aainstitutetext: Institut für Theoretische Physik, Westfälische Wilhelms-
Universität Münster, Wilhelm-Klemm-Straße 9, 48149 Münster,
Germanybbinstitutetext: Max-Planck-Institut für Astrophysik, Karl-
Schwarzschild-Straße 1, 85741 Garching, Germany

# Singlet–doublet fermion and triplet scalar dark matter with radiative
neutrino masses

Juri Fiaschi a Michael Klasen a, b and Simon May<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We present a detailed study of a combined singlet–doublet fermion and triplet
scalar model for dark matter. These models have only been studied separately
in the past. Together, they form a simple extension of the Standard Model that
can account for dark matter and explain the existence of neutrino masses,
which are generated radiatively. This holds even if singlet–doublet fermions
and triplet scalars never contribute simultaneously to the dark matter
abundance. However, this also implies the existence of lepton flavour
violating processes. In addition, this particular model allows for gauge
coupling unification. The new fields are odd under a new $\mathbb{Z}_{2}$
symmetry to stabilise the dark matter candidate. We analyse the dark matter,
neutrino mass and lepton flavour violation aspects both separately and in
conjunction, exploring the viable parameter space of the model. This is done
using a numerical random scan imposing successively the neutrino mass and
mixing, relic density, Higgs mass, direct detection, collider and lepton
flavour violation constraints. We find that dark matter in this model is
fermionic for masses below about $1\text{\,}\TeV$ and scalar above. The narrow
mass regions found previously for the two separate models are enlarged by
their coupling. While coannihilations of the weak isospin partners are
sizeable, this is not the case for fermions and scalars despite their often
similar masses due to the relatively small coupling of the two sectors,
imposed by the small neutrino masses. We observe a high degree of
complementarity between direct detection and lepton flavour violation
experiments, which should soon allow to fully probe the fermionic dark matter
sector and at least partially the scalar dark matter sector.

###### Keywords:

Beyond Standard Model, Cosmology of Theories beyond the SM, Neutrino Physics,
Discrete Symmetries

††preprint: MS-TP-18-33††arxiv: 1812.11133

##  1 Introduction

Evidence for dark matter (DM) is accumulating from astrophysical observations
at many different length scales Klasen:2015uma . Since no particle in the
Standard Model (SM) of particle physics has the required properties of being
uncharged under the electromagnetic and strong interactions and to be
sufficiently cold to explain the observed large scale structure in the
Universe, DM is widely believed to provide a strong indication for physics
beyond the SM.

The simplest solution to the DM puzzle consists in adding a small number of
new particles to the SM, out of which the lightest is neutral and stable, e.
g. due to a discrete $\mathbb{Z}_{2}$ symmetry. Weakly interacting massive
particles (WIMPs), such as heavy scalars or neutrinos, are among the most
promising candidates, as they have the right cross section to be produced as
thermal relics with the observed density
$\Omega_{\text{c}}^{\text{obs}}h^{2}=$0.120\pm 0.001$$ Aghanim:2018eyx . Here,
$h$ denotes the present Hubble expansion rate in units of
$100\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}\text{\,}{\mathrm{Mpc}}^{-1}$.
Many of these so-called minimal models have been studied in the past,
including those with scalar singlets Burgess:2000yq , doublets
LopezHonorez:2006gr ; Klasen:2013btp , and triplets Araki:2011hm ;
Fischer:2011zz ; JosseMichaux:2012wj ; Ayazi:2015mva ; Khan:2016sxm , fermion
singlets Esch:2013rta ; Klasen:2013ypa , doublets Cohen:2011ec ;
Cheung:2013dua ; Calibbi:2015nha , and triplets Dedes:2014hga , as well as
with higher scalar Hambye:2009pw  and/or fermion multiplets Cirelli:2005uq .
In particular, triplet scalar DM requires hypercharge $Y=0$ such that the
$\operatorname{SU}(2)_{\text{L}}$ component with $T_{3}=0$ is neutral
Restrepo:2013aga . On the other hand, mixing fermion singlets and doublets
reduces the coupling to weak gauge bosons and can transform DM from a Dirac
into a Majorana particle, yielding both the correct relic density and still
allowed direct detection cross sections Cohen:2011ec .

Particularly well motivated models are those that do not only provide a DM
candidate, but also solve other SM problems such as the smallness of neutrino
masses. This is possible when the $d=5$ Weinberg operator is realised at one
loop Bonnet:2012kz , such that the particles in the loop have opposite
$\mathbb{Z}_{2}$ parity to the SM particles and include a neutral DM candidate
Restrepo:2013aga . One of the most popular so-called radiative seesaw models
is the scotogenic model with only one additional scalar (inert Higgs)
$\operatorname{SU}(2)_{\text{L}}$ doublet and a (right-handed neutrino)
fermion singlet Ma:2006km , for which we recently demonstrated the importance
of coannihilations between the scalar dark matter and the right-handed
neutrinos Klasen:2013jpa . Many variants with general multiplets containing
two scalars and one fermion have subsequently been proposed. In particular,
the observed relic density was found to require a scalar triplet DM mass of
$2.5\text{\,}\mathrm{TeV}$ and couplings of $\mathcal{O}(${10}^{-3}$)$ to an
additional charged scalar singlet and vector-like doublet fermion to reproduce
the observed neutrino mass differences Law:2013saa ; Brdar:2013iea , whereas
the scalar triplet DM mass could vary from
$1.7\text{\,}\mathrm{TeV}2.6\text{\,}\mathrm{TeV}$ when a neutral scalar
singlet and vector-like doublet fermion were added Farzan:2010mr ; Law:2013saa
. Unfortunately, these masses/couplings are too large/small to explain the
observed tension in the muon’s $g-2$ between theory and experiment
Okada:2015vwh .

The models above all belong to the one-loop topology T3 with two scalars and
one fermion Bonnet:2012kz ; Restrepo:2013aga . In this paper, we study a model
of topology T1-3 with one scalar and two fermions, one of which is vector-
like. In contrast to the first of these models (T1-3-A with hypercharge
parameter $\alpha=0$), where the scalar DM was a singlet, had to be lighter
than $600\text{\,}\mathrm{GeV}$, and where the parameter space will soon be
explored by $\mu\to 3e$ and $\mu$–$e$ conversion experiments Klasen:2016vgl ,
we investigate here a model (T1-3-B, also with $\alpha=0$) where the scalar DM
is the neutral component of a triplet. In both cases, (at least) two
generations of scalars must be introduced, so that not only the constraints
from the DM relic density and observed Higgs mass, but also two non-zero
neutrino mass differences and mixing angles can be imposed Tanabashi:2018oca .
As for the model T1-3-A, we restrict ourselves to two scalar generations,
which implies that the lightest neutrino is massless. Both models, like a
previously studied model with both singlet–doublet scalars and fermions
(T1-2-A with $\alpha=0$) Esch:2018ccs , have the additional advantage that
they allow for gauge coupling unification at a scale
$\Lambda=\mathcal{O}(${10}^{13}\text{\,}\mathrm{GeV}$)$, at variance with most
of the other models mentioned above including the original scotogenic model
Hagedorn:2016dze . Like all models classified in Restrepo:2013aga , these
models feature Majorana neutrinos.

From the list of models with DM, radiative neutrino masses and gauge coupling
unification which have not been studied previously, our model features the
smallest number (six) of new interaction terms, which makes it possible to
fully explore its parameter space. We do so by constructing the most general
Lagrangian from the field content, eliminating a redundant term, generating
model files for SPheno and micrOMEGAs with SARAH 4.13.0 Staub:2013tta ,
imposing the neutrino masses using the Casas–Ibarra parametrisation
Casas:2001sr , and calculating the physical particle spectrum and relevant
precision observables with SPheno 4.0.3 Porod:2011nf  as well as the DM relic
density and direct detection cross sections with micrOMEGAs 4.3.5
Barducci:2016pcb . The constructed Lagrangian was checked and template SARAH
model files were generated by the newly-developed tool minimal-lagrangians,
which is available upon request.

The remainder of this paper is organised as follows: In section 2 we introduce
the model under examination, giving details of its field content and the full
Lagrangian, describing the mixing of fundamental particles into physical
states in the fermion and scalar sectors as well as the mass splitting between
neutral and charged states. In section 3, we discuss the features of the two
distinct DM candidates within our model and update and clarify contradictory
results in the literature. In section 4 we connect the dark sector to the
radiative generation of neutrino masses and explain how we impose neutrino
constraints on the parameter space of the model. In section 5 we study their
impact on the nature, parameter space and direct detection prospects of the DM
candidates and make predictions for the automatically generated lepton flavour
violating (LFV) processes. Our conclusions are given in section 6.

##  2 Description of the model

Figure 1: Loop topology T1-3 for the radiative generation of neutrino masses at the one-loop level. Table 1: New fields and their quantum numbers in the model T1-3-B with $\alpha=0$. Field | Generations | Spin | Lorentz rep. | $\operatorname{SU}(3)_{\text{C}}$ | $\operatorname{SU}(2)_{\text{L}}$ | $\operatorname{U}(1)_{\text{Y}}$ | $\mathbb{Z}_{2}$
---|---|---|---|---|---|---|---
$\Psi$ | 1 | $\nicefrac{{1}}{{2}}$ | $(\nicefrac{{1}}{{2}},0)$ | $\mathbf{1}$ | $\mathbf{1}$ | $0$ | $-1$
$\psi$ | 1 | $\nicefrac{{1}}{{2}}$ | $(\nicefrac{{1}}{{2}},0)$ | $\mathbf{1}$ | $\mathbf{2}$ | $-1$ | $-1$
$\psi^{\prime}$ | 1 | $\nicefrac{{1}}{{2}}$ | $(\nicefrac{{1}}{{2}},0)$ | $\mathbf{1}$ | $\mathbf{2}$ | $1$ | $-1$
$\phi_{i}$ | 2 | $0$ | $(0,0)$ | $\mathbf{1}$ | $\mathbf{3}$ | $0$ | $-1$

Following the notation in ref. Bonnet:2012kz ; Restrepo:2013aga , the model
T1-3-B with hypercharge parameter $\alpha=0$ is defined by extending the SM
with the colour-singlet fields in table 1. T1-3 labels the general one-loop
topology generating radiative neutrino masses,111The three topologies T1-1,
T1-2 and T1-3, collectively T1-$n$, differ only in the number $n$ of fermion
lines (as opposed to scalar lines) in the loop. as shown in figure 1, while
T1-3-B is a specific $\operatorname{SU}(2)_{\text{L}}$ multiplet structure of
the new fields. The parameter $\alpha$ is simply a common offset to the
hypercharge values given in table 1 – for example, in the model T1-3-B with
$\alpha=2$, every value in the column $\operatorname{U}(1)_{\text{Y}}$ is
increased by $2$ compared to T1-3-B with $\alpha=0$.

The present model therefore combines the $\operatorname{SU}(2)_{\text{L}}$
triplet scalars $\phi_{i}$ of zero hypercharge
($\operatorname{U}(1)_{\text{Y}}$) with an $\operatorname{SU}(2)_{\text{L}}$
singlet fermion $\Psi$ and doublet fermions $\psi,\psi^{\prime}$. They are all
odd under the discrete symmetry $\mathbb{Z}_{2}$, while the SM fields are
even. The components of the new fields are given by

$\Psi=\Psi^{0},\quad\psi=\begin{pmatrix}\psi^{0}\\\
\psi^{-}\end{pmatrix},\quad\psi^{\prime}=\begin{pmatrix}\psi^{\prime+}\\\
\psi^{\prime
0}\end{pmatrix},\quad\phi_{i}=\begin{pmatrix}\frac{1}{\sqrt{2}}\phi_{i}^{0}&\phi_{i}^{+}\\\
\phi_{i}^{-}&-\frac{1}{\sqrt{2}}\phi_{i}^{0}\end{pmatrix},$ (1)

where superscripts indicate electric charges. To obtain two non-zero neutrino
mass differences, two generations of scalar triplets are required. Since the
scalar triplets have zero hypercharge, they are treated as real,
$(\phi^{0}_{i})^{\dagger}=\phi^{0}_{i},(\phi^{+}_{i})^{\dagger}=\phi^{-}_{i}$.
$\Psi$ has the same quantum numbers as a $\mathbb{Z}_{2}$-odd right-handed
neutrino, whereas $\psi$ and $\psi^{\prime}$ together form a
$\mathbb{Z}_{2}$-odd vector-like lepton doublet, which makes the model
automatically anomaly-free. In principle, all neutral field components are
possible DM candidates. The $\mathbb{Z}_{2}$ symmetry not only stabilises the
lightest new particle against decay into SM fields, but also forbids neutrino
masses from a tree-level type-I seesaw mechanism.

The most general renormalisable Lagrangian for the model is222Here and in the
following, products of field multiplets are always defined to be Lorentz- and
gauge-invariant. For example, the product of the two
$\operatorname{SU}(2)_{\text{L}}$ Weyl spinors $\psi$ and $\psi^{\prime}$
should be interpreted as
$\psi\psi^{\prime}=\varepsilon^{ji}\varepsilon^{j^{\prime}i^{\prime}}\psi_{ii^{\prime}}\psi^{\prime}_{jj^{\prime}}$,
where the indices represent the $\operatorname{SU}(2)_{\text{L}}$ and spinor
components.

$\displaystyle\mathcal{L}$
$\displaystyle=\mathcal{L}_{\text{SM}}+\mathcal{L}_{\text{kin}}-\frac{1}{2}(M_{\phi}^{2})^{ij}\operatorname{Tr}(\phi_{i}\phi_{j})-\left(\frac{1}{2}M_{\Psi}\Psi\Psi+\text{H.\,c.}\right)-\left(M_{\psi\psi^{\prime}}\psi\psi^{\prime}+\text{H.\,c.}\right)$
$\displaystyle\quad-(\lambda_{2})^{ij}H^{\dagger}\phi_{i}\phi_{j}H-(\lambda_{3})^{ijkm}\operatorname{Tr}(\phi_{i}\phi_{j}\phi_{k}\phi_{m})$
$\displaystyle\quad-\left(\lambda_{4}(H^{\dagger}\psi^{\prime})\Psi+\text{H.\,c.}\right)-\left(\lambda_{5}(H\psi)\Psi+\text{H.\,c.}\right)-\left((\lambda_{6})^{ij}L_{i}\phi_{j}\psi^{\prime}+\text{H.\,c.}\right),$
(2)

where $H$ is the SM Higgs field (with vacuum expectation value $v$ and quartic
coupling $\lambda$)333We have used a value of $\lambda=$0.2612$$ as an input
to SPheno throughout the analysis. and we have combined the term

$-(\lambda_{1})^{ij}(H^{\dagger}H)\operatorname{Tr}(\phi_{i}\phi_{j})$ (3)

with the one proportional to $\lambda_{2}$ using the identity

$H^{\dagger}\\{\phi_{i},\phi_{j}\\}H=(H^{\dagger}H)\operatorname{Tr}(\phi_{i}\phi_{j}).$
(4)

However, for the remaining analysis, we will instead use the $\lambda_{1}$
term (3) for easier comparison with the previous literature on models with one
scalar triplet, which is completely equivalent as long as there are no CP-
violating phases. The couplings $\lambda_{4}$ and $\lambda_{5}$ have the
function of Yukawa terms, which link the fermion singlet and doublets to the
SM Higgs boson. The coupling $\lambda_{6}$ connects the SM lepton doublet $L$
to the new fields, so that it will be involved in the process of radiative
neutrino masses generation.444Except for
$(M_{\phi}^{2})^{ij},\lambda_{1}^{ij},\lambda_{3}^{ij}\in\mathbb{R}$, the
parameters can, in principle, have a complex CP-violating phase. Note that not
all of the parameters are independent:
$(M_{\phi}^{2})^{ij}=(M_{\phi}^{2})^{ji}$,
$(\lambda_{1})^{ij}=(\lambda_{1})^{ji}$,
${(\lambda_{2})^{ij}}^{*}=(\lambda_{2})^{ji}$,
$(\lambda_{3})^{ijkm}=(\lambda_{3})^{jkmi}=(\lambda_{3})^{kmij}=(\lambda_{3})^{mijk}$.

In order to ensure that the theory has a stable vacuum, the scalar potential
in (2) should have a minimum, i. e. it should be bounded from below. For the
SM Higgs field with a single (inert) scalar triplet (one generation),
constraints on the parameters following from this requirement are given in
Hambye:2009pw ; Araki:2011hm . In general, determining whether a function is
bounded from below is not a trivial task and expressing the constraints on the
parameters as analytic conditions is usually only possible for specific
directions in the variables. Constraints analogous to those for one generation
can be obtained:

$\lambda_{3}^{iiii}\geq 0,\quad
2\sqrt{\lambda\lambda_{3}^{iiii}}\geq-\lambda_{1}^{ii}.$ (5)

In the direction where all fields are taken to infinity equally with only some
possible signs between them, the condition

$\lambda+\lambda_{1}^{11}+\lambda_{1}^{22}+\lambda_{3}^{1111}+\lambda_{3}^{2222}+\lambda_{3}^{1212}+2\lambda_{3}^{1112}\geq
2|\lambda_{1}^{12}|+4(|\lambda_{3}^{1112}|+|\lambda_{3}^{1222}|)$ (6)

must hold. For $H=0$, the potential has the same form as that in Gunion:2002zf
(with $d=0$), so additional conditions can be obtained analogously, such as:

$\begin{gathered}\sqrt{\lambda_{3}^{1111}\lambda_{3}^{2222}}\geq\lambda_{3}^{1212}-2\lambda_{3}^{1122},\\\
\lambda_{3}^{1111}+\lambda_{3}^{2222}+\lambda_{3}^{1212}+2\lambda_{3}^{1122}\geq
4(|\lambda_{3}^{1112}|+|\lambda_{3}^{1222}|),\\\
\lambda_{3}^{1111}\lambda_{3}^{2222}+\frac{1}{2}\sqrt{\lambda_{3}^{1111}\lambda_{3}^{2222}}\left(\lambda_{3}^{1212}+2\lambda_{3}^{1122}\right)\geq
2(\lambda_{3}^{1111}|\lambda_{3}^{1112}|+\lambda_{3}^{2222}|\lambda_{3}^{1222}|).\end{gathered}$
(7)

To summarize, vacuum stability can be controlled through the parameter
$\lambda_{3}$, which does however not affect the DM phenomenology (see below).
Apart from loop corrections, the vacuum structure is otherwise unchanged since
the imposed exact $\mathbb{Z}_{2}$ symmetry prevents the new scalars
$\phi_{i}$ from acquiring nonzero vacuum expectation values.

After electroweak symmetry breaking, the mass terms of the neutral fermions
take the form

$\mathcal{L}_{\text{f},0}=-\frac{1}{2}M_{\Psi}\Psi\Psi-
M_{\psi\psi^{\prime}}\psi^{0}\psi^{\prime
0}-\frac{\lambda_{4}v}{\sqrt{2}}\psi^{\prime
0}\Psi-\frac{\lambda_{5}v}{\sqrt{2}}\psi^{0}\Psi+\text{H.\,c.},$ (8)

where $v$ is the vacuum expectation value of the SM Higgs field $H$.
Diagonalisation of the mass matrix Cohen:2011ec ; Cheung:2013dua

$M_{\text{f},0}=\begin{pmatrix}M_{\Psi}&\frac{\lambda_{5}v}{\sqrt{2}}&\frac{\lambda_{4}v}{\sqrt{2}}\\\
\frac{\lambda_{5}v}{\sqrt{2}}&0&M_{\psi\psi^{\prime}}\\\
\frac{\lambda_{4}v}{\sqrt{2}}&M_{\psi\psi^{\prime}}&0\end{pmatrix}$ (9)

with the unitary mixing matrix $U_{\chi}$ allows then to rewrite the neutral
mass Lagrangian

$\mathcal{L}_{\text{f},0}=-\frac{1}{2}(\Psi,\psi^{0},\psi^{\prime
0})M_{\text{f},0}\begin{pmatrix}\Psi\\\ \psi^{0}\\\ \psi^{\prime
0}\end{pmatrix}+\text{H.\,c.}=-\frac{1}{2}(\chi^{0})^{\mathsf{T}}U_{\chi}^{*}M_{0}U_{\chi}^{\dagger}\chi^{0}+\text{H.\,c.}$
(10)

with

$U_{\chi}^{*}M_{\text{f},0}U_{\chi}^{\dagger}=\operatorname{diag}(m_{\chi_{1}^{0}},m_{\chi_{2}^{0}},m_{\chi_{3}^{0}})$
(11)

in terms of the physical Majorana fermions

$\chi^{0}=U_{\chi}\begin{pmatrix}\Psi^{0}\\\ \psi^{0}\\\ \psi^{\prime
0}\end{pmatrix}.$ (12)

For the two generations of scalar triplets, electroweak symmetry breaking
induces a shift in the mass matrices

$M_{\phi^{0}}^{2}=M_{\phi^{\pm}}^{2}=M_{\phi}^{2}+\lambda_{1}v^{2}.$ (13)

Note that at tree level, the charged and neutral mass matrices are identical.
Similarly to the neutral fermions, the scalar mass eigenstates

$\eta^{0,\pm}=O_{\eta}\begin{pmatrix}\phi_{1}^{0,\pm}\\\
\phi_{2}^{0,\pm}\end{pmatrix}$ (14)

are obtained by diagonalising the scalar mass matrices with (in this case
orthogonal) rotation matrices $O_{\eta}$, leading to

$O_{\eta}M_{\phi^{0,\pm}}^{2}O_{\eta}^{\mathsf{T}}=\operatorname{diag}(m_{\eta_{1}^{0,\pm}}^{2},m_{\eta_{2}^{0,\pm}}^{2}).$
(15)

At the one-loop level, gauge interactions introduce mass splittings

$\Delta
m_{\eta_{i}}=m_{\eta_{i}^{\pm}}-m_{\eta_{i}^{0}}=$166\text{\,}\mathrm{MeV}$,$
(16)

which, together with the new interactions, render the neutral components
slightly lighter than the corresponding charged ones Cirelli:2005uq , making
them potential dark matter candidates.

##  3 Dark matter relic density

Depending on the spin of the lightest neutral $\mathbb{Z}_{2}$-odd mass
eigenstate, our model allows for both singlet–doublet fermion and triplet
scalar DM. As mentioned in the introduction, these models have been studied
separately in the past, which, however, prohibits the radiative generation of
neutrino masses. In this section, we briefly comment on both scenarios
individually, updating and clarifying the results in the literature. We will
limit ourselves to one generation of scalar triplets in this section, which
can be viewed as decoupling the second generation via

$(M_{\phi}^{2})^{22}=($1000\text{\,}\mathrm{TeV}$)^{2},\qquad(M_{\phi}^{2})^{12}=(\lambda_{1})^{i2}=(\lambda_{6})^{i2}=0,$
(17)

since the scalars are decoupled in section 3.1 anyway and the additional
generation does not introduce any qualitatively new phenomena in section 3.2.

###  3.1 Singlet–doublet fermions

Figure 2: The observed relic density contours
($\Omega_{\text{c}}^{\text{obs}}h^{2}$), separating regions of too small and
too large relic density, as a function of the singlet–doublet fermion DM mass
$m_{\chi_{1}^{0}}$ and the Yukawa couplings
$\sqrt{\lambda_{4}^{2}+\lambda_{5}^{2}}$ assuming
$M_{\Psi}=M_{\psi\psi^{\prime}}$ for different values of
$\nicefrac{{\lambda_{4}}}{{\lambda_{5}}}$.

We first assume that DM is composed of the lightest neutral fermion
($\chi_{1}^{0}$). The scalar sector is decoupled by setting

$(M_{\phi}^{2})^{11}=($1000\text{\,}\mathrm{TeV}$)^{2},\qquad\lambda_{1}=\lambda_{3}=\lambda_{6}=0.$
(18)

Choosing $M_{\Psi}=$200\text{\,}\mathrm{GeV}$$,
$M_{\psi\psi^{\prime}}=$300\text{\,}\mathrm{GeV}$$ and $\lambda_{5}=$0.36$$
and neglecting loop corrections, we generally reproduce the relic density and
direct detection cross sections in figure 2 of ref. Cohen:2011ec . In
particular, depending on the value of $\lambda_{4}$ and thus the
singlet–doublet fermion mixing, blind spots of spin-independent and -dependent
direct detection experiments with no DM couplings to Higgs and $Z$ bosons
appear. Note, however, that our relic density exhibits a slightly higher peak
value when the (now known) Higgs mass of $125.08\text{\,}\mathrm{GeV}$
Tanabashi:2018oca  is used. Also, our spin-independent direct detection cross
section is smaller by about a factor of two, which we can trace to updated
nuclear form factors in micrOMEGAs (version 4.3.5 vs. 2.4). A similar
observation also holds for the fermion mass scans in figure 5 of ref.
Cheung:2013dua , where $\nicefrac{{\lambda_{4}}}{{\lambda_{5}}}\in\\{\pm 2,\pm
10\\}$ and $\sqrt{\lambda_{4}^{2}+\lambda_{5}^{2}}=$0.3$$. While we reproduce
the relic density and spin-dependent detection contours, our spin-independent
detection contours are shifted by about $0.1\text{\,}\mathrm{TeV}$ even after
adjusting for the nuclear form factors. Furthermore, the latest XENON1T
results Aprile:2018dbl  almost double the size of the excluded low-mass
regions, so that, e. g., $M_{\Psi}\simeq
M_{\psi\psi^{\prime}}>$1\text{\,}\mathrm{TeV}$$ for
$\nicefrac{{\lambda_{4}}}{{\lambda_{5}}}=2$. Below masses of
$1\text{\,}\mathrm{TeV}$, the observed relic density requires $M_{\Psi}\simeq
M_{\psi\psi^{\prime}}$ independently of
$\nicefrac{{\lambda_{4}}}{{\lambda_{5}}}$. For higher singlet masses, DM
becomes a pure doublet with constant mass $1\text{\,}\mathrm{TeV}$. For other,
in particular negative values of $\nicefrac{{\lambda_{4}}}{{\lambda_{5}}}$,
even the new XENON1T limits become as weak as $200\text{\,}\mathrm{GeV}$.

Figure 2 shows another example illustrating the behavior of the
singlet–doublet fermion model for the special case
$M_{\Psi}=M_{\psi\psi^{\prime}}$. For small Yukawa couplings, both mixing
between the neutral fermions and cross sections for annihilation into gauge
bosons (through the mixing angles) and the Higgs boson are small. On the other
hand, for $\sqrt{\lambda_{4}^{2}+\lambda_{5}^{2}}\geq$0.4$$, the DM mass must
rise quickly to compensate the increasingly efficient annihilation channels.
The mass splitting depends on the relative size of the couplings
$\nicefrac{{\lambda_{4}}}{{\lambda_{5}}}$, controlling the impact of
coannihilation processes.

###  3.2 Triplet scalars

The literature in the case when DM is composed of the lightest neutral scalar
($\eta_{1}^{0}$) is unfortunately contradictory, and we take the opportunity
to clarify the situation. We therefore now decouple the fermion sector by
setting

$M_{\Psi}=M_{\psi\psi^{\prime}}=$1000\text{\,}\mathrm{TeV}$,\qquad\lambda_{4}=\lambda_{5}=\lambda_{6}=0.$
(19)

Furthermore, we find that the parameter $\lambda_{3}$ inducing scalar co-
annihilations and conversions does not play an important role, so that we not
only neglect loop corrections, but also set $\lambda_{3}=0$ in this section.

Figure 3: The observed relic density contour
($\Omega_{\text{c}}^{\text{obs}}h^{2}$), separating regions of too small
(above) and too large (below) relic density, as a function of the triplet
scalar DM mass $m_{\eta_{1}^{0}}$ and its Higgs coupling $(\lambda_{1})^{11}$.
Here, we assume only one generation of scalars.

For one generation, the so-called inert triplet model was first studied in
ref. Cirelli:2005uq  (although _only_ including gauge interactions) with the
result that DM had to be in a very narrow mass region of $2.0\pm
0.05\text{\,}\mathrm{TeV}$ (see also ref. Hambye:2009pw ; Fischer:2011zz ;
JosseMichaux:2012wj ; Khan:2016sxm ) and that the spin-independent cross
section was $1.3\text{\times}{10}^{-9}\text{\,}\mathrm{pb}$. For large
couplings $\lambda_{1}\simeq 1$ of DM to Higgs bosons, a DM mass increase by
about $20\text{\,}\mathrm{\char 37\relax}$ was predicted due the opening of
Higgs channels in addition to the dominant DM annihilation into weak gauge
bosons. Figure 3 confirms that
$m_{\eta_{1}^{0}}\simeq$2\text{\,}\mathrm{TeV}$$ for $\lambda_{1}\leq$0.2$$,
but also shows that the Higgs channels then become so efficient that the DM
mass rapidly increases into the multi-$\mathrm{TeV}$ region (cf. figure 5 of
ref. Hambye:2009pw  and figure 4 of ref. JosseMichaux:2012wj ). Qualitatively,
this behaviour was also found in figure 1 of ref. Araki:2011hm , but with a
significant shift to a larger DM mass of $5.5\text{\,}\mathrm{TeV}$.
Correcting the normalisation of the squared neutral mass in their equation (7)
by a factor of two can partially explain this discrepancy. With the same
(incorrect) normalisation, an even larger DM mass of $7\text{\,}\mathrm{TeV}$
was found, but not commented upon in ref. Ayazi:2015mva . Surprisingly, and
despite using more recent nuclear form factors, we can reproduce the spin-
independent detection cross sections in figure 2 of ref. Araki:2011hm , which
also agree roughly with the numbers given in ref. Cirelli:2005uq ;
Hambye:2009pw  for sufficiently small $\lambda_{1}\simeq$0.01$$. As described
there, gauge interactions increase the cross section at one loop by
$0.86\text{\times}{10}^{-9}\text{\,}\mathrm{pb}$, while the Sommerfeld effect
induces a shift towards slightly larger DM masses (cf. figure 6 of ref.
Hambye:2009pw ).

##  4 Radiative neutrino masses

Figure 4: Feynman diagram for the radiative generation of neutrino masses at
the one-loop level after electroweak symmetry breaking and mixing
($k\in\\{1,\dots,n_{\text{f}}\\}$, $l\in\\{1,\dots,n_{\text{s}}\\}$). The
convention for Feynman diagrams is the same as in ref. Dreiner:2008tw .

After electroweak symmetry breaking, neutrino masses in our model arise from
the one-loop diagram shown in figure 4. Only the $n_{\text{s}}$ neutral scalar
and $n_{\text{f}}$ neutral fermion fields contribute to mass generation,
whereas the charged fields enter only into the propagator correction. In our
model, the $n_{\text{f}}=3$ fermions are superpositions of
$\operatorname{SU}(2)_{\text{L}}$ singlets and doublets, while the
$n_{\text{s}}=2$ scalars are superpositions of the two generations of scalar
triplets required for two non-zero neutrino masses. Evaluating the two-point
function in dimensional regularisation and summing over all contributions
leads to

$\displaystyle(M_{\nu})_{ij}$
$\displaystyle=\frac{1}{32\pi^{2}}\sum_{l=1}^{n_{\text{s}}}\lambda_{6}^{im}\lambda_{6}^{jn}(O_{\eta})_{lm}(O_{\eta})_{ln}\sum_{k=1}^{n_{\text{f}}}{(U_{\chi})^{*}_{k3}}^{2}\frac{m_{\chi_{k}^{0}}^{3}}{m_{\eta^{0}_{l}}^{2}-m_{\chi_{k}^{0}}^{2}}\ln\left(\frac{m_{\chi_{k}^{0}}^{2}}{m_{\eta^{0}_{l}}^{2}}\right)$
$\displaystyle=\frac{1}{32\pi^{2}}\sum_{l=1}^{n_{\text{s}}}A_{l}\lambda_{6}^{im}\lambda_{6}^{jn}(O_{\eta})_{lm}(O_{\eta})_{ln},$
(20)

where the ultraviolet divergences have cancelled as expected for a
renormalisable model with vanishing neutrino masses at tree level. As evident
from equation 20, the structure of the mass matrix is chiefly determined by
the number of scalar generations, which must be at least as large as the
desired number of non-zero neutrino masses.

The symmetric neutrino mass matrix $M_{\nu}$ in equation 20 is diagonalised by
the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix $U_{\nu}$,

$U_{\nu}^{\dagger}M_{\nu}U_{\nu}=D_{\nu}=\operatorname{diag}(\pm
m_{\nu_{1}},\pm m_{\nu_{2}},\pm m_{\nu_{3}}).$ (21)

Since the neutrino mass matrix is of the same form as in model T1-3-A with
hypercharge parameter $\alpha=0$ Klasen:2016vgl , it can similarly be expanded
in the limit of small Yukawa couplings $\lambda_{4},\lambda_{5}\ll 1$. At
leading order, we obtain

$M_{\nu}\approx$100\text{\,}\mathrm{meV}$\
\frac{M_{\Psi}}{$1\text{\,}\mathrm{TeV}$}\
\left(\frac{\lambda_{6}^{ij}\lambda_{4,5}}{${10}^{-5}$}\right)^{2},$ (22)

i. e. for $\mathrm{meV}$-scale neutrinos only the product
$\lambda_{6}^{ij}\lambda_{4,5}$ has to be of order ${10}^{-5}$. As an example,
we show in table 2 the dependence of the two non-zero neutrino masses
$m_{\nu_{i}}$ ($i\in\\{2,3\\}$) on the scalar-fermion coupling
$\lambda_{6}^{i2}$. The mass parameters and other couplings used in this
figure are listed in table 2.

Figure 5: Neutrino masses $m_{\nu_{i}}$ in the model T1-3-B ($\alpha=0$) with
two generations of scalar triplets as a function of $\lambda_{6}^{i2}$. The
used parameters are given in table 2. The experimentally determined neutrino
masses are reproduced for $\lambda_{6}^{i2}=$0.0044$$.
Table 2: Mass parameters and couplings used in table 2.

Parameter | Value
---|---
$(M_{\phi})^{11}$ | $1.4\text{\,}\mathrm{TeV}$
$(M_{\phi})^{22}$ | $3\text{\,}\mathrm{TeV}$
$(M_{\phi})^{12}$ | $0$
$M_{\Psi}$ | $1\text{\,}\mathrm{TeV}$
$M_{\psi\psi^{\prime}}$ | $1.5\text{\,}\mathrm{TeV}$
$\lambda_{1}$, $\lambda_{3}$ | $0$
$\lambda_{4}$, $\lambda_{5}$ | $5\text{\times}{10}^{-3}$
$\lambda_{6}^{i1}$ ($i\neq 3$) | $3\text{\times}{10}^{-4}$
$\lambda_{6}^{3l}$ | $3.05\text{\times}{10}^{-3}$

The neutrino mass matrix $M_{\nu}$ depends explicitly on the couplings

$\lambda_{6}=\begin{pmatrix}\lambda_{6}^{e1}&\lambda_{6}^{e2}\\\
\lambda_{6}^{\mu 1}&\lambda_{6}^{\mu 2}\\\ \lambda_{6}^{\tau
1}&\lambda_{6}^{\tau 2}\end{pmatrix}$ (23)

and on the masses $m_{\eta^{0}_{l}}$ and $m_{\chi^{0}_{k}}$ of the neutral
scalars and fermions, while the dependence on the other couplings
$\lambda_{1,4,5}$ remains implicit in the mixing matrices. The Casas–Ibarra
parametrisation Casas:2001sr  then allows to obtain $\lambda_{6}$ from the
experimental neutrino data, once the other couplings and masses have been
fixed. The procedure is as follows: First, one factorises the explicit
dependence on the couplings $\lambda_{6}$ from the neutrino mass matrix in
equation 20,

$M_{\nu}=\lambda_{6}M\lambda_{6}^{\mathsf{T}}=\lambda_{6}U_{M}^{\dagger}D_{M}U_{M}\lambda_{6}^{\mathsf{T}},$
(24)

where in the last step the $n_{\text{s}}\times n_{\text{s}}$ matrix $M$ has
been diagonalised to the matrix $D_{M}$ using the unitary (here: orthogonal,
as we have neglected complex phases) matrix $U_{M}$. On the other hand, the
neutrino mass matrix can be diagonalised using the PMNS matrix as in equation
21. Combining these expressions, one obtains

$D_{\nu}=U_{\nu}^{\dagger}\lambda_{6}U_{M}^{\dagger}D_{M}U_{M}\lambda_{6}^{\mathsf{T}}U_{\nu}$
(25)

or, equivalently ($m_{\nu_{1}}=0$),555For a diagonal matrix $D$,
$D^{\pm\frac{1}{2}}=\operatorname{diag}\Big{(}D_{11}^{\pm\frac{1}{2}},D_{22}^{\pm\frac{1}{2}},\dots\Big{)}$.

$\operatorname{diag}(0,1,1)=D_{\nu}^{-\frac{1}{2}}U_{\nu}^{\dagger}\lambda_{6}U_{M}^{\dagger}D_{M}U_{M}\lambda_{6}^{\mathsf{T}}U_{\nu}D_{\nu}^{-\frac{1}{2}}.$
(26)

The left-hand side can be rewritten as
$\operatorname{diag}(0,1,1)=R^{\dagger}R$ with the ‘rotation matrix’

$R=\begin{pmatrix}0&\cos(\theta)&\sin(\theta)\\\
0&-\sin(\theta)&\cos(\theta)\end{pmatrix},$ (27)

which is thus given by

$R=D_{M}^{\frac{1}{2}}U_{M}\lambda_{6}^{\mathsf{T}}U_{\nu}D_{\nu}^{-\frac{1}{2}}.$
(28)

Inverting this equation then leads to the Casas–Ibarra parametrisation

$\lambda_{6}=U_{\nu}^{*}D_{\nu}^{\frac{1}{2}}R^{\mathsf{T}}D_{M}^{-\frac{1}{2}}U_{M}^{*}.$
(29)

The couplings $\lambda_{6}$ of the dark scalar and fermion sectors are
therefore partially determined by the other couplings and masses, encoded in
$U_{M}$ and $D_{M}$, and by the measured neutrino mass differences (here:
masses, since we assume $m_{\nu_{1}}=0$ and normal hierarchy) $D_{\nu}$ and
mixing angles in $U_{\nu}$ Gonzalez-Garcia:2014bfa , but they can still vary
with the arbitrary angle $\theta$ in the rotation matrix $R$.

##  5 Dark matter direct detection and lepton flavour violation

In this section, we now connect singlet–doublet fermions with triplet scalars
using the couplings $\lambda_{6}$ with the aim to not only explain the
observed small neutrino masses as described in the previous section, but also
in order to study the effect of the neutrino mass constraints on the nature,
allowed parameter space and direct detection prospects of the two DM
candidates in this combined model. In addition, we make predictions for the
automatically generated lepton flavour violating (LFV) processes and
investigate the corresponding current and future experiments with respect to
their potential to further restrict the viable parameter space of the model.

Figure 6: The relic density as a function of the DM mass for singlet–doublet
fermion ($\chi^{0}_{1}$) and triplet scalar ($\eta^{0}_{1}$) DM. The used
parameters are given in table 3.
Table 3: Mass parameters and couplings used in table 3.

Parameter | Fermion DM | Scalar DM
---|---|---
$(M_{\phi})^{11}$ | $3\text{\,}\mathrm{TeV}$ | $[$0.5\text{\,}\mathrm{TeV}$,$3.5\text{\,}\mathrm{TeV}$]$
$(M_{\phi})^{22}$ | $1000\text{\,}\mathrm{TeV}$ | $1000\text{\,}\mathrm{TeV}$
$(M_{\phi})^{12}$ | $0$ | $0$
$M_{\Psi}$ | $[$0.5\text{\,}\mathrm{TeV}$,$2.8\text{\,}\mathrm{TeV}$]$ | $3\text{\,}\mathrm{TeV}$
$M_{\psi\psi^{\prime}}$ | $M_{\Psi}$ | $2.8\text{\,}\mathrm{TeV}$
$\lambda_{1}$, $\lambda_{3}$, $\lambda_{4}$, $\lambda_{5}$ | ${10}^{-2}$ | ${10}^{-2}$
$\lambda_{6}^{i2}$ | $0$ | $0$

For simplicity, we start in table 3 with the dependence of the relic density
on $\lambda_{6}^{i1}$, i. e. for only one generation of scalars, assuming the
couplings to be equal for all three lepton families. The results in table 3
therefore do not yet yield the correct neutrino masses, but are shown for
illustrative purposes only. The mass parameters and other couplings used in
this figure are listed in table 3. As described in section 3, two distinct DM
candidates give the observed relic density: singlet–doublet fermions
$\chi_{1}^{0}$ with masses up to about $1\text{\,}\mathrm{TeV}$, and triplet
scalars with masses around $2\text{\,}\mathrm{TeV}$. As we can now see, this
observation holds also in the case that the masses in the other sector are
only larger by $0.2\text{\,}\mathrm{TeV}2.5\text{\,}\mathrm{TeV}$. Only for
small mass splittings and very large couplings $\lambda_{6}^{i1}$, fermion-
scalar coannihilation processes can reduce the relic density, as one can
clearly see for triplet scalar DM, which is otherwise insensitive to
variations of $\lambda_{6}^{i1}$. In the heavy scalar mass region,
annihilation into weak bosons becomes ineffective, while the one into Higgs
bosons remains small due to $\lambda_{1}=$0.01$$, so that coannihilations can
have a significant effect. For fermion DM, it is even more pronounced and
starts already at a mass of $1.65\text{\,}\mathrm{TeV}$, whereas below this
value the effect is the opposite and small except for very large
$\lambda_{6}^{i1}=$0.8$$.

In the following two subsections, we present our main results of a numerical
random scan in the parameter space that remains when the observed neutrino
mass and mixing constraints as well as the Higgs boson mass
($\pm$2.5\text{\,}\mathrm{GeV}$$) are imposed. In particular, we vary the
couplings $\lambda_{1,4,5}$ in the range $[10^{-6},1]$ and the parameters
$M_{\phi}$, $M_{\Psi}$ and $M_{\psi\psi^{\prime}}$ in the range
$[$10\text{\,}\mathrm{GeV}$,$10\,000\text{\,}\mathrm{GeV}$]$.666Figures 7 to
12 show the combined results of two random scans: an exploratory one where the
mass parameters $M_{\phi}$, $M_{\Psi}$, $M_{\psi\psi^{\prime}}$ were varied in
the range $[$10\text{\,}\mathrm{GeV}$,$2000\text{\,}\mathrm{GeV}$]$ and one
using the full range given above. We also add a random sign for the couplings
$\lambda_{1,5}$ while keeping $\lambda_{4}$ always positive Cheung:2013dua .
All entries of $\lambda_{1}$ were set to the same value for simplicity (the
off-diagonal element only introduces additional mass mixing to the scalar
triplets). The coupling $\lambda_{3}$ has no phenomenological implications,
and the couplings $\lambda_{6}$ are obtained from the Casas–Ibarra
parametrisation as a function of the other parameters. The free parameter
$\theta$ to determine $\lambda_{6}$ is taken randomly in the range $[0,2\pi]$.
Except for the Casas–Ibarra angle $\theta$, the values of all parameters have
been generated from the exponential of randomly-generated numbers with a
uniform distribution in the appropriate ranges.

Beyond the results presented in the following, we also checked the impact of
electroweak precision data in the form of the $T$ parameter Tanabashi:2018oca
on the model’s parameter space. $T$ is defined as exactly zero in the SM and
is given there (at tree level) by the expression

$\alpha T=\rho-1=\frac{m_{W}^{2}}{m_{Z}^{2}\cos(\theta_{\text{w}})}.$ (30)

For the points which survived all selection criteria (figures 7 and 10),
SPheno was used to calculate $T$ to one-loop level. The result is that the
vast majority of points yields $T<${10}^{-20}$$, with a maximum value of
$T=$1.2\text{\times}{10}^{-3}$$. Compared to the current experimental value of
$T=$0.07\pm 0.12$$ Tanabashi:2018oca , our results are orders of magnitude
below the current bound and therefore $T$ does not currently restrict the
parameter space.

###  5.1 Singlet–doublet fermion dark matter

Figure 7: The spin-independent direct detection cross section as a function of
the DM mass for singlet–doublet fermion DM. The colours show the branching
ratios for the LFV process $\mu\to e\gamma$. Also shown are the LEP limits on
light neutral and charged particles (shaded areas) as well as current (full
lines) and future (dashed lines) exclusion limits for the DM relic density
from XENON1T Aprile:2018dbl  and XENONnT Aprile:2015uzo ; oberlack , and for
$\mu\to e\gamma$ Adam:2013mnn ; Baldini:2013ke .

About one third of all models with the observed neutrino masses and mixings
feature singlet–doublet fermion DM, but only a fraction of order
$0.02\text{\,}\mathrm{\char 37\relax}$ yield the correct DM relic density
$\Omega_{\text{c}}^{\text{obs}}h^{2}=$0.120\pm 0.001$$ and Higgs mass. These
models are shown in figure 7 as a function of the DM mass, together with their
spin-independent direct detection cross section and the branching ratio for
the usually most sensitive LFV process $\mu\to e\gamma$. The models
accumulating at a DM mass of about $1\text{\,}\mathrm{TeV}$ feature mostly
doublet fermions, whereas lighter fermionic DM is generally a superposition of
singlet and doublet (cf. section 3). A large doublet component below
$\nicefrac{{m_{Z}}}{{2}}$ (dark shaded area) is excluded by the fact that the
LEP measurement of the invisible $Z$ boson decay width is consistent with
three generations of active neutrinos Tanabashi:2018oca . We assume this
conservative exclusion limit independently of the size of the singlet–doublet
mixing. Furthermore, the accompanying, only slightly heavier charged fermions
are excluded below $102\text{\,}\mathrm{GeV}$ by largely model-independent
searches with the OPAL detector at LEP (light shaded area) Abbiendi:2003yd .
The LHC limits for heavy long-lived charged particles from ATLAS and CMS reach
currently up to $440\text{\,}\mathrm{GeV}490\text{\,}\mathrm{GeV}$,
respectively, but are more model-dependent ATLAS:2014fka ; Khachatryan:2016sfv
. LHC signatures and constraints, also from indirect DM detection, up to a few
hundred GeV on singlet–doublet DM without triplet fermions have been discussed
in Calibbi:2015nha . The spin-independent direct detection cross section is
compared to the current XENON1T exclusion limit (full line) Aprile:2018dbl
and the expectation for 20 ton-years with the XENONnT experiment (dashed line)
Aprile:2015uzo , which was extrapolated linearly above
$1\text{\,}\mathrm{TeV}$ oberlack . XENON1T excludes most of the models with
small scalar-fermion couplings $\lambda_{6}$ and therefore also little LFV.
These models are therefore similar to those in the pure singlet–doublet
fermion DM model. The combination with the scalar sector opens up a
considerable parameter space of leptophilic DM, i. e. with nuclear recoil
cross sections way below even the expected XENONnT sensitivity. Interestingly,
one observes a strong complementarity with LFV experiments, which already
probe the models with the smallest spin-independent direct detection cross
section Adam:2013mnn .

Figure 8: The lightest neutral scalar vs. the lightest neutral fermion mass
for viable models with singlet–doublet fermion DM. Also shown is the branching
ratio for the LFV process $\mu\to e\gamma$ with current (full line) and future
(dashed line) exclusion limits Adam:2013mnn ; Baldini:2013ke .

Figure 8 shows the models that remain after imposing the XENON1T and LEP
limits of $\nicefrac{{m_{Z}}}{{2}}$ for neutral scalars Tanabashi:2018oca  and
$98\text{\,}\mathrm{GeV}$ for charged scalars Abbiendi:2003yd  in the mass
plane of the lightest neutral fermion (the DM particle $\chi_{1}^{0}$) and
scalar ($\eta_{1}^{0}$). For most models, we observe a near degeneracy of the
masses up to $1.1\text{\,}\mathrm{TeV}$, followed by a line of models with
this constant fermion mass and scalar masses up to the decoupling region.
These models will not be probed soon by the process $\mu\to e\gamma$
Baldini:2013ke . Interestingly, there are a few models beyond a fermion mass
of $1.1\text{\,}\mathrm{TeV}$, and these are in fact already excluded by LFV.

Figure 9: Correlations of the branching ratios for the LFV processes $\mu\to
e\gamma$, $\mu\to 3e$ and $\tau\to e\gamma$ for viable models with
singlet–doublet fermion DM. Also shown are current (full lines) and future
(dashed lines) exclusion limits Adam:2013mnn ; Baldini:2013ke ;
Bellgardt:1987du ; Blondel:2013ia ; Aubert:2009ag ; Aushev:2010bq .

In figure 9 we study which of the LFV processes $\mu\to e\gamma$, $\mu\to 3e$
and $\tau\to e\gamma$ has the best prospects of further constraining the
remaining models. We observe that the three LFV processes are strongly
correlated, but that the process $\mu\to e\gamma$ Adam:2013mnn ;
Baldini:2013ke , which is traditionally the most sensitive process, is
slightly outrivaled by its competitor $\mu\to 3e$ Bellgardt:1987du ;
Blondel:2013ia . In contrast, the process $\tau\to e\gamma$ has currently no
sensitivity to our model Aubert:2009ag ; Aushev:2010bq .

###  5.2 Triplet scalar dark matter

Figure 10: The spin-independent direct detection cross section as a function
of the DM mass for triplet scalar DM. The colours show the branching ratios
for the LFV process $\mu\to e\gamma$. Also shown are the LEP limits on light
neutral and charged particles (shaded areas) as well as current (full lines)
and future (dashed lines) exclusion limits for the DM relic density from
XENON1T Aprile:2018dbl  and XENONnT Aprile:2015uzo ; oberlack , and for
$\mu\to e\gamma$ Adam:2013mnn ; Baldini:2013ke .

About two thirds of all models with the observed neutrino masses and mixings
feature triplet scalar DM, but only a fraction of order
$0.02\text{\,}\mathrm{\char 37\relax}$ yield the correct DM relic density
$\Omega_{\text{c}}^{\text{obs}}h^{2}=$0.120\pm 0.001$$ and Higgs mass. These
models are shown in figure 10 as a function of the DM mass, together with
their spin-independent direct detection cross section and the branching ratio
for the LFV process $\mu\to e\gamma$. Similarly to the fermionic DM case, LEP
constraints already rule out light DM candidates. The precise measurements of
the $Z$ width exclude charged scalars below $\nicefrac{{m_{Z}}}{{2}}$ (dark
shaded area) and, by virtue of the small mass splitting between charged and
neutral components within the same triplet, we can assume this exclusion limit
valid also for the neutral scalars, although they do not couple directly to
the SM neutral boson because of their vanishing hypercharge. Likewise, the
exclusion limit from the aforementioned searches for charged scalars with the
OPAL detector is adopted for the neutral components as well, ruling out light
scalar DM below $98\text{\,}\mathrm{GeV}$ Abbiendi:2003yd  (light shaded
area). As for a pure triplet scalar model, we observe an accumulation of
points around a mass of $2\text{\,}\mathrm{TeV}$. Many of these models have
only very small couplings $\lambda_{6}$ to the fermion sector and thus very
little LFV. As $\lambda_{1}$ increases, so must the DM mass beyond
$2\text{\,}\mathrm{TeV}$ to compensate for the stronger Higgs annihilation.
However, most of these models will soon be probed by XENONnT, and those that
will not can soon be excluded by the process $\mu\to e\gamma$. While the mass
region from $1\text{\,}\mathrm{TeV}2\text{\,}\mathrm{TeV}$ with leptophilic
fermion DM, that was opened up by coupling the fermion and scalar sectors, was
already excluded by LFV limits (see above), the corresponding models with
scalar DM are still allowed, but will soon be probed by the process $\mu\to
e\gamma$. Note that there exists in principle also a region of very light
triplet scalar DM of about $6\text{\,}\mathrm{GeV}$ mass, which is however
excluded by the LEP limits on light non-sterile neutral (dark shaded area) and
charged (light shaded area) particles.

Figure 11: The lightest neutral fermion vs. the lightest neutral scalar mass
for viable models with triplet scalar DM. Also shown is the branching ratio
for the LFV process $\mu\to e\gamma$ with current (full line) and future
(dashed line) exclusion limits Adam:2013mnn ; Baldini:2013ke

Figure 11 shows the models that remain after imposing the XENON1T and LEP
limits in the mass plane of the lightest neutral scalar (the DM particle
$\eta_{1}^{0}$) and fermion ($\chi_{1}^{0}$). As for fermion DM, we observe a
near degeneracy of the masses from
$1\text{\,}\mathrm{TeV}2\text{\,}\mathrm{TeV}$, i. e. the two competing DM
candidates remain close in mass, but change spin around
$1\text{\,}\mathrm{TeV}$. Despite the small mass difference, coannihilation is
rare due to the small couplings $\lambda_{6}$. Once the triplet scalar mass
has reached the favoured value of $2\text{\,}\mathrm{TeV}$, the fermion mass
can increase up to the decoupling region. As discussed above, models with
higher scalar mass are possible, in particular when the Higgs coupling
$\lambda_{1}$ is large, but will soon be probed by either XENONnT or LFV.

Figure 12: Correlations of the branching ratios for the LFV processes $\mu\to
e\gamma$, $\mu\to 3e$ and $\tau\to e\gamma$ for viable models with triplet
scalar DM. Also shown are current (full lines) and future (dashed lines)
exclusion limits Adam:2013mnn ; Baldini:2013ke ; Bellgardt:1987du ;
Blondel:2013ia ; Aubert:2009ag ; Aushev:2010bq .

Finally, we study the correlation of the three LFV processes $\mu\to e\gamma$,
$\mu\to 3e$ and $\tau\to e\gamma$ in figure 12 for the case of triplet scalar
DM. As for singlet–doublet fermion DM, we observe a strong correlation of the
three processes, a similar sensitivity of the two muon decay processes with a
small advantage for the one to three electrons, and no sensitivity for the
$\tau$ decay. Therefore, in contrast to the fermion DM case, the parameter
space for scalar DM can only be fully probed with additional experimental
effort.

##  6 Conclusion

In this paper, we have combined the singlet–doublet fermion model with the
triplet scalar model in order to explain not only the observed DM relic
density, but also the neutrino masses and mixings, which were generated
radiatively. This model allows in addition for the correct Higgs boson mass,
couplings of natural size, masses in the $\mathrm{TeV}$ range and gauge
coupling unification at a scale of
$\mathcal{O}(${10}^{13}\text{\,}\mathrm{GeV}$)$.

We first updated and clarified some discrepancies in the literature for the
two separate DM models, showing that the new XENON1T limits roughly double the
excluded parameter space for the singlet–doublet fermion model and confirming
that the triplet scalar DM mass is around $2\text{\,}\mathrm{TeV}$ for small
Higgs couplings, but reaches into the multi-$\mathrm{TeV}$ region for large
Higgs couplings.

With two generations of scalars, we generated two non-zero neutrino masses and
implemented the experimental mass and mixing constraints using the
Casas–Ibarra parametrisation. Analytically, we found that the product of
Yukawa and new scalar-fermion couplings had to be of
$\mathcal{O}(${10}^{-5}$)$ for a DM mass of $1\text{\,}\mathrm{TeV}$.

We found that DM in our model is fermionic up to the $\mathrm{TeV}$ scale and
scalar beyond with small mass splittings not only among the partners of the
weak isospin multiplets, but also the lightest fermions and scalars. The
scalar-fermion couplings opened the parameter space, so that leptophilic
singlet–doublet fermion DM below $1\text{\,}\mathrm{TeV}$, but above the LEP
limits of $\nicefrac{{m_{Z}}}{{2}}$ for active neutral fermions from the
invisible $Z$ boson decay width and $102\text{\,}\mathrm{GeV}$ from largely
model-independent searches for charged fermions, became again viable below the
XENON1T exclusion limit, as did triplet scalar DM between
$1\text{\,}\mathrm{TeV}2\text{\,}\mathrm{TeV}$. In both regions, we observed
an interesting complementarity between the expectations for XENONnT and for
LFV experiments. For the latter, the process $\mu\to 3e$ has the largest
sensitivity, followed by $\mu\to e\gamma$, whereas or model is insensitive to
$\tau\to e\gamma$. For triplet scalar DM, we found a viable region at masses
of a few $\mathrm{GeV}$, which is, however, excluded by LEP limits below
$\nicefrac{{m_{Z}}}{{2}}$ for neutral and $98\text{\,}\mathrm{GeV}$ for
charged scalars. While current LHC limits reach to considerably higher masses
of $440\text{\,}\mathrm{GeV}490\text{\,}\mathrm{GeV}$, they are generally more
model-dependent. Triplet scalar DM models with masses above
$2\text{\,}\mathrm{TeV}$ require a large Higgs coupling leading to a large
spin-independent cross section that will soon be probed by XENONnT.

## Acknowledgements

This work has been supported by the BMBF under contract 05H18PMCC1 and the DFG
through the Research Training Group 2149 ‘Strong and weak interactions – from
hadrons to dark matter’. We thank U. Oberlack for communication on the XENONnT
expectations and C. Yaguna for helpful comments on the manuscript.

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