Phenomenological profile of scotogenic fermionic dark matter

We consider the possibility that neutrino masses arise from the exchange of dark matter states. We examine in detail the phenomenology of fermionic dark matter in the singlet-triplet scotogenic reference model. We explore the case of singlet-like fermionic dark matter, taking into account all co-annihilation effects relevant for determining its relic abundance, such as fermion-fermion and scalar-fermion co-annihilation. Although this in principle allows for dark matter below 60 GeV, the latter is in conflict with charged lepton flavour violation (cLFV) and/or collider physics constraints. We examine the prospects for direct dark matter detection in upcoming experiments up to 10 TeV. Fermion-scalar coannihilation is needed to obtain viable fermionic dark matter in the 60-100 GeV mass range. Fermion-fermion and fermion-scalar coannihilation play complementary roles in different parameter regions above 100 GeV.


INTRODUCTION
The existence of neutrino masses [1] and cosmological dark matter [2][3][4] constitute two of the main pillars indicating the need for new physics.While they can be associated to totally unrelated sectors, the idea that dark matter mediates neutrino mass generation proposed independently by Ma [5] and by Tao [6] has by now become a paradigm [7][8][9][10][11][12][13] for neutrino mass generation and for explaining cold dark matter in terms of weakly interacting massive particles (WIMPs).The main idea of these scotogenic models [5][6][7][8][9][10][11][12][13] is to have neutrino masses generated radiatively (in our case at the one loop-level), mediated by a TeV-scale sector that can also account for WIMP dark matter (bosonic or fermionic).A Z 2 symmetry is imposed in order to stabilize dark matter, ensuring also the radiative origin of neutrino masses.By making one parameter small (namely λ 5 ) the symmetry of the model is enhanced, making it natural in the sense of 't Hooft [14].In other words, neutrino masses are symmetry-protected.Together with the loop-suppression in Fig. 1 this protection makes it possible to keep the neutrino masses low in the presence of sizeable Yukawa couplings O(10 −1 ) for the new fermion mediators.This makes the scotogenic scenario phenomenologically very attractive in comparison with vanilla-type seesaw schemes.Indeed, the simplest type-I seesaw scenarios based on the Standard Model [15,16] require either the neutrino mass mediators to be superheavy or the Yukawa couplings too small, in order to fit the small observed neutrino masses.Hence they do not lead to observable phenomena, e.g. in colliders.Finally, they require some "external" mechanism to account for dark matter, making these unrelated phenomena.
In this paper we focus on the singlet-triplet scotogenic model, proposed in [7].This offers a theoretically consistent picture of scotogenic dark matter in which the underlying Z 2 symmetry can be preserved up to high energies after renormalization group evolution of the couplings in the scalar sector [8].Moreover, it offers a very rich, yet manageable phenomenology of direct dark matter detection, charged lepton flavour violation (cLFV) as well as collider physics.
Since the possibility of scalar scotogenic dark matter has already been considered [12,13], here we focus on the detailed study of fermionic scotogenic dark matter and its phenomenological features.The latter is more intricately connected to the neutrino mass generation than its scalar dark matter counterpart, where Higgs-portal-mediated annihilation processes play the dominant role, irrespective of the Yukawa couplings determining neutrino mass generation.In contrast, fermionic dark matter is closely related with neutrino mass generation, and also involves the mixing of singlet and triplet neutral fermions.We note that singlet-like dark matter has a richer phenomenological profile than that of the pure triplet dark matter, as the latter requires masses ∼ 3 TeV in order to have enough co-annihilation to satisfy the relic density requirement.We explore in detail the singlet-like fermionic dark matter, updating the experimentatal status and substantially improving upon the early work in [7] through a refined discussion of the relic abundance, in which we stress the important role of scalar-fermion dark matter co-annihilation effects, which are properly taken into account.Other novel features of our present work include a discussion of direct fermionic dark matter detection constraints, combining with the restrictions of cLFV and collider experiments on dark matter phenomenology.We explore the general feasibility of scotogenic fermionic dark matter, examining the prospects for direct dark matter detection in the next round of experiments all the way up to 10 TeV.We find that the possibility of light dark matter below 60 GeV or so is inconsistent with experimental cLFV and/or collider constraints.Fermionscalar coannihilation is required in order to obtain viable fermionic dark matter within the mass range of 60 GeV to 100 GeV.Beyond 100 GeV, fermion-scalar and fermion-fermion coannihilation play complementary roles at different regions of parameter space.

SINGLET TRIPLET SCOTOGENIC MODEL
In this section we discuss the basic features of the singlet-triplet scotogenic model.Apart from the Standard Model (SM) particles, this model contains four colour-singlet states, two of which are fermions (Σ, F ), the others are scalars (η, Ω).The new fermionic states Σ and F are SU (2) L triplet and singlet respectively, with no hypercharge, while the extra scalars η and Ω are doublet and real triplet under the SU (2) L gauge group, with hypercharges 1/2 and zero respectively.In addition to the SU(3) c ⊗ SU(2) L ⊗ U(1) Y gauge symmetry, a discrete Z 2 symmetry is also imposed on the Lagrangian in order to ensure the stability of the lightest Z 2 -odd particle and its role as a suitable dark matter candidate.Under this Z 2 symmetry, all the SM particles along with the triplet scalar Ω are even, whereas the remaning new states are taken to be odd.The particle assignments (excluding the quarks) are given in Table I, where L, e and Φ denote the usual SM states corresponding to the left-handed lepton doublets, right-handed charged lepton singlets and the scalar doublet.The hypercharge is assigned following the convention: The relevant part of the Lagrangian involving the leptonic interactions for this model is written as: where Y αβ are the usual Yukawa couplings for the SM leptons and Y α F , Y α Σ , Y Ω are the Yukawa couplings involving the new fermions and scalars of the model.Here the α, β are SM lepton family indices, while M F and M Σ denote Majorana mass terms for the additional fermions F and Σ, respectively.In the adjoint representation of SU (2) L the fermionic triplet Σ and its charge conjugate Σ c are expressed as: where the charged components form Dirac fermions.

The scalar sector
The scalar potential of this model, invariant under , is given by: where µ's are mass parameters and λ's are dimensionless quartic couplings amongst different scalar fields.
After electroweak symmetry breaking (EWSB), the neutral components of Φ and Ω acquire non-zero vacuum expectation values (VEVs), i.e. 22 GeV.In contrast, the neutral component of the "dark" scalar η does not get any VEV due to the conservation of Z 2 symmetry.Note that the triplet self-quartic interactions in general contain two contractions, [Tr (∆ † ∆)] 2 and Tr [(∆ † ∆) 2 ].However, since in our case the scalar triplet Ω is real and hyperchargeless, these terms are equivalent: Therefore it suffices to keep just one of them.Note also that Tr[Ω † σ i Ω] vanishes in our case for the same reason, which also eliminates quartic terms from the Higgs potential.
The scalar fields in the singlet-triplet scotogenic model can be expressed after EWSB as follows: Here ϕ ± , Ω ± and η ± are the charged partners of the neutral CP-even states ϕ 0 , Ω 0 and η 0 R , and neutral CP-odd states A 0 ϕ and η 0 I .Note that, in contrast to Φ and η, the neutral component of Ω has no pseudoscalar (CP-odd) particle, since Ω is a real triplet.Moreover, the charged components (Ω ± ) obey the relation (Ω + ) * = Ω − .On the other hand, the pseudoscalar component A 0 ϕ is the Goldstone boson eaten up by the Z-boson.Minimizing the scalar potential in Eq. ( 3), one gets two tadpole equations which allow us to express µ 2 ϕ and µ 2 Ω in the following fashion: The squared mass matrix for the CP-even neutral components of Z 2 -even scalars, i.e. ϕ 0 and Ω 0 , is given by: Diagonalization of this matrix leads to mixing of the scalars ϕ 0 and Ω 0 yield the mass eigenstates h 0 and H 0 , one of which is the SM Higgs boson.In our analysis, we have identified the lighter CP-even state h 0 as the SM Higgs boson with mass of 125.5 GeV.Likewise, the squared mass matrix for the Z 2 -even charged scalars ϕ ± and Ω ± take the form: where, g is the SU(2) gauge coupling and ξ W ± is the parameter related to the gauge fixing term of W ± boson.The charged bosons ϕ ± and Ω ± mix with each other through the diagonalization of M 2 ϕ ± matrix to provide the chargedscalar mass eigenstates, one of which gets absorbed as the longitudinal W ± Goldstone boson, while the second is a physical charged scalar state H ± .Note that although the mass of Goldstone boson, m G ± = 1 4 g 2 ξ W ± (v 2 ϕ + 4v 2 Ω ), depends on the choice of the gauge fixing parameter ξ W ± , the mass of the physical charged scalar does not, After EWSB, the squared masses for the Z 2 -odd or "dark" scalars and pseudoscalars are given by: Note that the squared masses of η 0 R and η 0 I differ by the term λ 5 v 2 ϕ , signaling the conservation of lepton number symmetry in the limit of vanishing λ 5 .

Neutrino mass generation
From the Lagrangian in Eq. (1) one sees that, once Ω 0 acquires a VEV, the neutral component of the dark fermionic triplet, i.e.Σ 0 , mixes with dark singlet fermion F through the term Y Ω T r[ ΣΩ]F c .The corresponding mass matrix representing the mixing of F and Σ 0 is given by: Due to the Pauli principle, the matrix M χ is in general symmetric, but complex, and can always be diagonalized by a unitary matrix V involving an angle θ and one Majorana phase [15].Here for simplicity we assume CP conservation in the dark fermion sector, so that M χ is diagonalized by an orthonormal transformation, as follows: where the mixing angle θ and the masses of physical states m χ 0 1,2 can be expressed as: The lightest of these two dark Majorana fermions χ 0 j is satibilized by the Z 2 symmetry, serving as our fermionic dark matter candidate.The mixing angle θ determines the amount of singlet or triplet composition of our WIMP dark matter candidate.These states are analogous to the Bino and neutral Wino dark matter candidates in R-parity conserving supersymmetic models.
Following the scotogenic paradigm, the neutrino mass-matrix is generated at one-loop level, as shown in Fig. 1.Majorana masses for neutrinos are mediated by dark sector exchange, and extracted from: where, Note that in the limit of vanishing λ 5 lepton number symmetry would be restored, the states η 0 R,I become degenerate, making the functions I j and hence the neutrino mass matrix M ν to vanish.Thus the choice of λ 5 ≪ 1 is natural in the sense of 't Hooft [14] as the limit of λ 5 → 0 enhances the symmetry of the model.
In the singlet-triplet scotogenic model the lightest neutrino is massless, as in all missing partner (radiative) seesaw schemes [15,17].In the same spirit as Ref. [18]) one can extract the Yukawa matrix Y ν in such a way as to automatically satisfy the neutrino oscillation data as: where M ν is the neutrino mass matrix in diagonal form and U is the lepton mixing matrix, both measured to a large extent.On the other hand, the complex matrix ρ can be paramatrized for normal and inverted neutrino mass-ordering through a complex angle ω as [17]:

CONSTRAINTS
As mentioned above, it is useful to extract the Yukawa matrix Y ν in terms of the measured neutrino oscillation data as in Eq. ( 13).This will expedite the scan procedure we perform in order to determine the viable model parameter space.Before doing this however, let us first compile the restrictions we have imposed on the numerical scan of the model parameters.

• Theoretical Constraints
In order to prevent the potential acquiring large negative values at large field values, the following conditions [8,19] should be obeyed by the quartic couplings of this model:  [20,21], the above mentioned values are pretty common choices in the literature [22].We also mention that the renormalization group running of different parameters of this model may lead to the breaking of Z 2 symmetry [8].This can be avoided by choosing µ η Ω ≲ O(1 TeV).

• S, T, U parameters
It is well-known that the presence of Higgs triplet affects the ρ-parameter at tree-level [15].Indeed, our triplet Ω contributes to the mass of W boson (but not to the Z boson mass), so that the ρ-parameter at tree-level becomes [23] Here ϕ 0 i is the VEV of the neutral component of ϕ i , T i and Y i are the weak isospin and the hypercharge of ϕ i , and c i is a constant that equals to 1/2 or 1 depending on the scalar being in real or complex representations (note that Ref. [23] uses the convention: Hence the precise measurement of the ρ-parameter puts a severe restriction on v Ω .The current global fit for ρ-parameter is ρ = 1.00038 ± 0.00020 [24], nearly 2σ away from the tree-level SM expectation.Taking the 3σ range of the fit, one can restrict the VEV of triplet as: v Ω < ∼ 4 GeV.This maximum value of v Ω can push the mass of W-boson up to 80.389 GeV which is consistent with the PDG value [24] at 1σ. The current global fit yields the following S, T and U parameter values1 : S = −0.02± 0.10, T = 0.03 ± 0.12, U = 0.01 ± 0.11, (17) which are in good agreement with SM prediction.However, the error in the fits leave lots of room for different new physics scenarios.Indeed, the recent CDF W-mass measurement [25] can weaken the above limits2 .

Collider Constraints
The new scalars as well as dark fermions present in the singlet-triplet scotogenic model are subject to restrictions arising from existing collider searches, which we now summarize.

• Direct Search Constraints
• Several searches have been performed to detect charged scalars at LEP and LHC.The LEP bound [34] on charged scalars (m H ± ≥ 80 GeV) assumes that the charged scalar is part of a SU (2) L doublet (weak isospin 1/2) and that the branching fractions of charged scalar to τ + ν and cs add up to unity [24].On the other hand ATLAS [35] and CMS [36] exclude charged scalars in the [80 -140] GeV and [90 -155] GeV ranges, respectively.These searches place constraints on scenarios where the quarks directly couple to the new scalar, which is not true for our case regarding both (η ± and H ± ).Therefore, their production cross-section at the LHC will be very suppressed and these bounds cannot be applied strictly to our scenario.However, the searches for supersymmetric particles, especially slepton-pair-production, followed by their decay to SM leptons and neutralinos, at LEP [37,38] and the LHC [39,40] can be used to constrain the masses of η ± and H ± .The LHC result for slepton decaying to lepton and massless neutralino rules out H ± below the mass of 400 GeV (considering a conservative limit [39,40]).Again, the constraints from the LEP require η ± to be heavier than 70 GeV [41].
• Additional neutral scalars have also been searched for at LEP and LHC, mainly in the context of 2-doublet Higgs models.Using the di-boson (γγ, W W * and ZZ * ) decay channels they exclude some mass range for the extra neutral scalars.The L3 collaboration provides the most severe bound on the lowest mass allowed for this fermiophobic scalar as 107 GeV [42], while the Tevatron reports the exclusion of any such scalar in the [100-116] GeV mass range [43].Di-photon decay modes in ATLAS and CMS rule out such scalars in [110-121] [44] and [110-147] GeV [45], respectively.On the other hand, assuming that the second Higgs decays invisibly, ALEPH puts the bound on the lowest allowed mass of any additional neutral scalar to be 114 GeV [46] using the single production channel of such scalar associated with a Z-boson.These constraints may be applicable to H 0 pushing its mass above 150 GeV or so.
Note however that these constraints are not relevant for the Z 2 -odd neutral scalars (η 0 R and η 0 I ).Indeed, the Z 2 -symmetry prevents these to be singly produced and also to decay to di-boson channels.Nevertheless, LEP searches [37,38] for neutralinos constrain the masses of η 0 R and η 0 I .For instance, using the LEP data in the case of inert two Higgs doublet model Ref. [47] excludes the region where the three conditions of m η 0 R < 80 GeV, m η 0 I < 100 GeV and m η 0 I − m η 0 R > 8 GeV are satisfied simultaneously.• LEP searches set a lower limit on the mass of heavy charged lepton as 102 GeV [48,49] assuming it to be long-lived or stable.Therefore, we take this into account while considering mass of Σ + , i.e.M Σ .For the heavy neutral fermions, LEP puts lower limit on fermion masses at 102 GeV and 90 GeV [49] for Dirac and Majorana fermions respectively, assuming their decay to SM charged leptons plus the W -boson.In our model there are two neutral leptons, χ 0 1,2 , where the lightest is the dark matter candidate and hence stable by the Z 2 -symmetry.Even the heavier Z 2 -odd neutral fermion cannot decay to two SM particles, so the LEP bound is not directly applicable.Moreover, since m χ 0 2 is always bigger than M Σ , imposing the lower limit on M Σ as 102 GeV automatically sets the same limit for m χ 0 2 .It is interesting to mention that the LHC [50] rules out Z 2 -even heavy triplet fermion up to 790 GeV of mass.But this result does not directly affect our Z 2 -odd triplet Σ since the final states considered at the LHC are forbidden in our model.
• Z/W Widths and Higgs Invisible Decay SM gauge boson (W/Z) decays have been measured with a great precision.New light particles could show up in Z and W decays. Whenever kinematically allowed, such additional decay channels will affect the widths of the Higgs, Z and W bosons.Likewise, Higgs decays involving dark channels contribute to the invisible width, and can also be constrained to some extent.The expressions for the Z, W and Higgs boson partial decay widths are listed in Appendix A. For our analysis, we have used following constraints: the branching fraction of the Higgs boson to the invisible modes should be less than 13% [51].
new contributions to the total Z-boson width should be less than 5 MeV (i.e.roughly within 2σ), as the error in the measurement is 2.3 MeV [24].
new contributions to the total W -boson width should be less than 90 MeV (i.e.almost within 2σ), as the error in the measurement is 42 MeV [24].
Here we have adopted a conservative approach to accommodate new contributions within the 2σ bound of the experimentally measured values.These uncertainties in the widths of W/Z help in restricting the BSM parameters of the model while keeping the W mass fixed at:

PARAMETER SIMULATION PROCEDURE
Besides the SM quark and lepton Yukawa couplings, the model contains many free parameters describing its Yukawa ( Eq. (1)) as well as scalar ( Eq. ( 3)) sector.They are tabulated in Table.II.
However, not all of these parameters are independent, since some can be cast in terms of known quantities, thereby reducing the number of independent parameters.For example the Higgs VEV and mass are determined, reducing two parameters, whereas neutrino oscillation data determine the two light neutrino masses (there are only two in our scheme), the three mixing angles and one of the CP phases, though rather poorly so far.Thus the Yukawa couplings Y F and Y Σ can be traded only in terms of complex angle ω using Eq.(13).Neglecting the Majorana phase this results in sixteen independent parameters to deal with.

Complex Yukawa Real Yukawa
Scalar mass terms Scalar quartic couplings Fermionic mass terms couplings a couplings b and trilinear couplings a Each complex quantity counts as two parameters.
b To simplify our analysis we assume Y Ω to be real.
TABLE II: Free model parameters consist of five µ i parameters, eight λ i , three complex Y α F , three complex, Y α Σ , Y Ω , and two fermionic masses M F and M Σ .
Instead of working directly with the parameters in Table II, it is useful to use other parameters which can be expressed as combinations of the above.For example, instead of working with µ's, it is more helpful to work with physical scalar mass parameters and VEVs.In fact, since we will be studying co-annihilation effects, the mass differences or squared differences are more useful parameters than the masses themselves.Therefore we define the parameters ∆m ΣF , ∆m η + F and ∆m 2 η 0 I η + in the following way: Two of the µ's can be removed in favor of the measured electroweak VEV and the SM Higgs mass (using Eq. ( 5) and Eq. ( 6)), and two more of them can be traded in favor of v Ω and ∆m η + F (with the help of Eq. ( 6) and Eq. ( 8)).This way we keep only one of the five µ's, say µ η Ω , as independent.Furthermore, λ 4 can be replaced easily by ∆m 2 η 0 I η + using Eq.(8).Then the set of independent parameters to deal with become: µ η Ω , v Ω , Y Ω , seven λ i (λ 4 is not independent), four parameters related to Z 2 -odd particle masses, i.e.M F , ∆m ΣF , ∆m η + F , ∆m 2 η 0 I η + and the complex angle ω.

M F
∆m ΣF ∆m η + F ∆m 2 [1, 1000] 200 500 1000 400 4.0 2.0 π/4 π/4 0.2626 0.5 0.5 10 −8 0.5 0.5 0.5 Our fermionic scotogenic dark matter scans require a multi-parametric setup.For definiteness we choose a benchmark point BP0 specified by the parameter values given in Table III.We change each parameter separately and study the effect on the various observables, implementing also the constraints discussed above.With this guidance we proceed with the parameter scanning.
The values of Y Ω , λ i and ω in BP0 satisfy all the conditions required in Sec.3.1.The chosen value of µ η Ω preserves the Z 2 symmetry at high scales, while particle masses satisfy the relevant collider constraints mentioned in Sec.3.3.We choose the second Z 2 -even scalar of our model (H) as heavier than the SM Higgs (h), resulting in the non-dark charged scalar (H + ) being heavier than h 0 .One can also check that the benchmark point BP0 does not add new contributions to the Z/W width or Higgs invisible decay.Since the mass differences between charged and neutral components of Ω and η are not too big, the T-parameter does not change much.As the triplet scalar is heavy (m H ∼ 400 GeV), the S-parameter remains also almost the same.Since U is the coefficient of the dimension-eight operator in the EFT approach, it usually remains very small for a wide class of BSM scenarios.All in all, BP0 is a valid benchmark point for study.
We first examine the prospects of cLFV and fermionic scotogenic dark matter using the benchmark point BP0.Our main goal with BP0 is to study the dependence of cLFV processes and dark matter relic density on the parameters of this model, as this serve as guide towards the scanning of the parameter space.To obtain our numerical results, the model has been implemented in SARAH [52][53][54].The cLFV prospects have been studied using the package SPheno [55,56] that employs the FlavourKit [57] component of SARAH.On the other hand, the dark matter analyses have been performed using micrOMEGAs [58,59].TABLE IV: The experimental bounds on cLFV processes involving µ decays or conversions.
The conventional charged current mechanism leading to charged lepton flavour violating processes such as µ → eγ is highly suppressed due to the GIM mechanism.However, the mediators responsible for neutrino mass generation also mediate rare processes with charged lepton flavour violation, such as µ → eγ.Indeed, we expect these new Feynman diagrams (see Fig. 2) to lead to enhanced rates for these rare decays, as examined in detail in Ref. [9].
The most stringent constraints on cLFV come from the measurement of the branching fractions of the decays µ → eγ, µ → 3e and the conversion rate of muon to electron in the muonic gold.The current experimental bounds and future sensitivity on these observables are given in Table IV.
It is interesting to mention that, although the current sensitivity of C(µ, T i → e, T i) is 4.3×10 −12 [69], which is much weaker than the sensitivity of C(µ, Au → e, Au), it is expected to reach up to O(10 −18 ) [66] in the future.Both Mu2e [67] and COMET [68]  In Fig. 3 we display the rates for these cLFV observables.Apart from a mild dependence on the parameters ∆m ΣF and ∆m η + F which determine Σ + and η + masses running in the loops, these processes are mainly controlled by M F ,  III.The blue points in all the plots correspond to BP0, while the orange points are generic.The horizontal red dashed lines indicate the current experimental bounds, whereas the brown dotted lines correspond to future sensitivities.λ 5 and Im(ω).In the top row we present these observables versus the dark matter mass m DM for two different values of λ 5 : 1.0 × 10 −8 (blue) and 1.0 × 10 −7 (yellow).The bottom row shows the same for two different values of Im(ω): π/4 (blue) and π/2 (yellow).These cLFV observables depend mainly on the Yukawa couplings Y α F and Y α Σ , which are governed by λ 5 and Im(ω).It is interesting to mention that Re(ω) contributes to the Yukawa couplings as trigonometric 'sin' and 'cos' functions, with mild variation, whereas Im(ω) affects the Yukawa couplings as hyperbolic functions, with a steeper variation.
In summary, one sees that the attainable values for these cLFV observables can all exceed the current experimental limits for low dark matter mass values.In fact, µ → eγ rules out the benchmark point BP0 up to 800 GeV of the dark matter mass.This fact plays an important role in ruling out the viability of low-mass dark matter in this model.
Before closing this section, let us also present the current sensitivities of cLFV tau decays, along with their future projections in Table V 3 .For the benchmark values taken above we find that the numerically computed branching ratios for these processes stay well within their current experimental limits.Only by taking very small values of λ 5 ≤ 10 −10 , these tau decays become competitive with other cLFV observables, in agreement with the results obtained in Ref. [9].
In summary, we conclude that BR(µ → eγ), BR(µ → 3e) and Rate(µ, Au → e, Au) remain as the three main cLFV observables to be explored in our fermionic scotogenic dark matter scenario.However, cLFV tau decays may also provide alternative probes to envisage in the future.TABLE V: Experimental bounds on rare leptonic cLFV τ decay processes.

FERMIONIC DARK MATTER PHENOMENOLOGY
In this section, we perform a detailed study of fermionic dark matter in the singlet-triplet scotogenic model, highlighting some major phenomenological aspects.The possibility of scalar scotogenic dark matter was explored previously in Ref. [12,13].Some fermionic dark matter results have also been given in a recent work in Refs.[10].
We briefly discuss two major astrophysical constraints on the dark matter sector: the constraints imposed by the measured relic abundance as well as the direct detection experiments.

• Relic Density
When the dark matter falls out of thermal equilibrium in the early Universe, the remnant dark matter remains frozen out.The relic dark matter density given by the Planck measurement [75] is Ωh 2 = 0.120 ± 0.001.

• Direct Detection Constraints
Several experiments have performed direct searches for dark matter by attempting to detect nuclear recoil from scattering of dark matter off given target nuclei.In the light mass regime from 0.1 GeV to 4 GeV, the most stringent upper bound on the spin-independent dark matter-nucleon scattering cross-section comes from DarkSide-50 [76].In the window of 4 GeV to 10 GeV, the strongest constraints are by XENON1T 8 B [77] and PandaX-4T 8 B [78].In the range of 10 GeV to 10 TeV, experiments like XENON1T [79], XENONnT [80], PandaX-4T [81] impose similar constraints on the spin-independent cross-section.Nevertheless, the strongest restriction in this mass range comes from LZ [82].
Note that the "coherent scattering of neutrinos" can also produce nuclear recoil and act as background to the WIMP searches.This effect produces the "neutrino floor" [83,84], providing an intrinsic limitation on the discovery of dark matter.Thus the direct detection experiments allows a narrow window for the spin independent neucleon-DM cross-section.
In what follows we study the dependences of these two observables on the parameters of the triplet-singlet scotogenic model.All the dark matter analyses in this section are performed using BP0 as the benchmark point, using the micrOMEGAs tool [58,59] 6.1.

Cosmological Relic Density
The annihilation channels for a fermionic dark matter candidate χ 0 1 in the singlet-triplet scotogenic model are given in Fig. 4. The annihilation processes can be divided into three broad categories, with distinct phenomenological implications for the dark matter sector.
FIG. 4: The η, χ 0 1,2 and Σ ± -mediated fermionic t-channel dark matter annihilation processes.The three s-channels on the right panels correspond to resonant annihilation through the neutral Z 2 -even scalars h 0 , H 0 .
We now discuss some of their salient features.
• A pair of fermionic dark matter particles annihilate to visible sector final states, through t-channel (and uchannel) processes mediated by dark scalars η 0 R,I , η ± and fermions χ 0 1,2 , Σ ± .The strength of these annihilation involves Yukawa couplings that determine radiative neutrino mass generation.The relevant Feynman diagrams are shown in Fig. 4 (first three diagrams).
• Another key dark matter annihilation mechanism in this scenario is through scalar resonances.This includes the Higgs portal as usual, while a triplet-like scalar H 0 is also present as a portal of dark matter annihilation.The H 0 mediated processes happen only due to the presence of ϕ 0 − Ω 0 mixing.The χ 0 1 χ 0 1 → h 0 /H 0 → SM SM modes have a resonant enhancement at m χ 0 1 ∼ m h 0 /H 0 /2.The relevant Feynman diagrams are given in Fig. 4 (last three diagrams).
• Another dark matter annihilation channel takes place when there is a mass degenerate dark sector.This can happen due to the presence of dark particles only marginally heavier than the lighest one.In this case the other particles in the nearly degenerate dark sector take part in the annihilation, hence termed as dark matter co-annihilation.Dark matter co-annihilation was first discussed in Ref. [85].The thermal-averaged dark matter co-annihilation cross section is parametrized as In our case the relevant Feynman diagrams are given in Appendix B, see Fig. 14 and Fig. 15.
In Fig. 5 we depict the variation of the dark matter relic density with its mass for the benchmark point BP0.Since ∆m ΣF and ∆m η + F are large in BP0 (see Table III), co-annihilation is negligible, so the relic density is fully determined by the diagrams shown in Fig. 4. As the dark matter mass increases, different annihilation channels open up, and the relic density decreases gradually, as can be seen in Fig. 5.One can notice two sharp resonant dips at 62.5 GeV and 200 GeV (i.e.m h 0 /2 and m H 0 /2).These correspond to s-channel dark matter annihilation through the neutral scalars h 0 and H 0 respectively, i.e. the processes χ 0 1 χ 0 1 → h 0 /H 0 → SM SM .Moreover, at m DM = (m W ± + m H ± )/2 ≈ 242 GeV and m DM = (m h 0 + m H 0 )/2 ≈ 263 GeVs respectively, two new annihilation channels χ 0 1 χ 0 1 → W ± H ± and χ 0 1 χ 0 1 → h 0 H 0 open up, reducing the relic density.This corresponds to the fall-off in the region around ∼ 250 GeV.Another sharp drop is seen at 400 GeV, illustrating the opening of the phase space for χ 0 1 χ 0 1 → H 0 H 0 and χ 0 1 χ 0 1 → H + H − respectively.Beyond 400 GeV, the relic abundance of χ 0 1 is mainly controlled by the annihilation channels to H 0 H 0 and H + H − .We now turn to the dependencies of the dark matter relic density on the parameters of our model.As we saw in the above paragraph, the relic density is very sensitive to the mass of the heavy Higgs particles, m H 0 ≈ m H ± .In our parametrization the value of m H 0 mainly depends on three 4 parameters: λ 1 , λ ϕ Ω , v Ω .In the three panels of Fig. 6 we present the individual impact of these three parameters on the relic density.We choose three different values for each of these parameters, keeping the undisplayed parameters fixed as BP0.One sees that, except the first dip at m h 0 /2, all others present in Fig. 6 shift towards the right as λ 1 or λ ϕ Ω increases, or as v Ω decreases, corresponding to a larger value of m H 0 .The most sensitive parameter dependence however, appears to be on λ 1 .With increasing m H 0 , the relic density dips at m DM ∼ m H 0 /2 move upwards, as the annihilation at resonance gradually decreases with increasing H 0 mass and its Breit-Wigner width.Ω and v Ω with the remaining parameters fixed as in BP0, Table III.These parameters control the mass of the second neutral scalar H 0 .The blue curve represents BP0.The red dashed line indicates the observed Planck value.
The other important parameters governing the relic density are the Yukawa couplings Y F , Y Σ and Y Ω .While the first t-channel diagram in Fig. 4 depends on Y F and Y Σ , the last three diagrams depend on the coupling Y Ω .In order to satisfy the neutrino oscillation data, the Yukawa couplings Y F and Y Σ are mainly determined by λ 5 and Im(ω).Therefore, in Fig. 7 we present the dependece of the relic density density on λ 5 , Im(ω) and Y Ω .In the first two panels of Fig. 7 one sees that by decreasing λ 5 or increasing Im(ω) one enhances Y F and Y Σ , which in turn reduce the relic density.Since these couplings do not affect the H 0 H 0 and H + H − channels that determine the relic density for m DM > m H 0 , these Yukawas do not affect the relic density in that region.In contrast, one sees from the third panel in Fig. 7 that the relic density decreases sharply with increasing Y Ω for higher DM mass.This happens since 4 Though Eq. ( 6) suggests additional dependence of m H 0 on µ ϕ Ω and λ Ω Ω , we note that µ ϕ Ω is not an independent parameter, while λ Ω Ω comes with v Ω , making its contribution negligible.
the parameter Y Ω determines the couplings for χ 0 i χ 0 j H 0 , χ 0 i χ 0 j h 0 and χ 0 i Σ + H − interactions.Note however, that the positions of the dips do not shift in Fig. 7, as the mass m H 0 does not depend on these three parameters.The remaining parameters that affect the relic density are the mass-difference parameters ∆m ΣF and ∆m η + F , as illustrated in Fig. 8.When ∆m ΣF and ∆m η + F are very small, i.e. the dark sector becomes almost mass degenerate, the dark matter annihilation is assisted by the other dark particles, significantly enhancing the annihilation cross section.This reduces the relic density significantly in the region m DM < m H 0 , due to the activation of new fermionfermion (Fig. 14) and fermion-scalar (Fig. 15) co-annihilation channels.However, the positions of the dips does not change.)  III.The blue curve represents BP0.Fermion-fermion as well as scalar-fermion co-annihilation effects are visible for smaller values of these parameters.The red dashed line indicates the observed Planck value.

Direct Detection Prospects
In the triplet-singlet scotogenic model the spin-independent nucleon-dark matter scattering occurs through a tchannel diagram mediated by the neutral non-dark scalars h 0 and H 0 .As seen in Fig. 9, in order for this nucleon recoil process to take place, we need both the mixing in the Z 2 -even scalar sector (ϕ 0 and Ω 0 ) as well as in the Z 2 -odd sector (F and Σ 0 ).
In the limit of negligible Mandelstam variable t, this cross-section can easily be estimated as [10,11]: where m N is the mass of nucleon, f N (≈ 0.3) is the nucleon form factor and µ red is the reduced mass of nucleon-dark FIG.9: Direct detection process diagram for singlet-triplet mixed fermionic dark matter χ 0 1 .The Feynman diagram is presented in terms of before mixing fermionic singlet F and triplet Σ.
Here the parameter θ is the fermionic mixing angle defined in Eq. ( 11), whereas β is the scalar mixing angle, expressed as The spin-independent dark matter nucleon scattering cross section, given by Eq. ( 20), depends on the parameters Clearly, the nuclear recoil cross section goes as σ SI DM−N ∝ Y 2 Ω .Moreover, the first three parameters govern the heavy Higgs (H 0 ) mass as well as the scalar mixing angle β, while the others control the mixing angle θ.It is worth mentioning that the direct detection cross section is almost independent of m DM , except in the region of light dark matter.This is evident from Eq. (20), where σ SI DM−N depends on m χ 0 1 only through µ red , which reduces to m N for large m χ 0 1 values.

DARK MATTER PHENOMENOLOGY: RESULTS
We now perform a thorough parameter scan in this model, aiming to extract its most salient features concerning fermionic dark matter.The lightest of the dark triplet-singlet scotogenic fermion mediator will be our fermion dark matter candidate χ 0 1 , mainly a singlet.The triplet-like dark fermion χ 0 2 can potentially assist the co-annihilation processes.Since v Ω in this model plays a crucial role in the mixing of both scalars and fermions, we broadly divide our analysis in two cases characterized by different v Ω values.We dub these as Scenario-I and Scenario-II, respectively, with the parameters specified as in Table VI.
While scanning for each scenario, three different situations are identified, depending on the assumptions made on the dark particle mass-differences, as follows: (i) no co-annihilation (both ∆m ΣF and ∆m η + F are large, denoted with no superscript), (ii) fermion-fermion co-annihilation (∆m ΣF is small but ∆m η + F is large, denoted with superscript 'FF') and (iii) fermion scalar co-annihilation (∆m ΣF is large but ∆m η + F is small, indicated by superscript 'FS').The relevant parameters for each of these cases are specified in Table VI, and their range of variation is shown as numbers inside square brackets.The values of λ 1 , λ ϕ Ω and v Ω are kept fixed, in order to fix the mass of the heavy Higgs H 0 throughout the scan.For definiteness we fix the values of the remaining parameters e.g.µ η Ω , λ 2 , λ 3 , λ Ω Ω and λ η Ω as they hardly affect the results.
First we implement the restrictions from Sec. 3 including those from neutrino oscillations, encoded in the associated neutrino observables, along with the relevant theoretical constraints.It is interesting to note that once we adopt the parameterization for the Yukawa couplings Y F and Y Σ given in Eq. ( 13), all the parameter-points generated lie automatically in the region close to the solution of the neutrino oscillation restrictions.We then implement the constraints arising from collider physics, cLFV and the measured relic density, successively.Finally there are points with under-abundant relic density obeying all of the bounds mentioned.Amongst these points, we collect those providing relic density of 1% to 100% of the required Planck value in order to study their direct detection prospects.We present the corresponding results for all the cases mentioned in Table VI, which capture the essential features, as follows.

Relic Density
Here we explore the singlet-like fermionic scotogenic dark matter relic density and the associated phenomenology, for masses ranging from 3 GeV to 10 4 GeV.While Fig. 10 describes the behaviour of the relic density for the no coannhilation case in Scenario-I and Scenario-II, Fig. 11 and Fig. 12 illustrate the same when fermion-fermion and fermion-scalar co-annihilations are present.In these figures, the points excluded by the successive implementation of collider physics, cLFV and relic density constraints are indicated by gray, blue and orange colours, respectively.In contrast, points obeying all of these restrictions are displayed in green colour.
As discussed in Sec.6.1, depending on the heavy Higgs boson mass m H 0 (whose values are ∼ 400 GeV and ∼ 1100 GeV for Scenario-I and Scenario-II, respectively) one can clearly notice four different dips in relic density in all of the six plots from Fig. 10 to Fig. 12.Although the qualitative features of the rsults remain similar, scenario-II appears somewhat more constrained than the Scenario-I concerning the allowed regions given by the green points.

• No Co-annihilation
In the no co-annihilation case, as indicated by BP 1 and BP 2 in Table VI, we vary both ∆m ΣF and ∆m η + F in the range of 100 GeV to 500 GeV, keeping ∆m 2 η 0 I η + fixed 5 at 1000 GeV 2 .This makes all Z 2 -odd particles very heavy compared to χ 0 1 , and hence the annihilation of singlet-like dark matter alone determines the freeze-out time and the relic density.As a result, all of the points in this case trivially satisfy the collider bounds mentioned in Sec.3.3.However, λ 5 and Im(ω) (which control Y F and Y Σ ) play a key role in determining the branching fractions for cLFV processes as well as the relic fermioninc dark matter density in the region m DM < m H 0 .This can be seen from Fig. 3 and Fig. 7. Therefore, the constraints on cLFV processes restrict λ 5 and Im(ω) in a severe way, ruling out most of the parameter-points below m H 0 .Indeed, in the left and right panels of Fig. 10 one can see a cluster of blue points 5 Choosing such ∆m η + F values always keeps the mass of η + above 100 GeV.Fixing ∆m 2 η 0 I η + at 1000 GeV 2 then corresponds to a mass mass difference of less than 5 GeV between the charged and neutral components of η.
out by the collider constraints only, see Fig. 11.These points are mainly eliminated by the constraint on M Σ (m Σ + ) and the h 0 and Z 0 boson decay widths.Above m h 0 /2 dark matter mass values the restrictions on cLFV processes and relic density also play an important role.Nevertheless, new fermion-fermion dark matter co-annihilation channels open up, as shown in Fig. 14.This helps keeping the cLFV processes (or equivalently Y F and Y Σ ) under control.Thus, in addition to the resonant region around m h 0 /2, we start getting allowed green points above 80 GeV dark matter mass in both Scenario-I and Scenario-II.As in the no co-annihilation case, we do not find allowed points much above 10 4 GeV dark matter mass.within the range from 1 GeV to 30 GeV with ∆m 2 η 0 I η + varying from 1 GeV 2 to 1000 GeV 2 , keeping very large χ 0 2 masses.In this case, the light dark matter region of m DM < m h 0 /2 is disfavoured because of collider constraints on the h 0 and Z 0 boson decay widths.Above m h 0 /2, bounds from cLFV and relic density also dismiss plenty of points.Yet, in both panels of Fig. 12, one can see a broad allowed region of green points above 62.5 GeV.The presence of various processes depicted in Fig. 15 assists in the lessening of the relic density in this case.It is interesting to mention that, unlike the earlier two cases, here dark matter masses above 5000 GeV become ruled out.

Direct Detection
Now we discuss the direct detection prospects of fermionic scotogenic dark matter in various experiments such as LZ, Xenon-nT, PandaX.In Fig. 13 we give direct detection results for all cases of no co-annhilation, fermion-fermion and fermion-scalar co-annihilation, both in Scenario-I and Scenario-II.The purple region indicates the neutrino floor arising from coherent scattering of neutrinos, whereas the greenish area represents the region ruled out by the LZ direct detection experiment, that currently provides the strongest restriction.In our study, we collect the green points from Fig. 10, Fig. 11 and Fig. 12 giving the observed relic density or providing at least 1% of the entire observed relic density.The resulting spin-independent dark matter-neucleon scattering cross-section is presented in Fig. 13.The points ruled out by direct detection experiments are coloured in black, while the undetectable points below the neutrino floor are displayed in brown.All the other points are viable for fermionic scotogenic dark matter in our triplet-singlet model.
We start in the left column in Fig. 13 by depicting the direct detection prospects for Scenario-I with v Ω = 4 GeV, while Scenario-II with v Ω = 1.5 GeV is examined in the right column in Fig. 13.One sees that in all cases the clusters of points follow a similar pattern in both scenarios.We also notice that the maximum dark matter-nucleon scattering cross section decreases for all the three cases in Scenario-II, when compared to that in the Scenario-I.Indeed, according to Eq. ( 11) and Eq. ( 21), a lower v Ω pushes both scalar and fermion mixing angles (β and θ, respectively) to smaller values.This reduces the direct detection cross section to a great extent, see Eq. (20).
One sees that dark matter detection is possible in Scenario-I (left column in Fig. 13) in all cases of no co-annihilation (BP 1 ), fermion-fermion co-annihilation (BP FF 1 ) as well as fermion-scalar co-annihilation (BP FS 1 ).Nevertheless we find the last case of fermion-scalar co-annihilation to be most promising.Indeed, one obtains a broad well-distributed set of points within the dark matter detectability region (white) all the way from 62.5 GeV (m h 0 /2) upto 5 TeV.In contrast, for the no co-annihilation case, most of the detectable points lie above 400 GeV (i.e.m H 0 ).On the other hand, in the case of fermion-fermion co-annihilation, most of the points get ruled out by direct detection experiments.Due to the smaller value of ∆m ΣF characteristic of this case, the dark fermion fermionic mixing angle θ increases.This leads to a substantial enhancement in the direct detection cross section σ SI DM−N .Although one gets detectable points for dark matter masses between 100 GeV to 10 TeV, they are not very well-distributed inside the white region.In contrast, in the fermion-scalar co-annihilation case, the larger ∆m ΣF pushes σ SI DM−N below the direct detection upper limits.Moreover, co-annihilation effects help to keep the relic density small.This makes fermion-scalar co-annihilation into a viable option for fermionic scotogenic dark matter in the case of higher v Ω .
We now turn to the possibility of having lower value of v Ω , as in Scenario-II, i.e. 1.5 GeV, shown in the right column in Fig. 13.Clearly the fermion-fermion co-annihilation option appears to be the best option for having detectable fermionic scotogenic dark matter in Scenario-II.The combined effect of having large ∆m ΣF and small v Ω in the case of no co-annihilation or fermion-scalar co-annihilation leads to smaller cross sections σ SI DM−N in Scenario-II, that go below the neutrino floor, making a big chunk of the parameter space undetectable.In contrast, the smaller value of ∆m ΣF enhances σ SI DM−N in the fermion-fermion co-annihilation.This ensures good detectability prospects for the dark matter masses above 100 GeV in this case.Thus for lower v Ω fermion-fermion co-annihilating fermionic scotogenic dark matter becomes a perfect candidate to be explored.

CONCLUSIONS AND OUTLOOK
We have explored in a dedicated manner the possibility that neutrino masses arise from the exchange of dark matter states, as in Fig. 1.The same Z 2 symmetry that makes the neutrino masses calculable also stabilizes the dark matter.We examined the rich phenomenological profile of scotogenic fermionic dark matter within the singlettriplet model proposed in [7].In this reference model we stressed the possibility of having a singlet-like fermion χ 0 1 as dark matter candidate 6 .This updates and substantially extends the original work in [7] and also complements the work in Refs.[10,11] and [12,13], the latter dealing with the phenomenology of scotogenic scalar dark matter.We determined the allowed parameter region in this model consistent with the imposition of various theoretical and experimental restrictions.These include consistency conditions for the scalar potential, electroweak precision observables, constraints from neutrino oscillations, collider experiments, as well as cosmological relic density of dark matter.Charged lepton flavour violation proceeds vis the exchange of dark states, see Figs. 2. Taking into account all of these we have determined the prospects for cLFV searches, see Fig. 3.
In order to get a fuller picture of the phenomenological profile of fermionic dark matter in our scenario we combined our relic density results with our direct detection analysis.The cosmological relic density is set by the processes indicated in Fig. 4. Our relic density results are given in Figs. 5, 6, 7 and 8, under various specific parameter assumptions.To obtain a more global picture we have performed a dedicated numerical scanning procedure.To make it efficient we have made a number of reasonable assumptions so as to reduce the number of independent new physics parameters.Our results are summarized in Figs. 10, 11 and 12, see detailed discussion in the text.For example, we noted that low-mass dark matter (≲ m h 0 /2) with viable relic density is in conflict with cLFV and/or collider limits.Our analysis indicates that, in the absence of co-annihilation, Figs. 14 and 15, allowed fermionic dark matter masses typically require m DM ≳ m H 0 .In fact, dark matter with masses upto 100 GeV will be viable only in the presence of scalar-fermion co-annihilation, see lower left and right panel in Fig. 13.
In contrast to the original scotogenic constructions [5,6] dark matter detection in nuclear recoil experiments proceeds at the tree level [7], through the diagram in Fig. 9. Our present study indicates that the generalized fermionic scotogenic dark matter scenario leads to promising direct detection prospects, as summarized in Fig. 13.
In summary, we conclude that fermionic scotogenic dark matter brings in novel features with respect to vanilla fermionic dark matter in supersymmetry [2][3][4].Indeed, we note that our scenario does mimic neutralino dark matter without having an underlying supersymmetric framework.For example, our singlet-like fermionic dark matter is very much analogous to the "bino-like" fermionic supersymmetric WIMP.However, for bino-like fermionic dark matter, even if viable relic densities are obtained through co-annihilation [86][87][88], direct detection cross sections typically remain below the neutrino floor [89,90], in contrast with results we have obtained in Fig. 13.
Moreover, fermionic scotogenic dark matter is accompanied by the presence of cLFV phenomena.Although absent in the case of supersymmetric dark matter, these constitute an intrinsic feature in the scotogenic dark matter construction.As a final comment we stress that the "missing-partner" nature of the radiative seesaw mechanism mediated by the dark fermions, see Table I, implies the presence of a massless neutrino, with a very clear implication concerning neutrinoless double beta decay.Indeed, the amplitude for latter has a lower bound [13,91,92] which could make it detectable in the next round of experiments, for the case of inverse neutrino mass-ordering.Together with cLFV phenomena, the possible observation of neutrinoless double beta decay would provide another distinctive feature of scotogenic fermionic dark matter with respect to vanilla supersymmetric dark matter.All in all, our generalized scotogenic framework provides a richer picture of particle dark matter than its supersymmetric counterpart.Upcoming experiments will help testing the two approaches in a quantitative manner.
where, (i, j) ∈ {1, 2}, k ∈ {R, I} corresponding to real and imaginary parts, c k is ±1 for η R,I respectively and d ij equals to one or two depending on whether i = j or i ̸ = j respectively.Here R χ αβ and R h αβ denote the (α, β) elements of the rotation matrices diagonalizing the neutral fermionic mass matrix (M χ ) and the squared neutral scalar mass matrix (M 2 h ) respectively.In the above expressions, one must impose the condition that mass of the decaying particle is greater than the sum of masses for the final state particles.

FIG. 1 :
FIG. 1: Feynman diagrams for neutrino mass generation in one loop.

7 m
FIG.3: B(µ → eγ), B(µ → 3e) and C(µ, Au → e, Au) versus λ 5 and Im(ω).The other parameters are fixed as in BP0 benchmark defined in TableIII.The blue points in all the plots correspond to BP0, while the orange points are generic.The horizontal red dashed lines indicate the current experimental bounds, whereas the brown dotted lines correspond to future sensitivities.

m)FIG. 5 :
FIG. 5: Relic density versus dark matter mass for the benchmark point BP0.The red dashed line indicates the observed Planck value.

m)FIG. 6 :
FIG.6: Relic density versus dark matter mass for different λ 1 , λ ϕ Ω and v Ω with the remaining parameters fixed as in BP0, TableIII.These parameters control the mass of the second neutral scalar H 0 .The blue curve represents BP0.The red dashed line indicates the observed Planck value.

m)FIG. 7 :
FIG. 7: Relic density versus dark matter mass for different λ 5 , Im(ω) and Y Ω fixing the other parameters as in BP0, Table III.The blue curve represents BP0.(The parameter Re(ω) has very minor effects on the relic density).The red dashed line indicates the observed Planck value.

m)FIG. 8 :
FIG.8: Relic density versus dark matter mass for different mass-difference parameters ∆m ΣF , ∆m η + F fixing the other parameters as in BP0, TableIII.The blue curve represents BP0.Fermion-fermion as well as scalar-fermion co-annihilation effects are visible for smaller values of these parameters.The red dashed line indicates the observed Planck value.

FIG. 12 :
FIG. 12: Relic density versus dark matter mass for the fermion-scalar co-annihilation case in Scenario-I (BP FS 1 ) and Scenario-II (BP FS 2 ).The red dashed line indicates the observed Planck value.

TABLE I :
Quantum number assignemnts for the singlet-triplet scotogenic model.The quark sector is standard.

TABLE III :
Specification of the benchmark scenario BP0.

•
Fermion-Scalar Co-annihilationThe case of fermion-scalar co-annihilation is represented by BP FS 1 and BP FS 2 in TableVI.Now ∆m η + F is varied