\sqrt{m_{1}}\left(c_{12}c_{23}e^{-i\delta}s_{13}-s_{12}s_{23}\right)\left(s_{\beta}^{*}(z_{1}T_{11}+z_{2}T_{12})z_{2}T_{22}c_{\beta}^{*}\right)}{\sqrt{2f}\sqrt{\omega_{2}}}\\\
+\frac{e^{-i\eta 2}\sqrt{m_{2}}\
T_{11}s_{\beta}^{*}\left(c_{23}e^{-i\delta}s_{12}s_{13}+c_{12}s_{23}\right)-c_{13}c_{23}e^{-i\eta
3}\sqrt{m_{3}}\left(T_{12}s_{\beta}^{*}-T_{22}c_{\beta}^{*}\right)}{\sqrt{2f}\sqrt{\omega_{2}}}\,,$
(46)
$Y^{R}_{11}=-e^{-i(\eta_{2}+\eta_{3})}\frac{s_{\beta}^{*}c_{12}c_{13}\sqrt{m_{1}}e^{i(\eta_{2}+\eta_{3})}(z_{1}K_{12}T_{11}+z_{2}K_{12}T_{12}-z_{1}T_{12}+z_{2}T_{11})}{\sqrt{2f}\sqrt{\omega_{1}}\
T_{11}T_{22}}\\\
+e^{-i(\eta_{2}+\eta_{3})}\frac{s_{\beta}^{*}\left(-c_{13}e^{i\eta_{3}}\sqrt{m_{2}}s_{12}(K_{12}T_{11}+T_{12})+\sqrt{m_{3}}s_{13}e^{i(\delta+\eta_{2})}(T_{11}-K_{12}T_{12})\right)}{\sqrt{2f}\sqrt{\omega_{1}}\
T_{11}T_{22}}\\\
+e^{-i(\eta_{2}+\eta_{3})}\frac{T_{22}c_{\beta}^{*}\left(c_{13}e^{i\eta_{3}}\left(\sqrt{m_{2}}s_{12}-c_{12}e^{i\eta_{2}}\sqrt{m_{1}}(z_{1}-z_{2}K_{12})\right)+\sqrt{m_{3}}s_{13}K_{12}e^{i(\delta+\eta_{2})}\right)}{\sqrt{2f}\sqrt{\omega_{1}}\
T_{11}T_{22}}\,,$ (47)
$Y^{R}_{21}=\frac{s_{\beta}^{*}\sqrt{m_{1}}(z_{1}K_{12}T_{11}+z_{2}K_{12}T_{12}-z_{1}T_{12}+z_{2}T_{11})\left(c_{23}s_{12}+c_{12}e^{-i\delta}s_{13}s_{23}\right)}{\sqrt{2f}\sqrt{\omega_{1}}\
T_{11}T_{22}}\\\
-e^{-i(\eta_{2}+\eta_{3})}\frac{s_{\beta}^{*}\sqrt{m_{1}}\left(e^{i\eta_{3}}\sqrt{m_{2}}(K_{12}T_{11}+T_{12})\left(c_{12}c_{23}-e^{-i\delta}s_{12}s_{13}s_{23}\right)+c_{13}e^{i\eta_{2}}\sqrt{m_{3}}s_{23}(T_{11}-K_{12}T_{12})\right)}{\sqrt{2f}\sqrt{\omega_{1}}\
T_{11}T_{22}}\\\
+e^{-i(\eta_{2}+\eta_{3})}\frac{T_{22}c_{\beta}^{*}\left(\sqrt{m_{1}}e^{i(\eta_{2}+\eta_{3})}(z_{1}-z_{2}K_{12})\left(c_{23}s_{12}+c_{12}e^{-i\delta}s_{13}s_{23}\right)+c_{13}e^{i\eta
2}\sqrt{m_{3}}s_{23}K_{12}\right)}{\sqrt{2f}\sqrt{\omega_{1}}\
T_{11}T_{22}}\\\ +e^{-i(\eta_{2}+\eta_{3})}\frac{T_{22}c_{\beta}^{*}\ e^{i\eta
3}\sqrt{m_{2}}\left(c_{12}c_{23}-e^{-i\delta}s_{12}s_{13}s_{23}\right)}{\sqrt{2f}\sqrt{\omega_{1}}\
T_{11}T_{22}}\,,$ (48)
$Y^{R}_{31}=\frac{-\sqrt{m_{1}}\left(s_{12}s_{23}-c_{12}c_{23}e^{-i\delta}s_{13}\right)\left(s_{\beta}^{*}(T_{11}(z_{1}K_{12}+z_{2})-T_{12}(z_{1}-z_{2}K_{12}))\right)}{\sqrt{2f}\sqrt{\omega_{1}}\
T_{11}T_{22}}\\\
-\frac{\sqrt{m_{1}}\left(s_{12}s_{23}-c_{12}c_{23}e^{-i\delta}s_{13}\right)T_{22}c_{\beta}^{*}(z_{1}-z_{2}K_{12})}{\sqrt{2f}\sqrt{\omega_{1}}\
T_{11}T_{22}}\\\
+\frac{e^{-i\eta_{2}}\sqrt{m_{2}}\left((K_{12}T_{11}+T_{12})s_{\beta}^{*}-T_{22}c_{\beta}^{*}\right)\left(c_{23}e^{-i\delta}s_{12}s_{13}+c_{12}s_{23}\right)}{\sqrt{2f}\sqrt{\omega_{1}}\
T_{11}T_{22}}\\\
+\frac{c_{13}c_{23}e^{-i\eta_{3}}\sqrt{m_{3}}\left((T_{11}-K_{12}T_{12})s_{\beta}^{*}+K_{12}T_{22}c_{\beta}^{*}\right)}{\sqrt{2f}\sqrt{\omega_{1}}\
T_{11}T_{22}}\,,$ (49)
$Y^{R}_{12}=-\frac{c_{\beta}c_{12}c_{13}\sqrt{m_{1}}(z_{1}K_{12}T_{11}+z_{2}K_{12}T_{12}-z_{1}T_{12}+z_{2}T_{11})}{\sqrt{2f}\sqrt{\omega_{2}}\
T_{11}T_{22}}\\\
-e^{-i(\eta_{2}+\eta_{3})}\frac{c_{\beta}\left(-c_{13}e^{i\eta_{3}}\sqrt{m_{2}}s_{12}(K_{12}T_{11}+T_{12})+\sqrt{m_{3}}s_{13}e^{i(\delta+\eta_{2})}(T_{11}-K_{12}T_{12})\right)}{\sqrt{2f}\sqrt{\omega_{2}}\
T_{11}T_{22}}\\\
-e^{-i(\eta_{2}+\eta_{3})}\frac{T_{22}s_{\beta}\left(c_{13}e^{i\eta_{3}}\left(\sqrt{m_{2}}s_{12}-c_{12}e^{i\eta_{2}}\sqrt{m_{1}}(z_{1}-z_{2}K_{12})\right)+\sqrt{m_{3}}s_{13}K_{12}e^{i(\delta+\eta_{2})}\right)}{\sqrt{2f}\sqrt{\omega_{2}}\
T_{11}T_{22}}\,,$ (50)
$Y^{R}_{22}=\frac{\sqrt{m_{1}}\left(c_{23}s_{12}+c_{12}e^{-i\delta}s_{13}s_{23}\right)(T_{22}s_{\beta}(z_{2}K_{12}-z_{1})+c_{\beta}(z_{1}K_{12}T_{11}+z_{2}K_{12}T_{12}-z_{1}T_{12}+z_{2}T_{11}))}{\sqrt{2f}\sqrt{\omega_{2}}\
T_{11}T_{22}}\\\
-e^{-i(\eta_{2}+\eta_{3})}\frac{e^{i\eta_{3}}\sqrt{m_{2}}(c_{\beta}(K_{12}T_{11}+T_{12})+T_{22}s_{\beta})\left(c_{12}c_{23}-e^{-i\delta}s_{12}s_{13}s_{23}\right)}{\sqrt{2f}\sqrt{\omega_{2}}\
T_{11}T_{22}}\\\
+e^{-i(\eta_{2}+\eta_{3})}\frac{c_{13}e^{i\eta_{2}}\sqrt{m_{3}}s_{23}(c_{\beta}(T_{11}-K_{12}T_{12})-K_{12}T_{22}s_{\beta})}{\sqrt{2f}\sqrt{\omega_{2}}\
T_{11}T_{22}}\,,$ (51)
$Y^{R}_{32}=\frac{-\sqrt{m_{1}}\left(s_{12}s_{23}-c_{12}c_{23}e^{-i\delta}s_{13}\right)(T_{22}s_{\beta}(z_{2}K_{12}-z_{1})+c_{\beta}(z_{1}K_{12}T_{11}+z_{2}K_{12}T_{12}-z_{1}T_{12}+z_{2}T_{11}))}{\sqrt{2f}\sqrt{\omega_{2}}\
T_{11}T_{22}}\\\
+e^{-i(\eta_{2}+\eta_{3})}\frac{e^{i\eta_{3}}\sqrt{m_{2}}(c_{\beta}(K_{12}T_{11}+T_{12})+T_{22}s_{\beta})}{\sqrt{2f}\sqrt{\omega_{2}}\
T_{11}T_{22}}\\\
+e^{-i(\eta_{2}+\eta_{3})}\frac{\left(c_{23}e^{-i\delta}s_{12}s_{13}+c_{12}s_{23}\right)+c_{13}c_{23}e^{i\eta_{2}}\sqrt{m_{3}}(c_{\beta}(T_{11}-K_{12}T_{12})-K_{12}T_{22}s_{\beta})}{\sqrt{2f}\sqrt{\omega_{2}}\
T_{11}T_{22}}\,.$ (52)

In these expressions $c_{kl}\equiv\cos\theta_{kl}$ and
$s_{kl}\equiv\sin{\theta_{kl}}$ ($k=1,2$ and $l=2,3$; $l\neq k$) are the usual
neutrino mixing angles measured in oscillation experiments, while
$m_{j}\equiv\sqrt{m_{1}^{2}+\Delta m^{2}_{j1}}$ ($j=2,3$) are neutrino mass
eigenvalues. Moreover, $m_{1}$ is the lightest neutrino mass, $\eta_{i}$
($i=2,3)$ are CP violating Majorana phases and $\delta$ is the CP violating
Dirac phase. We also use the notation $s_{\beta}\equiv\sin\beta$ and
$c_{\beta}\equiv\cos\beta$. In our numerical scans we have fixed
$T_{11}=T_{22}=1$ and
$\beta=T_{12}=K_{12}=z_{1}=z_{2}=m_{1}=\eta_{i}=\delta=0$ unless otherwise
stated.

## References

  * [1] A. Zee, “A Theory of Lepton Number Violation, Neutrino Majorana Mass, and Oscillation,” Phys. Lett. B 93 (1980) 389. [Erratum: Phys.Lett.B 95, 461 (1980)].
  * [2] T. P. Cheng and L.-F. Li, “Neutrino Masses, Mixings and Oscillations in $\rm SU(2)\times U(1)$ Models of Electroweak Interactions,” Phys. Rev. D 22 (1980) 2860.
  * [3] A. Zee, “Quantum Numbers of Majorana Neutrino Masses,” Nucl. Phys. B 264 (1986) 99–110.
  * [4] K. S. Babu, “Model of ’Calculable’ Majorana Neutrino Masses,” Phys. Lett. B 203 (1988) 132–136.
  * [5] Y. Cai, J. Herrero-García, M. A. Schmidt, A. Vicente, and R. R. Volkas, “From the trees to the forest: a review of radiative neutrino mass models,” Front. in Phys. 5 (2017) 63, arXiv:1706.08524 [hep-ph].
  * [6] D. Restrepo, O. Zapata, and C. E. Yaguna, “Models with radiative neutrino masses and viable dark matter candidates,” JHEP 11 (2013) 011, arXiv:1308.3655 [hep-ph].
  * [7] E. Ma, “Verifiable radiative seesaw mechanism of neutrino mass and dark matter,” Phys. Rev. D 73 (2006) 077301, arXiv:hep-ph/0601225.
  * [8] T. Toma and A. Vicente, “Lepton Flavor Violation in the Scotogenic Model,” JHEP 01 (2014) 160, arXiv:1312.2840 [hep-ph].
  * [9] A. Vicente and C. E. Yaguna, “Probing the scotogenic model with lepton flavor violating processes,” JHEP 02 (2015) 144, arXiv:1412.2545 [hep-ph].
  * [10] J. Kubo, E. Ma, and D. Suematsu, “Cold Dark Matter, Radiative Neutrino Mass, $\mu\to e\gamma$, and Neutrinoless Double Beta Decay,” Phys. Lett. B 642 (2006) 18–23, arXiv:hep-ph/0604114.
  * [11] D. Aristizabal Sierra, J. Kubo, D. Restrepo, D. Suematsu, and O. Zapata, “Radiative seesaw: Warm dark matter, collider and lepton flavour violating signals,” Phys. Rev. D 79 (2009) 013011, arXiv:0808.3340 [hep-ph].
  * [12] D. Suematsu, T. Toma, and T. Yoshida, “Reconciliation of CDM abundance and $\mu\to e\gamma$ in a radiative seesaw model,” Phys. Rev. D 79 (2009) 093004, arXiv:0903.0287 [hep-ph].
  * [13] A. Adulpravitchai, M. Lindner, and A. Merle, “Confronting Flavour Symmetries and extended Scalar Sectors with Lepton Flavour Violation Bounds,” Phys. Rev. D 80 (2009) 055031, arXiv:0907.2147 [hep-ph].
  * [14] C. Hagedorn, J. Herrero-García, E. Molinaro, and M. A. Schmidt, “Phenomenology of the Generalised Scotogenic Model with Fermionic Dark Matter,” JHEP 11 (2018) 103, arXiv:1804.04117 [hep-ph].
  * [15] N. G. Deshpande and E. Ma, “Pattern of Symmetry Breaking with Two Higgs Doublets,” Phys. Rev. D 18 (1978) 2574.
  * [16] R. Barbieri, L. J. Hall, and V. S. Rychkov, “Improved naturalness with a heavy Higgs: An Alternative road to LHC physics,” Phys. Rev. D 74 (2006) 015007, arXiv:hep-ph/0603188.
  * [17] L. Lopez Honorez, E. Nezri, J. F. Oliver, and M. H. G. Tytgat, “The Inert Doublet Model: An Archetype for Dark Matter,” JCAP 02 (2007) 028, arXiv:hep-ph/0612275.
  * [18] L. Lopez Honorez and C. E. Yaguna, “The inert doublet model of dark matter revisited,” JHEP 09 (2010) 046, arXiv:1003.3125 [hep-ph].
  * [19] M. A. Díaz, B. Koch, and S. Urrutia-Quiroga, “Constraints to Dark Matter from Inert Higgs Doublet Model,” Adv. High Energy Phys. 2016 (2016) 8278375, arXiv:1511.04429 [hep-ph].
  * [20] P. Escribano, M. Reig, and A. Vicente, “Generalizing the Scotogenic model,” JHEP 07 (2020) 097, arXiv:2004.05172 [hep-ph].
  * [21] E. Ma and V. De Romeri, “Radiative Seesaw Dark Matter,” arXiv:2105.00552 [hep-ph].
  * [22] M. Hirsch, R. A. Lineros, S. Morisi, J. Palacio, N. Rojas, and J. W. F. Valle, “WIMP dark matter as radiative neutrino mass messenger,” JHEP 10 (2013) 149, arXiv:1307.8134 [hep-ph].
  * [23] P. Rocha-Moran and A. Vicente, “Lepton Flavor Violation in the singlet-triplet scotogenic model,” JHEP 07 (2016) 078, arXiv:1605.01915 [hep-ph].
  * [24] I. M. Ávila, V. De Romeri, L. Duarte, and J. W. F. Valle, “Phenomenology of scotogenic scalar dark matter,” Eur. Phys. J. C 80 no. 10, (2020) 908, arXiv:1910.08422 [hep-ph].
  * [25] M. Aoki, S. Kanemura, and K. Yagyu, “Doubly-charged scalar bosons from the doublet,” Phys. Lett. B 702 (2011) 355–358, arXiv:1105.2075 [hep-ph]. [Erratum: Phys.Lett.B 706, 495–495 (2012)].
  * [26] D. Binosi and L. Theussl, “JaxoDraw: A Graphical user interface for drawing Feynman diagrams,” Comput. Phys. Commun. 161 (2004) 76–86, arXiv:hep-ph/0309015.
  * [27] D. Binosi, J. Collins, C. Kaufhold, and L. Theussl, “JaxoDraw: A Graphical user interface for drawing Feynman diagrams. Version 2.0 release notes,” Comput. Phys. Commun. 180 (2009) 1709–1715, arXiv:0811.4113 [hep-ph].
  * [28] G. ’t Hooft, “Naturalness, chiral symmetry, and spontaneous chiral symmetry breaking,” NATO Sci. Ser. B 59 (1980) 135–157.
  * [29] F. Staub, “SARAH 4 : A tool for (not only SUSY) model builders,” Comput. Phys. Commun. 185 (2014) 1773–1790, arXiv:1309.7223 [hep-ph].
  * [30] A. Vicente, “Computer tools in particle physics,” arXiv:1507.06349 [hep-ph].
  * [31] W. Porod, “SPheno, a program for calculating supersymmetric spectra, SUSY particle decays and SUSY particle production at e+ e- colliders,” Comput. Phys. Commun. 153 (2003) 275–315, arXiv:hep-ph/0301101.
  * [32] W. Porod and F. Staub, “SPheno 3.1: Extensions including flavour, CP-phases and models beyond the MSSM,” Comput. Phys. Commun. 183 (2012) 2458–2469, arXiv:1104.1573 [hep-ph].
  * [33] W. Porod, F. Staub, and A. Vicente, “A Flavor Kit for BSM models,” Eur. Phys. J. C 74 no. 8, (2014) 2992, arXiv:1405.1434 [hep-ph].
  * [34] G. Bélanger, F. Boudjema, A. Goudelis, A. Pukhov, and B. Zaldivar, “micrOMEGAs5.0 : Freeze-in,” Comput. Phys. Commun. 231 (2018) 173–186, arXiv:1801.03509 [hep-ph].
  * [35] P. F. de Salas, D. V. Forero, S. Gariazzo, P. Martínez-Miravé, O. Mena, C. A. Ternes, M. Tórtola, and J. W. F. Valle, “2020 global reassessment of the neutrino oscillation picture,” JHEP 02 (2021) 071, arXiv:2006.11237 [hep-ph].
  * [36] I. Cordero-Carrión, M. Hirsch, and A. Vicente, “Master Majorana neutrino mass parametrization,” Phys. Rev. D 99 no. 7, (2019) 075019, arXiv:1812.03896 [hep-ph].
  * [37] I. Cordero-Carrión, M. Hirsch, and A. Vicente, “General parametrization of Majorana neutrino mass models,” Phys. Rev. D 101 no. 7, (2020) 075032, arXiv:1912.08858 [hep-ph].
  * [38] MEG Collaboration, A. M. Baldini et al., “Search for the lepton flavour violating decay $\mu^{+}\rightarrow\mathrm{e}^{+}\gamma$ with the full dataset of the MEG experiment,” Eur. Phys. J. C 76 no. 8, (2016) 434, arXiv:1605.05081 [hep-ex].
  * [39] SINDRUM Collaboration, U. Bellgardt et al., “Search for the Decay $\mu^{+}\to e^{+}e^{+}e^{-}$,” Nucl. Phys. B 299 (1988) 1–6.
  * [40] SINDRUM II Collaboration, W. H. Bertl et al., “A Search for muon to electron conversion in muonic gold,” Eur. Phys. J. C 47 (2006) 337–346.
  * [41] Particle Data Group Collaboration, P. A. Zyla et al., “Review of Particle Physics,” PTEP 2020 no. 8, (2020) 083C01.
  * [42] Planck Collaboration, N. Aghanim et al., “Planck 2018 results. VI. Cosmological parameters,” Astron. Astrophys. 641 (2020) A6, arXiv:1807.06209 [astro-ph.CO].
  * [43] XENON Collaboration, E. Aprile et al., “Dark Matter Search Results from a One Ton-Year Exposure of XENON1T,” Phys. Rev. Lett. 121 no. 11, (2018) 111302, arXiv:1805.12562 [astro-ph.CO].
  * [44] M. Klasen, C. E. Yaguna, and J. D. Ruiz-Alvarez, “Electroweak corrections to the direct detection cross section of inert higgs dark matter,” Phys. Rev. D 87 (2013) 075025, arXiv:1302.1657 [hep-ph].
  * [45] Fermi-LAT Collaboration, M. Ackermann et al., “Searching for Dark Matter Annihilation from Milky Way Dwarf Spheroidal Galaxies with Six Years of Fermi Large Area Telescope Data,” Phys. Rev. Lett. 115 no. 23, (2015) 231301, arXiv:1503.02641 [astro-ph.HE].
  * [46] H.E.S.S. Collaboration, H. Abdallah et al., “Search for dark matter annihilations towards the inner Galactic halo from 10 years of observations with H.E.S.S,” Phys. Rev. Lett. 117 no. 11, (2016) 111301, arXiv:1607.08142 [astro-ph.HE].
  * [47] AMS Collaboration, M. Aguilar et al., “Antiproton Flux, Antiproton-to-Proton Flux Ratio, and Properties of Elementary Particle Fluxes in Primary Cosmic Rays Measured with the Alpha Magnetic Spectrometer on the International Space Station,” Phys. Rev. Lett. 117 no. 9, (2016) 091103.
  * [48] A. Reinert and M. W. Winkler, “A Precision Search for WIMPs with Charged Cosmic Rays,” JCAP 01 (2018) 055, arXiv:1712.00002 [astro-ph.HE].
  * [49] A. Cuoco, J. Heisig, M. Korsmeier, and M. Krämer, “Constraining heavy dark matter with cosmic-ray antiprotons,” JCAP 04 (2018) 004, arXiv:1711.05274 [hep-ph].
  * [50] ATLAS Collaboration, M. Aaboud et al., “Search for doubly charged scalar bosons decaying into same-sign $W$ boson pairs with the ATLAS detector,” Eur. Phys. J. C 79 no. 1, (2019) 58, arXiv:1808.01899 [hep-ex].
  * [51] ATLAS Collaboration, M. Aaboud et al., “Search for doubly charged Higgs boson production in multi-lepton final states with the ATLAS detector using proton–proton collisions at $\sqrt{s}=13\,\text{TeV}$,” Eur. Phys. J. C 78 no. 3, (2018) 199, arXiv:1710.09748 [hep-ex].
  * [52] CMS Collaboration, A. M. Sirunyan et al., “Search for invisible decays of a Higgs boson produced through vector boson fusion in proton-proton collisions at $\sqrt{s}=$ 13 TeV,” Phys. Lett. B 793 (2019) 520–551, arXiv:1809.05937 [hep-ex].
  * [53] LUX Collaboration, D. Akerib et al., “Results from a search for dark matter in the complete LUX exposure,” Phys. Rev. Lett. 118 no. 2, (2017) 021303, arXiv:1608.07648 [astro-ph.CO].
  * [54] LUX Collaboration, D. S. Akerib et al., “Results of a Search for Sub-GeV Dark Matter Using 2013 LUX Data,” Phys. Rev. Lett. 122 no. 13, (2019) 131301, arXiv:1811.11241 [astro-ph.CO].
  * [55] PandaX-II Collaboration, X. Cui et al., “Dark Matter Results From 54-Ton-Day Exposure of PandaX-II Experiment,” Phys. Rev. Lett. 119 no. 18, (2017) 181302, arXiv:1708.06917 [astro-ph.CO].
  * [56] DarkSide Collaboration, P. Agnes et al., “DarkSide-50 532-day Dark Matter Search with Low-Radioactivity Argon,” Phys. Rev. D 98 no. 10, (2018) 102006, arXiv:1802.07198 [astro-ph.CO].
  * [57] DEAP Collaboration, R. Ajaj et al., “Search for dark matter with a 231-day exposure of liquid argon using DEAP-3600 at SNOLAB,” Phys. Rev. D 100 no. 2, (2019) 022004, arXiv:1902.04048 [astro-ph.CO].
  * [58] J. Billard, L. Strigari, and E. Figueroa-Feliciano, “Implication of neutrino backgrounds on the reach of next generation dark matter direct detection experiments,” Phys. Rev. D 89 no. 2, (2014) 023524, arXiv:1307.5458 [hep-ph].
  * [59] LUX-ZEPLIN Collaboration, D. S. Akerib et al., “Projected WIMP sensitivity of the LUX-ZEPLIN dark matter experiment,” Phys. Rev. D 101 no. 5, (2020) 052002, arXiv:1802.06039 [astro-ph.IM].
  * [60] XENON Collaboration, E. Aprile et al., “Projected WIMP sensitivity of the XENONnT dark matter experiment,” JCAP 11 (2020) 031, arXiv:2007.08796 [physics.ins-det].
  * [61] GADMC Collaboration, C. Galbiati et al., “Future Dark Matter Searches with Low-Radioactivity Argon,” Input to the European Particle Physics Strategy Update 2018-2020 (2018) . https://indico.cern.ch/event/765096/contributions/3295671/attachments/1785196/2906164/DarkSide-Argo_ESPP_Dec_17_2017.pdf.
  * [62] M. Schumann et al., “Dark matter sensitivity of multi-ton liquid xenon detectors,” JCAP 1510 no. 10, (2015) 016, arXiv:1506.08309 [physics.ins-det].
  * [63] DARWIN Collaboration, J. Aalbers et al., “DARWIN: towards the ultimate dark matter detector,” JCAP 1611 (2016) 017, arXiv:1606.07001 [astro-ph.IM].
  * [64] J. Billard et al., “Direct Detection of Dark Matter – APPEC Committee Report,” arXiv:2104.07634 [hep-ex].
  * [65] C. Garcia-Cely and A. Ibarra, “Novel Gamma-ray Spectral Features in the Inert Doublet Model,” JCAP 09 (2013) 025, arXiv:1306.4681 [hep-ph].
  * [66] F. Giacchino, L. Lopez-Honorez, and M. H. G. Tytgat, “Scalar Dark Matter Models with Significant Internal Bremsstrahlung,” JCAP 10 (2013) 025, arXiv:1307.6480 [hep-ph].
  * [67] PAMELA Collaboration, O. Adriani et al., “An anomalous positron abundance in cosmic rays with energies 1.5-100 GeV,” Nature 458 (2009) 607–609, arXiv:0810.4995 [astro-ph].
  * [68] PAMELA Collaboration, O. Adriani et al., “Cosmic-Ray Positron Energy Spectrum Measured by PAMELA,” Phys. Rev. Lett. 111 (2013) 081102, arXiv:1308.0133 [astro-ph.HE].
  * [69] AMS Collaboration, L. Accardo et al., “High Statistics Measurement of the Positron Fraction in Primary Cosmic Rays of 0.5–500 GeV with the Alpha Magnetic Spectrometer on the International Space Station,” Phys. Rev. Lett. 113 (2014) 121101.
  * [70] AMS Collaboration, M. Aguilar et al., “Towards Understanding the Origin of Cosmic-Ray Positrons,” Phys. Rev. Lett. 122 no. 4, (2019) 041102.
  * [71] A. Cuoco, J. Heisig, L. Klamt, M. Korsmeier, and M. Krämer, “Scrutinizing the evidence for dark matter in cosmic-ray antiprotons,” Phys. Rev. D 99 no. 10, (2019) 103014, arXiv:1903.01472 [astro-ph.HE].
  * [72] I. Cholis, T. Linden, and D. Hooper, “A Robust Excess in the Cosmic-Ray Antiproton Spectrum: Implications for Annihilating Dark Matter,” Phys. Rev. D 99 no. 10, (2019) 103026, arXiv:1903.02549 [astro-ph.HE].
  * [73] J. Heisig, M. Korsmeier, and M. W. Winkler, “Dark matter or correlated errors: Systematics of the AMS-02 antiproton excess,” Phys. Rev. Res. 2 no. 4, (2020) 043017, arXiv:2005.04237 [astro-ph.HE].
  * [74] Fermi-LAT Collaboration, E. Charles et al., “Sensitivity Projections for Dark Matter Searches with the Fermi Large Area Telescope,” Phys. Rept. 636 (2016) 1–46, arXiv:1605.02016 [astro-ph.HE].
  * [75] CTA Consortium Collaboration, B. S. Acharya et al., Science with the Cherenkov Telescope Array.  WSP, 11, 2018.  arXiv:1709.07997 [astro-ph.IM].
  * [76] A. Sommerfeld, “Über die Beugung und Bremsung der Elektronen,” Annalen der Physik 403 no. 3, (Jan., 1931) 257–330.
  * [77] J. Hisano, S. Matsumoto, and M. M. Nojiri, “Explosive dark matter annihilation,” Phys. Rev. Lett. 92 (2004) 031303, arXiv:hep-ph/0307216.
  * [78] J. Hisano, S. Matsumoto, M. M. Nojiri, and O. Saito, “Non-perturbative effect on dark matter annihilation and gamma ray signature from galactic center,” Phys. Rev. D 71 (2005) 063528, arXiv:hep-ph/0412403.
  * [79] N. Arkani-Hamed, D. P. Finkbeiner, T. R. Slatyer, and N. Weiner, “A Theory of Dark Matter,” Phys. Rev. D 79 (2009) 015014, arXiv:0810.0713 [hep-ph].
  * [80] T. A. Chowdhury and S. Nasri, “The Sommerfeld Enhancement in the Scotogenic Model with Large Electroweak Scalar Multiplets,” JCAP 01 (2017) 041, arXiv:1611.06590 [hep-ph].
  * [81] A. Merle and M. Platscher, “Parity Problem of the Scotogenic Neutrino Model,” Phys. Rev. D 92 no. 9, (2015) 095002, arXiv:1502.03098 [hep-ph].