Minimalistic scotogenic scalar dark matter

We reexamine the minimal Singlet + Triplet Scotogenic Model, where dark matter is the mediator of neutrino mass generation. We assume it to be a scalar WIMP, whose stability follows from the same $\mathbb{Z} _{2}$ symmetry that leads to the radiative origin of neutrino masses. The scheme is the minimal one that allows for solar and atmospheric mass scales to be generated. We perform a full numerical analysis of the signatures expected at dark matter as well as collider experiments. We identify parameter regions where dark matter predictions agree with theoretical and experimental constraints, such as neutrino oscillations, Higgs data, dark matter relic abundance and direct detection searches. We also present forecasts for near future direct and indirect detection experiments. These will further probe the parameter space. Finally, we explore collider signatures associated with the mono-jet channel at the LHC, highlighting the existence of a viable light dark matter mass range.


I. INTRODUCTION
After the discovery of the Higgs boson, the particle physics community is eager to discover new phenomena, that would imply physics beyond the Standard Model (SM). Together with the evidence for dark matter, neutrino physics remains as the most solid indication of new physics. Neutrino experiments point towards two different neutrino mass squared differences, associated to solar and atmospheric oscillations. Hence, at least two of the three active neutrino species must be massive. Here we adopt the minimal picture in which one of the neutrinos is (nearly) massless. This is achieved in "missing partner" seesaw mechanisms [1] where one of the "left-handed" neutrinos fails to pair-off 1 . The presence of a massless neutrino has a very simple and clear implication concerning neutrinoless double beta (0ν2β) decay.
Indeed, if the lightest neutrino is massless, there is only one physical Majorana phase, and the effective mass parameter characterizing the amplitude for 0ν2β decay has a lower limit, even for a normal neutrino mass ordering, as currently preferred by oscillation data [3,4]. This is in sharp contrast to the standard three-massive-neutrino-scenario in which there can be in general a destructive interference amongst the three light neutrinos (such cancellation in 0ν2β decay may also be avoided in the three-massive-neutrino case in the presence of specific family symmetries [5][6][7]).
Our minimal scenario is a generalization of the scotogenic model initially proposed in [8]. The basic idea of this approach is that dark matter is the mediator of neutrino mass generation, and that the same Z 2 symmetry that makes the neutrino mass to have radiative origin also serves to stabilize dark matter. We reexamine the Singlet + Triplet Scotogenic Model extension proposed in [9], which generalizes the original idea introduced in [8], making its phenomenology viable and substantially richer. The presence of Singlet and Triplet fermions in such a scotogenic model extension automatically leads to two oscillation lengths associated to solar and atmospheric oscillations, that can be traced to each of the dark fermion types, leaving a massless neutrino. Compared to the simple scotogenic model, the unwanted spontaneous breaking of the Z 2 parity symmetry can be naturally avoided due to the effect of the new couplings, as discussed in [10]. Two dark matter candidates can be envisaged within the scotogenic framework, either the lightest dark fermion or the isodoublet dark scalar boson. Either possibility corresponds to that of a weakly interacting massive particle (WIMP), produced thermally in the early Universe, similarly to the SM particles. Either will constitute what is called cold dark matter. Following [11] here we focus on the case of scalar WIMP dark matter. The fermionic dark matter case mimics neutralino dark matter in supersymmetry [9] and has been recently re-visited in Refs. [12][13][14]. As we will see, scalar WIMP dark matter phenomenology in the Singlet + Triplet Scotogenic Model provides a rich scenario, sharing common features with other dark matter models such as the Inert Higgs Doublet Model [15][16][17][18] and with discrete dark matter models [19,20].
The paper is organised as follows. In section II we introduce the model, detailing the new fields and new interactions present. We describe in detail the scalar and fermionic sector, as well as the radiative neutrino mass generation, emphasizing that the lightest neutrino is massless and discussing the resulting lower bound for neutrinoless double beta decay. Section III describes the numerical analysis used to study the dark matter sector of the model, listing the main constraints included. Here we have assumed that the dark matter is a scalar particle. In Section IV we present the main results concerning the relic scalar dark matter density, direct and indirect detection. In section V we deal with the implications for the Large Hadron Collider (LHC) searches, taking into account the main results of the previous section. We focus on the E T + jet (mono-jet) signal. Finally, we give our conclusions in section VI.

II. THE SINGLET + TRIPLET SCOTOGENIC MODEL
In this section we will review the Singlet + Triplet Scotogenic Model. This generalization of the scotogenic model [8] was proposed in [9] and further studied in several papers [10][11][12][13]. In addition to the SM gauge symmetry there is a discrete Z 2 symmetry, whose role is to make the lightest Z 2 -odd or "dark" particle stable and to ensure the radiative generation of neutrino masses. The SM particle content is augmented by the inclusion of a Majorana fermion triplet Σ and a Majorana fermion singlet F , both odd under the Z 2 symmetry. Moreover, the model includes a new scalar doublet η -odd under the Z 2 symmetry, which does not acquire a vacuum expectation value (VEV) -and a triplet scalar Ω, which allows for the mixing of the neutral parts of the new fermions. This triplet scalar field has a zero hypercharge and it is even under the Z 2 symmetry, thus, its neutral component can acquire a nonzero VEV. The full particle content of the model is given in Table I, with the corresponding charge assignment under the different symmetry groups.
Taking into account the new fields and symmetries of the model, the relevant terms of the Lagrangian read whereη = iσ 2 η * . The first Yukawa term Y αβ is the standard model interaction for leptons, which we can assume to be diagonal in flavor (Greek indices stand for family indices).  Generations  3  3  1  1  1  1  1   SU(3)C  1  1  1  1  1  1  1   SU(2)L  2  1  2  3  1  2

A. Scalar sector
The scalar potential V invariant under the SU(2)×U(1)×Z 2 symmetry is Tr where we make the conservative assumption that m 2 φ , m 2 η and m 2 Ω are all positive, so that the spontaneous electroweak symmetry breaking will be driven by φ and (sub-dominantly) by the neutral component of Ω, while η cannot acquire a VEV. Notice that we are using the standard 2 × 2 matrix notation for the SU(2) L triplets: The other couplings appearing in Eq. 2 are constrained by a number of theoretical considerations. First, they must comply with the condition that the potential is bounded from below in order to have a stable minimum. This requirement leads to the following conditions [10,11] λ3 2λ1λ2λ It is worth noticing that while these conditions ensure that V is consistently bounded from below at the electroweak scale, the running of the RGEs may lead to breaking of the Z 2 symmetry at some higher energy scale. Another theory restriction comes from the requirement that the expansion of the potential V around its minimum must be perturbatively valid. In order to ensure this we require that the scalar quartic couplings in Eq. 2 are 1.
As mentioned before, η does not acquire a VEV and therefore the symmetry breaking is driven only by φ and Ω, which have non-zero vevs: The fields η, φ and Ω are written as follows where Ω 0 is real and does not contribute to the CP-odd scalar sector. After symmetry breaking there are three charged scalar fields (only two of which are physical, since one is absorbed by the W boson), plus three CP-even neutral fields, and one physical CP-odd neutral field (since the other is absorbed by the Z boson). The VEVs in Eq. 8 are restricted by the following tadpole equations or minimization conditions which we solve for m 2 φ and m 2 Ω . As for the neutral sector, the mass matrix of the CP-even (and Z 2 −even) neutral scalars in the basis (φ 0 , Ω 0 ) reads The lightest of the neutral scalar mass eigenstates is identified with the SM Higgs boson, h 0 with mass ∼ 125 GeV, while the second state, H is a heavier neutral scalar.
On the other hand, the mass matrix for the charged scalars is given as Note that, while the Z boson gets its longitudinal component only from the Higgs doublet φ and not from the triplet (because Ω 0 is real), the charged Goldstone boson is instead a linear combination of φ + and Ω + . The VEV of Ω will then contribute to the W boson mass, thus leading to an upper limit v Ω 5 GeV [21,22]: The mass of the new charged scalar bosons will be Because of the conservation of the Z 2 symmetry, the Z 2 -odd scalar field η does not mix with any other scalar. It proves convenient to write it in terms of its CP-even and CP-odd components: The physical masses of the neutral η field are easily determined as The difference m 2 η R − m 2 η I depends only on the parameter λ 5 which, as we shall see in the next paragraph, is also responsible for the smallness of neutrino masses. In the limit λ 5 → 0 lepton number conservation is restored. Hence, by construction, neutrino masses are "natural", in 't Hooft's sense [23], i.e. they are "symmetry-protected". Moreover, the Z 2 symmetry conservation also makes the lightest of the two eigenstates η R,I a viable dark matter candidate, as we will discuss in detail in section IV.
where θ is the mixing angle between the neutral fermion triplet Σ 0 and F , M Σ and M F are the Majorana mass terms for the triplet and the singlet, respectively. Although we will not consider this case here, it is interesting to notice that the lightest neutral eigenstate, χ 0 1 or χ 0 2 may also play the role of the dark matter [9], for more recent analyses see [12,13].

C. Neutrino masses
The previous subsection has been dedicated to the spectrum of the new fermions. Let us now comment on neutrino masses. By construcion, in the scotogenic approach, the dark matter candidate acts as a messenger for neutrino mass generation. Since the Z 2 symmetry is exact, all vertices including new particles must contain an even number of Z 2 -odd fields. For this reason neutrinos cannot acquire a tree-level mass term, their masses arising only at the loop level as portrayed in Fig. 1.
The relevant interactions for the generation of neutrino masses arise from equations 1 and 2. The expression for the neutrino mass matrix is [9][10][11]  where α and β are generation indices (α, β = 1, 2, 3), m χσ are the masses of the χ 0 1,2 fermion fields and Y ν αβ are the new neutrino Yukawa couplings introduced as a 3 × 2 matrix 2 , The matrix V (θ) is a 2 × 2 orthogonal matrix that diagonalizes the fermionic mass matrix M χ given in Eq. (18). As already noticed before, in the limit λ 5 → 0 the two eigenstates m η R and m η I are degenerate, hence neutrino masses are zero and the lepton number symmetry is restored. This limit would correspond to an exact cancellation between the η R and η I loops. The expression I σ (m 2 χσ , m 2 η R , m 2 η I ) in Eq. 23 involves differences of Passarino-Veltman functions B 0 [24], evaluated in the limit of vanishing external momentum.
We can then rewrite Eq. 18 more compactly as: where This recalls the structure of the standard type-I seesaw neutrino mass relation, with the Dirac mass term given by where F includes the loop functions. In order to compare with the current determination of neutrino oscillation parameters [3], we will apply a Casas-Ibarra parametrization [25]: where U ν is the lepton mixing matrix, m ν are the neutrino masses (whose squared differences are constrained as in [3]) and the matrix ρ is an arbitrary 2 × 3 rotation matrix that can be parametrized as [9] 2 The new fermions Σ and F match exactly the minimum set needed to describe neutrino oscillations. Indeed, if only one of them is present, the neutrino mass matrix would have only one nonzero eigenvalue, hence unable to account for the solar and atmospheric scales.
An interesting prediction of this model is that the lightest neutrino is massless. This feature is reminiscent of the "missing partner" nature of this "radiative" seesaw mechanism, in which one of the "right-handed" fermions is missing (there is only one Σ and one F ). As a consequence one of the "left" neutrinos can not pair-off and hence remains massless [1].

D. Neutrinoless double beta decay
Within the symmetrical parametrisation of the lepton mixing matrix [1] the 0ν2β effective mass parameter can be neatly expressed as [26] where m i are the three neutrino masses and θ 1x are the neutrino mixing angles measured in oscillation experiments. Note that in our case the lightest neutrino is massless (m 1 = 0), so that there is only one physical Majorana phase (φ ≡ φ 12 − φ 13 ). Since there is currently no restriction on its value, this phase is a free parameter. Except for this, all other parameters are well measured in oscillation experiments.
We show in Fig. 2 the dependence of m ee on this phase. One sees that, in contrast to the general case where the three active Majorana neutrinos are massive, here the effective mass parameter describing the 0ν2β decay amplitude has a lower limit [27,28].
The pink (light green) band refers to the 3σ C.L. region allowed by current oscillation experiments [3] for normal (inverted) mass ordering. The black lines correspond to the best fit values for both cases. We also show for comparison the 90% C.

III. NUMERICAL ANALYSIS
We now confront the model with current (and future) observations associated both with the primordial cosmological abundance of dark matter, as well as various pheomenological constraints, including the experimental prospects for direct and indirect dark matter detection.

A. Parameter scan
We have developed a numerical code using Python, to perform a scan varying randomly the main free parameters which characterize the model. This code is connected to some public computer tools used in particle physics in order to examine the constraints on the model parameters and also quantify the expected sensitivities of future experiments. In particular, our Singlet + Triplet Scotogenic Model is first implemented in SARAH 4.9.1 [36,37], which calculates all vertices, mass matrices, tadpole equations, one-loop corrections for tadpoles and self-energies. The physical particle spectrum and low-energy observables are computed with SPheno 4.0.3 [38,39] and FlavorKit [40]. In order to perform the dark matter analysis, we use Micromegas 5.0.2 [41] to compute the thermal component to the dark matter relic abundance as well as the dark matter-nucleon scattering cross sections. For the calculation of the cross sections relevant for the collider analysis, we have used MadGraph5 [42], importing the UFO files generated with SARAH 4.9.1. Our numerical scan was performed with 60000 points, varying the input parameters as given in table II. In particular, in the ranges of variation for the values of m 2 η and |λ 5 |, the lower limits considered were 100 GeV and 10 −5 respectively, to ensure good behaviour for the Z 2 symmetry [10].

Parameter
Range This model has in principle three potentially viable dark matter candidates: η R , η I or χ 0 . In the following we will fix λ 5 < 0 so as to ensure η R to be the dark matter candidate. This choice is made for definiteness, having in mind that the opposite case with λ 5 > 0 and η I as the lightest neutral scalar would also be potentially viable. Notice that the parameters that are not shown in the table are calculated from the ones displayed. For example, m 2 φ and m 2 Ω are obtained from the tadpole equations 10 and Y ν αβ is calculated via Eq. 27. Note that the smallness of neutrino masses does not preclude these Yukawas from being sizeable, since the neutrino masses are controlled by λ 5 and they are further suppressed by their radiative origin.

B. Constraints
The presence of new particles, absent in the Standard Model, will induce departures from the SM predictions for a number of observables. Throughout our analysis, we take into account the following constraints.
a. Theoretical constraints As already discussed in Sec. II, the coupling and mass parameters appearing in the Lagrangian Eq. 1 are subject to several theoretical constraints. First of all, we must ensure that the scalar potential is bounded from below, which we do by applying the conditions summarised in Eqs. 4. Moreover we must ensure the perturbativity of the couplings, i.e. the scalar quartic couplings are assumed to be O(1). Another theoretical consideration concerns the validity of the Z 2 parity symmetry which is an essential ingredient of this model. Its role is indeed twofold: it stabilises the dark matter candidate η R and it justifies the one-loop radiative seesaw mechanism which gives mass to neutrinos. Compared to the simple scotogenic model initially proposed in [8], the spontaneous breaking of the Z 2 parity symmetry can be naturally avoided in this extension thanks to the effect on the running of the couplings in the scalar sector induced by the inclusion of Z 2 -even scalar triplets. Nevertheless, even if the scalar potential is Z 2 -preserving at the electroweak scale, the RGEs running could lead to situations where the Z 2 is broken at some higher energy scale [10]. While a thorough analysis of the RGE running is beyond the scope of the present work, we have not ignored this restriction. Following the prescriptions in [10] we have avoided this problem by fixing the ranges of variation of the relevant parameters so that the Z 2 parity symmetry is in principle safe up to some high energy scale. Moreover, we have checked numerically that m 2 η is positive at all energy scales for both benchmark points chosen for the collider study -see Sec. V. Finally, although experimental constraints place no upper limit on the mass of the heavy neutral scalar H, we require that its decay width should comply with the perturbative unitarity condition, i.e. Γ H m H < 1 2 . b. Neutrino oscillation parameters One of the main motivation of our scotogenic model is to provide an explanation to the generation of neutrino masses. To ensure this, throughout our analysis we require compatibility with the best-fit ranges of the neutrino oscillation parameters. This is enforced via Eq. 27, where the mixing angles and squared mass differences are fixed according to Ref. [3]. For simplicity, the yet unknown Dirac and Majorana phases in U ν are set to zero. We further assume the currently preferred normal ordering of the light neutrino masses. Interestingly, as already mentioned, this model predicts the lightest active neutrino to be massless.
c. Lepton flavour violation This model could be in principle probed through the observation of charged lepton flavour violation, for example, at high intensity muon facilities [43]. However, the negative results of charged lepton flavor violation searches can be used to set constraints on the parameters of the model, in particular on λ 5 which controls the magnitude of the Yukawa couplings. We apply the most stringent limits to date on the branching fraction of some of such rare processes, namely BR(µ → eγ) < 4.2 × 10 −13 [44], BR(µ → eee) < 1. × 10 −12 [45], CR(µ − , Au → e − , Au) < 7 × 10 −13 [46].

d. Electroweak precision tests
The presence of new physics will affect the gauge boson self-energies, parameterised by the oblique parameters [47]. The most important constraint is expected from the T parameter, which is sensitive to the mass splitting between the neutral and charged components of the scalar fields. We require consistency with electroweak precision data by requiring v Ω 5 GeV, in order to get a negligible deviation of the ρ parameter from one [10], namely we impose −0.00018 δρ 0.00096 (3σ). Moreover, we fix the Higgs VEV v φ in order to get the correct mass of the W boson, inside its experimental range.
e. Invisible decay widths of the Higgs boson If the new neutral scalar masses m η R,I are small enough, there can appear new invisible decay channels -at tree level -of the Higgs boson into the lighter stable particles. In the region of parameters where these new invisible decays are possible we enforce that BR(h 0 → inv) 24% [48]. At the loop level, the decay of the Higgs boson into two photons may also be modified by its coupling to the charged scalars. We require consistency at the 3σ level, that is 0.62 BR(h 0 → γγ)/BR(h 0 → γγ) SM 1.7.
f. Dipole moments of leptons At the one-loop level the charged scalars present in our model may also induce sizeable contributions to the magnetic dipole moments of leptons. We have required that the contributions to the anomalous muon magnetic dipole moment induced by the new physics do not exceed the allowed discrepancy between the measured value and the one predicted within the SM [48], ∆(a µ ) = a exp µ − a SM µ = 268(63)(43) × 10 −11 . Contributions to the electric dipole moments arise instead only at the two-loop level, so they are suppressed [49].
g. Dark matter and cosmological observations In the following, we assume a standard cosmological scenario, where the dark matter candidate, the scalar η R , was in thermal equilibrium with the SM particles in the early Universe. If η R is the only candidate contributing to the cosmological dark matter, its relic density must comply with the cosmological limits for cold dark matter derived by the Planck satellite data [50,51]: 0.1126 ≤ Ω η R h 2 ≤ 0.1246 (3σ range). Values of Ω η R h 2 ≤ 0.1126 are also allowed, if η R is a subdominant component of the cosmological dark matter and allowing for the existence of another candidate. Moreover, our scenario can be tested at direct detection (DD) experiments, which are meant to probe the nuclear recoil in the scattering of galactic η R off-nuclei inside the detector. We apply the current most stringent limit on WIMP-nucleon spin-independent (SI) elastic scattering cross section, which has been set by the XENON1T experiment at LNGS [52].
h. Colliders Existing searches for new charged particles at colliders such as LEP and LHC, already set lower limits on their masses in the region below 100 GeV or so [48]. In our analysis we apply the following limits: m H ± ≥ 80 GeV and 122 GeV ≤ m h 0 ≤ 128 GeV, the latter to take into account numerical uncertainties.

IV. PHENOMENOLOGY OF SCALAR SCOTOGENIC DARK MATTER
In this section we collect the results of our analysis of dark matter in the Triplet + Singlet Scotogenic Model. As already commented before, this model can harbor either fermionic or bosonic WIMP dark matter. Detailed studies of the phenomenology of the fermionic dark matter candidate χ 0 [9] have been presented in Ref. [12,13]. Here we will assume the Z 2 -odd scalar η R to be the dark matter candidate and investigate its phenomenology. The latter has common features with those of the simplest scotogenic constructions [8] as well as the Inert Higgs Doublet Model [16,17,53].

A. Relic density
We show in Fig. 3 the expected dark matter relic abundance as a function of the mass of the scalar dark matter candidate η R . The narrow black band depicts the 3σ range for cold dark matter derived by the Planck satellite data [50,51]. Only for solutions falling exactly in this band (cyan points) the totality of dark matter can be explained by η R . Blue points refer to solutions where η R would be subdominant, and another dark matter candidate would be required. Grey points are instead excluded by any of the constraints discussed in Sec. III B, mainly by the Planck constraint itself. Dark grey points are solutions in conflict with the current limit on WIMP-nucleon SI elastic scattering cross section set by XENON1T [52]. The features appearing in the plot can be explained by looking in detail into the η R annihilation channels. The first dip on the left depicts the Z-pole, that is where m η R ∼ M Z /2 and the annihilation via s-channel Z exchange becomes relevant. Similarly, the second depletion of the relic density around m η R ∼ 60 GeV corresponds to efficient annihilations via s-channel Higgs exchange. Notice that it is likely for solutions in this dip to be in conflict with current collider limits on BR(h 0 → inv). The latter depletion is more efficient than the Z-mediated one, which is momentum suppressed. For heavier η R masses, quartic interactions with gauge bosons become effective and, when kinematically allowed, also two-top final states. Annihilations of η R into W + W − via quartic couplings are particular important at m η R > ∼ 80 GeV thus explaining the third drop in the relic abundance. Finally, in the range m η R > ∼ 120 GeV η R can annihilate also into two Higgs bosons. At even heavier m η R the annihilation cross FIG. 3: Relic abundance Ω η R h 2 as a function of the η R mass. Blue points denote solutions with viable relic density, although leading to underabundant dark matter. Cyan points fall within the 3σ C.L. cold dark matter measurement by the Planck collaboration [50,51]. Grey dots are excluded by at least one of the bounds in Sec. III B. Dark grey points are in conflict with the current limit on WIMP-nucleon SI elastic scattering cross section set by XENON1T [52]. section drops as ∼ 1 m 2 η R and the relic density increases proportionally. Eventually, heavy η R mainly annihilate into W + W − , h 0 h 0 , HH. We collect all the Feynman diagrams contributing to η R annihilations and co-annihilations in Appendix A. We may also notice that the relic abundance constraint does not put any bound on the absolute value of the |λ 5 | parameter. On the other hand, cohannihilations with η I and η ± may occur in all regions of the parameter space with the effect of lowering the relic abundance.
We show in Fig. 4 the most relevant branching ratios (at tree level) for the annihilation cross section of η R into SM final states versus the mass of η R , from our numerical scan. Different kinematical regimes are visible from this figure: below M W , η R annihilates predominantly into bb, gluons or τ + τ − ; when the quartic coupling with W becomes kinematically accessible, η R annihilates mainly into W + W − . Similarly, annihilations into h 0 h 0 , HH and Z 0 Z 0 become relevant as soon as kinematically open.

B. Direct detection
Let us discuss now the η R direct detection prospects. The tree-level spin-independent η R -nucleon interaction cross section is mediated through the Higgs and the Z portals. The relevant Feynman diagrams for this process are summarised in Appendix A. Since the η doublet has nonzero hypercharge, the η R -nucleon interaction through the Z boson would in general exceed the current constraints from direct detection experiments. Nevertheless, in most of the solutions, λ 5 induces a mass splitting between the CP-odd partner η I and η R such that the interaction through the Z boson is kinematically forbidden, or leads to inelastic scattering. The η R -nucleon interaction via the Higgs is therefore dominant in most of the parameter space. As a consequence, the coupling between η R and the Higgs boson (which depends on the sum λ 3 +λ 4 +λ 5 and on v Ω , µ 2 and λ Ω η ) turns out to be the relevant quantity controlling both this cross section and the signals at LHC that we will discuss in section V. We show in Fig. 5 the spin-independent η R -nucleon elastic scattering cross section weighted by ξ = Ωη R Ω Planck versus the η R mass. The color code of displayed points is the same as in Fig. 3. The dark green plain line denotes the most recent upper bound from XENON1T [52]. Although we Orange points refer to annihilation into bb, dark cyan to τ + τ − , blue to gluons, dark red to W + W − , green to Z 0 Z 0 and magenta to h 0 h 0 . only show the most stringent up-to-date limit from XENON1T, we note that other leading liquid xenon experiments such as LUX [54] and PandaX-II [55] can also probe the spin-independent dark matter-nucleon elastic scattering cross section for dark matter heavier than ∼ 50 GeV. On the other hand, DarkSide-50 [56] and DEAP-3600 [57] are less competitive for medium and high-mass WIMPs, because of their higher thresholds and lower exposures. Finally, we also depict as for comparison the lower limit (dashed orange line) corresponding to the "neutrino floor" from coherent elastic neutrino-nucleus scattering (CEνNS) [58] and the projected sensitivity for LUX-ZEPLIN (LZ, green dot-dashed) [59]. Most of the allowed solutions with a relic abundance within the 3σ C.L. cold dark matter measurement by the Planck collaboration [50,51] lie in a tight vertical region around m η R ∼ 500 − 600 GeV. Lighter η R lead to viable dark matter, although under-abundant, hence it would then require the existence of an additional dark matter candidate.
Finally, it is worth commenting on how the phenomenology of η R dark matter compares to that of the scalar dark matter in the simple Scotogenic Model [8]. While the two candidates have similar properties, the presence of a scalar triplet in the Singlet + Triplet Scotogenic Model slightly changes the interaction of η R with the Higgs boson. As a consequence, both its Higgs-mediated annihilation cross section as well as the η R -nucleon interaction cross section contain a term dependent on µ 2 and on v Ω (see the relevant vertex in Appendix B). This is nonetheless weighted by the (small) mixing between h 0 and H. As a result the η R dark matter phenomenology turns out to be very similar in both models. The real advantage of the Singlet + Triplet Scotogenic Model comes from the enlarged viable parameter space, especially at low η R masses, as it avoids the unwanted spontaneous breaking of the Z 2 parity symmetry [10].

C. Indirect detection
If η R annihilates into SM products with a cross section near the thermal relic benchmark value, it may be detected indirectly. Among its annihilation products, γ rays are probably the best messengers since they proceed almost unaffected during their propagation, thus carrying both spectral and spatial information. First we consider prospects of detecting γ rays from η R annihilations by considering the continuum spectrum up to the η R mass which originates from decays of the annihilation products. We consider annihilations into bb, τ + τ − and W + W − to compare with ] FIG. 5: Spin-independent η R -nucleon elastic scattering cross section versus the η R mass. Colour code as in Fig 3. The dark green line denotes the most recent upper bound from XENON1T [52]. The dashed orange line depicts the lower limit corresponding to the "neutrino floor" from coherent elastic neutrino-nucleus scattering (CEνNS) [58], while the green dot-dashed one stands for the projected sensitivity for LZ [59]. current limits set by the Fermi Large Area Telescope (LAT) satellite [60] and HESS telescope [61]. We show in Fig. 6 the results of our numerical scan of the annihilation cross section (weighted by ξ 2 and by the correspondent branching ratio) versus the η R mass, for η R annihilating into bb (orange points), τ + τ − (dark cyan) and W + W − (dark red). Grey points are excluded by any of the constraints listed in section III B. Points in light red are solutions with relic abundance falling exactly within the 3σ band measured by Planck. In the same figure we also show the 95% C.L. upper limits currently set by the Fermi-LAT with γ-ray observations of Milky Way dwarf spheroidal satellite galaxies (dSphs) based on 6 years of data processed with the Pass 8 event-level analysis [60] (plain lines assuming annihilation into bb (orange), τ + τ − (dark cyan) and W + W − (dark red)). Moreover we show as a red dot-dashed curve the current upper limit obtained by H.E.S.S. using Galactic Center (GC) γ-ray data accumulated over 10 years [61], assuming a W + W − channel and an Einasto dark matter density profile. Finally, we also depict sensitivity projections for Fermi-LAT from a stacked analysis of 60 dSphs and 15 years of data, in the bb channel [62] (dashed orange) and for CTA, for the Milky way galactic halo target, W + W − channel and an Einasto dark matter density profile [63]. Although current limits lie a couple of orders of magnitude above the predicted signals in this model, future data from Fermi-LAT and CTA offer promising prospects, eventually allowing one to test part of the parameter space both in the low (∼ 70 GeV) and in the high ( > ∼ 500 GeV) mass regions 3 .
Besides γ rays, charged cosmic rays can be used to probe η R as a dark matter candidate. The positron fraction measured by PAMELA [69,70] and more recently by AMS-02 [71,72], allows us to place constraints on annihilating WIMPs, which are particularly stringent in the case of annihilations to the first two generations of charged leptons. In our scenario, light η R annihilate mainly to τ + τ − , as can be seen from Fig. 4. As a result bounds from cosmic FIG. 6: Predicted η R annihilation cross section into γ rays -weighted by the relative abundance -for annihilations to bb (orange), τ + τ − (dark cyan) and W + W − (dark and light red) final states. The orange, dark cyan and dark red plain lines refer to the corresponding 95% C.L. upper limits currently set by the Fermi-LAT with γ-ray observations of dSphs [60]. The dark red dot-dashed curve is the current upper limit obtained by H.E.S.S. using GC data [61]. We also compare with sensitivity projections for Fermi-LAT (bb, 60 dSphs and 15 years of data) [62] and for CTA (GC, W + W − ) [63]. See text for more details.
positrons are less relevant than those from γ rays. In addition to cosmic-ray positrons, AMS-02 has also provided a high-precision measurement of the cosmic-ray antiproton spectrum [73]. These can be translated into upper limits on hadronic dark matter annihilation, which can be a factor of few stronger than those from γ-ray observations of dSphs [74,75]. Since these results rely on a careful treatment of systematic uncertainties, namely the antiproton production cross-section, and the modelling of the effect of solar modulation we decided not to include them here and leave it for a dedicated work. Similarly, searches for anti-deuterium or anti-helium events could potentially provide a powerful probe of η R annihilations [76][77][78][79], although also affected by substantial uncertainties.

V. SCALAR DARK MATTER SIGNATURES AT THE LHC
In this section we confront our scalar dark matter candidate with the latest data from particle colliders, in particular from the LHC run at √ s = 13 TeV. As in any model with a dark matter candidate, the generic signature to be searched for is missing energy ( E T ), measured from the total transverse momentum recoil of the visible particles in the event (see for instance [80,81]). In the Triplet + Singlet Scotogenic Model typical signatures are E T + X, where X can be one or two jets [82,83], two leptons [84] or one photon [85]. Although all of them are in principle interesting, we have checked numerically that in our scenario the most promising one is E T + jet (mono-jet). In the following we will focus on mono-jet final states, arising from pp → η R η R + g and pp → η R η R + q processes. Here ones looks for events with one high-p T jet (higher than 100 − 200 GeV in the central region of the detector, with pseudorapidity |η| < 2.4) and E T above roughly 200 GeV in the 13 TeV analyses for the ATLAS and CMS detectors [82,86]. At leading order, the relevant Higgs-mediated Feynman diagrams for mono-jet events are shown in Fig. 7.
In all cases, the dark matter is produced via the decay of a neutral scalar (h 0 or H), produced from its interaction with quarks, or through its effective coupling to gluons. The latter involves a top quark loop and enters in gluongluon fusion (ggF) processes. An important point is that in ggF processes only the SM-like Higgs doublet couples with fermions. Indeed, since H is mainly a triplet, its coupling with quarks is suppressed. The interaction vertex between η R and h k is given in Appendix B. Note that if the mass difference between η I and η R is small, η I should also contribute to the invisible final states. In this case, η I would subsequently decay to η R plus soft fermions or jets which are not energetic enough to be detected. Besides Higgs mediation, the mono-jet signal can proceed also via Z-mediation. Therefore we also include the contributions shown in Fig. 8, which are described as pp → η R η I + g and pp → η R η I + q processes. Finally, we must mention that in this same scenario of small mass differences, a pair of η I can also be produced.

A. Benchmark Points
The constraints previously described in Section III B restrict the parameter space allowed by a vast array of experimental probes, among which are the relic density, direct detection and indirect detection analyses. Motivated by these preliminary studies, we now investigate using the CheckMATE 2 collaboration tools [87][88][89][90][91] whether the solutions that satisfy all experimental limits in section III B could lead to detectable dark matter mono-jet signals at LHC 13 TeV. This code allows us to determine whether or not a given parameter configuration of our model is excluded at 95% C.L. Indeed, for each signal region, CheckMATE 2 computes the expected number of signal events S after cuts, and directly compare it to the 95% C.L. upper limit S 95 exp , given a signal error ∆S. The most relevant analysis for our study is Ref. [82].
In this way, we identify two interesting benchmark points which survive the entire set of constraints described in Section III B and shown in Table III. Values of the relevant parameters and the corresponding scalar spectrum are summarised. We also show in this table the value of observables obtained in Section III B for each benchmark. The main difference between the two benchmark points is the value of H mass, which is governed by µ 2 and v Ω . However, because this heavy scalar is mainly triplet, its coupling with quarks in the ggF processes is suppressed, so that a significant change in its mass is not expected to lead to a large variation in the magnitude of the cross sections.   8) and Higgs-mediated processes (Fig. 7), respectively.
calculated by CheckMATE 2 4 , which translates into a significant number of signal events after the cuts, S. These specific cuts are implemented by the ATLAS analysis in order to map out the associated regions of consistent parameter choices, and will be described later.
Our dark matter candidate η R with mass around ∼ 50 − 60 GeV and chosen to satisfy all theoretical and experimental constraints of Sec. III B would lead to a signature in the E T +jet channel in the ATLAS experiment. For that we require, for both benchmark points, that the leading jet has p T > 250 GeV and |η| < 2.4, separation in the azimuthal plane of ∆φ(jet, p miss T ) > 0.4 between the missing transverse momentum direction and each selected jet. The difference between our benchmarks are the E T thresholds. While for Benchmark 2 a E T minimum of 500 GeV is required, in the other case we take E T > 600 GeV.
For larger η R masses we investigate the behaviour of the cross sections at √ s = 13 TeV and the projected signal events at √ s = 14 TeV. We assume the coupling |λ 345 | to lie in the range [0.02, 0.9] and we fix the other parameters according to Benchmark 1 in Tab. III. We analyse η R η R + j and η R η I + j separately because the rate of these processes depends on different parameters and we want to analyze their contributions to the total cross section separately.
In Fig. 9 we present the production cross section for E T + jet process at LHC √ s = 13 (14) TeV. Using Madgraph5 [42] we simulate events with an initial cut of p jet T > 100 GeV, according to the latest analyses in mono-jet searches [93,94]. Since the relevant processes leading to these events are mediated by mainly the SM Higgs (left panel) and Z boson (right panel), one has the characteristic peaks at m η R ∼ m h 0 /2 and at m η R ∼ m Z /2 respectively, providing larger cross sections in these mass ranges. Therefore, the Higgs boson mediated processes are dominant up to m η R ∼ 60 GeV and also contribute in the range ∼ [700 − 1400] fb (13 TeV). In addition, Z-mediated processes complements the search for pp → η R η R + jet process at the LHC. For this mass range, the cross sections are ∼ [190 − 80] fb while, for dark matter masses between [65 − 200] GeV, we have ∼ [70 − 5] fb, providing a sizeable contribution to the total mono-jet cross section, which could be within LHC sensitivity. At √ s = 14 TeV the cross section increases by a few fb. These results agree with expectations of other models, such as the Inert Higgs Doublet Model, whose contributions to this signal are very similar [93]. In summary, one sees that there are good prospects for probing the mono-jet signal at the LHC for dark matter masses up to ∼ 60 GeV.  (14) TeV. The left panel shows the Higgs boson mediated events from pp → η R η R + jet. The maximum value of the cross section is ∼ 1400(1800) fb for √ s = 13 (14) TeV respectively. The right panel is the Z-mediated process, pp → η R η I + jet, with peak contribution ∼ 190 (220) fb.
There are regions of parameters in which η I and η R are relatively close in mass, as shown in Figure 10. This as required for model consistency, as the mass difference between these particles is intimately connected with the smallness of neutrino mass as generated in the scotogenic picture. This requires the violation of lepton number through the value of λ 5 , as seen by Eqs. 16 and 17. Indeed, if m η R − m η I is small we can obtain neutrino mass square differences, as needed to account for neutrino oscillation data [4]. Moreover, the particles produced from the decay η I → η R +X are not energetic enough to have the trajectories reconstructed by the detector (soft particles), leading to our E T + jet final state signal.
As already commented in previous sections, as a result of its small coupling with quarks, the heavy neutral scalar H does not influence significantly our signal. As shown in Ref. [11], the production cross section of H at the LHC is 3 to 5 orders of magnitude smaller than the production of the SM Higgs boson, independent of the center-of-mass energy. Hence our results for the scalar dark matter jet + missing energy final states within the Singlet + Triplet Scotogenic Model should also hold within the simplest Scotogenic scenario of [8].

VI. SUMMARY AND CONCLUSIONS
In this work we have reexamined the generalized version of the minimal Singlet + Triplet Scotogenic Model, in which dark matter emerges naturally as the mediator of neutrino mass generation and its stability follows from the same Z 2 symmetry also responsible for the radiative origin of neutrino masses. In contrast with the simplest version of [8], our generalized scotogenic model is the minimal one fully consistent with oscillation data, since it allows for non-zero solar and atmospheric mass splittings to be generated consistently. We have assumed dark matter to be a scalar WIMP and we have presented a full numerical analysis of the signatures expected at dark matter detectors as well as collider experiments. We have identified the regions of parameters where dark matter predictions are in agreement with theoretical and experimental constraints, such as those coming from neutrino oscillation data, Higgs data, dark matter relic abundance and direct detection searches. We have also presented expectations for near future direct and indirect detection experiments. These will further probe the parameter space of our scenario. Finally, we have examined the collider signatures associated to the mono-jet channel at the LHC. In particular, we have found a viable light dark matter mass range in the region 50 − 60 GeV. This should encourage future studies at the upcoming high-luminosity run of the LHC.  Figure 13 shows the most important Feynman rules for the relevant scalar dark-matter-physics interactions in the Singlet + Triplet Scotogenic Model. These are important for all the signatures studied in this paper, like the η R -nucleon spin-independent elastic scattering and for the searches in the E T +jet channel at the LHC.
In contrast to the simplest Scotogenic Model, the interaction vertex with the Higgs is not fully determined by λ 345 , as it contains an extra contribution dependent on λ Ω η and µ 2 , involving the heavy neutral scalar H, although weighted by its mixing with h 0 . Instead, the interaction vertex with the Z 0 boson depends on the quadrimomenta p η R µ , p η I µ and on the electroweak couplings g 1 and g 2 .  FIG. 13: Relevant η R interaction vertices. All particle momenta are considered as incoming. In the interaction with neutral scalars, Z H k1 and Z H k2 are entries of the mixing matrix that diagonalizes the mass matrix in Eq. 11 (k = 1 refers to the SM Higgs h 0 and k = 2 to the heavy scalar H). In the interaction with the Z 0 boson, g 1 and g 2 are the electroweak coupling constants associated to the SM groups U (1) Y and SU (2) L , respectively.