Cold light dark matter in extended seesaw models
Abstract
We present a thorough discussion of light dark matter produced via freezein in twobody decays DM. If and are quasidegenerate, the dark matter particle has a cold spectrum even for keV masses. We show this explicitly by calculating the transfer function that encodes the impact on structure formation. As examples for this setup we study extended seesaw mechanisms with a spontaneously broken global symmetry, such as the inverse seesaw. The keVscale pseudoGoldstone dark matter particle is then naturally produced cold by the decays of the quasidegenerate righthanded neutrinos.
a,b]Sami Boulebnane, a]Julian Heeck, a,b]Anne Nguyen, a]Daniele Teresi
Prepared for submission to JCAP
Cold light dark matter in extended seesaw models

Service de Physique Théorique, Université Libre de Bruxelles, Boulevard du Triomphe, CP225, 1050 Brussels, Belgium

Centre de Physique Théorique, École Polytechnique, Université ParisSaclay, 91128 Palaiseau Cedex, France
Contents
 1 Introduction
 2 Freezein of dark matter from decays
 3 Extended seesaw models
 4 Conclusion
 A Decays with Majorana fermions
 B Derivation of Boltzmann equation
 C Perturbation theory for the extended seesaw
1 Introduction
Cosmological and astrophysical evidence for dark matter (DM) requires an extension of the Standard Model (SM) of particle physics, which lacks a candidate that is sufficiently stable, massive, and dark. Considerable attention has been devoted to the search for DM particles with electroweak masses and couplings, so far without any undisputed evidence. An interesting alternative comes in the form of light DM, having in mind masses at the keV scale. Such DM particles are typically required to be produced nonthermally, implying small couplings to the SM [1, 2]. They can nevertheless be searched for, especially if they are unstable and produce xray signals in their decay [3]. A hint for a linelike signal with photon energy was indeed observed recently in several astrophysical objects [4, 5, 6, 7, 8]; nonobservation in other objects [9, 10, 11] makes the relevance of this signal difficult to assess [12, 13]. More data is required to settle this issue, but it provides an interesting jumpoff point to speculate about its implications for new physics. Many possible models have been put forward, but arguably the simplest explanation of such an xray line would be the decay of a DM scalar (fermion) into ().
We are not the first to point out that such keV DM particles could endanger the formation of small structures in our Universe, seeing as they have freestreaming lengths of order of Mpc if produced thermally. There are in fact some problems in structureformation simulations that could be solved if DM would smear out smaller structures [14, 12, 15], but they could also be artifacts of the noninclusion of baryons [16]. This issue is far from settled, and we have nothing to add to the discussion. Aside from body simulations, we however also have astrophysical observations of distant quasars at our disposal, which can be used to study small structures [17]. Data from these Lyman spectral lines provides strong constraints on the freestreaming length of DM, typically quoted as a lower mass limit in the benchmark model of a thermal relic fermion. Translating these limits to other scenarios requires precise knowledge of the DM momentum distribution function, which is itself determined by the DM production mechanism. In particular, many of the popular production mechanisms of a DM particle are in tension with Lyman bounds [18, 19], making necessary more involved ways to produce cold light DM if the xray hint is taken seriously [20, 21, 22, 23, 24, 25].
As put forward recently by two of us (JH and DT), there exist several freezein production mechanisms that allow for light DM to be almost arbitrarily cold, and in particular easily accommodate a mass without violating Lyman constraints [23]. Our proposed scenarios require additional thermalized heavy particles which produce light DM either via scattering or decays . In this article we will study the latter scenario in more detail, in particular providing the transfer functions necessary to assess Lyman bounds. We will also study extended seesaw models that naturally accommodate this production mechanism and in addition solve the neutrino mass problem of the SM. As discussed in detail in [23], the putative xray line can be generated in these models without spoiling the coldenough production of DM.
The rest of this article is structured as follows: in Sec. 2 we discuss the momentum distribution function of light DM produced by freezein decay, to then study its impact on structure formation by means of its transfer function. In Sec. 3 we delve into simple extended seesaw models that naturally lead to a light pseudoGoldstone DM candidate coupled to quasidegenerate heavy neutrinos. We conclude and summarize our work in Sec. 4. A number of appendices provide additional information for the interested reader. App. A lists relevant decay width formulae for Majorana fermions. App. B gives a derivation of the most general Boltzmann equation relevant for freezein production via twobody decays of thermalized particles. Finally, in App. C we discuss the matrix perturbation theory for singular matrices such as the extended seesaw ones.
2 Freezein of dark matter from decays
In this section we will discuss the momentum distribution function of light freezein DM produced via twobody decays of thermalized particles. This generalizes the analysis of Ref. [23] for decays. We assume throughout that the new thermalized particles decay before Big Bang nucleosynthesis.
2.1 Dark matter momentum distribution
The distribution function of DM from the freezein decay is conveniently rewritten in terms of the dimensionless parameters and . A full derivation is given in App. B, here we only show the result using a Maxwell–Boltzmann distribution for the thermalized mother particle (with internal degrees of freedom):
(2.1) 
Here, is the partial decay width of in the rest frame of and
(2.2) 
the total threemomentum of the DM particle in that frame. is a symmetry factor that is equal to 2 if and 1 otherwise; is the rescaled Planck mass.
The DM distribution at “time” can be obtained from Eq. (2.1) by integrating over from to , which is in general not possible analytically, but straightforward numerically. The final DM abundance today () then follows as a further integral over all momenta, normalized to the critical density [26, 23],
(2.3) 
Similarly, the mean DM momentum at production time can be obtained from
(2.4) 
Since our derivation assumes a constant , we will let in order to obtain the mean momentum today, taking care of the entropy dilution in the SM bath by hand [27]:
(2.5) 
The production temperature is around – [1, 28], which we will assume to be above the electroweak scale for the most part. Since the mother particle (and potentially ) is in equilibrium with the SM by assumption, it will contribute to , but the effect is mild unless many new particles are introduced. If more than one DM production process exists, e.g. several decay channels , their effect on is simply additive because we are in the freezein regime [29, 1, 2] where the backreaction of on the thermal plasma is negligible.
We are interested in the limit of very light DM, and in particular . Taking the limit in Eq. (2.1) allows us to perform the integration over , resulting in a simple DM distribution function today [23],
(2.6) 
and equally simple expressions for DM abundance and mean momentum,
(2.7) 
In particular, the mean DM momentum becomes smaller the more degenerate and are, which implies that DM becomes colder [23]. Before we verify this statement by looking at structure formation, let us first argue that it is sufficient to work with the analytical approximation of Eq. (2.6).
In Fig. 1 we show the ratio of the full distribution function over the analytical approximation of Eq. (2.6) for various values of and . As can be seen, keeping the DM mass nonzero cuts off the distribution function both for low and high . This is not particularly important in practice, since these extremal values are anyway suppressed in the relevant function . Interestingly, the analytical approximation typically matches the numerical result in the region of highest probability . Furthermore, one can see that the full solution will actually lead to a smaller mean momentum, making DM colder still (Fig. 2). However, the effect is negligible unless the hierarchy is not realized. Notice that the limit leads to a vanishing DM momentum in the rest frame of (Eq. (2.2)), but not in the thermal bath frame, where is thermally distributed [23]. Numerically, we find the finite average DM momentum in this limit. Obviously the DM production becomes arbitrarily inefficient for , making this limit rather uninteresting. Therefore, as expected, the DM mass effects are negligible except for the region of total phasespace closure of the decay , , in which case the DM momentum is even further suppressed. We will therefore use the massless DM approximation of Eq. (2.6) for the distribution function in the following.
2.2 Structure formation
Thermalized DM particles in the keV range are usually considered dangerous for structure formation due to their large freestreaming length, which implies a washout of small scale structures. A popular probe for this comes from the Lyman forest, i.e. light from distant quasars [17]. These are typically stronger than other limits, for instance from satellite counting. Limits are usually derived for the benchmark model of a thermal relic fermion with mass , modeled after the known neutrinos. If such a particle makes up all of the DM abundance, current limits range from [30], [31, 32] to [33], depending on the combination of data sets and the peculiarities of the analysis. In this article we will use as a fairly conservative limit, but our results can be easily rescaled.
As emphasized by many authors before us, structure formation mass limits depend strongly on the DM momentum distribution function, which in turn depends on the DM production mechanism. A naive way of translating the thermal relic limits on to other models (with DM mass ) is to set equal their freestreaming lengths [34], which effectively depends on the mean DM momentum . This leads to the simple formula [35, 36, 12, 23]
(2.8) 
Using our result from Eq. (2.7) and assuming DM production above the electroweak scale, this implies . Since can be made arbitrarily small, the DM mass can in principle be lowered even far below the keV scale. If DM is a fermion, there still exists a lower mass bound of order hundreds of eV that holds independently of the production mechanism simply due to Fermi–Dirac statistics [37] (for an updated analysis using various assumptions see, e.g., [38]); for bosonic DM on the other hand, we can apparently make DM arbitrarily light if is tiny. We stress that this is qualitatively different from the socalled misalignment mechanism, popularized through axion DM [39], which uses classical field oscillations to obtain cold DM [40].
Independent of the production mechanism, one can still obtain a lower mass limit around [41, 42, 43] for bosonic DM from structure formation and Lyman due to the macroscopic de Broglie wavelength of such a light particle. This fuzzy DM scenario has recently been popularized as an alternative to warm DM in solving smallscale structure issues [44, 45]. With our production mechanism it is not possible to reach the fuzzy DM regime, at least not with our approximation of TeVscale ; this is because we cannot compensate an arbitrarily small in (Eq. (2.7)) by a larger width . Demanding gives , so fuzzy DM values unavoidably require rather light and particles, which introduces many additional constraints. Demanding for simplicity to be heavier than TeV to satisfy all our assumptions then gives as the lowest DM mass achievable with our production mechanism. This is already very optimistic, seeing as would imply large couplings that make possible other DM production mechanisms, e.g. via scattering, as we are going to show explicitly below.
More importantly, DM with mass below keV requires a large DM number density in order to achieve . This is in conflict with our freezein approximation, i.e. that the DM abundance is negligible and we can ignore inverse reactions such as . To estimate the region of freezein validity, we calculate the thermallyaveraged reaction rate for the inverse process relative to , assuming that the DM distribution is still given by the freezein formula of Eq. (2.1). This ratio peaks at , so we have to demand conservatively
(2.9) 
for freeze in. Here, denotes the appropriate phasespace integration measure including energy–momentum conservation and the matrix element, which is the same for both directions. Using Eq. (2.7) to translate the partial width into the DM abundance, we find the inequality for freezein DM. Below this value, the DM distribution will start to differ from Eq. (2.1) and require a full solution of the integrodifferential Boltzmann equations, with is beyond the scope of this article. In combination with the Lyman bound, this implies that our calculations are trustworthy down to DM masses of . Note that both rates can still be smaller than the Hubble rate when the above inequality is violated, so DM is not automatically in equilibrium with the SM. The shape of the DM distribution will however move towards a thermal one when becomes relevant.
We have yet to verify our translation formula from above. Eq. (2.8) relies on the freestreaming length as a good measure of structure wash out. Implicitly, it assumes that the DM momentum distribution possesses a relevant mean value that characterizes it. While this might be valid for the distribution of Eq. (2.6) we use as an approximation, it is obviously not valid in general, since the distribution function can in principle be arbitrarily complicated and without useful mean [46]. Even sticking to our freezein production via decays, as soon as several channels with different and branching ratios contribute, can have a complicated structure with several peaks [23]. It is in those situations that it is necessary to go beyond the freestreaminglength approximation and delve into the actual calculation of structure formation [34, 47, 22, 46].
It is beyond our scope to perform body simulations to study the impact of our models on structure formation. Instead, we implemented our distribution function of Eq. (2.6) in the Boltzmann solver CLASS [48, 49] (Cosmic Linear Anisotropy Solving System) to obtain the transfer function for a given wavenumber ,
(2.10) 
and being the power spectra for our DM model and cold DM, respectively. The necessary cosmological parameters have been taken from Planck, specifically the dataset combination “Planck 2015 TT, TE, EE+lowP” [50]. This transfer function is then compared to the function we obtain for a thermal relic of mass , which we take as an exclusion region. Following Ref. [22], we regard a model as excluded if for all smaller than the halfmode , defined via . This procedure is very robust in our case, since the distribution function is very close to a rescaled thermal one (see Fig. 3) and, consequently, the shape of our transfer function is almost identical to the thermal relic one (see Fig. 4). In particular, a thermal relic of mass 4.65 gives the same transfer function as our distribution function with and , in accordance with Eq. (2.8). For between and , we obtain an approximate expression for the halfmode of the form
(2.11) 
which slightly improves on Eq. (2.8), with 5% discrepancy. The difference turns out to be minor, which shows that the meanmomentum and freestreaming length are useful quantities in the singledecay scenario.
If multiple decay channels are open, the DM distribution function is simply the sum of the different channels. For simplicity we will still consider only one mother particle , but with different decay channels characterized by and branching ratios . The function then has a multipeak form with moments
(2.12) 
the last equation with the Gamma function being valid for , far more general than what is needed here. We can also define the standard deviation . If one decay channel dominates, it is easy to verify that , so the mean momentum is a useful quantity, as explicitly verified above. In the presence of several decays, on the other hand, one can have , making the mean less useful [21]. This happens essentially when channels exist which have the same but vastly different . In this case one indeed finds that Eq. (2.11), with , is a bad estimator for . Remarkably, the shape of remains the same independent of the number of decay channels, only the position changes. This implies that or only depends on one quantity to be build from . Replacing in Eq. (2.11) with
(2.13) 
turns out to be an excellent ansatz to generalize Eq. (2.11), with obtained from a fit. We have verified this formula for many points with up to three decays and found a percentlevel agreement.
To sum up our results for this part, we have seen that for light DM produced from several decays with and , the Lyman forest sets the constraint
(2.14) 
with from Eq. (2.13) with . In other words, in our model of DM genesis via the decays of a massive particle in equilibrium, we can go down to DM masses
with  (2.15) 
without violating the Lyman bound that corresponds to a thermal relic. Eq. (2.15) is the appropriate generalization of Eq. (2.11) in the presence of several decay channels. In the following we will explore particlephysics models that lead to a small in order to allow for DM masses at the keV scale.
3 Extended seesaw models
In Sec. 2 we have discussed how light DM produced by freezein decays affects structure formation and Lyman limits. We have established that bosonic DM can have keV masses without violating Lyman constraints, as long as and are somewhat degenerate and much heavier than the DM particle. In the second part of this article we will discuss simple particlephysics models that realize this scenario. The obvious choice for a light bosonic DM particle is a pseudoGoldstone boson of some global symmetry, as this can ensure a small mass without finetuning [28]. Plenty of candidates have been discussed in the literature already, be it majorons [51, 52, 53, 54, 28, 55, 56, 57], connected to the lepton symmetry [58, 59], familons [28], connected to family symmetries [60, 61], and axions (or axionlike) particles [62, 63], connected to the Peccei–Quinn symmetry [64]. Our examples below will be modeled after majorons in order to simplify the discussion. Note that we will not concern ourselves with the origin of the DM mass, but simply assume an explicit breaking term in the scalar potential. In all the cases below, the radiative decay of DM into photons can be generated via mixing with either the SM scalar, boson or anomalous couplings to photons. The important point is that the couplings involved in the radiative decay do not spoil the successful production of cold DM [23].
In the standard singlet majoron model [58, 59, 57, 23], the decays among the heavy neutrino states are suppressed compared to the decays into the light neutrinos . The dominant part of the majoron DM is then produced with , too warm for our purposes. A solution was already put forward in Ref. [23] in the form of extended seesaw mechanisms by assigning different charges to some righthanded neutrinos, thereby inducing faster decays.
We are interested in minimal models, extending the SM only by gauge singlet fields for simplicity. To keep the number of parameters small, we also assume only one complex scalar field to break the symmetry, with being the Goldstone boson of interest for DM.^{1}^{1}1One can also aim at generating all mass entries spontaneously, which requires additional scalars [65, 66]. In this setup, we have to introduce some righthanded neutrinos that carry the same charge as the SM neutrinos in order to generate neutrino masses. To get a different phenomenology from the singlet majoron scenario, we further introduce a number of righthanded fermions with different charge. Playing with the charges, one can identify three interesting cases:

Extended Inverse Seesaw (EIS): assigning , , and , we find
(3.2) For , this is an extended inverse seesaw mechanism and works with the same number of states. The majoron of this model with was already discussed in Ref. [69].

Extended Seesaw (ES): assigning , , we find
(3.3) Although at first sight just a special case of the EIS, the above is not an inverse seesaw mechanism at tree level. The case , has been dubbed minimal extended seesaw [72, 73, 74],^{2}^{2}2The main motivation to study in the past was the occurrence of light sterile neutrinos in the limit [75, 76, 72, 73, 74, 12].and so we will refer to the more general scenario as Extended Seesaw. The number of massless states obtained by diagonalizing is . Even though the active neutrinos do not carry a charge in this scenario, we will still refer to the Goldstone boson as a majoron.
At oneloop level, one actually does obtain neutrino masses even for ; these are proportional to , making it an inverse seesaw [77]. We will omit a discussion of this interesting and rather minimal case, as it requires a calculation of the majoron couplings at loop level for consistency.
All three cases have in common a heavyneutrino mass submatrix of the form that does not commute with the coupling matrix of . In other words, the mass matrix consists of bare terms plus breaking terms, which ensures that will have “flavor changing” couplings, i.e. offdiagonal couplings to the heavy states. This is the main difference compared to the singletmajoron model, where these offdiagonal terms only arise at higher order in the seesaw expansion and are therefore very suppressed. A further common feature of all three scenarios above is the existence of a pseudoDirac limit: for , the heavy states will form quasidegenerate pairs, which is precisely the situation of interest for our phasespace suppressed decays. Below we will discuss the models in more detail. The diagonalization of the mass matrices and calculation of the majoron couplings is described in detail in App. C, here we will simply quote the results.
3.1 Inverse Seesaw
For simplicity we will for now work in the onegenerational limit, i.e. . We assign charges and , allowing us to write the couplings of interest as
(3.4) 
with . Here we already replaced the SM scalar doublet by its vacuum expectation value to obtain the Dirac mass , as in the standard seesaw mechanism. Upon symmetry breaking , we obtain the IS mass matrix of Eq. (3.1) with . For , this gives three Majorana states with masses
(3.5) 
Taking to be positive without loss of generality, is the heaviest state, but almost degenerate with . corresponds to the activeneutrino eigenstate with wellknown IS mass proportional to [67, 68, 69, 70].
The couplings of the majoron to the neutrino mass eigenstates to lowest nonvanishing order are given by
(3.6) 
Note that the offdiagonal coupling of the quasidegenerate states is proportional to the mass splitting . The decay rates of interest to us can be immediately obtained with the formulae of App. A. In the limit they take the simple form
(3.7) 
The last one corresponds to DM decay into active neutrinos and is of the same form as in the singletmajoron model, i.e. proportional to the neutrinomass squared, but bigger by a factor of 4 due to the larger coupling. Since we assume subMeV DM masses, it will be incredibly difficult to directly search for such monochromatic neutrinos [57], but one can still obtain a lower bound of – on this coldDM decay from its cosmological impact [78, 79]. For a majoron, this corresponds to a lower limit on the breaking scale , assuming normal neutrino hierarchy.
It is convenient to split the discussion of DM production according to whether the production temperature is above or below the electroweak phase transition of : above, we can set the electroweak vacuum expectation value to zero, which also turns off the decays ; below, the neutrino masses are so small compared to that the scattering processes become negligible. More details are given below.
3.1.1 DM production below the electroweak scale
Dark matter is produced entirely below the EW phase transition for GeV. In this case, the optimal channel for cold DM production is , which gives a small average DM momentum . In order to make the competing production channel , subdominant, we would naively need to impose the hierarchy . This is not the usual hierarchy in IS, but poses no problems. We can be actually be more precise about this inequality; we have shown in Eq. (2.15) that the quantity of interest for Lyman is , so we should actually demand
(3.8) 
in order to forbid the channels to contribute to the transfer function. In the last relation we have approximated for simplicity. Eq. (3.8) is a much stronger requirement than the naive guess and shows how dangerous even a small subcomponent of warm DM can be when it comes to structure formation. But if this relation is satisfied, Eq. (2.15) reduces back to the single decay case and the Lyman limit depends only on the decay, i.e. on , which can be made small.
Overall, our onegeneration IS scenario has five parameters of interest, , which have to reproduce the DM abundance and neutrino mass scale, as well as satisfy numerous inequalities to ensure compliance with Lyman and DM stability. Fixing the DM abundance and neutrino mass , we can express and in terms of ,
(3.9)  
(3.10) 
as well as determine the quantities
(3.11)  
(3.12)  
(3.13) 
The last quantity has to be smaller than (see Eq. (3.8)) in order for to be a reliable mean DM momentum that can be plugged into Eq. (2.8) to check consistency with Lyman constraints. We see that one can easily realize cold keV DM for heavy neutrinos in the range. Note that these heavy neutrinos decay fast, before BigBang nucleosynthesis, via the Yukawa coupling GeV).
The decay via 2loop neutrinoinduced might be too slow to lead to an observable signature for MeV. However, other possibilities exist [23]. For instance, a mixing with the Higgs boson can easily lead to an observable flux, in particular for the putative 3.5 keV line. In any case, this would not affect the coldenough production of dark matter. We stress here that this is true for all scenarios discussed below.
3.1.2 DM production above the electroweak scale
If the DM production temperature is above the electroweak scale, i.e. , the decays are forbidden by invariance. Nevertheless, there exist scattering processes that compete with our coldDM production channel that need to be taken into account to ensure that DM is cold enough.
If is close to , the dominant DM production channel that competes with comes from the scattering , seeing as the majoron couples most strongly diagonally to . The cross section for this process in the limit of massless takes on the form
(3.14) 
with . This matches the expression derived in Ref. [80]; as shown there, this scattering process would thermalize if . Even if this annihilation rate is too slow to reach equilibrium, it can still freezein a population of [28] with mean momentum . To push DM below the keV scale, we therefore have to demand that this process gives only a subcomponent of DM. The DM abundance can be readily calculated with the formulae from Ref. [23]; numerically, the ratio can be approximated by
(3.15) 
which we demand to be smaller than in order to not mess up the small mean DM momentum, by extrapolating the argument of the decay case (3.8).
For , the scattering channels , , and can become relevant, as they are only suppressed by . Similar to the scattering case above, we can calculate the resulting abundance of rather warm DM from all these processes as
(3.16) 
being the Yukawa coupling that leads to the neutrino Dirac mass . We can once again express all relevant quantities in terms of :
(3.17)  
(3.18)  
(3.19)  
(3.20)  
(3.21)  
(3.22) 
In addition, we should demand and in order to avoid having nonperturbatively large Yukawa couplings in the model, but this turns out to be a much weaker constraint than the above inequalities. Notice, however, that these Yukawa couplings are anyway large enough to guarantee the thermalization of with the SM bath, as assumed in the analysis of Sec. 2.
For a given DM mass, provides a lower bound on to ensure DM stability. together with Lyman then gives a lower bound on , which at some point leads to a large DM production rate via . From the above it is clear that righthanded neutrinos can generate cold keV DM without running into problems with Lyman.
The above shows that the inverse seesaw mechanism can not only generate light neutrino masses, but also provide the necessary ingredients to produce cold majoron DM. Aside from the unavoidable decay channel of DM into neutrinos, additional interactions could lead to a detectable decay into without endangering our production mechanism [23]. So far we have only considered the onegenerational IS, but the above discussion can be generalized straightforwardly. Let us remark on one particularly interesting consequence of more generations, that is lepton flavor violation. The IS with hierarchy has the feature of keeping neutrino masses small without tiny active–sterile mixing angles. This makes it a popular model to discuss rare flavor violating decays such as , induced at oneloop level by the nottooheavy neutrinos. This feature is diluted when adopting our hierarchy of interest, , which looks more like a normal seesaw mechanism when it comes to active–sterile mixing. Due to possible matrix cancellations in the threegenerational case it is of course still possible to have detectable rates for e.g. , but there is little predictivity from the DM side. Rare decays involving majorons, e.g. , are similarly expected to be suppressed if forms DM, but this might be discussed elsewhere in more detail. Finally, notice that whereas in principle the quasidegenerate states could be used to generate the baryon asymmetry of the Universe via resonant leptogenesis [81, 82, 83] (if their mass is above the weak scale), or via the Akhmedov–Rubakov–Smirnov [84, 85] and Higgsdecay mechanisms [86] (if below), their mass splitting, essentially given by , turns out to be too large (see (3.9)) to have a successful generation of the asymmetry.
3.2 Extended Inverse Seesaw
The second model of interest is the Extended Inverse Seesaw, which we obtain by assigning , , and , leading to the allowed couplings for one generation
(3.23) 
which leads to the EIS mass matrix of Eq. (3.2) upon symmetry breaking, where . Taking all mass terms to be real and positive for simplicity,^{3}^{3}3A more general expression allowing for complex is given in Eq. (C.90). with hierarchy , the mass spectrum is
(3.24) 
and the majoron couplings to lowest order are
(3.25) 
Compared to the IS case from above, none of the couplings here are large, but rather of order or even further suppressed by powers of . As a result, the decay automatically dominates the DM production compared to (suppressed by ) and (suppressed by phase space). The decays relevant for DM production and decay are
(3.26) 
Fixing the relic abundance with Eq. (2.7), we can express in terms of as
(3.27)  
(3.28)  