Fermion Dark Matter and Radiative Neutrino Masses from Spontaneous Lepton Number Breaking

In this paper, we study the viability of having a fermion Dark Matter particle below the TeV mass scale in connection to the neutrino mass generation mechanism. The simplest realization is achieved within the scotogenic model where neutrino masses are generated at the 1-loop level. Hence, we consider the case where the dark matter particle is the lightest $\mathcal{Z}_2$-odd Majorana fermion running in the neutrino mass loop. We assume that lepton number is broken dynamically due to a lepton number carrier scalar singlet which acquires a non-zero vacuum expectation value. In the present scenario the Dark Matter particles can annihilate via $t$- and $s$-channels. The latter arises from the mixing between the new scalar singlet and the Higgs doublet. We identify three different Dark Matter mass regions below 1 TeV that can account for the right amount of dark matter abundance in agreement with current experimental constraints. We compute the Dark Matter-nucleon spin-independent scattering cross-section and find that the model predicts spin-independent cross-sections"naturally"dwelling below the current limit on direct detection searches of Dark Matter particles reported by XENON1T.

In this paper, we study the viability of having a fermion Dark Matter particle below the TeV mass scale in connection to the neutrino mass generation mechanism. The simplest realization is achieved within the scotogenic model where neutrino masses are generated at the 1-loop level. Hence, we consider the case where the dark matter particle is the lightest Z2-odd Majorana fermion running in the neutrino mass loop. We assume that lepton number is broken dynamically due to a lepton number carrier scalar singlet which acquires a non-zero vacuum expectation value. In the present scenario the Dark Matter particles can annihilate via t-and s-channels. The latter arises from the mixing between the new scalar singlet and the Higgs doublet. We identify three different Dark Matter mass regions below 1 TeV that can account for the right amount of dark matter abundance in agreement with current experimental constraints. We compute the Dark Matter-nucleon spinindependent scattering cross-section and find that the model predicts spin-independent cross-sections "naturally" dwelling below the current limit on direct detection searches of Dark Matter particles reported by XENON1T.

I. INTRODUCTION
The observed fundamental particles as well as their interactions via the strong and electroweak forces are well described under the Standard Model (SM) picture. However, the SM predicts massless neutrinos contradicting neutrino oscillation experiments which indicate that at most one active neutrino can be massless [1][2][3][4]. In addition, so far there is no experimental evidence on the exact mechanism chosen by nature to generate neutrino masses. In this regard, the most popular idea to circumvent this mismatch between the SM and neutrino oscillation data is to assume that neutrinos are Majorana particles and invoke the so-called seesaw mechanism [5][6][7][8][9][10]. Furthermore, the SM does not provide a candidate to account for the dark matter (DM) relic abundance in the Universe. The dark matter constitutes about 80% of the matter content of the Universe and its presence is strongly supported by observational evidence at multiple scales, through gravitational effects, its role in structure formation and influence in the features of the Cosmic Microwave Background (CMB). By looking at the CMB and other observables, the Planck collaboration has put the following limit on the dark matter relic abundance [11], Theoretically, it is very tempting to think that the DM sector and neutrino mass generation mechanism are linked. This connection appears naturally when the neutrino masses are generated at the loop level [12]. In such scenarios, the smallness of the neutrino masses is due to a loop suppression and the additional particles carry a non-trivial charge under an unbroken symmetry which is responsible for DM stability. The simplest idea in this regard is the so-called scotogenic model [12], where the neutrino masses are generated at the 1-loop level. In this model, the DM candidate happens to be the lightest particle running inside the loop with an odd charge under a Z 2 discrete symmetry. It could be either bosonic, a CP-even (odd) scalar, or fermionic, a heavy Majorana particle.
Here we have considered the case where the neutrino mass is generated after the spontaneous breaking of lepton number in the scotogenic model [13] leading to the existence of the Majoron, J, a physical Nambu-Goldstone boson [9,14]. As a consequence, an invisible Higgs decay channel opens up contributing to its total decay width [15][16][17][18]. On top of that, in this model there are two DM annihilation channels when the DM is a Majorana fermion. One is mediated by Z 2 -odd particles (t-channel) [19] and the other one (s-channel) [13,20] coming from the mixing between the scalar singlet and the SM model Higgs after the spontaneous breaking of lepton number and electroweak symmetries. The latter helps to explain DM relic abundance in the Universe for DM masses below the TeV region.
We organized the paper as follows: we introduce the model in the next Section. All the constraints used in our analysis are given in Section III. We describe how the analysis is made and we present our results in Section IV. Finally, we conclude in Section V. Table I. Particle content and charge assignments of the model.

II. THE MODEL
We consider a model where a scalar singlet σ, a SU (2) L scalar doublet η with hypercharge 1/2, and three generations of Majorana fermions N i (with i = 1, 2, 3) are added to Standard Model. It is assumed that the scalar doublet η = (η + , η 0 ) T and the Majorana fermions have an odd charge under an unbroken discrete Z 2 symmetry. This setup can be seen as an extension of the scotogenic model [12]. Hence, the lightest Z 2 -odd particle turns out to be a stable DM candidate. Furthermore, we consider the case where the masses of the heavy Majorana fermions are dynamically generated when the scalar singlet gets a vacuum expectation value σ . This requires that the scalar singlet σ has a non-trivial charge under lepton number and is responsible of the neutrino mass generation after spontaneous symmetry breaking. The particle content and charge assignments of the model are shown in Table I.
Considering the particle content and additional symmetries, the renormalizable SM ⊗ U (1) L ⊗Z 2 invariant Lagrangian for leptons is given by: whereη = iτ 2 η * , L i = ( Li , ν Li ) T with i, j = e, µ and τ . The scalar fields denote the usual SM Higgs doublet and the inert doublet respectively. On the other hand, the scalar potential of the model reads For simplicity, the dimensionless parameters λ i (with i = 1, ..., 8) in the last equation are assumed to be real. The scalar singlet σ and the neutral component of the doublet Φ in eq. (4) can be shifted as follows where v a (with a = σ, Φ) are the vacuum expectation values and v Φ = 246 GeV; R j and I j (with j = 1, 2) represent the CP-even and CP-odd parts of the fields.

A. Mass spectrum
Computing the second derivatives of the scalar potential in eq. (4) and evaluating them at the minimum of the potential, one gets the CP-even and CP-odd mass matrices, M 2 R and M 2 I respectively. There are two CP-odd massless fields, one of them corresponds to the longitudinal component of the Z boson and the other one is a physical Nambu-Goldstone boson resulting from the spontaneous breaking of the U (1) L symmetry, the Majoron J [9,14]. Hence, For the CP-even part, one can define the two mass eigenstates h i through the rotation matrix O R as follows, The angle α is interpreted as the doublet-singlet mixing angle. Then, we have that where M 2 R is the squared CP-even mass matrix whose eigenvalues are given by, where the "−" ("+") sign corresponds to h 1 (h 2 ). Notice that one of these scalar has to be associated to the SM Higgs boson with a 125.09 GeV mass [21]. Furthermore, the masses of the CP-even and CP-odd components of the inert doublet, η, turn out to be The mass of the charged scalar field is given by, Notice that the masses of the CP-even and CP-odd fields satisfy the relation As it was mentioned before, neutrino masses are generated dynamically like the rest of the SM fermions. That is, the Majorana masses of N i as well as the light neutrinos arise after the spontaneous breaking of the global U (1) L symmetry. From eq. (2) follows that the mass matrix for the N i fields is given by The one-loop neutrino mass generation is depicted in Fig. 1. After the electroweak symmetry breaking one gets that the light neutrino mass matrix is given by the following expression [12,13] (

III. SUMMARY OF CONSTRAINTS
Before analyzing the sensitivities of the experimental searches for WIMPs, we first discuss the theoretical and experimental restrictions that are implemented in our analysis.

A. Boundedness conditions
In order to ensure that the theory is perturbative, the quartic couplings in the scalar potential, eq. (4), as well as the Yukawa couplings in eq. (2) are limited to be, Furthermore, the consistency requirements of the scalar potential demand that the dimensionless parameters in eq. (4) have to fulfill the following conditions [22], From the last relations it is guaranteed that the scalar potential is bounded from below.

B. Searches of new physics
As we described in the previous section, there are 6 physical scalars in the model: three CP-even h i (i = 1, 2) and η R ; two CP-odd η I and the Majoron J; and a charged scalar η ± . Therefore, one has to impose the constraints on the scalar masses coming from the LEP results [23] and the latest reports from the LHC on the Higgs properties [24]. Notice that the invisible Higgs decay channel is always present, namely the Higgs decay into Majorons h → JJ, where in our case the SM Higgs h will be identified with either h 1 or h 2 . Then, this decay mode coexists with the Higgs decay into the fermion dark matter, N 1 , when it is kinematically allowed, i.e. h → N 1 N 1 when m N1 < m h /2. Therefore, we consider [24] B inv ≡ BR(h → invisible) < 0.28 at 95% C.L.
On the other hand, the LEP collaboration studies on the invisible decays of W ± and Z 0 gauge bosons [23] provide bounds on the masses of the inert scalars η R (η I ) and η ± . From these searches, the following conditions must be fulfilled [25] The LEP reports also established disallowed mass regions for the mass splitting given by, and m η ± > 80 GeV.
Finally, it is important to mention that the oblique parameters S, T and U are also sensitive to new physics [26,27]. Then, it has to be considered that values of these parameters in the model lie within the following regions [24]. S = 0.02 ± 0.10, T = 0.07 ± 0.12 and U = 0.0 ± 0.09.

C. Dark matter searches
The abundance of DM in the Universe, given in terms of the cosmological abundance parameter, eq. (1), provides restrictions on the parameter space of DM models. Furthermore, there exist constraints coming from searches of DM by experiments using (in)direct detection techniques. The direct dark matter detection experiments have set bounds, for DM masses above 6 GeV, on the dark matter-nucleon spin-independent scattering cross section. The most stringent bounds are set by the XENON1T experiment, that is σ SI 4.1 × 10 −47 cm 2 for a DM mass of 30 GeV at 90% C.L [28]. On the other hand, the astronomical gamma ray observations constrain the velocity averaged cross section of dark matter annihilation into gamma rays σv γ . The Fermi-LAT satellite has performed this indirect DM search and constraint the cross section to be σv γ 10 −29 cm 3 s −1 [29].

D. Neutrino oscillation parameters
The neutrino masses are obtained after diagonalization of the mass matrix given in eq. (13). The relation between m ν and the diagonal mass matrix is given by, The matrix U L is defined as the lepton mixing matrix, m νi are the neutrino masses, θ ij are the mixing angles and δ CP corresponds to the Dirac CP-violating phase. U ν and U are the matrices that diagonalize the neutral and charged mass matrices, M ν M † ν and M M † , respectively. The lepton mixing angles θ ij are determined by neutrino oscillation experiments. From global fits of neutrino oscillation parameters [4] (for other fits of neutrino oscillation parameters we refer the reader to [30,31]) the best fit values and the 1σ intervals for a normal neutrino mass ordering (NO) are

IV. NUMERICAL ANALYSIS
We have mentioned that the nature of the DM candidate in this model could be either fermionic or scalar. This is the lightest particle with odd charge under the Z 2 symmetry and running in the neutrino mass generation loop as shown in Fig. 1. In our study we will focus in the case in which the DM is the lightest Majorana particle 1 , i.e. N 1 . Therefore, in this case the DM annihilates via the t-and s-channel In Fig. 2 . The former is mediated by a Majorana fermion N i and by the inert scalars. It has been shown that the bounds on lepton flavor violation (LFV) processes, e.g. µ → eγ, demand small neutrino Yukawas (namely, Y ν 1) and hence the t-channel mediated by the inert scalars becomes suppressed inducing DM overabundance [32,33]. However, we show that we can keep suppressed the inert scalar mediated t-channel and thanks to the s-channel mediated by the singlet σ it is possible to account for the right amount of DM relic abundance. As a result, it is crucial to have a non-vanishing mixing angle between CP-even parts of the Higgs doublet Φ and the iso-singlet σ, eq. (7), and in agreement with current experimental data.
For the numerical analysis we have used the MicrOMEGAS [36] and performed a scan over all free parameters of the model. For the dimensionless parameters in the scalar sector we took the following intervals: and λ 1 being determined by the SM Higgs mass. We are taking h 1 as the SM Higgs then m h1 = 125 GeV. On the other hand, we are varying the mass of h 2 within the range m h2 ∈ [20,2000] GeV.
Notice that we are taking masses below 125 GeV which is in perfect agreement with both the LHC and LEP constraints as long as the doublet-singlet mixing given by sin α in eq. (7) is less than 20% [18]. For the masses of the inert scalars we considered the following ranges, and mass of the CP-odd part η I is determined by using the relation For the lepton number breaking scale, namely the singlet's vev v σ , we have used v σ ∈ [500, 10000] GeV. Bear in mind that this vev provides the mass of the heavy Majorana fermions, N i , whose masses (taken to be diagonal) are varied in the following ranges, m N1 ∈ [8, 1000] GeV and m N2,3 ∈ [100, 5000] GeV. (24) Since N 1 2 is the DM candidate of the theory we have to impose m N1 < m N (2,3) < m η (R,I,±) 3 . The above considerations are made in such a way that they all satisfy the theoretical and experimental constraints described in Sec. III. We computed the value of the S and T parameters using the expressions given in [37,38], taking U = 0 [24] and keeping those solutions that are in agreement with the bounds given in eq. (19). It is worth to mention that we considered only S and T within the 90% level shown in Fig. (10.6) from reference [24]. In addition, we calculated the light neutrino masses feeding the neutrino mass expression given in eq. (13) and assumed normal ordering for neutrino masses 4 . Then, we took as valid only the points that satisfy the best fit values from the global fit of neutrino oscillation parameters 5 [4].
Our last requirement is that the annihilation cross section of the fermion DM candidate into Majorons (see Fig. 2) is subdominant at the moment of the freeze-out in order to guarantee detectability in DM direct detection experiments and to avoid direct detection cross sections in regions far below the neutrino floor.

A. Viable dark matter mass regions
Following the considerations that we stated previously, we show in Fig. 3 the nucleon-dark matter spin-independent cross section σ SI as a function of the fermion DM mass, m N1 . From the numerical analysis we have found three different viable mass regions for a fermion DM candidate within the model. We refer as viable to those solutions that fulfill the theoretical and experimental bounds given in Section III. These are: • the low mass region, with an approximate DM mass range 8 GeV m N1 20 GeV and bb as dominant annihilation channel; • the resonant region, where m N1 m h /2 (with m h = 125.09 GeV); and The latest bound on direct dark matter detection is set by the XENON1T experiment [28] (top shaded area). The dashed lines represent the expected sensitivities in forthcoming experimental searches such as XENONnT [39], LUX-ZEPLIN [40], DarkSide-20k [41], DARWIN [42] and PandaX-4T [43].
• the high mass region, for DM masses above 80 GeV where the fermion DM annihilates efficiently into the gauge bosons, i.e.
In all these domains, the DM annihilation into Majorons (N 1 N 1 → JJ) at the moment of the freezeout is always below 10%. The latest bound coming from direct detection searches of dark matter particles is set the XENON1T experiment [28] and is defined by the top shaded area in Fig. 3. The dark red points showed in the m N1 −σ SI plane account for 100% of the DM relic abundance while the solutions in purple and pink correspond only to a fraction of the DM abundance. Notice that in the high mass region it is most likely a fermion DM with a mass around 500 GeV accounting for the whole amount of DM in the Universe. There are few points around m N1 ∼ 10 GeV and m N1 ∼ 100 GeV that could not be distinguished from the neutrino floor background (bottom shaded area). Fig. 3 also displays the future sensitivities for dark matter searches in direct detection experiments such as XENONnT [39], LUX-ZEPLIN [40], DarkSide-20k [41], DARWIN [42], and PandaX-4T [43]. For completeness we provide three benchmarks in Appendix A within each mass neighborhood and their corresponding outputs. Fig. 4 shows the predictions for the velocity averaged cross section of dark matter annihilation into gamma rays σv γ as function of dark matter mass m N1 . We have found that the annihilation cross section of dark matter into gamma rays is up to two orders magnitude below the limit set by Fermi-LAT satellite results [29] on the indirect DM search (cyan dashed line in Fig. 4). This is the case for fermion DM with a mass inside the low mass region. One can see that there are solutions in the high mass region that are ruled out by observations. In particular, the indirect search of DM excludes some points where the fermion DM represent only a fraction of the DM relic abundance. As before, all points satisfy the theoretical and experimental constraints listed in Section III and the dark red points correspond to the solutions that account for the whole amount of DM in the Universe. The lighter (purple and pink) colors would require the existence of other DM candidates to explain observations, eq. (1).

V. CONCLUSIONS
In this work we have studied the scotogenic model with spontaneous breaking of lepton number. We have shown that it is possible to account for the whole amount of DM relic density thanks to the scalar singlet used to break lepton number which mixes with the CP-even part of the SM Higgs doublet. Notice that this DM annihilation portal is absent in the simplest version of the scotogenic model, where lepton number is explicitly broken by the Majorana mass term, N i N i . In our analysis the constraints coming from LFV processes are suppressed because the neutrino Yukawas are kept small. We present a numerical analysis of the parameter space of the model and the predictions for the nucleon-dark matter spin-independent cross section σ SI . We show that there are three different DM mass regions that can explain the DM relic abundance, satisfy current experimental constraints as well as the limits on σ SI reported XENON1T. We also included the future sensitivities of experiments that are devoted to search for the direct dark matter detection.

ACKNOWLEDGMENTS
The work of C. Here we present three benchmarks (BM1, BM2, BM3) corresponding to the different mass regions described in Section IV A where the fermion DM satisfy all experimental and theoretical constraints summarized in Section III, see Tables II and III. Additionally, these representative points are such that the DM particle N 1 constitute 100% of the relic abundance in the Universe, see Table IV.
The values for the dimesionless parameters in the Lagrangian as well as the dimensionful paramaters in the scalar potential are shown in Table II. Notice that the neutrino Yukawas Y ν i (with i = 1, 2, 3) are small and as a result LFV processes are suppressed. We include as example in Table III the branching fractions of µ → eγ for each benchmark.
Finally, Table IV shows the main DM annihilation channels in the model and prediction in the DM sector.