PLB138282138282S0370-2693(23)00616-010.1016/j.physletb.2023.138282The Author(s)PhenomenologyFig. 1One-loop Feynman diagram contributing to the Majorana neutrino mass in Eq. (13).Fig. 1Fig. 2Parameter space fulfilling −10 keV ≤μ ≤ 10 keV, for the DM masses indicated above each plot. The color in each plot, from light to dark, represents yN = 0.01,0.05 and 0.1, respectively. Here we assume that mΩ2≫mΩ. The discontinuity appears when a degenerate mass spectrum is reached in Eq. (15).Fig. 2Fig. 3Diagrams (a)-(g) are relevant for the freeze-out of Ω. The diagram (h) is relevant for direct detection, with N representing the nucleons. Here to simplify notation we have used σ to denote either of (h1,h2,χ).Fig. 3Fig. 4Left-panel: Relic abundance as a function of yΩ for mΩ = 200 and 500 GeV. Here we have set mh2=120 GeV and θ = 0.1, with the rest of the parameters specified in the text. Right-panel: Direct detection cross section as a function of mh2 and different combinations of (mΩ[GeV],tan⁡θ). The horizontal red and orange lines are the current DD upper limits for each value of mΩ, whereas the grey curves are the XENONnT projections. The solid curves depicted here fulfill the correct relic abundance.Fig. 4Fig. 5Parameter space restrictions from direct and indirect dark matter. The thermal value of 〈σv〉ΩΩ→h2χ is given by the red band on top of the plots. It is assumed that h2 decays into either bb¯ (blue solid line) or W+W− (green solid line). The dashed blue horizontal line represents the latest bound on the DM annihilation cross section into bb¯ from AMS-02. The green dashed and dotted-dashed are the projected sensitivities of CTA depending on the DM profile. The dark and light orange areas correspond to the LZ exclusion region and the future sensitivity of XENONnT, respectively. The dark gray area in the left panels satisfies mh2>2mχ and is then forbidden.Fig. 5Fig. 6Branching ratio BR(μ→eγ) as a function of the mass of the lightest RH Majorana neutrino NR. The shadowed region is excluded by MEG [48]. The black star corresponds to the prediction of the best-fit point of the model, for mΩ = 200 GeV (left-panel) and mΩ = 500 GeV (right-panel). The green points are compatible with current neutrino oscillation experimental limits at 3σ. The orange points are out of the 3σ range and, hence, excluded by neutrino oscillation data.Fig. 6Table 1Charge assignments of scalar and lepton fields. Here i = 1,2,3 and k = 1,2.Table 1ΦσηLLilRiνRkNRkΩRkSU(2)L21121111U(1)Y1/200−1/2−1000U(1)X0−11/2−1−1−11−1/2Table 2Neutrino oscillation parameters from global fits [51].Table 2ObservableΔm212[10−5eV]Δm312[10−3eV]sin2⁡θ12/10−1sin2⁡θ23/10−1sin2⁡θ13/10−2δCP/∘Best fit ±1σ7.50−0.20+0.222.55−0.03+0.023.18 ± 0.165.74 ± 0.142.200−0.062+0.069194−22+243σ range6.94 − 8.142.47 − 2.632.71 − 3.694.34 − 6.102.00 − 2.405128 − 359Table 3Dimensionless parameters used to compute BR(μ → eγ) compatible with neutrino oscillation data. Case (a) considers mΩ = 200 GeV and case (b) is for mΩ = 500 GeV.Table 3Dimensionless parameters (a)yν11=0.0109e1.87iyν31=0.0124e1.49iyν12=0.0105e−1.31iyν32=0.0808e1.77iyν21=0.0270e1.70iyN = 2.04 × 10−2yν22=0.0463e1.56iθˆ=1.303radDimensionless parameters (b)yν11=0.00525e−1.51iyν31=0.0177e1.572iyν12=0.0151e−1.59iyν32=0.00424e1.61iyν21=0.0166e1.57iyN = 1.22 × 10−2yν22=0.0354e−1.58iθˆ=−4.98radTable 4Best fit values of the model. Case (a) considers mΩ = 200 GeV and case (b) is for mΩ = 500 GeV.Table 4Observableμ1[keV]mNR[GeV]Δm212[10−5 eV2]Δm312[10−3 eV2]sin⁡θ12(l)/10−1sin⁡θ13(l)/10−3sin⁡θ23(l)/10−1δCP(l)(∘)Best fit case (a)−0.562700.57.532.513.532.115.66195.3Best fit case (b)−0.4099401.37.502.553.252.225.64174.8LetterDark matter from a radiative inverse seesaw majoron modelCesarBonillaa⁎cesar.bonilla@ucn.clA.E.Cárcamo Hernándezbcdantonio.carcamo@usm.clBastiánDíaz Sáezebastian.diaz.s@usach.clSergeyKovalenkofcdsergey.kovalenko@unab.clJuanMarchant Gonzálezghdjuan.marchant@upla.claDepartamento de Física, Universidad Católica del Norte, Avenida Angamos 0610, Casilla 1280, Antofagasta, ChileDepartamento de FísicaUniversidad Católica del NorteAvenida Angamos 0610Casilla 1280AntofagastaChileDepartamento de Física, Universidad Católica del Norte, Avenida Angamos 0610, Casilla 1280, Antofagasta, ChilebUniversidad Técnica Federico Santa María, Casilla 110-V, Valparaíso, ChileUniversidad Técnica Federico Santa MaríaCasilla 110-VValparaísoChileUniversidad Técnica Federico Santa María, Casilla 110-V, Valparaíso, ChilecCentro Científico-Tecnológico de Valparaíso, Casilla 110-V, Valparaíso, ChileCentro Científico-Tecnológico de ValparaísoCasilla 110-VValparaísoChileCentro Científico-Tecnológico de Valparaíso, Casilla 110-V, Valparaíso, ChiledMillennium Institute for Subatomic Physics at High-Energy Frontier (SAPHIR), Fernández Concha 700, Santiago, ChileMillennium Institute for Subatomic Physics at High-Energy Frontier (SAPHIR)Fernández Concha 700SantiagoChileMillennium Institute for Subatomic Physics at High-Energy Frontier (SAPHIR), Fernández Concha 700, Santiago, ChileeDepartamento de Física, Universidad de Santiago de Chile, Casilla 307, Santiago, ChileDepartamento de FísicaUniversidad de Santiago de ChileCasilla 307SantiagoChileDepartamento de Física, Universidad de Santiago de Chile, Casilla 307, Santiago, ChilefDepartamento de Ciencias Físicas, Universidad Andrés Bello, Sazié 2212, Piso 7, Santiago, ChileDepartamento de Ciencias FísicasUniversidad Andrés BelloSazié 2212Piso 7SantiagoChileDepartamento de Ciencias Físicas, Universidad Andrés Bello, Sazié 2212, Piso 7, Santiago, ChilegDepartamento de Física, Universidad Técnica Federico Santa María, Casilla 110-V, Valparaíso, ChileDepartamento de FísicaUniversidad Técnica Federico Santa MaríaCasilla 110-VValparaísoChileDepartamento de Física, Universidad Técnica Federico Santa María, Casilla 110-V, Valparaíso, ChilehDepartamento de Matemática, Física y Computación, Facultad de Ciencias Naturales y Exactas, Universidad de Playa Ancha, Subida Leopoldo Carvallo 270, Valparaíso, ChileDepartamento de Matemática, Física y ComputaciónFacultad de Ciencias Naturales y ExactasUniversidad de Playa AnchaSubida Leopoldo Carvallo 270ValparaísoChileDepartamento de Matemática, Física y Computación, Facultad de Ciencias Naturales y Exactas, Universidad de Playa Ancha, Subida Leopoldo Carvallo 270, Valparaíso, Chile⁎Corresponding author.Editor: A. RingwaldAbstractWe propose a Majoron-like extension of the Standard Model with an extra global U(1)X-symmetry where neutrino masses are generated through an inverse seesaw mechanism at the 1-loop level. In contrast to the tree-level inverse seesaw, our framework contains dark matter (DM) candidates stabilized by a residual Z2-symmetry surviving spontaneous breaking of the U(1)X-group. We explore the case in which the DM is a Majorana fermion. Furthermore, we provide parameter space regions allowed by current experimental constraints coming from the dark matter relic abundance, (in)direct detection, and charged lepton flavor violation.Data availabilityData will be made available on request.1Introduction1For a recent review on neutrino mass mechanisms at different energy scales see for instance [8].It took half a century to experimentally confirm with stunning accuracy every single part of what constitutes a unified description of the strong and electroweak interactions dubbed the Standard Model (SM). Despite the revolution and success in particle physics, the SM is far from being the final description of our universe, leaving many of its fundamental properties unexplained. For instance, it does not account for the origin of neutrino masses, lacks any hint of about 85% of the “dark” matter budget of the universe and what is the nature of this hidden sector, and does not explain the baryon asymmetry of the universe. These, among other issues, lead us to think that the SM is at best a low-energy effective field theory that belongs to a bigger framework.The simplest and most popular realization to explain the smallness of neutrino masses is through the Type-I seesaw mechanism [1–7]. In this approach, Majorana right-handed neutrinos (RHN), NR, are added to the SM. This implies to extend the Yukawa Lagrangian by including terms yνL¯NRH˜+MRNRNRc‾. If we consider order one Yukawa interactions, i.e. yν∼O(1), the size of neutrino masses get determined by RHN masses according to mν≈vΦ2/MR, where vΦ is the Higgs vacuum expectation value (vev). Then, the new physics energy scale is expected to be at fifteen orders of magnitude above the electroweak one for mν∼0.1 eV. This feature makes the Type-I seesaw mechanism inaccessible to current experimental sensitivities.One of the main motivations to explore other scenarios accounting for neutrino masses1 is that some manifest new physics signatures at energies around the TeVs, i.e. at the reach of current or upcoming experimental searches [9]. One example of this is the so-called inverse seesaw model which is characterized by predicting significant lepton flavor violating rates [10–13]. This model introduces two Majorana fermion pairs to the SM, NRi and SLj (i,j=1,2,3). These fields transform as singlets under the SM gauge group and carry a lepton number of +1. Then, after electroweak symmetry breaking the Lagrangian in the neutrino sector is given by(1)Lν=mDν‾LNR+MN‾RSL+μSLTC−1SL+h.c., where mD and M are Dirac 3×3 matrices, while μ is a Majorana 3×3 matrix breaking lepton number explicitly. The latter can have a dynamical origin [4]. As usual, C denotes the charge conjugation matrix. The neutrino mass matrix in the (νL,NR,SL) basis turns out to be(2)Mν=(0mD0mDT0M0MTμ).Taking the limit μij<<(mD)ij<<Mij (i,j=1,2,3) leads to a 3×3 matrix for light neutrinos given by(3)mν≃mD1Mμ1MTmDT.Note that the lightness of neutrinos could be associated only to the smallness of μ, which is naturally protected from large radiative corrections by the U(1)L-symmetry of lepton number conservation, restored in the limit μ→0 in Eq. (1). Obviously, neutrinos become massless in this limit. Nevertheless lepton flavor violating processes are still allowed [14]. Non-zero but small μ can be generated via radiative corrections opening up the intriguing possibility for relation of neutrino mass generation to the dark sector [15–29].In this paper we propose a variant of the inverse seesaw model where the μ term is generated at the 1-loop level after the spontaneous breaking of a global U(1)X symmetry. This Abelian continuous symmetry breaks down to a discrete subgroup Z2 which is an exact low energy symmetry which stabilizes the Dark Matter (DM) particle candidates of our model. Then, we explore the viability of having as DM the lightest Majorana fermion involved in generation of the μ term in Eq. (3) at 1-loop level. We analyze the case in which this thermal relic communicates to the SM mainly via Higgs portal and provide constraints and prospects for direct and indirect detection in DM searches. Let us point out that here, we focus only on the neutrino and DM sectors, leaving the possibility of including the quark sector for subsequent studies.This paper is organized as follows. In section 2 we provide the details of the model such as the particle content, charge assignments and symmetry breaking. In addition, we describe the scalar potential and mass spectrum. In section 3 the phenomenology of the fermionic dark matter candidate is studied. The implications of the model in charged lepton flavor violation are discussed in section 4. We state our conclusions in section 5.2The modelWe consider a model that adds to the SM, two complex scalars σ and η and six Majorana fermions νRk, NRk and ΩRk (k=1,2). All these new fields are SU(2) gauge singlets and carry neutral electric charge. In addition, the existence of a global U(1)X symmetry is assumed. This symmetry breaks down to a Z2 symmetry when the singlet scalar gets a vacuum expectation value (vev) 〈σ〉=vσ. Table 1 shows the charge assignments of scalars and leptons under the SU(2)L⊗U(1)Y⊗U(1)X symmetry.In fact, after electroweak symmetry breaking the unbroken symmetry is SU(3)C⊗U(1)EM⊗Z2 where Z2 turns out to be the symmetry that stabilizes the dark matter candidate of the theory. Schematically, the symmetry breaking chain goes as follows,(4)G=SU(3)C⊗SU(2)L⊗U(1)Y⊗U(1)X⇓vσSU(3)C⊗SU(2)L⊗U(1)Y×Z2⇓vΦSU(3)C⊗U(1)Q⊗Z2 where the Higgs vev is represented by 〈Φ0〉=vΦ.Given the charge assignments shown in Table 1 we have that the singlet's vev is invariant under the following transformation, e2πiXˆ〈σ〉=〈σ〉, where Xˆ is the U(1)X charge operator. This implies the existence of a residual discrete symmetry (−1)2Xˆ∈Z2 surviving spontaneous breaking of the global U(1)X group. Therefore, to all fields are assigned the corresponding Z2-parities (−1)2QˆX according to their U(1)X charges QX in Table 1. The particles η and ΩRk (k=1,2) have odd Z2-parities and form the dark sector of the model.2.1Scalar sectorThe scalar potential invariant under the symmetry group G is given by(5)V(Φ,σ,η)=−μΦ22|Φ|2+λΦ2|Φ|4−μσ22|σ|2+λσ2|σ|4+μη22|η|2+λη2|η|4+λ1|Φ|2|σ|2+λ2|Φ|2|η|2+λ3|σ|2|η|2+μ42ση2+h.c.,where the quartic couplings λa are dimensionless parameters whereas the μa are dimensionful. In our analysis we will impose perturbativity (λa<4π) and the boundedness conditions given in Appendix A.The singlet σ and the neutral component of the doublet Φ=(ϕ+,ϕ0)T acquire vacuum expectation values (vevs). Here the singlet's vev vσ is responsible for the breaking of the global U(1)X symmetry while the double's vev vΦ triggers electroweak symmetry breaking. Therefore we shift the fields as(6)ϕ0=12(vΦ+ϕR+iϕI),σ=(vσ+σR+iσI2).Evaluating the second derivatives of the scalar potential at the minimum one finds the CP-even, MR2, and CP-odd, MI2, mass matrices. The CP-even mass matrix MR2 mixes ϕR and σR and its eigenvalues correspond to the squared masses of the physical scalar states. They are given by(7)mh1,h22=12(λσvσ2+λΦvΦ2∓λσvσ2−λΦvΦ2cos⁡2θ),where we identify h1 with the 125 GeV Higgs boson, vΦ=246 GeV, and the mixing angle θ fulfilling(8)tan⁡2θ=2λ1vΦvσλσvσ2−λΦvΦ2.Moreover, the flavor and physical bases are connected through out the following relations,(9)σR=−h1sin⁡θ+h2cos⁡θ,ϕR=h1cos⁡θ+h2sin⁡θ. The CP-odd mass matrix MI2 has two null eigenvalues. One of them corresponds to the would-be Goldstone boson which becomes the longitudinal component of the Z-boson by virtue of the Higgs mechanism. The other one is the physical Goldstone boson resulting from spontaneous breaking of the global U(1)X symmetry, similar to the singlet Majoron model in Ref. [4]. Notice that given that this Goldstone is an SM singlet, it is invisible, and its phenomenological impact in the model is not dangerous. It includes any cosmological manifestations. Let us also stress that the mass of this field is non-vanishing, at least due to the well-known quantum gravity effects. It can also gain mass due to non-perturbative QCD effects in an extended version of our model in which the fields in the quark sector have non-trivial U(1)X assignments entailing a mixed (SU(3)C)2U(1)X-anomaly. This is what defines our Majoron model variant. The masses of Z2-odd scalar components η=ηR+iηI turn out to be(10)mηR2=μη2+12(λ2vΦ2+λ3vσ2)+μ4vσ,(11)mηI2=μη2+12(λ2vΦ2+λ3vσ2)−μ4vσ,where the mass splitting of these two components can be recasted as μ4=(mηR2−mηI2)/(2vσ). Note that ηR and ηI are degenerate when μ4→0. Here, the lightest of these two components, ηR and ηI, can be the stable dark matter.2.2Neutrino sectorUsing Table 1, the invariant lepton Yukawa Lagrangian is given by(12)−LY(l)=∑i=13∑j=13(yl)ijL‾LilRjΦ+∑i=13∑k=12(yν)ikL‾LiνRkΦ˜+∑n=12∑k=12Mnkν‾RnNRkc+∑n=12∑k=12(yN)nkN‾RnΩRkcη+∑n=12∑k=12(yΩ)nkΩ‾RnΩRkcσ+h.c., where ψc=Cψ‾T and Φ˜=iσ2Φ⁎. After spontaneous symmetry breaking (SSB), the neutrino mass matrix has the form,(13)Mν=(03×3mD03×2mDT02×2M02×3MTμ),where mD is the tree-level Dirac mass term(14)(mD)ik=(yν)ikvΦ2, with i=1,2,3 and k=1,2. The submatrix μ in Eq. (13) is generated at one-loop level,(15)μsp=∑k=12(yN)sk(yNT)kpmΩk16π2[mηR2mηR2−mΩk2ln⁡(mηR2mΩk2)−mηI2mηI2−mΩk2ln⁡(mηI2mΩk2)], with s,p=1,2. The Feynman diagram of μ is depicted in Fig. 1. One can see from Eq. (15) that the μ term vanishes when the scalars ηR and ηI are degenerate. This implies that neutrino masses go to zero in the limit μ→0.Then one has that active light neutrino masses are generated via an inverse seesaw mechanism at the one-loop level. Physical neutrino mass matrices are given by22The diagonalization of the neutrino mass matrix in Eq. (13) can be followed from Ref. [30].:(16)M˜ν=mD(MT)−1μM−1mDT,(17)Mν(−)=−12(M+MT)+12μ,(18)Mν(+)=12(M+MT)+12μ.Now M˜ν is the mass matrix for active light neutrinos (νa), whereas Mν(−) and Mν(+) are the mass matrices for sterile neutrinos. From Eq. (16) one can see that active light neutrinos are massless in the limit μ→0 which implies that lepton number is a conserved quantity. Eqs. (17) and (18) tell us that the smallness of the parameter μ (small mass splitting) induces pseudo-Dirac pairs of sterile neutrinos.From Eq. (16), one can see that a sub-eV neutrino mass scale can be linked to a small lepton number breaking parameter μ which depends on the Yukawas yN2, yΩ and the masses of the particles running in the loop (mηR,mηI,mΩ). This parameter is further suppressed by the loop factor, see Eq. (15). Fig. 2 shows the allowed parameter space regions for fixed Yukawa couplings yN and masses of the Z2-odd Majorana fermions Ωk. Each plot is generated using Eq. (15), varying the masses (mηR,mηI), fixing mΩ and yN. Then, from left to right, Fig. 2 shows the parameter space that fulfills −10 keV ≤μ≤10 keV in the (mηR,mηI)-plane, considering mΩ=100,500, and 1000 GeV, respectively. In all panels, yN=0.01,0.05, and 0.1, the smaller the Yukawa value, the lighter the region. The discontinuity appears when the mass spectrum in Eq. (15) is degenerate.As we have mentioned, the dark sector is formed by the Z2-odd particles, see Table 1. The dark matter candidate of the model is the lightest component of either the singlet scalar η or the Majorana fermion Ω. The phenomenological consequences of having the lightest component of the scalar singlet η as dark matter candidate is similar to what have been discussed in Refs. [20,23,28,31,32]. For this reason, in what follows we discuss only the constraints and projections of the model for the case in which the DM candidate is the Majorana fermion Ω.3Fermion dark matterFor simplicity, we consider the case in which yΩ in Eq. (12) is a diagonal matrix and assume that ΩR1 is the lightest Z2-odd state. That is, ΩR1 is the fermion DM candidate accounting for the 80% of the matter content of the universe. According to the Planck collaboration the DM relic abundance is [33](19)Ωch2=0.1200±0.0012at68%C.L. In our setup the Lagrangian providing the relevant interactions of ΩR1 is given by(20)L⊃yN1N¯RΩR1cη+yΩ1Ω‾R1ΩR1cσ+h.c..After SSB Eq. (20) becomes(21)L⊃(yN1N¯RΩ1Rcη+h.c.)+mΩ1Ω‾Ω+yΩ1Ω‾Ω(−h1sin⁡θ+h2cos⁡θ)+yΩ1iΩ‾γ5Ωχ, where mΩ1=yΩ1vσ/2. We have defined Ω≡(Ω1R)c+Ω1R, NR≡(−NR++NR−)/2 and χ≡σI.Taking into account the assumptions above, the relic abundance of Ω is determined by the annihilation channels shown in Fig. 3. Given that Ω is the DM candidate of the theory then mΩ<(mNR,mηR,mηI). In this case, the main annihilation channels are those s-channels depicted by diagrams (a), (b) and (c) in Fig. 3. We further simplify the analysis by considering that the annihilation channels mediated by the Higgs via the dimensionless parameters λ2 and λ3 are subleading. That is, we set λ2=λ3=0. Therefore, the independent parameters to be used in the numerical analysis turn out to be (mΩ,mηR,mηI,mh2,mNR,yN1,yΩ1,θ).Let us note that in this inverse seesaw model, the Majorana dark matter candidate can interact with the atomic nucleons at tree-level. For this reason, our model gets restricted by direct detection constraints. As a matter of fact, these constraints come from the t-channel exchange of h1 and h2 shown by Fig. 3-(h). Then, here the spin-independent (SI) tree-level DM-nucleon scattering cross section is, approximately, [34,35](22)σΩ≈fp2mN4mΩ24πvΦ2(mΩ+mN)2(1mh12−1mh22)2(yΩ1sin⁡2θ)2, where mN denotes the nucleon mass and the nuclear elements fp≈0.27. The approximation given in Eq. (22) does not take into account the finite width of both Higgs scalars, although the outputs of this expression match the numerical results from Micromegas code v5.3.35 [36] which do include these finite widths.3.1Analysis and resultsIn what follows, we compute the relic abundance of the Majorana fermion Ω assuming freeze-out mechanism, the direct detection via non-relativistic scattering, and indirect detection prospects today. For our calculation, we make use of Micromegas code v5.3.35 [36].As mentioned, here we explore the case where λ1≠0, λ2=λ3=0. Therefore, the contribution to the relic abundance coming from the annihilation of Ω into SM particles happens only via the Higgs portal associated to λ1. The left panel in Fig. 4 shows the relic abundance of Ω as a function of yΩ, for mΩ=200 (solid blue) and 500 GeV (solid greed), assuming mh2=120 GeV and θ=0.1.33Collider searches of additional scalars restrict the doublet-singlet mixing angle to be θ≲0.2 [37]. This benchmark considers yN=0.1, mηR=2000 GeV, mηI=2001 GeV and mN=300 GeV (these parameters will be fixed to such values from now on). The limit of the DM relic abundance given in Eq. (19) is represented by the red dashed line. One can see from Fig. 4 that the relic abundance has a strong dependence on the DM mass and the parameter yΩ. For a small Yukawa coupling yΩ≲10−2 and mΩ>mN, the process ΩΩ→NN is kinematically allowed and dominates over the other annihilation channels. In the case of mΩ<mN, the leading contributions to the relic abundance are those processes which involve the fields (χ,h1,h2) (diagrams b and c in Fig. 3). In such a case, the relic abundance turns out to be inversely proportional to yΩ2. For this reason, as shown on the left-panel in Fig. 4, the solid blue (green) curve decreases when the value of Yukawa increases. It is evident that, for a given yΩ, the relic abundance grows as the DM mass decreases. This behavior is expected since ΩΩh2 is inversely proportional to the annihilation cross section which depends on the center-of-mass energy of the colliding non-relativistic DM particles, i.e. s≈4mΩ2. Here, we are focusing on the case of small doublet-singlet mixing θ, i.e. ≲0.1. For this reason, the DM relic abundance turns out to be “blind” to this parameter and is completely determined by the interaction of fields belonging to the dark sector (namely, χ and/or h2). The DM annihilation channel into a pair of χ is always present unless further assumptions are made to suppress it.In contrast to the situation previously described, the DM direct detection given by the SI cross section σΩ, Eq. (22), is sensitive to the values of the doublet-singlet mixing θ. This is shown by the plot on the right-hand side of Fig. 4 where σΩ is depicted as a function of the mass of second CP-even scalar mh2 (considering mΩ=200 and 500 GeV, and each case with θ=0.1,0.01). The blue and green curves represent the points in the parameter space fulfilling the correct relic abundance while the red, yellow and gray dashed horizontal lines correspond to the experimental limits provided by XENON1T [38] and LUX-ZEPLIN (LZ) [39], and the projections by XENONnT [40], respectively. From Eq. (22) one can notice that the cross section rests on the existence of a mixing between the scalar doublet Φ and the singlet σ. This dependence can be observed from Fig. 4 which shows the sensitivity of the cross section to θ variations. Furthermore, we can see that when the CP-even scalars h1 and h2 are (semi)-degenerate, i.e. mh2≈mh1, there is a numerical cancellation that generates the inverted peak in the SI cross section. This allows to elude the experimental bounds when h1 and h2 are close in mass. Another possibility to relax the experimental constraints, including the one coming from XENONnT, happens by shrinking the value of doublet-singlet mixing as depicted in Fig. 4. This parameter space limit is reached, for instance, when lepton number gets broken at energies much higher than the electroweak scale.Let us note that this model is characterized by the presence of the process ΩΩ¯→χh1,2, with h1,2 decaying into SM particles [34]. These s-wave processes are velocity independent channels not present in other models [20,23] and give good prospects for indirect DM detection especially when χ is a pseudo-Goldstone boson, i.e. mχ≠0. For this reason, we look into regions of the parameter space where the processes ΩΩ¯→χh1,2 dominate the DM annihilation and provide the limits coming from the Alpha Magnetic Spectrometer (AMS) experiment as well as the future sensitivities of the Cherenkov Telescope Array (CTA) experiment. For convenience, we assume mχ>mh2/2 so that the decay h2→2χ is kinematically disallowed. In this way, the DM candidate does not annihilate primarily into invisible channels.Using Eq. (9), one can express the h2 branching fraction into SM particles as [41],(23)BR(h2→SM)=sin2⁡θ[Γ(h2→SM)Γtot] where Γ(h2→SM) corresponds to the partial decay width of the scalar boson h2 (with mass mh2) into SM states, and the total decay width is given by(24)Γtot=sin2⁡θ×ΓtotSM+Γ(h2→2h1)+Γ(h2→2Ω)+Γ(h2→2χ). Here ΓtotSM corresponds to the total decay width of h2 into SM states [42]. For simplicity, we focus on the case in which mh2<2mΩ. Furthermore, in order to assure observability, via h2 decaying into SM particles, we assume mχ≠0 as well as mh2<2mχ. Therefore, the third and fourth terms in Eq. (24) are not present in our study.Taking into account the considerations previously stated, we perform a numerical analysis. Fig. 5 shows the predictions for indirect detection signals coming from the DM annihilation into a pseudo-Goldstone χ and h2. The CP-even scalar subsequently decays into SM particles, see Eq. (23). Then, the DM annihilation would be ΩΩ¯→χh2→χSM. The assumed thermal value 2×10−26 cm/3s≤〈σv〉ΩΩ¯→χh2≤3×10−26 cm/3s is depicted by the red region at the top of each panel in Fig. 5. The left-panels consider mΩ=200 GeV for θ=0.1 and 0.01, while the panels on the right take mΩ=500 GeV for the same values of θ. In all panels, the Yukawa coupling is yN=0.1, and the dark scalar and pseudoscalar masses are fixed as mηR=2000 GeV and mηI=2001 GeV, respectively. The solid blue (green) line corresponds to the DM matter annihilation into bb¯ (W+W−). The dashed blue horizontal line is the upper bound of AMS-02 [43] for DM annihilation into bb¯, whereas the dashed green horizontal one represents the future sensitivity of CTA [44] in the W+W− channel. We also consider the bound projected by CTA for DM annihilation searches with W+W− in the final state assuming a gNFW profile with a slope parameter γ=1.26 [44]. Fig. 5 includes direct detection bounds at fixed θ and DM mass. The dark orange is the exclusion region coming from the LZ results on DD searches. The light orange area represents the future sensitivity of XENONnT. Therefore, only the light orange and white parts in each panel are allowed by current DM direct detection constraints. In addition, the dark gray area in the left panels of Fig. 5 satisfies mh2>2mχ and is then forbidden. This is because we are working under the assumption that mh2<2mχ or mχ=2mΩ−mh2>2/3mΩ, i.e. h2 does not decay into 2χ. This guarantees that the fermion DM candidate Ω annihilates into observable modes.Fig. 5 shows the parameter space fraction that CTA would be able to test in the future by looking for W+W− products. One can see that the CTA projections do not reach the model predictions if the DM density distribution follows an Einasto profile. On the other hand, if the DM posses a gNFW profile, CTA searches could prove masses mh2≳150 GeV. In addition, we have that AMS-02 data impose parameter space restrictions over the parameter space from DM annihilation into bb¯ searches. Notice that, as expected from Eq. (22), the direct detection constraints get weaker when the singlet-doublet mixing takes smaller values.4Charged lepton flavor violationIn this section we analyze charged lepton flavor violation (cLFV) processes present due to the mixing between active and heavy sterile neutrinos. Here we focus in the one-loop decays li→ljγ whose branching ratios are given by [45–47](25)BR(li→ljγ)=αW3sW2mli5256π2mW4Γi|Gij|2(26)Gij≃∑k=13([(1−RR†)Uν]⁎)ik((1−RR†)Uν)jkGγ(mνk2mW2)Gij≃+2∑l=12(R⁎)il(R)jlGγ(mNRl2mW2),Gγ(x)=10−43x+78x2−49x3+18x3ln⁡x+4x412(1−x)4, where Γμ=3×10−19 GeV is the total muon decay width, Uν is the matrix that diagonalizes the light neutrinos mass matrix which, in our case, is equal to the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix since the charged lepton mixing matrix is equal to the identity Uℓ=I. In addition, the matrix R is given by(27)R=12mD⁎M−1, where M and mD are the heavy Majorana mass matrix and the Dirac neutrino mass matrix, respectively. We provide in Appendix B all assumptions made for computing R in Eq. (27). Table 3 contains the benchmarks points used to compute μ→eγ.Then, feeding Eq. (25) with the values of the dimensionless parameters and masses given in Table 3, one gets the following branching fractions,(28)BR(a)(μ→eγ)≃2.02×10−13andBR(b)(μ→eγ)≃1.13×10−13, where (a) is for mΩ=200 GeV and (b) is for mΩ=500 GeV.Fig. 6 shows the correlation between the branching ratio BR(μ→eγ) and the mass of the lightest RH Majorana neutrino NR. One observes that the branching ratio decreases as the mass of NR increases. In both plots, the red horizontal line and the shaded region represent the latest experimental constraint provided by the MEG [48] collaboration,(29)BR(μ→eγ)exp<4.2×10−13.The black stars in Fig. 6 correspond to the branching ratio predicted, Eq. (28), by the best-fit points of the model for mΩ=200GeV (left-panel) and mΩ=500GeV (right-panel). The scatter plots come from a random variation of the dimensionless parameters up to 30% around the best-fit value. The green points are compatible with current neutrino oscillation experimental limits at 3σ. One can see that neutrino oscillation data restrict the lightest RH neutrino mass to be in the range 436.9GeV≤mNR≤996.5GeV for mΩ=200GeV, and 204.5GeV≤mNR≤649.9GeV for mΩ=500GeV. All orange points are out of the 3σ range and, hence, excluded by neutrino oscillation data.5ConclusionsWe have built an SM extension where the tiny masses of the light active neutrinos are generated from an inverse seesaw mechanism at one-loop level. This model adds two scalar singlets and six right handed Majorana neutrinos to the SM particle content. In addition, the SM gauge group is enlarged by a global U(1)X symmetry, which is spontaneously broken down to an Abelian discrete subgroup, Z2⊂U(1)X. The latter is an exact low-energy symmetry guarantying a one-loop realization of the inverse seesaw mechanism and the existence of stable scalar and fermionic dark matter candidates. We focused our study on the case in which the dark matter is the stable Majorana fermion.We have found that our model, besides providing a dynamical origin of the inverse seesaw mechanism, successfully reproduces the measured values of the DM relic density. We have identified the parameter space regions sensitive to (in)direct DM searches. In the case of direct detection searches, we show the parameter space restrictions of the model (imposed by XENON1T and LZ results) and future prospects of XENONnT experiment. Furthermore, we compare the model predictions coming from the Majorana fermion annihilation into bb¯ and W+W− with AMS-02 bounds and CTA projections. We found that our benchmarks could be tested by CTA if the DM has a gNWA profile.We have also analyzed the lepton flavor violating μ→eγ process and provided the model predictions in simplified scenarios that are in agreement with current neutrino oscillation data. Declaration of Competing InterestThe authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Cesar Bonilla reports financial support was provided by National Agency for Research and Development. A.E. Carcamo Hernandez reports financial support was provided by National Agency for Research and Development. Bastian Diaz Saez reports financial support was provided by National Agency for Research and Development. Sergey Kovalenko reports financial support was provided by National Agency for Research and Development. Juan Marchant Gonzalez reports financial support was provided by National Agency for Research and Development. Cesar Bonilla reports financial support was provided by Abdus Salam International Centre for Theoretical Physics.AcknowledgementsThe work of C.B. was supported by FONDECYT grant No. 11201240. A.E.C.H. is supported by ANID-Chile FONDECYT 1210378, ANID PIA/APOYO AFB220004 and Milenio-ANID-ICN2019-044. S.K. is supported by ANID-Chile FONDECYT 1230160 and Milenio-ANID- ICN2019-044. J.M. is supported by ANID Programa de Becas Doctorado Nacional code 21212041. B.D.S. has been founded by ANID Grant No. 74200120. C.B. would like to acknowledge the hospitality and support from the ICTP through the Associates Programme (2023-2028).Appendix ABoundedness conditionsThe boundedness conditions of the scalar potential in Eq. (5) are derived assuming that the quartic terms dominate over at high energies. In order to do so, we define the following bilinears,(A.1)a=|Φ|2;b=|σ|2;c=|η|2, and rewrite the quartic terms of the scalar potential. Then, using the expressions in Eq. (A.1) one gets(A.2)Vq=12(λΦa−λσb)2+12(λΦa−ληc)2+12(λσb−ληc)2+(λ1+λΦλσ)ab+(λ2+λΦλη)ac+(λ3+λσλη)bc−12(λΦa2+ληc2).Following Refs. [49,50] the boundedness conditions of the model turn out to be,(A.3)λΦ≥0;λσ≥0;λη≥0(A.4)λ1+λΦλσ≥0;λ2+λΦλη≥0;λ3+λσλη≥0Appendix BBenchmarks for μ→eγThe cLFV process μ→eγ is computed using Eq. (26) and taking as inputs the model outputs that minimize the χ2 function given by,(B.1)χ2=[Δm212(exp⁡)−Δm212(th)]2σΔm2122+[Δm312(exp⁡)−Δm312(th)]2σΔm3122+∑i<j[sij(exp⁡)−sij(th)]2σsij2+[δCP(exp⁡)−δCP(th)]2σδCP2, where sij≡sin⁡θij (with i,j=1,2,3), δCP is the leptonic CP violating phase, the label (th) is used to identify the model outputs, while the ones with label (exp) correspond to the experimental values, and σa represent the experimental errors. Table 2 shows best fit values and 1σ−3σ intervals reported by neutrino oscillation global fits [51].44Table 2 corresponds to normal neutrino mass ordering. The inverted mass ordering can be consulted in [51]. For other fits of neutrino oscillation parameters we refer the reader to Refs. [52,53].In order to compute all neutrino oscillation parameters we perform a random scan of the free parameters in the lepton sector and make assumptions about the flavor structure of the Dirac mass matrix mD, and the Majorana matrices M and μ in Eq. (13). Using the Casas–Ibarra parametrization [54], the matrix mD in Eq. (14), reads as follows [54–59],(B.2)mD=yνvΦ2=UPMNS(mˆν)1/2Rˆμ−1/2M,where UPMNS≡Uℓ†Uν, mˆν=diag(m1,m2,m3) is the diagonal neutrino mass matrix and Rˆ is a rotation matrix given by,(B.3)Rˆ=(00cos⁡θˆsin⁡θˆ−sin⁡θˆcos⁡θˆ)withθˆ∈[0,2π].The matrices M and μ are assumed to be diagonal,(B.4)M=(M100M2)andμ=(μ100μ2).For the matrix M in Eq. (B.4) we varied M1 within the range 100GeV≤M1≤1TeV and considered M2=10M1, where M1 is the mass of the lightest RH neutrino, M1=mNR. In the case of the μ matrix we used Eq. (15) to compute μ1 and assumed μ2=10μ1. This matrix depends on yN, mΩ, mηR and mηI. 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