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Connecting neutrino physics with dark matter

The origin of neutrino masses and the nature of dark matter are two of the most pressing open questions of the modern astro-particle physics. We consider here the possibility that these two problems are related, and review some theoretical scenarios which offer common solutions. A simple possibility is that the dark matter particle emerges in minimal realizations of the see-saw mechanism, like in the majoron and sterile neutrino scenarios. We present the theoretical motivation for both models and discuss their phenomenology, confronting the predictions of these scenarios with cosmological and astrophysical observations. Finally, we discuss the possibility that the stability of dark matter originates from a flavour symmetry of the leptonic sector. We review a proposal based on an A_4 flavour symmetry.


Introduction
It is by now a well-established fact that a wide variety of cosmological observations provide strong support for the ΛCDM model, that can thus be regarded as the Standard Cosmological Model (SCM). These observations include measurements of the temperature and polarization anisotropies of the cosmic microwave background (CMB) [1,2,3,4,5,6,7,8], of the distribution of large scale structures [9,10,11], of the abundance of light elements [12,13,14], of the present value of the Hubble constant H 0 [15,16], of the magnitude-redshift relationship for Type Ia supernovae [17,18]. According to the SCM, the Universe is spatially flat and its present energy density is dominated by non-relativistic matter (roughly 30% of the total), in the form of baryons and cold dark matter (CDM), and dark energy (the remaining 70%), in the form of a cosmological constant, that is responsible for the present accelerated expansion. Photons and neutrinos are subdominant today but their energy density drove the expansion of the Universe during the early radiation-dominated phase. The structures that we observe today have grown from primordial adiabatic, nearly scale-invariant fluctuations generated after a phase of inflationary expansion, in the simplest models driven by the energy density of a scalar field. The recent observations of the primordial B-modes of the CMB polarization made by the Bicep2 telescope [19] represent, if confirmed, a strong evidence in favour of this inflationary paradigm.
However, in spite of its phenomenological success, one of the striking features of the SCM, from a theoretical point of view, is that nearly 95% of the total matter-energy content of the Universe has to be explained by some physics beyond the Standard Model (SM) of elementary particles, if one assumes that gravity is correctly described by General Relativity. In particular, we know that only a part of the total matter density can be provided by SM particles. The density of baryons can be inferred in several ways, mainly through the knowledge of light elements abundances (that are extremely sensitive to the baryon-to-photon ratio) and from the CMB anisotropies (since the presence of baryons induces a peculiar alternating pattern in the peaks of the power spectrum). On the other hand the total density of matter affects, through gravity, both the background expansion and the evolution of perturbations, and thus can be constrained, among others, by measuring the clustering properties of galaxies, or again by CMB observations. All the observational evidence points towards a coherent picture where baryons constitute roughly 20% of the total cosmological matter content, while the remaining 80% is provided by an electromagnetically neutral component dubbed dark matter. Since the gravitational evidence for dark matter comes from observations at different scales, it is difficult to explain these anomalies in terms of a modified theory of gravity.
Dark matter also drives the process of formation of cosmological structures, as it creates the potential wells were luminous matter -i.e., baryons -fall once they are free from the support of photon pressure. In fact, the observed clustering properties of galaxies also constrain the velocity dispersion of the dark matter component, since this defines the free-streaming length below which perturbations in the dark matter density are erased and clustering is suppressed. This rules out "hot" dark matter candidates (HDM), like the SM neutrinos themselves, whose large velocity dispersion results in a cut off in the matter power spectrum well above the galactic scale (i.e., well above a few comoving Mpcs). Thus dark matter has to be "cold" (CDM) or "warm" (WDM), namely, with a damping length below or around the galactic scale, respectively. The predictions of the CDM and WDM scenarios at the largest scales are identical, however WDM has been often invoked as a possible solution for the shortcomings of CDM at small scales, like those related to the abundance of dwarf satellites and to the inner density profile of galaxies. On the other hand, complex baryonic physics could also be responsible for this, and this issue is still matter of intense debate in the structure formation community.
The exact nature of dark matter is still a mystery to date. Many candidates have been proposed for the role of dark matter, with different particle physics motivation. Compiling a comprehensive list would be impossible, however popular examples include the supersymmetric particles, Kaluza-Klein particles, the axion. Unfortunately, the search for these or any other candidate has been unfruitful so far, and all theoretical possibilities are still open. In this review we will focus on the theoretically appealing possibility that dark matter is somewhat related to neutrinos, and in particular to the mechanism of neutrino mass generation, which is another open question of theoretical physics.
We know from the observation of neutrino oscillations that neutrinos have masses [20,21]; however their origin and nature (Dirac or Majorana) is still unknown. Moreover, the smallness of neutrino masses with respect to those of the SM charged fermions, remains a puzzle. An elegant solution to these issues is provided by the see-saw mechanism [22,23,24,25,26,27], described in Sec. 2, that allows to generate neutrino masses that naturally lie much below the electroweak scale. In this scenario neutrinos are Majorana particles, implying that the lepton number is violated. Therefore, searches of lepton-number violating processes, in particular neutrinoless double β decay, are a decisive test for this hypothesis.
The general idea behind the see-saw mechanism can be embedded in many distinct scenarios, that often provide in turn viable dark matter candidates. Here we describe two simple and direct connections between the neutrino mass generation and dark matter. In Sec. 3 we focus on the possibility that dark matter is the Goldstone boson associated to the spontaneous breaking of the ungauged lepton number. This particle, called Majoron, can acquire a mass through quantum gravity effects that explicitly break global symmetries, and thus play the role of dark matter. In the case of sterile neutrinos dark matter, that we review in Sec. 4, three right-handed singlets are added to the particle content of the SM; dark matter is made from the lightest sterile state, while the two heaviest states generate the active neutrino masses through the see-saw mechanism. In these two scenarios, the mass of the dark matter particle is not predicted by the theory; however, the keV mass range is favoured by observations, especially in the case of the sterile neutrino.
The majoron and sterile neutrino scenarios that we are considering, are both based on a minimal implementation of the seesaw mechanism. Dark matter candidates can also arise when the seesaw is embedded in more complex models, for instance those in which neutrino masses are generated radiatively (e.g. [28,29,30,31,32]), in Rparity violating SUSY models [33,34,35], or in different types of low-scale seesaw scenarios (e.g. [36,37,38,39]). The resulting candidates can lie at different mass scales and therefore exhibit a completely distinct phenomenology with respect to the sterile neutrino and the majoron.
We give a concrete example, in Sec. 5, considering a scenario where the dark matter candidate is not only related to the generation of neutrino masses but also to the presence of a discrete flavour symmetry, which at the same time approximately accounts for the observed pattern of neutrino oscillations and stabilizes the dark matter particles.

Generation of neutrino masses
The neutrino masses can be generated at non-renormalizable level by the effective dimension-five operator [40]: where L i denotes an electroweak doublet (i, j = e, µ, τ ) and H is the SM higgs doublet. This operator violates lepton number by two units and, after the electroweak symmetry is broken and the higgs acquires a vacuum expectation value v 2 ≡ H , generates Majorana neutrino masses m ν ∝ v 2 2 . Assuming that O 5 is generated at tree level by the exchange of some heavy degree of freedom with mass M , one has where λ is a dimensionless coupling, and the last equality follows from the fact that v 2 sets the mass scale m D of the charged fermions. The mass of the heavy particles corresponds to the energy scale at which lepton number is violated. This can be pushed arbitrarily high, in order to obtain m ν m D and thus explain in a natural way the smallness of neutrino masses with respect to the other SM fermions. If lepton number violation is related to some gravitational Planck-scale effect, M M Pl , but the resulting neutrino mass (assuming λ 1) would be too small to explain the observed mass differences. This suggests that neutrino masses are generated by new physics below the Planck scale.

Seesaw models
In seesaw models [22,23,24,25,26,27], the effective operator (1) is generated at tree level by the exchange of heavy particles. This can be achieved in different ways. A first possibility is to add SU(2) L ⊗ U(1) Y singlet right-handed fermions ("sterile neutrinos") ν R to the particle content of the SM ‡. This allows to introduce the following term in the Yukawa lagrangian: that, after electroweak symmetry breaking, generates a Dirac mass term for neutrinos, with m D = Y 2 v 2 . Moreover, because the ν R are gauge singlets, we can also add to the lagrangian a gauge-invariant bare mass term for the right-handed neutrinos: This term can also be generated by interactions with a singlet scalar fiels, as in majoron models (see below). A similar bare mass term for the active neutrinos is forbidden as it would spoil the gauge invariance of the SM. Thus, the neutrino mass matrix in the (ν, ν R c ) basis reads: If the Majorana mass M R (that is related to some physics beyond the SM) is much larger than the Dirac mass m D ≡ Y 2 v 2 , then the mass eigenvalues are approximately M R and m 2 D /M R M R . This is the type-I seesaw [22,23,24,26]. In the type-II seesaw [26,41,25,27,42], instead, a scalar Higgs triplet ∆ is added to the theory, schematically coupling to the ordinary neutrinos through Once the triplet acquires a vacuum expectation value v 3 ≡ ∆ , it gives rise to a Majorana mass term for active neutrinos, with M L = Y 3 v 3 . The triplet vev induces a change in the electroweak ρ parameter, that is experimentally constrained to lie below a few GeVs. This requires in turn that v 3 v 2 . It is worth stressing that one can consider a more general seesaw model where both the right-handed singlets and the scalar triplet are present, so that the neutrino mass matrix has the full seesaw structure: with M L m D M R . We also note that the majorana mass terms violate lepton number conservation.
Another possibility to generate the effective operator 1 is through the tree-level exchange of SU(2) L fermion triplets Σ, as in the type-III seesaw [43].

The model
The basic idea behind the majoron model, originally proposed by Chikashige, Mohapatra and Peccei [44] in 1980, is that the lepton number symmetry of the standard model of particle physics is spontaneously broken globally. Indeed, as noted above, if neutrinos are massive majorana particles, lepton number is necessarily broken. If lepton number is a global symmetry, then after its breakdown a massless Nambu-Goldstone boson -the majoron -is generated.
In the simplest version of the majoron model, three singlet neutrinos ν R are added to the SM, allowing to generate a Dirac mass for the neutrinos through a term of the form (3). A complex scalar higgs singlet σ with lepton number 2 is also introduced, coupling to the singlet neutrinos through: As we shall see, σ is the parent field of the majoron. For the moment, we just note that when σ acquires a vacuum expectation value v 1 ≡ σ , lepton number is broken and a Majorana mass term for the singlet neutrinos is generated, like the one in Eq.
Thus, the neutrino mass matrix in the (ν, ν R c ) basis has the type-I seesaw structure: where the condition v 2 v 1 is required in order to ensure the smallness of neutrino masses. In the simplest implementation of the model, the absence of a higgs triplet implies that there is no Majorana mass term for the ordinary neutrinos.
Once σ has acquired a vacuum expectation value (and lepton number is broken), we can write where ρ and J are, respectively, a massive and massless boson field with zero vacuum expectation values. The field J is the majoron, the Goldstone boson associated to spontaneous breaking of lepton number. The model was generalized shortly after by Schechter & Valle [42] by allowing for the presence of all Higgs multiplets -singlet, doublet and triplet. Thus, with respect to the model sketched above, a Higgs triplet ∆ and the corresponding term (6) in the Yukawa lagrangian are also introduced. Once the triplet acquires a vacuum expectation value v 3 , a Majorana mass term for the active neutrinos is generated. The full neutrino mass matrix has thus the general seesaw structure: where this time, in writing the matrix, we have also taken into account that the Yukawa couplings are actually 3 × 3 matrices. The three vevs satisfy v 1 v 2 v 3 as well as a vev seesaw relation v 1 v 3 v 2 2 . The resulting light neutrino mass is The properties of the majoron in this more general scenario can be derived using the invariance of the potential under lepton number and weak hypercharge symmetries [42].
In particular, the majoron is given, apart from a normalization factor, by the following combination of the Higgs fields: where Φ 0 and ∆ 0 are the neutral components of the doublet and triplet, respectively. It is clear that the pure singlet model can be recovered by setting v 3 = 0. The hierarchy of the vevs anyway implies that the majoron has to be dominantly singlet. Again using the symmetry properties one can show that the majoron couples to the light neutrinos proportional to their mass and inversely proportional to v 1 [42] . The majoron, being a Goldstone boson, is massless. However, it has been conjectured that if gravity violates global symmetries (see e.g. Ref. [45]), then the majoron may acquire a mass through nonpertubative gravitational effects [46]. The value of its mass m J cannot be unambiguously calculated, as it depends on the details of how this explicit breaking of global symmetries occurs, which are rather unknown.
A massive majoron can decay at tree level to a pair of light neutrinos with a rate given by [46] Moreover, the majoron also possesses a subleading decay mode to two photons, induced at the loop level through its coupling to the charged fermions. The decay rate for this process is (in the limit m J m f ) [47] where N f , Q f , T f 3 and m f denote respectively the color factor, electric charge, weak isospin and mass of the SM electrically charged fermions f . The radiative decay rate is proportional to the triplet vev v 3 and is thus a peculiar feature of the more general seesaw model.

Majoron cosmology and astrophysics
A massive majoron has many astrophysical and cosmological implications that were first explored in Refs. [46,48,49,50]. In particular, in Refs. [48,50] it was suggested that the majoron could play be the role of the dark matter particle. The majoron could be produced in the early Universe either thermally or through some non thermal mechanism, like for example, a phase transition [50] or the evaporation of majoron strings [48].
In order for the majoron to be the dark matter, however, its lifetime must be large enough for it to be stable on cosmological timescales. CMB data have been used to constrain the majoron decay rate to neutrinos first in Ref. [51] and more recently in Ref. [52]; in particular, an analysis of the recent WMAP 9-year data [53] yields [52] Γ J→νν ≤ 6.4 × 10 −19 s −1 (95% C.L.) , corresponding to τ J ≥ 50 Gyr. From the same analysis, it is also found that the majoron energy density is: We stress that CMB observations cannot access directly (i.e., in a model-independent way) the exact value of the dark matter particle mass, unless this is so light to be hot, which is however forbidden by structure formation. Assuming that cosmological majorons have a thermal spectrum, and that they decoupled early enough, the inferred value of the majoron energy density translates into: The keV mass range is, in principle, very interesting as it would make the majoron a WDM candidate and could help to solve the problems of the standard CDM paradigm at the galactic scales; however, a mass in the sub-keV range is potentially problematic as well as it would probably lead, in the case of a thermal spectrum, to an excessive cancellation of small-scale perturbations. However, larger majoron masses are possible in the case of non-thermal production mechanisms, or if majorons do not make up for all the DM content of the Universe.  The bound (14) on the decay rate can be expressed in terms of the underlying particle physics parameter as This limit is shown in Fig. 1. For masses in the range 100 eV m J 100 GeV, the sub-leading majoron decay to two photons of Eq. (15), arising in the more general see-saw scenario, can also be constrained through a number of X-and γ-ray astrophysical observations [47,52]. This decay mode provide an interesting route to probe majoron dark matter. In particular, the J → γγ constraints from line emission searches already exclude part of the parameter space for models with v 3 larger than a few MeVs [52]. A summary of current astrophysical constraints and theoretical predictions is shown in Fig. 2.

Sterile neutrino as dark matter
As discussed in Section 2 sterile neutrinos are responsable for the generation of neutrino masses in the context of the type-I seesaw mechanism. These particles interact with matter via the mass mixing with the active neutrino states, which can be parametrized in terms of the active-sterile mixing angle θ 1: where the sum runs over all the sterile neutrino species. The mixing angle depends on the number of sterile neutrino states present in the model and the relative size and texture of the Dirac and Majorana mass matrices. In general terms, this quantity comes from the diagonalization mass matrix described in Eq. (5). The size of the mixing, usually expressed as sin 2 (2θ), and the sterile neutrino mass m s characterize the phenomenology of the sterile neutrino, i.e. its lifetime, decay modes and the production rate in the early Universe. If the mixing angle is sufficiently small, sterile neutrinos are long lived and they can act as decaying dark matter candidates. In order to account for the neutrino masses and explain dark matter at least three sterile neutrinos should be introduced. In the minimal realization, known as Minimal Neutrino Standard Model (νMSM) [54,55], the lightest sterile state acts as a dark matter candidate while neutrino masses and oscillations mainly depend on the other two heavier states. Interestingly, the decays of heavy sterile neutrinos in the early Universe could also explain the observed baryon-antibaryon asymmetry [56].

Lifetime and decay modes
Sterile neutrino dark matter decays into SM particles with the diagrams shown in Fig. 3. The lifetime can be expressed as: Small mixing angles are needed in order to push the lifetime beyond the age of the Universe. The loop-induced decays into γν induce an almost monochromatic photon emission which can be searched for with astrophysical observations. This decay mode provides therefore a way to test the sterile neutrino scenario, in analogy with the γγ signal for the majoron. For sterile neutrino masses in the keV range, the γν line falls in the X-ray band. Searches of these signals have been performed by different X-ray observatories, focussing on a large number of targets. The corresponding constraints on the sterile neutrino parameter space are summarized in Fig. 4.
Recently it has been reported an identified line signal at 3.5 keV [57], that can be interpretated as monochromatic emission from a decaying dark matter candidate with a mass of 7.1 keV. The analysis has been performed on stacked observations of 73 galaxy clusters with the XMM-Newton telescope, and the Perseus cluster with the Chandra Xray Space Telescope. A consistent signal has been identified by another group analyzing  . The interpretation of the 3.5 keV X-ray line in terms of sterile neutrinos is shown in brown [57]. In the red regions sterile neutrinos are either under-abundant or over-abundant [58]. The Green region corresponds to the Tremaine-Gunn bound [59]. The pink one shows the constraint obtained from observations of the stellar velocity dispersion in the Fornax dwarf galaxy [60]. The rest of the regions are based on analysis of different celestial bodies in X-rays: Milky Way and Ursa minor (dark blue) [61], Andromeda (M31) [62], Coma an Virgo clusters [63], cosmic X-ray background assuming Milky Way halo models (light blue including hatching region [64]), Milky Way SPI-INTEGRAL exclusion [65], Milky Way HEAO-I exclusion [66], and the diffuse X-ray background [64].
the X-ray spectra of the Andromeda galaxy and of the Perseus cluster [67]. Further observations and analyses are necessary to firmly confirm these results and to reject any standard astrophysical interpretation of this emission. Interestingly, in case this signal would be confirmed, sterile neutrino could offer a viable explanation, as shown in Fig. 4.

Production mechanism in the Early Universe
Sterile neutrinos are produced in the early Universe through oscillations with the active neutrino states, the so-called Dodelson-Widrow mechanism [68]. As discussed in the previous section, astrophysical observations and the requirement of a sufficient long lifetime force the active-sterile mixing angle to be small. For these small mixings, sterile neutrino dark matter does not reach a thermal equilibrium with the plasma in the Universe. These particles are produced relativistically and they become nonrelativistic during the radiation-dominated epoch, behaving therefore as WDM. Their free-streaming length determines the scale below which the matter powerspectrum is damped compared to the CDM scenario. This suppression can be probed using the Lyman-α observations [69]. Bounds on the free-streaming length from Lymanα can be traduced in lower limits on the mass of the sterile neutrino of the order of few keV [70,71]. Further lower bounds on the sterile neutrino mass are obtained from the phase-space dark matter density of dwarf-galaxies (the Tremaine-Gunn bound [59]) and their density profiles, e.g. [60]. Assuming the standard Dodelson-Widrow scenario, these constraints, combined with the bounds on X-rays observations, are in tension with the values of sterile neutrino masses and mixing angles for which the correct dark matter abundance is obtained [72].
However, the sterile neutrino production is modified in presence of a lepton asymmetry in the Universe significantly larger than the baryonic one. In this case, sterile neutrinos are produced via resonant neutrino oscillations, a mechanism known as resonant production. In this scenario sterile neutrinos can have the correct abundance in a much larger portion of the parameter space, namely the region between the red areas in Fig. 4. In a specific point the correct sterile neutrino abundance is obtained for a proper value of the lepton asymmetry. Moreover, in the case of resonant production the sterile neutrino velocity distribution is modified with respect to the non-resonant scenario and the bounds from Lyman-α observations become less stringent [73].
Summarizing, sterile neutrinos in the keV range are viable decaying dark matter candidates. The constraints on its parameters discussed in these sections are shown in Fig. 4.

Flavour symmetries and dark matter
A fundamental property of dark matter is its stability over cosmological times. Naively, its lifetime should be larger than the age of the Universe, ∼ 10 18 s. Astrophysical observations pushes even further this bound. For instance, the observed photon background and measurements of cosmic-rays fluxes constrain the lifetime of dark matter to be larger than 10 26 − 10 27 s for dark matter masses in the range ∼ 10 − 1000 GeV (see [74] and references therein).
This apparent stability could be the result of the the smallness of the couplings involved in the dark matter decays. This is the case, for example, of the Majoron and sterile neutrino, discussed in the previous sections, or the gravitino in R-parity violating SUSY models [75]. Another option is that dark matter is stabilized by some symmetry, the most simple solution is a Z 2 parity. In many dark matter models the stabilization symmetry is imposed by hand, without any explanation for its origin. However, its existence could be intimately related to the structure of the theory and it could also play a central role in determining the interactions of dark matter. Therefore it would be theoretically more appealing to motivate the presence of such a symmetry in dark matter models. Different mechanisms have been proposed for this purpose, for instance gauge symmetries [76,77,78] (e.g. the R-parity in SUSY models could arise from a U (1) B−L [79]), global symmetries and accidental symmetries [80,81] (see [82] for a review on the subject). Interestingly, the stability of dark matter could also originate from a discrete flavor symmetry §.
Non abelian discrete flavor symmetries have been extensively studied in order to explain the pattern of neutrino mixing [86,87]. These symmetries and their breaking patterns define the structure of the leptons (and possibly quark) mass matrix and in turns their mixing angles. In reference [88] it has been proposed that the breaking of a flavor symmetry could leave a remnant symmetry stabilizying the dark matter. A concrete example is presented, based on a A 4 flavor symmetry. A 4 is the group of even permutations of four objects and contains 12 elements. The generators of the group, in the 3 dimensional unitarity representations are: In this model the scalar potential of the SM is supplemented by three SU (2) scalar doublets, η = (η 1 , η 2 , η 3 ) which transform as an A 4 triplet, while the SM scalar SU (2) doublet, H, is an A 4 singlet. The minimization of the scalar potential gives the symmetry breaking pattern H = v H and η = (v η , 0, 0). This breaks A 4 into a residual Z 2 symmetry generated by S and acting on an A 4 triplet Ψ as Since in this scenario the SM particles are even under this Z 2 parity, the lightest Z 2 odd particle of the model is automatically stable and can play the role of dark matter. In this scheme neutrino masses are generated through a type-I seesaw mechanism introducing four heavy right handed neutrinos. With a suitable assignement of the A 4 charges of the RH neutrinos, the lepton doublets and the RH charged leptons, this model can accomodate the solar and atmospheric mixing angles and mass differences . During the last years, the connection between a possible flavour structure of the leptonic sector and the stability of the dark matter, has been investigated in other models, considering § The dark matter stability can be connected to flavor physics also in the context of Minimal Flavor Violation [83,84,85].
The original model was constructed in such a way to give a vanishing reactor mixing angle θ 13 . While this value was acceptable when the model was proposed, a non-vanishing θ 13 has now been measured [89,90,91,92,93]. Recently, it has been proposed that including radiative corrections, the model can reproduce the correct value of θ 13 [94].
also different flavor symmetries [95,96]. In short, models based on flavor symmetries can lead to different patterns of the neutrino mass matrix and give a remnant or accidental Z 2 stabilizying the dark matter (see also [97,98,99,100,101,102,103]).
The precise properties of the dark matter candidate depends on the details of the model under consideration. In the examples presented in refs. [88,95,96,97], the dark matter candidate is identified with a neutral Z 2 -odd scalar particle, more precisely the lightest mass eigenstate of the η 2 − η 3 scalar system presented above. Dark matter can communicate with the SM via "Higgs portal" interactions, induced by terms in the scalar potential like for instance η † ηH † H. In presence of "weak-scale" couplings, this dark matter candidate has a mass close to electro-weak scale and acts as a Weakly Interacting Massive Particle (WIMP). The correct cosmological relic abundance can be achieved through the standard thermal freeze-out mechanism. As typical in WIMPs models, there exist multiple strategies to detect these particles, namely searches at colliders, with underground dark matter detectors (direct searches) and with astrophysical observations (indirect searches). In particular, the phenomenology of the dark matter candidate proposed in ref. [88] has been extensively studied in [104]. Direct detection constraints exclude large regions of the parameter space while current indirect detection searches are sensitive to low dark matter masses.

Conclusions
We have reviewed the possibility that dark matter is somewhat related to the origin of neutrino masses. In particular, we have shown how see-saw models, already in their minimal implementation, could at the same time generate neutrino masses and provide viable dark matter candidates. Two simple and theoretically motivated examples are the sterile neutrino and the majoron. In the sterile neutrino scenario, the dark matter is provided by the lightest of the singlet right-handed neutral fermions, which are fundamental ingredients of the seesaw mechanism. In the majoron scenario, on the other hand, the dark matter is associated to the scalar field responsible for the dynamical generation of the majorana mass term of the right-handed neutrinos.
We have shown that both the sterile neutrino and the majoron can act, in some regions of the parameter space of the respective models, as decaying dark matter candidates. Moreover, requiring that their cosmological abundance matches the observed dark matter density, as inferred by CMB and other observations, roughly single out the keV mass range for both particles.
Interestingly enough, late decays of these candidates in monochromatic keV photons give an handle to identify them with astrophysical X-rays observations. Current searches, in combination with the bounds from structure formation, are testing some portions of the parameter space of these models, as we have reviewed in the previous sections. Prospects with future experiments are discussed in [105,106]. We also mention the recent proposal for a new fixed-target experiment at the CERN SPS accelerator that will use decays of charm mesons to search for heavy neutral leptons [107].
The generation of neutrino masses can be related to dark matter also in more complicated, non-minimal, models. The masses and properties of these dark matter candidates can be very different from those of the majoron or the sterile neutrino. As a concrete example, we have discussed the case of a WIMP-like, stable dark matter candidate. In particular we have considered the possibility that the stability of the dark matter particles are connected with the existence of a discrete A 4 flavour symmetry of the neutrino sector.