Type-II Majoron Dark Matter

We discuss in detail the possibility that the ``type-II majoron'' -- that is, the pseudo Nambu-Goldstone boson that arises in the context of the type-II seesaw mechanism if the lepton number is spontaneously broken by an additional singlet scalar -- account for the dark matter (DM) observed in the universe. We study the requirements the model's parameters have to fulfill in order to reproduce the measured DM relic abundance through two possible production mechanisms in the early universe, freeze-in and misalignment, both during a standard radiation-dominated era and early matter domination. We then study possible signals of type-II majoron DM and the present and expected constraints on the parameter space that can be obtained from cosmological observations, direct detection experiments, and present and future searches for decaying DM at neutrino telescopes and cosmic-ray experiments. We find that -- depending on the majoron mass, the production mechanism, and the vacuum expectation value of the type-II triplet -- all of the three decay modes (photons, electrons, neutrinos) of majoron DM particles can yield observable signals at future indirect searches for DM. Furthermore, in a corner of the parameter space, detection of majoron DM is possible through electron recoil at running and future direct detection experiments.


Introduction
After a decade of operations, the Large Hadron Collider (LHC) has given no conclusive evidence of new phenomena beyond those predicted by the Standard Model (SM) of particle physics.In particular, no sign of new particles at the TeV scale so far has been found.Similarly, Dark Matter (DM) direct detection experiments, that is, searches for nuclear recoils due to collisions with DM particles, have been only providing ever more stringent constraints on models where DM is made of Weakly Interacting Massive Particles (WIMPs).In combination with the LHC bounds, this significantly reduced the viable parameter space of typical WIMP models, see, e.g., Refs.[1,2].While the WIMP paradigm is far from being excluded, these experimental facts have recently motivated an increased interest in alternative DM candidates, such as very light, feebly-coupled -thus more elusive -particles, see, e.g., Refs.[3][4][5][6] and the references therein.
Paradigmatic examples of such light DM candidates are provided by the QCD axion [7][8][9] and, more in general, axion-like particles (ALPs), that is, pseudo Nambu-Goldstone bosons (pNGBs) of global U (1) symmetries spontaneously broken at high energies.These fields can thus be interpreted as light remnants of some high-energy new physics scale, usually related to outstanding problems of the SM, such as the strong CP problem [7][8][9], the flavour puzzle [10][11][12][13][14][15][16][17][18], the origin of neutrino masses [19][20][21].Besides a possible resolution of the DM problem [22][23][24], the possible connection to other open questions in particle physics makes this class of models very appealing.Furthermore, their rich phenomenology provides a unique way to test new physics at scales inaccessible to any foreseeable collider experiment.
In this paper, we consider one of the best motivated ALP DM candidates: the majoron, that is, the pNBG associated with the spontaneous breaking of the lepton number [19].This enables us to draw an appealing connection between the origin of neutrino masses, the DM puzzle, past and future searches for DM signatures in photon, electron, and neutrino fluxes from the cosmos, as well as searches for DM-electron scattering at direct-detection experiments such as XENONnT [25], PandaX [26], LZ [27].
After the discovery of neutrino oscillations [28,29], two decades of enormous experimental developments made it possible to measure neutrino masses and mixing with high precision, see Refs.[30][31][32] for the current status.However, it remains undetermined whether the neutrino mass terms are of the Dirac type, such as all the other SM fermions, or Majorana [33] that would imply that neutrinos and their own antiparticles coincide.This latter possibilitywhose simplest realisation is provided by the famous dimension-five Weinberg operator [34] necessarily requires violation of the lepton number.While this may occur through an explicit breaking, such as the one due to Majorana mass terms of right-handed neutrinos within the simplest realisation of the type-I seesaw mechanism [35][36][37][38], a spontaneous breaking is also a logical possibility.In general, the existence of a light majoron would be an unavoidable consequence of any mechanism of generation of neutrino masses involving a spontaneous breaking of the lepton number (as long as the latter is a global symmetry).
In short, within a fairly wide class of models, it is natural to add the majoron field to the particle content of the SM alongside the other fields involved in the generation of neutrino masses, thus providing a compelling candidate for ALP DM and a realistic target for searches, among others, of neutrino lines at present and future neutrino telescopes [39].
It is also interesting to notice that, in cosmology, majorons have recently gathered attention in the context of possible solutions of the Hubble tension: it has been shown that an additional radiation component due to light majorons as well as neutrino interactions mediated by majorons can relax the tension [40,41] -cf.Refs.[42][43][44], for explicit realisations.
In the following pages, we focus in particular on a majoron model [45,46] based on type-II seesaw [47][48][49][50][51][52], which consists in adding only two fields to the SM field content: a scalar SU (2) L triplet and a scalar singlet responsible for the spontaneous breaking of the lepton number.Our aim is to study the model's DM phenomenology that has received comparatively little attention in the literature. 1 In Section 2, we review the model and discuss how it is constrained by laboratory experiments, including searches for the triplet at the LHC.In Section 3, we discuss two possible majoron DM production mechanisms, freeze-in and misalignment, both during a standard radiation dominated epoch and an early matter-dominated era.In Section 4, we study the constraints on the parameter space relevant for majoron DM from cosmological surveys, direct detection experiments, and present and future searches for decaying DM at neutrino telescopes and other cosmic-ray probes.We summarise and draw our conclusions in Section 5, while some technical results and useful formulae are collected in the Appendices.

Type-II majoron model
One minimal extension to the SM, which can accommodate Majorana masses of neutrinos, consists in introducing an SU (2) L triplet scalar field with lepton number L = −2, which couples with two lepton doublets.Spontaneous breaking of the lepton number via the vacuum expectation value (vev) of the triplet scalar field induces neutrino Majorana mass terms and, at the same time, provides a Nambu-Goldstone boson, the majoron [20,21].This minimal scenario for Majorana neutrino masses was already ruled out long ago by the non-observation of majorons in the decays of Z bosons at LEP. 2The famous type-II seesaw mechanism also features the same scalar triplet [47][48][49][50][51][52], but the lepton number is violated explicitly by the scalar cubic interaction involving the triplet and two Higgs doublets.Therefore, no majoron appear and the phenomenological problems of the triplet scalar model related to the presence of the majoron field are circumvented.However, one can impose that the origin of the mass-dimensionful coupling of the scalar cubic term in the type-II seesaw model is the vev of an extra scalar field that spontaneously breaks the lepton number.The new scalar field should be a singlet under the SM gauge symmetries but it should be charged under the lepton number U (1) L .After the spontaneous breaking of the lepton number, a majoron state appears as in the triplet scalar model.However, this majoron field consists dominantly of the imaginary component of the singlet field and does not suffer the phenomenological problems that affect the original triplet majoron model, being in particular consistent with the measured invisible width of the Z boson.This possibility, which we call the type-II majoron model, was first proposed in Refs.[45,46], and various aspects of its phenomenology have been discussed in, e.g., Refs.[59][60][61][62].The model is described in detail in Appendix A. In this section, we discuss the most relevant aspects for our analysis.
The part of the Lagrangian of the type-II majoron model, which is responsible for neutrino masses, reads where L α,β are lepton doublets whose flavour is denoted by the indices α and β, and σ, Φ, and ∆ are scalar fields in the SU (2) L singlet, doublet, and triplet representation, respectively. 3he superscript c on the field L denotes charge conjugation.Notice that both the couplings Y ∆ and κ are mass-dimensionless.The SU (2) L triplet scalar field ∆ carries lepton number L = −2, and the singlet field σ has L = +2, while the doublet Φ, which is the SM Higgs scalar field, carries no lepton number.As a consequence, the above Lagrangian is invariant under the U (1) L lepton number transformation, but that symmetry is spontaneously broken by the vev ⟨σ⟩ = v 1 / √ 2 of the singlet field.This spontaneous breaking of the lepton number brings about a majoron field J stemming from the imaginary part of σ.
After the electroweak symmetry is broken by the vev of the doublet ⟨Φ 0 ⟩ = v 2 / √ 2, the neutral component of the triplet field acquires a vev ⟨∆ 0 ⟩ = v 3 / √ 2 too, which is mediated by the triplet scalar field itself: where M ∆ is the mass of the triplet. 4The combination of vevs relevant to the electroweak symmetry breaking is identified as 3) The vev of the triplet contributes to the masses of the weak gauge bosons and modifies the relation between them.The deviation from the relation predicted by the SM is measured in terms of the ρ parameter, which is given by (2.4) The most conservative global fit of the electroweak precision observables (EWPO) reported in Ref. [63] gives ρ = 1.0002 ± 0.0009, which sets an upper bound to the triplet vev, v 3 < 7 GeV at 95% confidence level. 5he Yukawa interaction in Eq. (2.1) in combination with the vev of the triplet provides neutrino with Majorana mass terms: Notice that the standard neutrino mass formula of the type-II seesaw mechanism is recovered considering that the coupling of the trilinear scalar interaction Φ T ∆Φ is given by µ = κ v 1 / √ 2. The scalar potential and the masses of the physical states are discussed in Appendix A. This sector of the model is subject to bounds from vacuum stability and perturbative unitarity [68][69][70][71][72].In order to have a viable DM candidate, the vevs of the scalar fields are required to exhibit the hierarchical pattern In fact, at least one vev needs to be ultralarge in order to suppress the majoron couplings and make it long-lived enough to be DM (see Section 4) and such a vev can only be v 1 , because v 3 is constrained by EWPO, as discussed above, and v 2 must provide the main source of the observed electroweak-symmetry breaking.This implies that the mixing among the CP-odd scalars is very small -as one can see by inspecting the mixing matrix O I in Eq. (A.11) for the regime of Eq. (2.6).Consistently with this observation, and for the sake of the perturbativity constraints mentioned above, we assume all couplings of the scalar potential to be ≪ 1 and small mixing to occur in the real scalar sector too.We stress that our conclusions do not depend on the details of the scalar potential, as long as the couplings are not too large.The small mixing regime also prevents the majoron (and the real component of the singlet) to thermalise with the hot bath in the early universe.This and other cosmological consequences of the κ term will be discussed in Section 3. In addition, as we will see below, κ is bounded from below by searches for extra Higgs bosons at the LHC.

Spectrum of the model and LHC bounds
Let us now turn to the spectrum of the model.The scalar sector comprises the following physical states: three CP-even Higgs bosons (H 1 , H 2 , H 3 ), a CP-odd Higgs boson A, two singly-charged Higgs bosons H ± , two doubly-charged Higgs bosons H ±± , and the majoron J, which -together with the three Nambu-Goldstone bosons (G 0 , G ± ) that become the longitudinal components of the electroweak bosons -add up to the 12 degrees of freedom present in the three complex scalar fields of the model.Due to the small mixing regime that we are working in, the (mainly) singlet Higgs boson H 1 decouples from the rest of the spectrum and is super heavy, M H 1 ∼ v 1 ≫ v EW .H 2 consists dominantly of the neutral component of the doublet, and we identify it with the SM Higgs boson, that is, we set M H 2 = M H SM ≃ 125 GeV.The other Higgs bosons mainly originate from the triplet, and their masses mainly stem from a common origin, which is the κ term in Eq. (2.1).Consequently, they have masses of similar size that, expressed in terms of κ and the vevs, read For the full description of the scalar sector, we again refer to Appendix A. The collider phenomenology of the triplet Higgs model has been extensively discussed in, e.g., Refs.[69,[72][73][74][75][76][77][78][79][80][81][82][83][84].Since fields charged under the electromagnetic U (1) can be produced through the Drell-Yan process, the LHC can set a robust bound to the masses of the charged Higgs bosons regardless of the details of the model.In particular, searches for the production of doubly-charged states pp → H ++ H −− followed by decays into W bosons, H ±± → W ± W ± , are the most relevant for scenarios with v 3 > O(10 −4 ) GeV [83].The null result of such searches at the LHC [85] provide a lower limit on the mass of the triplet, M ∆ ≳ 400 GeV.Following from Eq. (2.7), this limit can be interpreted in our case as a lower bound to κ: Finally, we need to discuss the mass M J of the majoron field.Indeed, the majoron can not be a genuine Nambu-Goldstone boson -that is, exactly massless -in order for it to be a candidate for DM.In other words, a majoron mass term must arise out of a (small) explicit breaking of the lepton number.Such breaking can originate from higher-dimensional Plancksuppressed operators, which is consistent with the commonly accepted notion that no global symmetry is preserved within quantum gravity -see, e.g., Ref. [86] for a recent review.A straightforward example is provided by operators only involving the scalar singlet [87,88]: where M Pl = 1.22 • 10 19 GeV is the Planck mass, d = m + n ≥ 5, λ m,n are dimensionless coefficients, and the lepton number is explicitly broken if m ̸ = n.As one can see, the above operators would induce contributions to the majoron mass M J as Pl . (2.10) As we will show in the following, majoron DM requires v 1 ≳ 10 7 GeV.In such a regime, d = 5 operators induce thus the mass range relevant for DM detection (10 keV ≲ M J ≲ 100 MeV, see Section 4) require small couplings λ m,n ≲ 10 −12 − 10 −4 for v 1 ≳ 10 7 GeV.If the dominant contributions to M J arise at dimension 6, we have instead (2.12) hence the whole of the mass range we are interested in is accessible for d = 6 operators with coefficients λ m,n = O(1).
Notice that, even if present, operators such as σ m σ * n Φ † Φ p and σ m σ * n ∆ † ∆ p (of dimension d = m + n + 2 p, with m + n ≥ 3 and p ≥ 1) would give subdominant contributions to M J compared to the operators in Eq. (2.9) of the same dimensionality, as a consequence of the vev hierarchy of Eq. (2.6).As we will see in the next subsection, the couplings of the majoron relevant to DM phenomenology (that is, those to photons and SM fermions) arise from mixing in the scalar sector.As a consequence, the explicit breaking of the lepton number provided by Eq. (2.9) -which does not substantially contribute to the mixing among singlet, doublet and triplet -does not affect the majoron couplings and thus DM production and decays. 6Hence M J can be safely treated as a free parameter with no consequence on the majoron interactions, and so will we do throughout the paper.
where (δM 2 )ij ≡ (OI )i1M 2 J (O T I )1j and we considered the mixing matrix OI of Eq. (A.11) in the regime v3 ≪ v2 ≪ v1.As we can see, for MJ ≲ 100 MeV, MA ≃ M∆ > 100 GeV, the operators responsible for M 2 J induce a relative correction ∼ M 2 J /M 2 A ≲ 10 −6 , thus completely negligible, to the singlet-triplet mixing (which is ∼ v3/v1 at leading order) and do not affect the mixing with the doublet. 7This obviously requires exploiting the freedom of adjusting the above-defined operator coefficients λm,n.

Majoron couplings and decays
Being dominantly part of the singlet σ, the type-II majoron inherits couplings to SM fields through mixing in the scalar sector, as shown in Appendix A. These can be written as where f denotes the SM fermion species: neutrinos, charged leptons, up and down quarks, f = ν, ℓ, u, d.The couplings to the neutrino mass eigenstates (ν i , i = 1, 2, 3) stem from the interaction Y ∆ of the triplet with lepton doublets in Eq. (2.1) and, for the hierarchical regime v 3 ≪ v 2 ≪ v 1 that we are considering, read The couplings to the other fermions are where α = 1, 2, 3 denotes the flavour of the fermion and, again, we restricted to the hierarchical regime shown in Eq. (2.6) -for the full expressions beyond this limit, see Appendix A. Notice that the couplings to quarks and charged leptons are ∝ v 2 3 /v 2 2 .Furthermore, they are flavourconserving, since they are all inherited from the fermion interactions with the Higgs doubletin fact, ∆ does not couple to quarks and its neutral component does not couple to charged leptons either -hence rotations of the fields to the mass basis diagonalise them too.
Finally, the effective coupling with photons arises from majoron-pion mixing as well as loops of charged fermions: where Q f and N f c are, respectively, electric charge and colour multiplicity of the fermion f , and τ f ≡ 4m 2 f /M 2 J .The expression of the loop function B 1 (τ f ) can be found in Appendix A. This function is such that B 1 (τ f ) → 0 for τ f → ∞, that is, the majoron decouples from photons for M J ≪ m e .This is because the lepton number is anomaly free hence there is no anomalous majoron coupling with photons. 8n the mass range relevant to DM direct detection and astrophysical probes that will focus on in Section 4, that is, M J ≈ 1 keV -100 MeV, the majoron can only decay into photons, neutrinos, and possibly electrons.The corresponding decay rates read (see, e.g., Refs.[91,92]) Cosmological and astrophysical consequences of these decays are discussed in Section 4.

Other laboratory constraints
The type-II seesaw triplet can mediate charged-lepton-flavour-violating (cLFV) processes, such as µ → eeē, cf. for instance Ref. [84].See also Ref. [93] for a discussion of distinctive cLFV signals of type-II seesaw within the context of grand unified theories.The current limit BR(µ → eeē) < 10 −12 [94] translates into a bound TeV, hence cLFV muon decays typically require (Y ∆ ) αβ ≲ 0.005 for M ∆ ∼ 1 TeV.As we will discuss in the next section, freeze-in production of majoron DM is only possible if indeed M ∆ ≲ 1 TeV.However, we have seen that neutrino masses require Y ∆ ∼ m ν /v 3 , hence we can not expect a signal at the upcoming cLFV search campaign (see, e.g., [95]) unless v 3 ≲ 10 −7 GeV, a regime where the majoron DM phenomenology is solely controlled by its decays into neutrinos, freezein DM production is not effective and the observed relic density can only be accounted for by the misalignement mechanism, cf.Sections 3 and 4. In addition, cLFV decays into majorons, such as µ → eJ, which can be powerful probes of the leptonic couplings of other kinds of majorons as well as generic ALPs -cf.Refs.[92,96] for an overview -can not occur for the type-II majoron, because its couplings to charged leptons are flavour conserving, as discussed above -see Eq. (2.15).Neutrinoless double beta decays can also be mediated by the doubly-charged Higgs boson, which is a component of the triplet scalar [97].The size of the contribution, given in terms of the vevs and the triplet mass ).This has to be compared with the standard Majorana neutrino mass contribution (m ν ) ee /p 2 ν , where p ν is the momentum carried by a neutrino in an atom, which is typically of the order of 0.1 GeV.Considering the values of the parameters required by majoron DM, in particular v 1 ≳ 10 7 GeV (see Section 4), one can find that the triplet contribution is smaller than the standard one (setting (m ν ) ee to a value compatible with the current limit from Ref. [98]) by several orders of magnitude.In addition, a recent result of the search for neutrinoless double beta decays with majoron emission provides a direct bound to the majoron-neutrino coupling at the level of |g Jνeνe | ≲ 10 −5 [99], which is also significantly larger than the size of the coupling in the parameters' range relevant for DM majorons, |g P Jν i ν i | ≃ m ν i /(2v 1 ) ≲ 10 −18 .Majoron-quark interactions can be constrained by searches for K + → π + + invisible at kaon experiments such as NA62 [100].In our case, the dominant contribution to processes of the kind K → πJ comes from a J-penguin diagram with a top and a W in the loop [101].See also Ref. [102], where this process was recently revisited.The authors of Ref. [101] discuss the decay occurring through the mixing of the Higgs with a light scalar χ and give a bound to the mixing parameter θ defined as (θm f /v)χ f f .In the type-II majoron model, the mixing parameter θ then corresponds to .15).According to Ref. [102], the NA62 limit translates into the bound θ ≲ 10 −4 , while for values of the vevs corresponding to a majoron DM we find θ ≲ 10 −8 .Similarly, we find that the parameter region where the majoron is a good DM candidate is far beyond the sensitivity reach of ALP searches at B-factories [103] and in π decays [104].

Majoron production in the early universe
Throughout this paper, we require that the parameters of the model, besides fulfilling the constraints discussed in the previous section, be such that the majoron is a good DM candidate.To be more specific, the majoron must be produced in the early universe as cold DM, and its relic abundance today must reproduce the value inferred from cosmological observa-tions, Ω DM h 2 = 0.12 [105].In this section, we review two possible production mechanisms for majoron DM and provide the formulae to estimate its relic density.

Misalignment mechanism
The misalignment mechanism, first introduced for the QCD axion [22][23][24], is one of the most popular DM production mechanisms within ALP scenarios in general [6].If the global symmetry (in our case, the lepton number) is broken before inflation, the ALP field attains a uniform value across all observable universe.If such initial value is misaligned from the minimum of the potential, the field starts falling to the minimum at the temperature T osc where the Hubble parameter H(T ) becomes comparable to the mass of the ALP, and the energy dissipated by the oscillation of the classical field about the minimum is converted into energy carried by the ALP, such that the resulting relic ALPs act as cold DM.
Assuming a standard evolution of the universe (with radiation dominating the energy budget of the universe when the field starts oscillating), one obtains the following estimate of the majoron relic density [106] (see also Ref. [6]) where h ≃ 0.68,9 ρ c0 is the critical density today, g * s (T ) is the effective number of entropy degrees of freedom at a temperature T , ρ J (T ) is the energy density of majoron at T , and T 0 is the temperature of radiation (that is, relic photons) today.With the initial misalignment of the majoron field parameterised as J 0 = v 1 θ 0 , the energy density at T osc is expected to be ρ Substituting this into Eq.(3.1), we find that the final relic density would be determined by the initial value of the majoron field, of the order of the scale of the lepton number violation as [106] Ω J h 2 ≃ 0.12 v 1 θ 0 1.9 where g * (T osc ) is the effective number of relativistic degrees of freedom at T osc .In Figure 1, we show the contours corresponding to Ω J h 2 = 0.12 as black lines on the M J −v 1 plane, assuming a standard history of the early universe.As we will see in Section 4, direct DM searches through electron recoil events, such as those performed at the XENONnT experiment [25], can be sensitive to majoron DM with GeV.For those values of the parameters, the formula in Eq. (3.2) suggests that the standard misalignment mechanism could only account for a tiny fraction of the observed DM relic density.As a consequence, if a DM signal will be discovered at XENONnT or other direct searches in the future, one would need another production mechanism.On the other hand, we will also see that searches for DM decaying into gamma rays or neutrinos can be sensitive to majoron DM produced through the misalignment mechanism.
Since the history of the universe before the big bang nucleosynthesis (BBN) is not experimentally probed yet, we are still allowed to assume various non-standard cosmological scenarios to occur between inflation and the BBN.For example, it is possible that the universe underwent an era in which the energy budget was dominated by non-relativistic matter 12 from the misalignment mechanism for majoron oscillations starting either during a standard radiation-dominated era (RD, black lines) or an early matter-dominated era (MD, red lines), and for different values of the initial misalignment angle θ 0 and the reheating temperature T R .
before the radiation-dominated era, so long as the reheating temperature T R (at which the two eras are switched) is set to be higher than the BBN temperature, which is in the MeV range [107][108][109][110].In such a setup, the oscillation of the majoron field would start during the early matter-dominated era.The formula for the relic density in Eq. (3.1) has to be modified by replacing T osc with the reheating temperature T R .The energy density at T R is related to that at , since the majoron is cold DM and H 2 ∝ a −3 in the early matter-dominated era.The initial energy density is as in the standard case, that is, . Substituting these expressions into the modified Eq. (3.1), we can find that the final relic density is [106] which, as we can see, does not depend on the mass of the majoron nor on g * s .The contours corresponding to Ω J h 2 = 0.12 for early matter domination are shown as red lines in Figure 1.
Reproducing the correct relic density with v 1 ∼ O(10 7 ) GeV would require too high a value of T R , in fact inconsistently T R ≫ T osc .Hence the misalignment mechanism in the early matter-dominated era would do not help explain a possible DM signal at the XENONnT experiment in the future -however, it further opens up the parameter space that can be tested by indirect DM searches, as we will see in the next section.As shown in the following, majoron DM with such low values of v 1 can be instead efficiently produced via the freeze-in mechanism.

Freeze-in mechanism
As we have seen in the previous section, LHC searches for extra Higgs bosons require non-zero κ, cf.Eq. (2.8), and therefore, there are inevitable scalar-majoron interactions, which lead to majoron production in the early universe, even if the majoron field is decoupled from the thermal bath due to its feeble couplings, through the so-called freeze-in mechanism [111].The possible production processes are10 The decay rates of these processes and the details of the relic density calculation are given in Appendix B. We find that the majoron pair-production processes are suppressed in comparison with the single production ones, and the rates of the decays of the SM-like Higgs boson (H 2 ) are smaller than those of a triplet Higgs boson (H 3 , A, H ± ) by orders of magnitude.Furthermore, the process H 3 → AJ is suppressed by the small mass splitting between H 3 and A. Therefore, the dominant production processes are the first three in Eq. (3.4) with the triplet scalar in the initial state.
Freeze-in production is at work approximately until the temperature of the universe drops below the mass of the parent particle, which is M ∆ in this scenario.Note that before the electroweak phase transition, the vevs v 2 and v 3 are null and the majoron interactions are turned off, i.e., the freeze-in production happens only after the EWPT.In other words, to make the triplet scalars massive and turn on the cubic interactions with a majoron at the temperature T > M ∆ , we have to assume that the electroweak phase transition (EWPT) occurs at a temperature higher than the triplet scalar mass, i.e., T ewpt > M ∆ > 400 GeV. 11Naively, the critical temperature of EWPT in the SM is expected to be around the mass of the SM Higgs boson [112], and therefore, freeze-in majoron production seems to require an extension of the scalar potential, such that the critical temperature is increased.In this work, we do not step into the issue of the phase transition.Nevertheless, we notice the the EWPT at a temperature higher than the SM Higgs mass is not forbidden by any phenomenological observation.For recent studies on high-temperature EWPT (and EW symmetry non-restoration) see, e.g., Refs.[113][114][115][116].
Employing Eqs.(2.2), (B.6), and (B.9), the relic density from the above decay processes can be estimated as12 such that the majoron is overproduced in the parameter region explored by direct detection experiments - Majoron overproduction is not the only problem.It is possible that the scalar quartic interactions from the κ term keep the majoron in equilibrium with the thermal bath, in contrast with the starting assumption of the freeze-in mechanism.To avoid thermal equilibrium (and thermal production) of majorons, we have to set the value of κ -or equivalently that of M ∆ once the vevs are fixed, see Eq. (2.7) -smaller than a certain value.This upper bound to κ can be estimated through the Gamow's criterion of thermalisation, i.e., comparing the rate Γ of a given process with the rate H of expansion of the universe.To be specific, we focus on the following quartic interaction contained in the κ term of Eq. (2.1), where I 1 is the imaginary part of the singlet scalar field, R 2 is the real part of the neutral component of the doublet field, and I 3 is the imaginary component of the triplet field.As discussed in Section 2, the majoron field consists dominantly of I 1 , and R 2 and I 3 are the main components of the SM Higgs boson H 2 and the CP-odd Higgs boson A, respectively.In short, this interaction induces majoron production through the process H 2 H 2 → AJ.The rate Γ(κ, T ) of this scattering process is estimated as where n R 2 is the number density of R 2 and σ(R 2 R 2 → I 3 I 1 ) is the cross section of the process, which is given by σ = |κ| 2 /(16πE 2 CM ) in the massless limit.The centre-of-mass energy E CM at a given temperature T can be estimated as 4 T , where the number and energy densities, n R 2 and ρ R 2 , of R 2 are given by the Bose-Einstein distribution.The thermalisation is assessed by comparing this interaction rate Γ with the Hubble parameter H(T ) at a given T .If Γ(κ, T ) < H(T ), the interaction stays out of equilibrium at T .The Hubble parameter in the standard radiation-dominated early universe is given by where M Pl = 1.22 • 10 19 GeV is the Planck mass, and the number of degrees of freedom can be set at g * ≃ 110 in the range of temperatures we are interested in.
In Figure 2, the boundary of the condition, Γ = H, is shown in the T -κ plane with a solid blue curve, where v 1 and v 3 are set at values compatible with the sensitivity of XENONnT, as discussed in Section 4. Note that since Γ/H ∝ 1/T , the scattering process comes into equilibrium only at a low temperature.This plot should be read as follows; Fixing a value of κ, i.e., M ∆ , we determine a horizontal line.Then, we start from the right end (high T ) of the line and move to the left (low T ) along the line as the universe evolves.When we hit the blue curve, the process H 2 H 2 → AJ comes into equilibrium and majorons are thermally produced through it.To avoid thermal production of majorons through the scattering process, the interaction rate Γ must be suppressed before we hit the blue curve.There are two possibilities to circumvent the thermalisation: (a) n R 2 is suppressed by the Boltzmann factor, or (b) σ is turned off because of the kinematical threshold of the process.The Boltzmann suppression is activated at T < M H SM in this case, and to have T < M H SM below the blue curve (before hitting the blue curve), we have to set κ < 1.5 • 10 −6 which corresponds to M ∆ ≲ 670 GeV with the vevs given in this example.In other words, taking into account the LHC bound, the possible range for the triplet mass is narrowed down to 400 GeV < M ∆ ≲ 670 GeV.Note that majorons are overproduced from the freeze-in processes with the value of κ suggested by this range of M ∆ , cf.Eq. (3.5), if we assume values of the vevs that can be tested at the XENONnT experiment as in this plot.The suppression of the process by the kinematical

��� �����
Figure 2: Constraints on κ: temperature at which a four-scalar interaction in the κ term enters thermal equilibrium is indicated with the solid blue curve, i.e., majorons are thermalised through the κ interaction in the light blue region.The extra Higgs searches at LHC set the lower bound to M ∆ , which is indicated with a gray line [83].By assuming an early matterdominated era and setting the reheating temperature as T R = {50, 100} GeV, the solid blue curve of the thermalisation condition is relaxed as indicated by the dashed curves.
threshold works for E CM < M ∆ , which corresponds to T < M ∆ /5.4.We find that this condition appears on the left of the solid blue curve in Figure 2, and therefore, the suppression from the kinematical threshold is not available for any choice of κ (hence M ∆ ).The other scattering processes induced by the κ term provide similar conditions.In summary, freeze-in production of type-II majorons testable at direct detection experiments has two problems (i) DM overproduction and (ii) the condition of non-thermalisation in combination with the LHC bounds.However, those two shortcomings can be lifted simultaneously by considering a non-standard cosmic history.When the BBN starts at T ∼ MeV, the universe must be dominated by radiation, however, there are no strict phenomenological requirements on the development of the universe before that, as we already mentioned in Section 3.1.We can suppose that non-relativistic matter, which is not coupled to radiation thermally, dominates the energy budget of the universe before the radiation-dominated era.The majoron relic density can be then adjusted by the dilution due to the late-time production of entropy, which is caused by the decay of matter into radiation.In addition, the Hubble parameter in the early matter-dominated era is modified, and it helps avoid the thermal production of majorons, as we will see later.
Following the discussion in Refs.[117,118], we can calculate the freeze-in production of majorons during the early matter-dominated era with the effect of the dilution due to the late-time entropy injection.The details of the calculation are presented in Appendix B.Here we give an approximate formula of the relic density from the decay processes: which is only valid for T R ≪ M ∆ .Setting the reheating temperature at O(10) GeV, the correct relic density can be reproduced with M ∆ ∼ 500 GeV and the vevs and M J that the XENONnT experiment is sensitive to.This size of M ∆ together with those reference values of the vevs suggests a relatively large value of κ, which enhances the rates of the scalar scattering processes leading to thermal production of majorons.However, in the early matter-dominated era, where T > T R , the Hubble parameter is modified as In short, the upper bound to κ is significantly relaxed in a scenario with an early matterdominated era, and we have a chance to reproduce the correct relic density within the parameter region that XENONnT can explore in future.This can be seen in Figure 3: in the left panel, the relic density Ω J h 2 is shown as a function of κ -and, through Eq. (2.7), equivalently M ∆ -as a black solid curve.Here, the vevs and the mass of the majoron field are fixed at v 1 = 5•10 7 GeV, v 3 = 5 GeV, and M J = 10 keV.The contribution from the scattering processes to the majoron production rate is included, although subdominant, cf.Eq. (B.16).As discussed above, for standard freeze-in production occurring during a radiation-dominated epoch, the triplet mass is constrained within the range indicated by the black arrows.The black curve clearly shows that majorons from the standard freeze-in mechanism are overproduced by orders of magnitude for the values of M ∆ in the allowed range.The orange curve is calculated by assuming an early matter-dominated era with T R = 30 GeV, which shows that the correct relic density can be reproduced between the LHC bound and the modified bound from the thermalisation condition indicated with the orange arrow.In the right panel of Figure 3, we can find the necessary reheating temperature T R in order to reproduce the correct relic density of majorons for a given triplet mass M ∆ .The combination of the relic density condition and the thermalisation condition places the upper bound to the mass of the triplet scalar field, M ∆ ≲ 1100 GeV.In the future, the LHC bound is expected to reach up to 640 GeV [83], and the parameter space will be significantly narrowed down.
Light DM particles produced by the freeze-in mechanism are not thermalised, but a significant amount of them may carry large enough momentum, and this may be in conflict with the bounds on warm DM from structure formation.Lyman-α observations translate into a lower limit on the DM mass in the 10-15 keV range, if standard cosmology is assumed [119][120][121].However, notice that this bound is not directly applicable to the scenario discussed above, which is based on majoron production during an early matter-dominated era.A reconsideration of the Lyman-α bound for such an exotic cosmic history would be needed.For definiteness, in the following, we will regard freeze-in production of majorons with O(keV) mass as marginally compatible with structure formation.

Constraints on type-II majoron dark matter 4.1 Present status
Besides the conditions for a substantial production in the early universe discussed in the previous section, a minimal requirement for the majoron to be a viable DM candidate is its stability on cosmological time scales.In other words, the majoron lifetime τ J should at least exceed the age of the universe t 0 ≈ 13.8 Gyr.τ J can be calculated from the decay rates given in Section 2.2.For a majoron with M J ≪ m e but still heavier than neutrinos, the decays into the three pairs of neutrino eigenstates dominate.From Eq. (2.18), one then gets for the majoron lifetime in this regime: (4.1) As we can see, for a given majoron mass, the requirement τ J > t 0 translates into a lower bound to v 1 .In other words, only a very large lepton-number breaking scale v 1 can make the majoron long-lived enough for it to be a DM candidate, which, together with the constraints on v 2 and v 3 from Eqs. (2.3) and (2.4), implies the vev hierarchy anticipated in Eq. (2.6).
A somewhat stronger model-independent constraint to the lifetime of majoron DM can be obtained from observations of the cosmic microwave background (CMB), because the late time decay of the DM particles would affect the density fluctuations and the spectrum of the CMB even if the decay products are invisible [122,123].Taking into account some tensions with other cosmological data, the most conservative limit is τ J > 63 Gyr [123].Unlike decays into neutrinos, majoron decay rates into γγ and e + e − have a substantial dependence on v 3 , scaling as ∼ v 4  3 .On the one hand, these modes can be suppressed for small values of v 3 , since the majoron decouples from fermions in the limit v 3 → 0, see Eq. (2.15).On the other hand, if v 3 is sizeable, they dominate and entail much stronger constraints on the majoron DM parameter space.For instance, the CMB bound on DM particles decaying into photons or electrons corresponds to lower limits on the partial lifetimes of the order of 10 24 − 10 25 s [124], while from Eq. (2. 19) we find that our model predicts (4.2) The above cosmological bounds are summarised in Figure 4 where the regions of the M J − v 1 plane excluded by various searches and constraints are shown for different values of v 3 : the portion of the plane corresponding to τ J < t 0 and that excluded by the CMB data [123,124] are indicated in blue. 13Evidently, the CMB constraints are particularly strong for large rates of J → γγ (that require M J ∼ m e and sizeable v 3 ) and if J → e + e − is kinematically allowed (and, again, v 3 is large enough to provide a substantial coupling of the majoron with electrons), reaching values of v 1 as large as ∼ 10 16 GeV.
The oblique lines in Figure 4 indicate where on the M J − v 1 plane the observed DM relic density is achieved, that is, Ω J h 2 = 0.12, through the two production mechanisms discussed in Section 3. The black line corresponds to the misalignment mechanism occurring in the radiation-dominated epoch with θ 0 = 1.As we can see from Figure 1, all of the parameter space above the black line could be compatible with the observed relic density either due to the standard misalignment mechanism or through misalignment during early matter domination.The yellow lines correspond to the freeze-in mechanism with M ∆ = 400 GeV during a radiation-dominated era (solid line) and an early matter-dominated era with T R = 20 GeV (dashed line).In other words, the whole region of the parameter space below the solid yellow line can yield Ω J h 2 = 0.12 through freeze-in with an appropriate lowscale value of T R .However, as we can see, the freeze-in mechanism ceases to provide sufficient DM production for v 3 ≲ 0.1 GeV.The vertical green line indicates the approximate lower bound on the mass of freeze-in DM from Lyman-α observations, M J ≳ 10 keV [119][120][121].As discussed at the end of the previous section, such a constraint can not be straightforwardly applied to freeze-in during an early matter-dominated epoch (dahsed yellow line) -although one should not expect that a majoron with M J ≪ O(keV) be viable.Furthermore, it is well known that mass limits of this kind have no meaning in the case of misalignment-produced ALPs that, being extremely non-relativistic, do not affect structure formation and are thus allowed to be much lighter.
Provided an efficient production mechanism and the fulfillment of the above constraints from cosmological observations, the type-II majoron is a good DM candidate that can be In the dark blue region the majoron lifetime is shorter than the age of the universe ("τ J < t 0 "); the light blue region is excluded by CMB and other cosmological observations ("CMB"); the purple region is excluded by X-ray and gamma-ray data ("γ-rays"); bounds on majoron DM decaying into electro-positron are shown in red ("e + e − "); the grey regions are excluded by neutrino experiments: Borexino ("Bo"), KamLAND ("KL"), Super-Kamiokande ("SK").The orange region represents the direct-detection limit on DM coupling to electrons recently reported by XENONnT ("Xe-nT").The vertical green line is an illustrative lower bound from structure formation on the mass on freeze-in produced majoron DM ("Lyα").The yellow lines correspond to Ω J h 2 = 0.12 from freeze-in mechanism in a radiation-dominated era (solid) and an early matter dominated-era with T R = 20 GeV (dashed); the black line corresponds to the misalignment mechanism with θ 0 = 1.See the text for details and references.
searched for and further constrained by direct-detection and indirect-detection DM experiments.Let us start discussing the latter class of probes.Being a decaying DM particle, the type-II majoron will yield signals that, depending on M J , can be searched for by surveys of astrophysical X-rays/gamma-rays, cosmic-ray electrons and positrons, and neutrino lines.
Stringent constraints on light DM decaying into photons can be obtained from the measurements of the diffuse X-ray and gamma-ray galactic and extra-galactic backgrounds performed by a variety of satellite telescopes.For the range of DM masses that we are considering here, these bounds translate into lower limits on the partial lifetime τ γγ ≡ 1/Γ(J → γγ) in the range 10 26 − 10 29 s.The purple-shaded regions in Figure 4 show the the impact of these constraints on the type-II majoron parameter space.For M J up to O(MeV) we employed the limits on τ γγ collected in Ref. [125], including the bound from the INTEGRAL satellite as recently reevaluated in Ref. [126].For heavier majorons, we adopted the results of Ref. [127].
Above the kinematic threshold M J = 2m e , a distinctive signature of majoron DM stems from the decay J → e + e − .The red regions in Figure 4 are excluded by constraints on DM decaying into e + e − as obtained in Ref. [128] considering the cosmic-ray data of the Voyager satellite, and in Ref. [129] analysing the X-ray data collected by XMM-Newton.This latter bound stems from the upscattering of low-energy galactic photons due to the electron pair that would generate an X-ray emission.We find that it is more stringent than the Voyager limit with the exception of the mass window 16 MeV ≲ M J ≲ 40 MeV.
Constraints on DM decaying into neutrinos can be obtained from the fluxes of cosmic neutrinos [130] measured by experiments such as Borexino [131], KamLAND [132], Super-Kamiokande [133,134].Here, we employ the bounds that these experiments set on potential neutrino lines as evaluated in Ref. [39] -see also Refs.[135][136][137].Since the majoron field directly couples to neutrino mass eigenstates, the neutrinos produced in majoron decays do not undergo oscillations [39].Consequently, the flavour composition of the neutrino flux is the same at the source and at the detection point, and is given by (where U ℓi are entries of the PMNS matrix and ℓ = e, µ, τ ) such that the neutrino and antineutrino fluxes for the flavour ℓ are ∝ α ℓ Γ(J → νν) [39].Hence the quantities α ℓ control the relative importance of searches targeting different neutrino flavours and how they compare with the CMB bound that depends only on Γ(J → νν).In Figure 4, in order to compare the type-II majoron decay rates with the limits reported in Refs.[39,135] (with the Super-Kamiokande limit adapted from Ref. [130]), we assume normal ordering, m ν 1 ≪ m ν 2 < m ν 3 , and take the best-fit values of the oscillation parameters from Ref. [31].The impact of other possible choices for the neutrino parameters will be discussed in the next subsection.The resulting excluded regions are depicted in grey.Notice that, for values of v 3 ≲ 10 −3 GeV, the only searches that are currently sensitive to our majoron DM parameter space are those performed employing neutrino data.In other words, for small values of v 3 , the majoron DM phenomenology is dominated by the J → νν decays that do not depend on v 3 itself, 14 hence the bounds shown in the last panel of the figure apply to lower values of v 3 as well.Future prospects of all kinds of searches for decaying DM are discussed in the next subsection.
The electron-majoron interaction can also give rise to an electron recoil signal detectable at DM direct-detection experiments. 15The currently most sensitive search has been performed by XENONnT with an exposure of 1.16 ton×year and interpreted in terms of ALP DM with mass in the range 1 − 130 keV, yielding upper limits on the ALP coupling with electrons of the order 10 −14 − 10 −13 [25].In our case, from Eq. (2.15), we get for the majoron-electron coupling Hence, XENONnT is sensitive to keV-scale majoron DM evading the CMB bound -which requires v 1 > 10 7 GeV, as we have seen above -provided that v 3 > 1 GeV, a value close ).The oblique lines correspond to Ω J h 2 = 0.12 as in Figure 4. See the text for further details.
to the current upper limit from electro-weak precision observables, v 3 < 7 GeV.In Figure 4, the region excluded by XENONnT is shown in orange.As discussed in the previous section, the corner of the parameter space that direct-detection experiments are sensitive to can be compatible with the observed DM relic density only if our type-II majoron is produced through the freeze-in mechanism occurring during an early matter-dominated era (yellow dashed line).
It is well known that a keV-scale (or lighter) long-lived ALP emitted by electrons in stars would provide an extra cooling process affecting stellar evolution [148].On the other hand, present astrophysical observations are largely compatible with the standard cooling process due to the emission of neutrinos and disfavour ALP coupling with electrons larger than O(10 −13 ) [149][150][151].Hence star cooling bounds are currently weaker than the limit provided by XENONnT in the region of the parameter space we are considering.

Future prospects
Let us now move to consider the prospect of testing type-II majoron DM at future experiments.As we have seen above, searches for DM-electron scattering at direct detection experiments recently started to test a corner of our majoron paremeter space, for M J ∼ 1 − 10 keV and large values of v 3 , that is, close to the EWPO limit.In the left plot of Figure 5, we show the projected sensitivity of XENONnT as obtained by naively rescaling the expected limit reported in Ref. [25] according to the final exposure goal of the experiment -20 ton×year [152].
A comparable exposure is to be expected at analogous direct-detection experiments such as PandaX [26] and LZ [27].As we can see, the limit on v 1 will be improved by roughly a factor of two, up to v 1 ∼ 10 8 GeV.That direct detection experiments can be to any extent sensitive Figure 6: Current and future expected limits on the type-II majoron DM parameter space from neutrino experiments for different choices of the neutrino mass spectrum.First line: normal mass ordering (NO); Second line: inverted mass ordering (IO).The excluded blue regions and the Ω J h 2 = 0.12 line from misalignment are as in Figure 4. Present limits are shown in gray, future limits from JUNO and Hyper-Kamiokande ("HK") are indicated by dashed purple lines and orange lines respectively.See the text for details.
to this instance of ALP DM is a remarkable consequence of the fact that the lepton number is anomaly free, hence the majoron decouples from photons for M J ≪ m e , see Eq. (2.17).Otherwise, this corner of the parameter space would be completely excluded by X-ray constraints, as discussed in the literature addressing the (now excluded) excess observed in the XENON1T experiment [138] -see, in particular, Refs.[139,140].
In the right panel of Figure 5 we show the future constraints on decaying DM for v 3 = 1 GeV.Decreasing the value of v 3 , the limit on v 1 just scales as ∼ v 2  3 , until sensitivity is lost for v 3 ≲ 10 −3 GeV, as shown in Figure 4.The purple line shows the reach of future searches for DM decaying into photons.For small M J the most sensitive probe will be provided by the instruments aboard the THESEUS mission [153], while the GECCO telescope [154] is expected to give the best prospects for M J ≳ 0.2 MeV.For a compilation of the future limits on decaying DM in the X-ray to soft gamma-ray region, see Ref. [125].GECCO will also be sensitive to DM decaying into electrons [154].As the orange dashed line shows, this can yield a factor of ten improvement on the bound on v 1 for 1 MeV ≲ M J ≲ 10 MeV.
The prospects of future neutrino experiments -adapted from the results of Ref. [136] are displayed in Figure 6 by dashed lines: orange for Hyper-Kamiokande (HK) [155], purple for JUNO [156].As in Figure 4, the lifetime/CMB bounds and the current neutrino constraints are shown in blue and gray respectively.For the former constraints, we take τ J = 1/Γ(J → νν), that is, they are valid as long as v 3 ≲ 10 −3 GeV, as we can see from Figure 4, while the bounds set by neutrino experiments do not depend on v 3 .As we mentioned, for small enough values of v 3 , these are the only relevant limits on the parameter space of type-II majoron DM.In Figure 6 we show values of M J up to 1 GeV, hence, compared to Figure 4, we can additionally display a limit from the measurement of the atmospheric muon neutrino flux at Super-Kamiokande ("SK atm") as derived in Ref. [130]. 16The relative importance of this search compared to the other limits from neutrino experiments (based on observations of νe ) and the bounds on the majoron lifetime depends on the flavour composition of the the neutrino flux given by Eq. (4.3).In order to gauge the dependence of α ℓ on the neutrino parameters, we show in Figure 6 four different possible neutrino mass spectra, still setting the oscillation parameters to the best-fit values reported in Ref. [31]: normal mass ordering (NO) with m ν 1 = 0 or 0.04 eV; inverted mass ordering (IO) with m ν 3 = 0 or 0.04 eV. 17Notice that the largest value of the absolute mass scale is chosen so as to be (marginally) compatible with the cosmological upper limit i m ν i < 0.13 eV [157] The accidental suppression of the electron portion of the neutrino flux for NO with m ν 1 = 0 explains why, in the top-left panel of Figure 6, the Borexino limit barely extends beyond the CMB limit and, in the large mass regime, the future HK sensitivity does not seem to be able to improve over the limit from atmospheric neutrinos.As we can see from the other plots, this is not the case for the other three considered spectra, for which, in particular, Hyper-Kamiokande may improve the current bound by about one order of magnitude.
Let us recall that, above the black lines in Figures 5 and 6, the whole parameter space can be compatible with the observed DM relic abundance through the misalignment mechanism.These figures thus show that depending on M J , v 3 , and the details of the production mechanism, all of the three considered decay modes (γγ, e + e − , νν) of majoron DM particles could yield observable signals at upcoming experiments. 16Notice that MJ = 1 GeV is above the kinematic thresholds of majoron decays into muons and hadrons, J → µ + µ − and J → πππ.Nevertheless, we checked that the resulting constraints are still subdominant as long as v3 ≲ 10 −4 GeV. 17 We remind that for NO, the spectrum is such that , where ∆m 2 atm and ∆m 2 sol are respectively the atmospheric and solar neutrino mass splittings.

Summary and conclusions
In this study, we have worked within the context of type-II seesaw, which is perhaps the most economical model to address the origin of neutrino masses, one of the outstanding questions in particle physics.Besides providing a simple UV completion to the neutrino Majorana mass terms, type-II seesaw enjoys other theoretically and phenomenologically desirable features.
To name a few, the triplet scalar in type-II seesaw can account for the observed baryon asymmetry through leptogenesis [158,159] and it can also play a fundamental role in gauge coupling unification within the context of minimal grand unified theories [93,[160][161][162][163][164][165][166].
Here, we considered a minimal extension of the type-II seesaw mechanism that dynamically addresses the breaking of the lepton number by introducing an additional scalar singlet [45,46].We showed that the resulting PNGB, the type-II majoron, is an excellent dark matter candidate, thus adding the nature of dark matter to the number of outstanding problems that type-II seesaw can account for.We performed the first systematic study of the production of type-II majoron DM in the early universe and its possible signals at direct and indirect detection experiments.The latter searches can be sensitive to our model, because the type-II majoron is an instance of DM that, depending on its mass, decays into e + e − , γγ, νν.
We have shown that type-II majorons can account for the measured DM relic abundance in its entirety, if produced through either the freeze-in mechanism or the misalignment mechanism.Freeze-in production can occur through the decay of the heavy states belonging to the scalar triplet into a majoron and a SM Higgs or gauge boson, see Eq. (3.4), which requires a triplet mass M ∆ ≲ 1 TeV, hence light enough to be tested at the LHC and/or future colliders.Provided that, freeze-in production can be effective, while the model can evade at the same time constraints on decaying dark matter, up to values of the lepton-number breaking vev v 1 of the order of 10 9 GeV if the triplet vev v 3 is O(1) GeV, see Figure 4.For lower values of v 1 , majoron relic density can be made consistent with cosmological observations either by decreasing v 3 or by taking a low value of the reaheating temperature T R , which would imply freeze-in production taking place during an early matter dominated era so causing a dilution of the final relic abundance, as extensively discussed in Section 3. In any case, Figure 4 shows that, below v 3 ≃ 0.1 GeV, freeze-in production ceases to be viable in particular due to CMB constraints.This latter bound, in combination with lower limits on the DM mass from structure formation, also implies that majoron DM requires v 1 ≳ 10 7 GeV.Provided that the above conditions are fulfilled, the freeze-in mechanism can effectively produce type-II majoron DM for majoron masses in the range 1 − 100 keV.
Misalignment production of type-II majorons can account for the entirety of the observed DM relic density while being compatible with bounds on decaying DM for v 1 ≳ 10 10 GeV.Below that value, majorons are always a subdominant DM component (unless another production mechanism is at work, such as freeze-in), while above it the majoron relic abundance can match the observed one if one decreases the value of the initial misalignment angle θ 0 or, again, if DM production occurs during an early matter dominated era, which requires a low value of the reheating temperature, cf.Section 3 and Figure 1.
In Section 4, we have extensively discussed the constraints on and discovery prospects of the type-II majoron in the regime where it is a viable DM candidate, as following from its decay modes and coupling with electrons.For small values of the triplet vev, v 3 ≲ 10 −3 GeV, the majoron phenomenology is dominated by its couplings with neutrinos, because those with other fermions (and consequently photons) are suppressed by a factor ∼ v 2 3 /v 2 2 ≃ v 2 3 /v 2 EW .In this regime, type-II majoron DM is subject to constraints from neutrino experiments and its decay J → νν can be a target for searches of monochromatic neutrino lines at upcoming neutrino telescopes as long as M J ≳ 10 MeV -see Figure 6.If v 3 > 10 −3 GeV, the decays J → e + e − and J → γγ can give rise to observable signals at future X-ray and soft gammaray probes such as GECCO [154], for values of M J as low as M J ≈ 10 keV, see Figures 4  and 5 (right).For lower majoron masses, we found a corner of the parameter spacev 1 ≈ 10 7 − 10 8 GeV, 1 keV ≲ M J ≲ 10 keV, v 3 > 1 GeV, a regime suitable for freeze-in DM production -where the type-II majoron can give an electron recoil signal observable at direct detection experiments such as XENONnT [25], see Figure 5 (left).This is a consequence of the fact that, being the lepton number free of electromagnetic anomalies, majorons enjoy suppressed coupling with photons for M J ≪ m e , which makes them a plausible target for direct detection experiments in a regime where other ALP DM candidates are excluded by X-ray data [139,140].
Finally, another distinctive feature of the type-II majoron that we found is that its interactions with SM fermions are flavour conserving, being inherited from mixing with the Higgs doublet.This gives rise to a different phenomenology compared to other ALPs of cosmological interest (including the majorons from type-I seesaw [91] and type-III seesaw [167]) that are instead tightly constrained by searches for two-body flavour-violating decays of mesons or leptons into an invisible ALP a, such as K → πa and µ → ea, see Refs.[18,92,96,125,[168][169][170].Hence flavour processes of such kind are not only a promising avenue to search for a wide class of ALP DM candidates but they can also provide a handle for model discrimination in case of positive signals.
where v 1,2,3 are the vacuum expectation values, R 1,2,3 and I 1,2,3 are the real and imaginary part of the neutral components.The triplet scalar carries lepton number −2 so that it can have a Yukawa interaction with two lepton doublets, cf.Eq. (A. 19).The doublet scalar, which is to be identified with the SM Higgs scalar, does not have lepton number charge.The lepton number assignment of the singlet scalar is +2 so that the σΦ T ∆Φ term becomes invariant under the U (1) lepton number transformation.In this setup, the lepton number is spontaneously broken by the vacuum expectation value of the singlet field, which provides a Nambu-Goldstone boson, the majoron, in the imaginary component of the singlet field. 19nvariance under the SM gauge symmetries and the lepton number determine the scalar potential as follows The mass matrices of the component fields follow from the quadratic terms in the potential: where the singly-charged scalars are labelled as and the mass matrices are (A.9) The mass eigenstates, H i , A i , H ± i , and H ±± , are given in terms of the mixing matrices O as The matrices O I and O ± can be explicitly expressed in terms of the vevs as where In the mass eigenbasis specified by the mixing matrices in Eq. (A.11), the Nambu-Goldstone bosons and the Higgs bosons are identified as follows: where J is the majoron field, G 0 and G ± become the longitudinal components of the weak gauge bosons, and A and H ± are CP-odd and charged Higgs bosons with mass Since the rank of the mass matrix M 2 R is three in general, there are three CP-even Higgs bosons.We are working in the small-mixing regime (β 2,3 ≪ 1 and κ ≪ 1), (O R ) ia ≃ δ ia ,20 and therefore, H 1 mainly consists of the singlet field, with mass M H 1 ≃ 2β 1 v 2 1 , and is decoupled from the other scalars.In addition, H 1 does not participate in any gauge interaction, hence it remains out of equilibrium during the whole history of the universe.The 125 GeV Higgs boson is identified to H 2 , while H 3 is an extra CP-even Higgs boson, which has a mass of Since the extra Higgs bosons except for H 1 , i.e., H 3 , A, H ± , and H ±± , are made dominantly of the triplet field ∆, and their masses stem mainly from a common origin, which is the κ term in Eq. (A.3), their masses are of a similar size: The gauge interactions of the scalars are listed in the appendix of Ref. [61].We will give the necessary scalar cubic and quartic interactions.The Yukawa interactions in this model are given as where ℓ R is a right-handed lepton singlet, Q is a quark doublet, u R and d R are up-and down-type quark singlets, respectively, and iσ 2 is an anti-symmetric tensor for the indices of SU (2) L fundamental representations.This provides the fermion masses and the interactions among the scalars and the fermions, where the masses of the fermions are given by in terms of the diagonal matrices Y diag X , obtained by diagonalising the Yukawa matrices in Eq. (A. 19).
Although the anomaly contribution to the Jγγ interaction cancels, the majoron in this model has an interaction with photons through (i) majoron-pion mixing and (ii) triangle diagrams that depends on the mass of the fermions in the loop.The contribution (i) is calculated in Sec.7 of Ref. [89].The coefficients in the formula Eq. ( 92) in Ref. [89] correspond to the parameters of the type-II majoron model as where c is the combination of vevs given in Eq. (A.12).The contribution (ii) is discussed in Sec. 4 of Ref. [89].For majorons with a mass in the range that we are interested in, only the contributions from first-generation fermions are relevant.The coefficients for the up-and down-type quarks are given above.For the electron-loop contribution, we need the coefficient c ee , which is given as 21 With the majoron interactions in Lagrangian Eq. (A.20), we obtain B1(τ f )−1 instead of B1(τ f ) in Eq. (47) of Ref. [89] as a contribution from the loop of the fermion f [171].However, the "−1" part cancels summing over the contributions from different fermions in each generation, because the lepton number is anomaly free.In the final expression Eq. (A.27), we only have B1(τ f ) as in Ref. [89], which correctly decouples in the heavy fermion limit, m f ≫ MJ . 22Depending on the mass hierarchy of the triplet scalars, H3 → AJ or A → H3J also produce majorons.However, the masses are almost degenerate and the decay rates are suppressed in comparison to the main production processes in Eq. (B.6).
where the cubic coupling g H i AJ is given as (B.5) In the parameter region we are interested in, the decay rates of Eqs.(B.1)-(B.3)can be approximated to which are the main engine of the freeze-in production of majorons.The decay rate of H 3 → JJ is significantly smaller than them.The SM Higgs boson also decays into majoron(s), and the decay rates are They only contribute to the majoron production sub-dominantly.The relic density of majorons produced through B 1 → B 2 J and B → JJ results [111] Ω where g * (T ) is the relativistic degrees of freedom at a temperature T , and g * s (T ) is the effective degrees of freedom for entropy at T .Following the procedure developed in Sec.6.3 in Ref. [111], we can calculate the relic density of majoron produced from 2 → 2 scattering processes with the scalars ∆, Φ 1 , Φ 2 , and J, where ∆ and Φ i denote the components of triplet and doublet scalar fields, respectively.Here we have to take into account the masses of the bath particles, which are neglected in the calculation in Ref. [111].Setting the mass M ∆ of the heaviest particle in the process as the lower limit of the momentum in the integration, the Boltzmann equation reads where K 1 (x) is the modified Bessel function of the first kind, and P ij is a function introduced in Ref. [111], which is defined as With Y J (T eq ) = Y J (T 0 ), the relic density resulting from the scattering processes is23 where s 0 , ρ c0 , and T 0 are the entropy, the critical density, and the temperature today.
The terms in the scalar potential, which contribute to the scattering processes are Freeze-in DM production in an early matter-dominated era is discussed in Ref. [117,118].Following Sec.B.2 in Ref. [118], here we derive the formula of the relic density of majoron produced from the decay process B 1 → B 2 J.Here we start with Eq. (B.20) in Ref. [118], which is the quantity X J ≡ n J a 3 at the reheating temperature T R ,24  where a R is the scale factor at T R and ρ M (a) is the energy density of the matter component, which are explicitly given in Appendix C. The collision term C B 1 →B 2 J (a) of the process B 1 → B 2 J is calculated to be and the temperature T is related to the scale factor a as T = T R a 3/8 R /a 3/8 in the early matter-dominated era, cf.Eq. (C.6).Substituting this into Eq.(B.18), we have

C Early matter-dominated era
The cosmology of an early matter-dominated era is discussed in, e.g., Refs.[117,118].Here we derive the equations and relations that we used in our discussion on the freeze-in production of majorons in the early matter-dominated era.Suppose that a non-relativistic matter field dominates the energy budget of the universe before the radiation-dominated era, and the matter field gradually decays to radiation and injects entropy into the universe -an example being provided by the oscillations of the inflaton field during the epoch of its decay.Energy-momentum conservation (T total ) µν ;µ = 0 determines the time-evolution of the energy densities of the matter and radiation components, ρ M and ρ R , as where Γ M is the decay rate of the matter particle and H ≡ ȧ/a is the Hubble parameter.
Since the decay of the matter field progresses gradually and the evolution of ρ M is dominantly described by the dilution due to the expansion of the universe, the matter component evolves as where ρ M in and a in are the "initial" values of the energy density of the matter component and the scale factor, i.e., the values at the time t in when the early matter-dominated era began.
In the evolution of the radiation component, the source term from the decay of the matter field is important, and the solution of the differential equation Eq. (C.where ρ Rin is the initial value of ρ R .Here we used the Friedmann equation in the early matter-dominated era. Since the radiation species are in thermal equilibrium and follow the statistical distribution, ρ R ∝ T 4 -more concretely, ρ R = (π 2 /30)g * (T )T 4 -from Eq. (C.4) we find ρ R ∝ a −3/2 in the early matter-dominated era.Combining these two observations, we can find that the temperature T is related to the scale factor a as T = T R a in the early matter-dominated era.Using the relation Eq. (C.6), they can be rewritten as functions of a, which are used in our calculations of the relic density of majoron.
The scale factor a R can be given as a function of T R with the relation T a =const, which is valid in the radiation-dominated era.The radiation-dominated era starts at T R and ends at T eq which is the temperature of the matter-radiation equality era.With a eq = 1/3400 and T eq = 0.75 eV, we have it is convenient to give T R as a free parameter.

Figure 3 :
Figure3:[Left] Relic densities of majoron with the standard freeze-in mechanism (black) and the freeze-in in the early matter-dominated era (orange).The reheating temperature is set T R = 30 GeV for the orange curve.The bound from extra Higgs boson searches at the LHC is indicated with the label "LHC excl".In the region indicated as "J thermalised", the scalar quartic interactions from the κ term thermalise at T = M HSM .This bound is denoted by a black arrow for the standard freeze-in case, and by an orange for the freeze-in in the early matter domination.[Right] Contour on which the correct relic density is reproduced in the plane of the triplet scalar mass M ∆ and the reheating temperature T R .Above the blue dashed curve, the κ quartic interaction comes into equilibrium and majorons are thermally produced.

. 10 )
cf. Eqs.(C.5) and (C.7) in Appendix C. By setting T R to be smaller than the masses of the scalars, the Hubble parameter is enhanced in comparison with the standard case, and the Gamow's criterion of thermalisation can be fulfilled with a larger value of κ.The values of κ and T which fulfill Γ = H with the reheating temperature T R = {50, 100} GeV are shown as dashed curves in Figure2: for those values of T R the light blue region would retreat above the dashed lines.

Figure 4 :
Figure4: Bounds on type-II majoron DM on the M J − v 1 plane for different values of v 3 .In the dark blue region the majoron lifetime is shorter than the age of the universe ("τ J < t 0 "); the light blue region is excluded by CMB and other cosmological observations ("CMB"); the purple region is excluded by X-ray and gamma-ray data ("γ-rays"); bounds on majoron DM decaying into electro-positron are shown in red ("e + e − "); the grey regions are excluded by neutrino experiments: Borexino ("Bo"), KamLAND ("KL"), Super-Kamiokande ("SK").The orange region represents the direct-detection limit on DM coupling to electrons recently reported by XENONnT ("Xe-nT").The vertical green line is an illustrative lower bound from structure formation on the mass on freeze-in produced majoron DM ("Lyα").The yellow lines correspond to Ω J h 2 = 0.12 from freeze-in mechanism in a radiation-dominated era (solid) and an early matter dominated-era with T R = 20 GeV (dashed); the black line corresponds to the misalignment mechanism with θ 0 = 1.See the text for details and references.

Figure 5 :
Figure 5: Prospected sensitivity to the type-II majoron of direct (left plot) and indirect (right plot) DM searches for two illustrative choices of v 3 .The current XENONnT bound is shown as an orange region, while the gray-shaded areas are excluded by the other constraints displayed in Figure 4.The dark orange dashed line depicts the sensitivity of XENONnT with the planned exposure of 20 ton×year.The purple (orange) dashed line shows the impact of future constraints on DM decaying into γγ (e + e − ).The oblique lines correspond to Ω J h 2 = 0.12 as in Figure 4. See the text for further details.
B.12)    with s denoting the centre-of-mass energy.Defining Y J (T ) ≡ n J (T )/s(T ), we can write the Boltzmann equation asY J (T eq ) = 3|κ| 2 128π 5 M ∆ ∞ Teq dT T 2 K 1 (M ∆ /T ) H(T )s(T ) , (B.13)where s(T ) is the entropy density at T , not the centre-of-mass energy.In the radiationdominated era T > T eq -where T eq is the temperature of the epoch of matter-radiation equality -the Hubble parameter is density is expressed in terms of the effective degrees of freedom as s(T ) = 2π 2 45 g * s (T )T 3 .(B.15)
a R and T R are the scale factor and the temperature when ρ M = ρ R .We call T R reheating temperature.With the relation Eq. (C.6), we can rewrite the evolution of ρ M Eq. (C.3) as a function of T ; The matter density is diluted due to the expansion of the universe as ρ M ∝ 1/a 3 , which means ρ M ∝ T 8 .From the temperature-dependence of ρ M and ρ R and the definition of the reheating temperature, we find ρ R (T ) = π 2 30 g * (T )T 4 , ρ M (T ) = π 2 30 g * (T )