A cosmological sandwiched window for lepton-number breaking scale

A singlet majoron can arise from the seesaw framework as a pseudo-Goldstone boson when the heavy Majorana neutrinos acquire masses via the spontaneous breaking of global U(1) L symmetry. The resulting cosmological impacts are usually derived from the effective majoron-neutrino interaction, and the majoron abundance is accumulated through the freeze-in neutrino coalescence. However, a primordial majoron abundance can be predicted in a minimal setup and lead to distinctive cosmological effects. In this work, we consider such a primordial majoron abundance from relativistic freeze-out and calculate the modification to the effective neutrino number N eff. We demonstrate that the measurements of N eff will constrain the parameter space from a primordial majoron abundance in an opposite direction to that from neutrino coalescence. When the contributions from both the primordial abundance and the freeze-in production coexist, the U(1) L -breaking scale (seesaw scale) f will be pushed into a “sandwiched window”. Remarkably, for majoron masses below 1 MeV and above the eV scale, the future CMB-S4 experiment will completely close such a low-scale seesaw window for f ∈ [1,105] GeV. We highlight that any new light particle with a primordial abundance that couples to Standard Model particles may lead to a similar sandwiched window, and such a general phenomenon deserves careful investigation.


Introduction
Over the past few decades, cosmological observations have reached unprecedented precision, which offers a promising avenue for probing new physics beyond the Standard Model (SM).For instance, nowadays the measurements of the effective number of neutrino species N eff have reached the accuracy of O(0.1) through the observations of big bang nucleosynthesis (BBN) and cosmic microwave background (CMB) [1,2].In general, any new light particles coupling to the SM particles might contribute to the relativistic degrees of freedom in the early universe and lead to significant deviations from the SM prediction of N eff , and therefore will receive strict constraints from cosmology.
In this work, we revisit the majoron-neutrino interactions under the cosmological constraints of N eff and study their possible implications for the low-scale seesaw scenario.A singlet majoron J can naturally arise as a pseudo-Goldstone boson [3] in the framework of the type-I seesaw model [4][5][6][7][8], where the heavy Majorana neutrinos acquire masses through the spontaneous breaking of the global U(1) L lepton-number symmetry.The lepton-number breaking scale f characterizes the mass scale of heavy neutrinos, i.e., the scale of new physics responsible for the origin of neutrino masses.After the electroweak gauge symmetry breaking, the majoron will couple with the SM neutrinos ν i through the mixing between active and sterile neutrinos, where the couplings are suppressed by m i /f (with m i the mass of ν i ).Therefore, the majoron is expected to be long-lived due to the large hierarchy between m i and f , which makes the majoron an interesting candidate of dark matter (DM) [9][10][11][12][13][14][15][16][17]. 1  For a relatively low breaking scale f ≲ 100 TeV, which corresponds to the low-scale seesaw scenario, the decay of the majoron into a pair of SM neutrinos could occur and contribute to N eff at some crucial epochs of the Universe, leaving observable imprints in, e.g., the BBN and CMB.Given the strict constraints on N eff from current and upcoming CMB experiments [1,[22][23][24][25], it is hopeful that we could probe the mechanism for neutrino mass generation from observables in the early Universe.
The constraints on the lepton-number breaking scale from N eff have been noticed for a while [26][27][28][29][30], mostly in a model-independent setup.In previous works, the primordial abundance of the majoron was usually neglected such that the majoron abundance was initialized via the effective interaction between the active neutrinos and the majoron.In particular, for a light majoron below the MeV scale, the majoron abundance is accumulated through the freeze-in 2ν → J process where the majoron-neutrino coupling is sufficiently small.In this case, the primordial majoron abundance is negligible before the SM neutrino decoupling, and the decay of the majoron into a pair of neutrinos J → 2ν after the SM neutrino decoupling will inject energies into neutrinos, causing an excess of N eff .Consequently, a smaller breaking scale f (equivalent to a larger majoron-neutrino coupling) brings about a larger majoron abundance, thereby leading to a larger deviation of N eff .Therefore, the constraints from the N eff measurements will put a lower bound on f in the freeze-in situation.
However, a primordial majoron abundance generated beyond the effective majoronneutrino interaction in general cannot be neglected before the SM neutrino decoupling.In fact, it can be predicted even in a minimal setup and more importantly, it may have a sizable effect to N eff .In this work, we investigate such an effect that has usually been neglected in the literature.We focus on the situation where the majoron is assumed to be in thermal equilibrium with SM particles in the early Universe, such that the primordial majoron abundance is inherited from relativistic freeze-out.As can be seen later, this situation is easy to realize, either via the interactions between the majoron and the heavy Majorana neutrinos or through the majoron-Higgs interactions in the scalar sector.We will also consider the situation where the primordial majoron abundance is accumulated by some other mechanism beyond the relativistic freeze-out.
The primordial majoron abundance will lead to several interesting phenomena.First of all, for a non-negligible primordial abundance, if it exists before the SM neutrino decoupling but is only depleted into radiation near the recombination epoch, N eff will be drastically increased by orders of magnitude (see Sec. 3.2).Therefore, the precision measurements of 1 If the U(1)L global symmetry is only broken spontaneously, then the majoron will be strictly massless as a real Nambu-Goldstone boson.However, to serve as a DM candidate, the majoron should acquire a nonzero mass where the global symmetry is broken explicitly [11,[18][19][20][21].
N eff at the CMB epoch will severely constrain the abundance, and put strict bounds on the ultraviolet (UV) physics that features the abundance.Furthermore, for nonrelativistic majoron decay, a larger breaking scale f leads to later decay, and hence a larger modification to N eff .So the constraints from N eff will put an upper bound on f , in contrast to the freeze-in situation.This is particularly interesting because when the freeze-in and primordial abundances coexist, the constraint from the N eff measurements will push f into a sandwiched window.Remarkably, the next-generation CMB experiments could further narrow or completely close such a sandwiched window [22][23][24][25].
The remaining part of this paper is organized as follows.In Sec. 2, we start with a brief review of the singlet majoron model.Then we perform a general analysis of the majoron evolution in the early Universe and its cosmological impacts.In Sec. 3, we derive N eff using an approximate analytical (but intuitive) method by assuming instantaneous majoron decay.A stricter calculation of N eff beyond instantaneous majoron decay is conducted in Sec. 4. The constraints on f from N eff are given in Sec. 5, which provide a cosmological sandwiched window for the low-scale seesaw scenario.We summarize our main results in Sec. 6.Finally, some technical details are included in the appendices.

The singlet majoron model
The singlet majoron model [3] introduces a complex scalar S, which is a singlet under the SM gauge symmetries.The relevant Lagrangian in the Yukawa sector is given by where ℓ L = (ν L , l L ) T and Φ ≡ iσ2 Φ * are the left-handed lepton doublet and the Higgs doublet, respectively.Y ν is the Dirac neutrino Yukawa coupling matrix, and Y N is the Yukawa coupling matrix for the right-handed (RH) neutrino singlets N R , where T has been defined with C = iγ 2 γ 0 the charge-conjugation matrix.
The Lagrangian in Eq. (2.1) owns a global U(1) L symmetry if we assign the lepton numbers of relevant particles to be: L(ℓ L ) = L(N R ) = +1, L(Φ) = 0, and L(S) = −2.This global symmetry is spontaneously broken after S acquires a non-zero vacuum expectation value (VEV) from the scalar potential (see Appendix A.2 for the discussion about a general scalar potential obeying the lepton-number conservation) In this work, we adopt the linear realization of the broken symmetry.That is, the complex scalar is parametrized as where ρ and J are two real degrees of freedom.In addition, f is the scalar VEV, corresponding to the lepton-number breaking scale, which provides a Majorana mass term M R = Y N f / √ 2 for the RH neutrinos.Note that f is also the seesaw scale provided O(Y N ) ≃ O(1), i.e., f ≃ O(M R ), which is usually considered as the UV completion of the canonical seesaw model [4][5][6][7][8].
From Eq. (2.2), it is easy to show that ρ will acquire a mass proportional to f after the lepton-number breaking while J remains massless.Therefore, the pseudo-scalar J (i.e., the majoron) is identified as the Goldstone boson of the spontaneous U(1) L symmetry breaking.In practice, a non-zero majoron mass m J can be generated by adding terms that explicitly violate the U(1) L symmetry, at either tree or loop levels [11,[18][19][20][21].A simple case of mass generation is discussed in Appendix A.2.In the following analysis, we will simply treat the majoron mass as a free parameter.
After the breaking of the electroweak gauge symmetry, the majoron can interact with active neutrinos through the flavor mixing between active and sterile neutrinos.The general interaction in the mass basis turns out to be [31] A detailed derivation is presented in Appendix A.1.In Eq. (2.4), n i denotes the neutrino mass eigenstate with mass m i , and the indices i = 1, 2, 3 (i = 4, 5, 6) correspond to the active (sterile) neutrino species.The mixing parameters C ij ≡ 3 k=1 U * ki U kj are defined by the 6 × 6 unitary matrix U which diagonalizes the neutrino mass matrix via with M D = Y ν v/ √ 2 the Dirac neutrino mass matrix and v ≃ 246 GeV the electroweak VEV.Note that O (M D ) ≪ O (M R ) is required to generate the tiny masses for active neutrinos through the seesaw mechanism.
To obtain the interaction between the majoron and the active neutrinos, one can simply take i, j = 1, 2, 3 in Eq. (2.4) and identify n i ≡ ν i (for i = 1, 2, 3).Then it follows that C ij ≃ δ ij , where the non-diagonal elements are suppressed by O M 2 D /M 2 R , leading to Therefore, the interaction between the majoron and the active neutrinos is approximately diagonal and is suppressed by feeble majoron-neutrino couplings g ν i ≡ m i /f .For the mass region of 1 eV ≲ m J ≲ 1 MeV which will be considered in this work, the majoron can decay to two active neutrinos, where the decay width is given by -4 - Figure 1.The decaying temperature of the majoron T J with different values of the majoron mass m J and the lepton-number breaking scale f .Some typical epochs of the majoron decay are shown in the plot where: the majoron is stable over the cosmological time scale τ U ≃ 4.4×10 17 s (above the blue long-dashed line), the majoron decays after the epoch of the matter-radiation equality T eq ≃ 1 eV (above the green short-dashed line), the majoron decays before the SM neutrino decoupling epoch T ν,dec ≃ 0.1 MeV (below the red dotted line), and the majoron decays as a relativistic particle (below the black solid line).The yellow-shaded region is the parameter space which interests us in this work, i.e., the majoron decays nonrelativistically with the decaying temperature T eq ≲ T J ≲ T ν,dec .The lepton-number breaking scale in the yellow-shaded region resides in 1 GeV ≲ f ≲ 100 TeV, which corresponds to the low-scale seesaw scenario.
In addition, the majoron can also decay to two photons at two-loop level with the width scaling as [15] where α is the fine-structure constant.The di-photon decay mode is severely constrained by the CMB, X-, gamma-and cosmic-ray observations [12,13,15].In fact, for the lowscale seesaw scenario (i.e., f ≲ 100 TeV) considered in this paper, the Yukawa couplings Y ν are suppressed by the tiny neutrino masses (recall that the seesaw relation gives ), thereby making J → 2ν the dominant decay mode of the majoron.

Majoron cosmology in the low-scale seesaw scenario
Due to the large hierarchy between m i and f , the majoron is expected to be long-lived [cf.
For relatively small f , e.g., around the electroweak scale, the majoron becomes unstable within the cosmological time scale and will dominantly decays to active neutrinos.
To see when the majoron decays, let us consider the temperature T J of nonrelativistic majoron decay, which can be defined as Here H(T ) ≃ 1.66 g ρ (T )T 2 /M Pl is the Hubble parameter at the radiation-dominated epoch, with g ρ (T ) the relativistic degrees of freedom for energy density and M Pl ≃ 1.22 × 10 19 GeV the Planck mass.Moreover, the decay width of the majoron Γ J can be calculated by Eq. (2.7) as an approximation.Note that for relativistic majoron decay, the lifetime τ J in Eq. (2.9) is enhanced by an additional Lorentz factor E J /m J > 1. Combining Eqs.(2.7) and (2.9), we obtain that the nonrelativistic decaying temperature T J scales as where g ρ in the right-handed side of Eq. (2.10) should be calculated at T J .In Fig. 1, we show the plane of (m J , f ) where the lifetime of the majoron is longer than the age of the Universe τ U ≃ 4.4 × 10 17 s, and the parameter space where the majoron decays as a nonrelativistic particle (i.e., T J < m J ).We also show the values of m J and f that lead to the decaying temperature of T J = 1 eV and T J = 0.1 MeV, which typically corresponds to the epochs of matter-radiation equality T eq and the completion of the SM neutrino decoupling T ν,dec , respectively.It can be seen from Fig. 1 that in the region of nonrelativistic majoron decay with the decaying temperature T eq ≲ T J ≲ T ν,dec , the corresponding lepton-number breaking scale resides in f ∈ [1, 10 5 ] GeV (i.e., the yellow-shaded region in Fig. 1), which happens to be the low-scale seesaw scenario provided that the masses of RH neutrinos arise from the spontaneous lepton-number violation.Before calculating the contribution of majoron decay to N eff , we need to specify the early evolution of the majoron, which determines the primordial majoron abundance.Generally speaking, there are two possibilities that the majoron can keep thermal equilibrium with SM particles in the early Universe: • First, the RH neutrinos can readily be thermalized in the SM plasma via twobody/inverse decay or rapid active-neutrino oscillations [32,33].Before the electroweak gauge symmetry breaking, the majoron couples to RH neutrinos via where the mass eigenstates of the RH neutrinos N R + N c R ≡ (N 1 , N 2 , N 3 ) T have been defined, and M i is the mass of N i .Furthermore, for Yukawa couplings Y N that are not too small, the scattering 2N → 2J via Eq.(2.11) can in turn thermalize the majoron for the temperature T ≳ M i .Note that there is no decaying channel of N i → N j + J (with M i > M j ) since the non-diagonal elements in Y N are not physical before the gauge symmetry breaking.
• Second, the majoron can couple with the Higgs boson via the following interaction which is not forbidden by the global U(1) L symmetry and should be included into a general scalar potential (see Appendix A.2 for more details).Moreover, it was shown [34] that even for a portal coupling λ ΦS as small as O(10 −6 ), the scattering 2Φ → 2S can still thermalize the scalar S.After the gauge symmetry breaking, the majoron couples with the SM Higgs boson h through the mixing (induced by λ ΦS ) between two CP-even bosons h and ρ, and can be thermalized via h → 2J or 2h → 2J.
Therefore, a thermalized majoron in the early Universe can easily be realized in the minimal setup that exhibits a spontaneous global U(1) L breaking in the scalar sector and at the same time, predicts an unstable majoron decaying at T eq ≲ T J ≲ T ν,dec .Since the majoron is much lighter than the RH neutrinos and the Higgs bosons, it is expected to undergo relativistic freeze-out.The freeze-out temperature T fo of the majoron depends on the details of the UV models, and will be treated as an input parameter in the following discussions.
After the gauge symmetry breaking, the active neutrinos can also generate the majoron abundance through the neutrino coalescence process 2ν → J [cf.Eq. (2.6)].Due to the suppression of the majoron-neutrino coupling g ν i ≡ m i /f , the production channel follows the freeze-in evolution [35][36][37][38] and culminates at T ≃ O(m J ).Therefore, the majoron abundance is, strictly speaking, not a constant after the relativistic freeze-out.As can be seen in Fig. 1, for 1 eV ≲ m J ≲ 1 MeV and a suppressed coupling g ν i , the majoron is expected to decay after the SM neutrino decoupling.This late-time decay will modify the effective number of neutrino species N eff due to the energy injection to active neutrinos.Such effects have been studied in Refs.[26][27][28][29][30] where the constraints of N eff from the CMB measurements have been applied to derive the upper bound of the majoron-neutrino coupling in terms of the majoron mass.
Nevertheless, previous studies have mainly focused on the majoron-neutrino effective interaction, where the majoron abundance is generated by and then depleted back to the SM neutrinos. 3It is the purpose of this work to detail the effects of the primordial abundance on N eff .While we concentrate on the low-scale seesaw scenario, it is worthwhile to emphasize that the analysis performed in subsequent sections can analogously be applied to other UV scenarios, where a new light particle coupling to active neutrinos or photons has a nonnegligible primordial abundance before the SM neutrino decoupling and decays after that epoch.

Analytical derivation of ∆N eff from majoron decay
In this section, we perform an approximate analytical calculation of the N eff excess from majoron decay (i.e., ∆N eff ≡ N eff − N SM eff ) by assuming that the majoron decays instantaneously, where N SM eff ≃ 3.045 denotes the SM prediction of the effective number of neutrino species [39][40][41][42][43][44][45][46].The calculation depends on the kinematic properties (that is, relativistic or nonrelativistic) of the majoron at decay.In the first two subsections, we compute ∆N eff from relativistic and nonrelativistic majoron decay respectively, with the primordial majoron abundance inherited from the relativistic freeze-out.In the last two subsections, we compute ∆N eff from the freeze-in production followed by majoron decay, where there is no primordial majoron abundance.

Relativistic freeze-out and relativistic decay
In order to be model-independent, we do not specify the freeze-out process of the majoron from UV scenarios (i.e., the majoron will be thermalized via the scattering either with the RH neutrinos or with the Higgs boson); rather, we treat the freeze-out temperature T fo as an input parameter.Then the yield of the majoron at T fo is given by where with g s (T ) the relativistic degrees of freedom for entropy in the SM.Therefore, the majoron freeze-out yield depends only on the relativistic degrees of freedom at T fo .
The observations of the light-element abundances generated during the BBN era put stringent constraints on the interactions of neutrinophilic particles [47,48].Here we would like to estimate the contribution of the majoron to N eff at the BBN epoch.It is usually stated that the BBN process starts after the deuterium bottleneck temperature T ≃ 0.078 MeV [49][50][51].Nevertheless, any extra radiation before the neutron-proton freeze-out at T ≃ 0.8 MeV will modify the Hubble expansion rate, leading to more neutron abundance at freeze-out and hence more 4 He.Observations of the primordial helium-4 synthesized at the BBN epoch will constrain the extra radiation, which is effectively parametrized by ∆N eff [51].If the majoron decays after T ≃ 0.1 MeV, the majoron itself contributes to the Hubble expansion rate before the BBN starts.Then ∆N eff at the BBN epoch can be approximated by 4 where g s,BBN and g s,fo denote the relativistic degrees of freedom at T BBN and T fo , respectively, and is the one-flavor neutrino energy density with T ν = T γ before neutrino decoupling.For definiteness, we will take T BBN ≃ 1 MeV to denote the temperature before the neutronproton freeze-out.∆N eff calculated from relativistic majoron decay at the epochs of BBN or CMB with different freeze-out temperature T fo (black short-dashed line).The regions above different horizontal lines correspond to the 2σ excluded regions of ∆N eff from the constraints of BBN+Y p +D (green dotted line), Planck (red short-dashed line), as well as the future projected sensitivities of SO (magenta dot-dashed line) and CMB-S4 (blue dashed line).The gray solid line denotes the minimal ∆N eff ≃ 0.027, which corresponds to the freeze-out temperature T fo ≳ O(100) GeV.
Next, let us consider the situation where the majoron decays relativistically after the BBN ends and prior to the matter-radiation equality epoch T eq ≃ 1 eV.The energy from the majoron would then inject into active neutrinos, acting as extra radiation at the recombination epoch T ≃ 0.1 eV and being constrained by the CMB measurement.In the approximation of instantaneous decay, we arrive at where ρ J→2ν denotes the neutrino energy density inherited from relativistic majoron decay, and the factor T ν /T γ = (4/11) 1/3 has been used for the (instantaneous) neutrino decoupling.
In addition, we have assumed that the decay is completed before T eq so that ∆N eff is evaluated at T eq .Given g s,BBN /g s,CMB = 11/4,5 we arrive at . This identity implies that when the majoron decays in the relativistic regime, the extra radiation contributes equally to N eff at the epochs of BBN and CMB, which only depends on the freeze-out temperature.It is clear that a lower T fo leads to a larger ∆N eff .This can be understood as follows.After the relativistic freeze-out of the majoron, the decoupling of other SM particles will inject energies into the plasma of photons and neutrinos, thereby reheating the SM bath and enhancing ρ SM ν .Therefore, the later the majoron decouples, the less enhancement ρ SM ν will receive, which implies a larger ∆N eff .
In Fig. 2, we show the behavior of ∆N eff calculated from relativistic majoron decay at the epochs of BBN or CMB with different freeze-out temperatures T fo , where the evolution of g s (T ) is taken from Ref. [52].We also show the constraints on ∆N eff from the Planck 2018 result [1]: N eff = 2.99 ± 0.17, the combination of BBN, helium (Y p ) and deuterium (D) abundances [2]: N eff = 2.889±0.229,and from the future sensitivities of Simon Observatory (SO) [22,23] and CMB-S4 [24,25].The 2σ upper bounds are given by Planck : ∆N eff < 0.285 , CMB-S4 : ∆N eff < 0.06 . (3.9) It can be seen that for the majoron that decays relativistically after the neutrino decoupling and before the matter-radiation equality epoch, the current constraints from BBN+Y p +D and Planck requires T fo > 64 MeV and T fo > 104 MeV, respectively, while the future sensitivities from the SO and CMB-S4 will further limit the decoupling temperature to T fo > 192 MeV and T fo > 397 MeV, respectively.Note that a minimal ∆N eff ≃ 0.027 is expected for the early-time decoupling of the majoron, i.e., T fo ≳ O(100) GeV, where all the SM particles are relativistic with g s,fo = 106.75.

Relativistic freeze-out and nonrelativistic decay
Now let us turn to the more interesting scenario, where the majoron decays in the nonrelativistic regime.We expect that ∆N BBN eff ̸ = ∆N CMB eff in this case.The former is still given by Eq. (3.3) while the latter is calculated as where the energy injection from the majoron into neutrinos at T eq is estimated by the value of m J n J at T J [defined in Eq. (2.9)].Note that we do not include the contribution to the majoron abundance from the freeze-in process 2ν → J, so the yield Y n J keeps constant after freeze-out.
From Eq. (3.10) it is clear that later decay of the majoron will cause a larger ∆N eff at the CMB epoch.This can also be understood by the dilution-resistant effect [30] (see Sec. 3.4 for more discussions): After the decoupling of neutrinos, we have ρ SM ν ∝ a −4 with a the scale factor, while the majoron energy density in the nonrelativistic regime scales as ρ J ∝ a −3 .As a result, later decay of the majoron implies that ρ SM ν suffers from more redshift (dilution) than the energy of the nonrelativistic majoron ρ J (resistance), thereby leading to a larger ∆N eff after majoron decay.
A key observation is that, since later decay of the majoron implies a larger leptonnumber breaking scale f or equivalently a smaller majoron-neutrino coupling, and leads to a larger ∆N CMB eff , it implies that the constraint from ∆N CMB eff will put an upper bound on f .This is in contrast to the freeze-in scenario without a primordial majoron abundance, where a lower bound of f can be obtained from the constraint of ∆N CMB eff , as we shall discuss below.

Freeze-in without primordial majoron
In the low-scale seesaw scenario, where f is around the electroweak scale, the neutrino coalescence 2ν → J can also contribute to the majoron abundance via the freeze-in mechanism.In this subsection, we calculate the excess of N eff from freeze-in production of the majoron, and make a comparison with the results derived in the previous subsection with a primordial majoron abundance.
To simplify the calculation of ∆N eff , we assume that the freeze-in production of the majoron is essentially completed before it decays.We first consider the case of nonrelativistic majoron decay.The Boltzmann equation of the majoron abundance before decay is given by where Y n J,fi ≡ n J /s SM denotes the freeze-in abundance of the majoron and the collision term Let us assume again that the decay occurs instantaneously at T J after the neutrino decoupling.Then the excess of N eff at the CMB epoch coming from the nonrelativistic majoron decay can be calculated by where g s,T J denotes the relativistic degrees of freedom at the decaying temperature.Note that the nonrelativistic factor m J /T J ∝ m J /M Pl f /m i [cf.Eq. (2.10)] also appears here as in Eq. (3.10).
At first sight, it seems that a heavier majoron and a larger lepton-number breaking scale lead to a larger ∆N CMB eff , which occurs in the freeze-out case (see Sec. 3.2).However, ∆N CMB eff in Eq. (3.12) also depends on m J and f through Y n J,fi .To see it more clearly, we calculate Y n J,fi with the following approximations: • We take the Boltzmann approximation of the neutrino distribution functions so that the collision term C n 2ν→J is given by Eq. (B.6).
• To obtain the freeze-in abundance of the majoron, one needs to integrate Eq. (3.12) over the temperature.Since the majoron decays nonrelativistically after the neutrino decoupling, we can further expand the collision term in the integral in terms of m J /T for T > T ν,dec and in terms of T /m J for T < T ν,dec .
With above approximations, we arrive at a very simple result where g s,T J ≃ 3.36 has been used, and for concreteness, we have taken the normal ordering of the neutrino masses with m 1 = 0 and 3 i=1 m 2 i ≃ (0.0086 eV) 2 + (0.05 eV) 2 [53].From Eq. (3.13) it is now clear that for the freeze-in case, a lighter majoron and a smaller f will lead to a larger ∆N eff .Therefore, the constraint from CMB measurements will put a lower bound on the lepton-number breaking scale.This is in contrast to the case of freeze-out with a primordial majoron abundance, as already discussed in Sec.3.2.

Comparison between freeze-out and freeze-in
From Eq. (3.13), one may naively expect that ∆N eff can keep increasing with the decreasing of m J and f .However, a lighter majoron and a larger coupling (i.e., a smaller f ) would make the majoron-neutrino system easier to reach thermal equilibrium, when the freeze-in formalism and consequently the result in Eq. (3.13) are no longer applicable.In fact, as pointed out in Refs.[30,42], when the majoron gets thermalized with the SM neutrinos, one obtains a maximal ∆N eff ≃ 0.12.
Based on the analysis in Sec.3.3, we can understand the maximal ∆N eff in a different way. 6Recall that the result in Eq. (3.13) is derived in the regime of nonrelativistic majoron decay, i.e., p J < m J with p J the magnitude of the majoron momentum.The averaged momentum for a nonrelativistic thermalized majoron is given by ⟨p J ⟩ ≃ √ 3m J T J . 7By requiring ⟨p J ⟩ < m J and taking advantage of Eq. (2.10), we obtain a lower bound on f : where 3 i=1 m 2 i ≃ (0.0086 eV) 2 + (0.05 eV) 2 and g ρ (T J ) ≃ 3.36 have been used.Substituting Eq. (3.14) back to Eq. (3.13), we arrive at ∆N CMB eff ≲ 0.14.This confirms the result in Refs.[30,42] that nonrelativistic majoron decay from the freeze-in mechanism without a primordial abundance can lead to a maximal ∆N eff at O(0.1).
Another point worth mentioning is that there is no excess of N eff from the relativistic majoron decay in the freeze-in scenario.In Sec.3.1, we have demonstrated that the relativistic majoron decay with a primordial abundance can generate a nonzero ∆N eff .However, this is not the case if the majoron abundance only comes from the freeze-in production, which instead predicts a vanishing ∆N eff .To see it more clearly, we can rewrite the Boltzmann equation of the majoron-neutrino system as [30] where P J denotes the pressure of the majoron.In the relativistic regime, we have P J = ρ J /3, so the comoving energy density (ρ J + ρ ν ) a 4 is a constant.In this case, if there is no primordial majoron abundance, the energy density will first transfer from neutrinos to the majoron through the freeze-in process 2ν → J and then back to neutrinos through the relativistic majoron decay J → 2ν, while the total energy density of the majoron-neutrino system in a comoving volume remains unchanged.Therefore, there would be no excess of N eff if the majoron decays in the relativistic regime.
In conclusion, if there is no primordial majoron abundance, a nonzero ∆N eff can only be generated if the majoron decays in the nonrelativistic regime.In addition, a smaller leptonnumber breaking scale f and majoron mass m J lead to a larger ∆N eff , which has a maximal value at O(0.1).On the other hand, when there is a primordial majoron abundance, both the relativistic and nonrelativistic majoron decay at late times can generate a nonzero ∆N eff .In particular, for the case of nonrelativistic decay, a larger f and m J will predict a larger ∆N eff , contrary to the freeze-in case where there is no primordial abundance.To have a more precise constraint on the lepton-number breaking scale from observations of N eff , in the next section, we proceed to perform a stricter calculation of ∆N eff in the nonrelativistic region by going beyond the approximation of instantaneous majoron decay used in Sec.3.2.

Calculation of ∆N eff beyond instantaneous majoron decay
In this section, we carry out a stricter calculation of ∆N eff from nonrelativistic majoron decay, which has a primordial abundance inherited from the relativistic freeze-out.We also make a comparison with the freeze-in production 2ν → J followed by nonrelativistic decay J → 2ν.For later reference, we use the following shorthand: FONR ≡ relativistic Freeze-Out + NonRelativistic decay , FINR ≡ Freeze-In + NonRelativistic decay .
We are interested in the case where the majoron decays after the SM neutrinos have decoupled from the plasma at around T ν,dec ≃ 0.1 MeV [41][42][43][44][45][46]54], so as to suppress the nontrivial impacts on the BBN processes as well as the neutrino decoupling.Under this circumstance, the neutrinos generated from majoron decay at temperatures below T ν,dec can no longer be thermalized via the SM weak interactions.Therefore, in the FONR case, we can treat majoron decay separately from the SM thermal bath.
In general, the neutrino coalescence 2ν → J also contributes to ∆N eff in the FONR case.Nevertheless, we will neglect it in the calculation of ∆N eff for two reasons.First, as shown in Sec.3.4, the contribution from neutrino coalescence to ∆N eff can only reach up to O(0.1), which is beyond the sensitivity of current CMB measurements [1].Second, as can be seen from Sec. 3.2 and Sec.3.3, the dependence of ∆N eff on m J and f is opposite between the FONR and the FINR cases, so it would be more clear to treat the two processes separately.The inclusion of neutrino coalescence in the FONR case will be discussed in Sec. 5. Without including the contribution from neutrino coalescence, the Boltzmann equations governing the nonrelativistic majoron decay are given by where C ρ J→2ν is the collision term responsible for the energy transfer rate.In the nonrelativistic regime, we have C ρ J→2ν ≃ m J n J Γ J with Γ J ≃ Γ J→2ν and the primordial majoron abundance n J = ζ(3)T 3 /π 2 coming from relativistic freeze-out. Defining SM , we can rewrite the Boltzmann equations as which lead to the solutions with an initial temperature T ini ≫ T J and H ini ≡ H(T ini ).For definiteness, we take T ini = 0.1 MeV, which is consistent with the requirement of T J ≲ T ν,dec .It should be pointed out that a higher T ini does not modify the result significantly, since the decay mainly occurs around T J determined by Γ J ≃ 2H(T J ), and Y n J is exponentially suppressed for T < T J . 8In addition, the initial abundance is determined by the freeze-out abundance in Eq. (3.1), i.e., In Tab. 1, we have listed the correspondence between T fo and Y n J,ini for some typical freeze-out temperatures.Alternatively, one can also use the contribution of the relativistic majoron to ∆N eff at the BBN epoch to characterize the primordial abundance (i.e., the third line of Tab. 1).Note that the first five values of ∆N BBN eff in Tab. 1 can be obtained from T fo using Eq.(3.3).For ∆N BBN eff < 0.027, it cannot be inherited from the relativistic freeze-out mechanism, where the thermal plasma at T > O(100) GeV only contains the SM degrees of freedom.Nevertheless, we can still establish a one-to-one correspondence between ∆N BBN eff and the primordial majoron abundance Y n J,ini .
Since the nonrelativistic majoron decay occurs after the neutrino decoupling and before the matter-radiation equality epoch, when the relativistic species include photons and neutrinos, it is reasonable to treat the relativistic degrees of freedom g ρ (in the Hubble parameter) and g s (in the entropy density) as constants.Taking into account the temperature ratio between photons and neutrinos derived from noninstantaneous neutrino decoupling: T /T ν ≃ 1.3985 [42], one obtains ) Note that ∆N eff calculated above is based on Eq. (4.1), where only the contribution from majoron decay J → 2ν is included.On the other hand, the heavy sterile neutrinos N i could also contribute to ∆N eff through decay (N i → J + ν j ) or 2-to-2 scattering (2J → 2ν mediated by N i ).In the following, we show that the contributions from sterile neutrinos to ∆N eff are negligible compared with that from majoron decay.
• We first consider the contribution from sterile neutrino decay N i → J + ν j (for i, j = 1, 2, 3).Since the masses of sterile neutrinos satisfy M i ∼ f ≫ T ν,dec , the decay occurs much earlier than the decoupling of active neutrinos.So the direct contribution from decay to ∆N eff is washed out as active neutrinos are still in thermal equilibrium with photons.On the other hand, the freeze-in production of majoron abundance from N i decay culminates at T ≃ O(M i ), at which the relativistic majoron freeze-out also takes place.At this epoch, the contribution to primordial majoron abundance is dominated by the thermal freeze-out, while the freeze-in contribution is negligible.
• Next we consider the contribution from 2J → 2ν mediated by N i (for i = 1, 2, 3).We can compute the cross section of 2J → 2ν using Eq.(A.17), with the coupling suppressed by C ij ∼ Y ν ≲ 10 −5 in the low-scale seesaw scenario.The cross section is s-wave dominated, so that we can estimate the collision term as follow: Here we have neglected the exact numerical factor from phase-space integration, which is smaller than that from majoron decay.Recall that the collison term from majoron decay is given by C ρ J→2ν = m J n J Γ J .So we obtain C ρ J→2ν /C ρ 2J→2ν ∼ m J f 2 /T 3 .Given m J ≳ T for nonrelativistic majoron decay/annihilation and f ≫ m J , we conclude that Figure 3. Allowed parameter space of the majoron mass m J and the lepton-number breaking scale f under the constraints of current CMB measurements on N eff .The pink-and green-shaded regions in the upper right corner are excluded by Planck 2018 at 2σ level [1], with different primordial majoron abundances characterized by the freeze-out temperatures T fo .The excluded regions are obtained from instantaneous decay in Sec.3.2 (dashed line) and noninstantaneous decay in Sec. 4 (solid line), respectively.In addition, the yellow-shaded region in the lower left corner corresponds to the relativistic decay regime, while the gray-shaded region in the upper left corner corresponds to the scenario where the majoron decays after the matter-radiation equality epoch (post-equality decay).
the contribution to ∆N eff from majoron annihilation 2J → 2ν is strongly suppressed compared with that from majoron decay J → 2ν.
Based on our calculations of ∆N eff discussed above, the excluded regions in the (m J , f ) plane from the current CMB measurements are shown in Fig. 3. To compare the calculation of ∆N eff using the approximation of instantaneous majoron decay in Sec.3.2 with noninstantaneous decay in Sec. 4, we have shown the excluded regions obtained from both in Fig. 3, which match quite well with each other.Therefore, the instantaneous majoron decay serves as a good approximation in our problem.
The excess of N eff from the majoron decay depends on the primordial majoron abundance, which is characterized by the freeze-out temperature T fo in Eq. (4.4).In Fig. 3, we have shown results from two typical freeze-out temperatures: T fo = 64 MeV and T fo > 100 GeV.The first one is the lowest freeze-out temperature that is allowed by BBN+Y p +D (see Fig. 2).A lower T fo corresponds to a larger primordial abundance and would lead to a larger ∆N eff excluded by the BBN measurements [cf.Eq. (3.3)].The second one is the case where the freeze-out temperature is sufficiently high and all the SM species are relativistic, corresponding to a minimal ∆N eff ≃ 0.027 at the BBN epoch.We can see that for the largest primordial abundance allowed by current BBN measurements, the majoron that decays nonrelativistically is severely constrained.In particular, a lepton-number breaking scale above 300 GeV is excluded when the majoron mass is within [10 −5 , 1] MeV.The constraint is less strict with a smaller primordial abundance or equivalently, a higher freeze-out temperature.For T fo > 100 GeV, which corresponds to the smallest primordial abundance that can be induced from relativistic freeze-out, we find that a breaking scale f > 2 TeV is already excluded by the current Planck measurements for m J ∈ [10 −5 , 1] MeV.
Note that for f < 1 GeV, the majoron decay generally occurs in the relativistic regime, as can be inferred from Fig. 3. Therefore, we concentrate on f > 1 GeV.Also, it is worthwhile to mention that for a lighter majoron around the eV scale, the nonrelativistic decay could occur after the epochs of matter-radiation equality and recombination when effects on, e.g., the structure formation and the neutrino free-streaming become significant [29].On the other hand, for a heavier majoron above the MeV scale, the decay channel to electron-positron pairs opens.Besides, the majoron decay to neutrinos and electron-positron pairs could occur during the epochs of the BBN and neutrino decoupling and have observable effects.In particular, for 1 MeV ≲ m J ≲ 100 MeV, the most stringent constraint on f comes from Supernova 1987A.The absence of observing 100 MeV-range neutrino events from Supernova 1987A puts a lower bound on the lepton-number breaking scale: f ≳ 0.1 GeV (m J /MeV) [55].See also [56,57] for further discussions.It would be interesting to further investigate these nontrivial effects outside the mass region shown in Fig. 3, which goes beyond the scope of the current work.

The sandwiched window from precision N eff measurements
We have seen from Fig. 3 that the current Planck measurements of N eff can already put strict upper bounds on the lepton-number breaking scale for the scenario where the majoron decays nonrelativistically with a primordial abundance (FONR).On the other hand, in Sec.3.3 we demonstrated that the majoron abundance can also be accumulated through the freeze-in production 2ν → J and later decays nonrelativistically back to the neutrinos (FINR).As discussed in Sec.3.4, the FINR case can lead to a maximal ∆N eff ∼ O(0.1), which is beyond the sensitivity of the current Planck measurements.
However, the forecast sensitives of ∆N eff measurements in the future CMB experiments will be increased by a factor of a few.For instance, the SO experiment [22,23] has a projected 2σ sensitivity ∆N eff < 0.1, while the CMB-S4 experiment [24,25] is expected to have a 2σ sensitivity ∆N eff < 0.06.Therefore, for future CMB experiments, the FINR case can also be probed.Furthermore, due to the opposite dependence of ∆N eff on m J and f between the FONR and the FINR cases, one can expect that the viable parameter space of the lepton-number breaking scale would be pushed into a narrow sandwiched window when the contributions from the FONR and FINR cases coexist.In particular, null signals of the N eff excess in future CMB experiments could even close such a window and completely exclude the model.
To see the sandwiched window more clearly, we first consider the bounds from the FONR and FINR cases separately.That is, the bound from one case is derived without the contribution from the other case.In Fig. 4, we show the sandwiched window for the lepton-number breaking scale from the constraints of future SO and CMB-S4 experiments  In each subfigure, the green-shaded regions denote the excluded regions from the FONR case with different primordial majoron abundances (characterized by ∆N BBN eff ).The shaded regions in the lower left corner denote the regime of relativistic decay (yellow) and the regions excluded from the FINR cases (purple).The white bands labeled by "low-scale seesaw window" are the viable parameter space where the contributions from FONR and FINR cases coexist.at 2σ level.Note that for the FONR case, the constraints depend on the primordial majoron abundances, which are characterized by ∆N BBN eff , i.e., the contributions of the primordial relativistic majoron at the BBN epoch (see Tab. 1).For the FINR case, we apply the results derived in Ref. [30] by replacing the effective majoron-neutrino coupling g ν with the breaking scale via f = 0.05 eV/g ν .It corresponds to mapping the model-independent results into the seesaw framework.
In the upper left panel of Fig. 4, we show the parameter space of m J and f under the constraints of SO with a primordial majoron abundance characterized by ∆N BBN eff = 0.06 (corresponding to T fo = 397 MeV).It can be seen that for such a primordial abundance, the excluded region from the FONR case happens to be adjacent to that from the FINR case, and there is no viable parameter space for nonrelativistic majoron decay.This implies that if an excess of ∆N CMB eff > 0.1 is not observed by the future SO experiment at 2σ level, then the scenario with the freeze-out temperature T fo < 397 MeV will be ruled out within the regime of nonrelativistic decay.
In the upper right panel of Fig. 4, the result is shown for a smaller primordial majoron abundance characterized by ∆N BBN eff = 0.027 (corresponding to T fo > 100 GeV).Since the primordial abundance is reduced, the excluded region from the FONR case is expected to be smaller than that in the upper left panel, thereby leaving a viable parameter space that is sandwiched by the constraints from the FONR and the FINR cases (denoted as "low-scale seesaw window").For such a low-scale seesaw scenario, this is the parameter space where a minimal primordial majoron abundance inherited from relativistic freeze-out can survive, provided that no excess of ∆N CMB eff ⩾ 0.1 is observed by the future SO experiment.In the lower left and lower right panels of Fig. 4, the results are shown under the constraints of CMB-S4 with a primordial majoron abundance characterized by ∆N BBN eff = 0.008 and ∆N BBN eff = 0.003, respectively. 9We can infer from the lower left panel that the future CMB-S4 experiment is able to fully close the window for the low-scale seesaw scenario, where the primordial majoron abundance comes from relativistic freeze-out and later decays nonrelativistically, provided that an excess of ∆N CMB eff ⩾ 0.06 is not observed by CMB-S4 at 2σ level.Nevertheless, the case with a smaller primordial abundance (∆N BBN eff < 0.008) is still possible to survive if the abundance comes from other mechanism rather than relativistic freeze-out, as is shown in the lower right panel.This implies that even if the majoron itself as a relativistic species at the BBN epoch only contributes to ∆N eff at the order of 0.1%, its late-time nonrelativistic decay can generate observable effects in the future CMB experiments.
Finally, it should be noted that the above analysis is based on the assumptions that the bounds from the FONR and FINR cases are treated separately.For the real situation, the contribution to ∆N eff from neutrino coalescence 2ν → J should also be taken into account in the FONR case.In this case, one expects that the sandwiched window in Fig. 4 will become somewhat narrower.This can be understood by the fact that the contribution to ∆N eff at the CMB epoch for the real situation can always be divided into two parts where ∆N FO eff (∆N FI eff ) denotes the contribution from the primordial majoron abundance (neutrino coalescence).Since both ∆N FO eff and ∆N FI eff are non-negative, one will always obtain a larger ∆N eff than taking into account ∆N FO eff and ∆N FI eff separately.Therefore, the constraints and exclusion regions that we obtained in Fig. 4 should be considered as the most conservative results.Nevertheless, since ∆N FI eff can reach a maximal value at the order of 0.1 and the dependence of ∆N FI eff on m J and f is opposite to ∆N FO eff , it justifies our treatment as a good approximation to visualize the sandwiched window.
Before closing this section, we briefly comment on the applicability of our results to other well-motivated scenarios that feature lepton-number violations.In the low-scale type-I seesaw framework concerned here, the Dirac Yukawa coupling is highly suppressed by the active neutrino mass, i.e., Y ν ∼ √ m i f /v ≲ 10 −5 .Such small couplings could be evaded, e.g., in the framework of the inverse seesaw model [58].In the inverse seesaw scheme, apart from three RH neutrinos N R , one introduces another three singlets N L .The Lagrangian is given by where M is the mass scale of heavy sterile neutrinos.The Majorana mass µ = Y ′ N f / √ 2 explicitly breaks lepton-number symmetry and thus can be very small in a technically natural way [59].Assuming the hierarchy µ ≪ Y ν v ≪ M , the active neutrino mass is given by: m ν ∼ Y 2 ν v 2 µ/M 2 , where we have suppressed the flavor indices for simplicity.Therefore, one can have an O(1) Yukawa coupling Y ν in the low-scale inverse seesaw scenario since the active neutrino mass is naturally suppressed by small µ.In this case, we find the coupling between majoron and active neutrinos turns out to be g ν ∼ Y ′ N m ν /f .Furthermore, if the lepton-number breaking scale f is around the electroweak scale, then we have g ν ∼ m ν /f , which is the same order as the case of the type-I seesaw model [see Eq. (2.6)].Therefore, our above calculation of ∆N eff from majoron decay is also applicable to the inverse seesaw scheme.
While the majoron cannot serve as dark matter in the low-scale type-I seesaw scenario, the dark matter candidate can be readily accommodated in the model-dependent setup.In particular, under the inverse seesaw framework, some of the sterile neutrinos could be at the keV-MeV scale and play the role of warm dark matter [60].For such warm dark matter, its contribution to ∆N eff is negligible due to the Lyman-α forest constraints [61].

Conclusions
In this work, we have focused on the low-scale seesaw scenario associated with a singlet majoron from the spontaneous global U(1) L breaking, and investigated the constraints on the lepton-number breaking scale f from the cosmological measurements of N eff .This provides a complementary approach to the collider searches for lepton-number violation and TeV-scale Majorana neutrinos.
For the low-scale seesaw scenario, the breaking scale f is expected to be not far above the electroweak scale, so the majoron can decay within the cosmological time scale.The majoron interactions in the early Universe have two possible effects on N eff at the epoch of CMB.The first is that the majoron abundance is accumulated through freeze-in production (2ν → J) after the electroweak gauge symmetry breaking, and later decays nonrelativistically back to neutrinos (J → 2ν).This possibility has been widely discussed in the literature, where lower bounds of f were obtained.In this work, we mainly focused on the second possibility that was usually neglected in previous studies.That is, the majoron possesses a non-negligible primordial abundance, which is inherited from relativistic freezeout and later depleted into neutrinos.This situation is quite common within the minimal framework, where the majoron gets thermalized in the SM bath through the interactions with RH neutrinos or the Higgs boson.
We have demonstrated that the primordial abundance has sizable modification to ∆N eff that can be probed by the current and future CMB experiments.If the majoron decays relativistically, the contribution to ∆N eff was shown in Fig 2, which only depends on the freeze-out temperature.Things become more interesting if the majoron decays in the nonrelativistic regime.As was shown in Fig. 3, the current Planck measurements could already put severe constraints on the lepton-number breaking scale.More importantly, in this case, a larger f would lead to a larger ∆N eff .Therefore, opposite to the freezein scenario, upper bounds of f are obtained from primordial majoron decay.Furthermore, when both the freeze-out and freeze-in abundances coexist, we obtain a sandwiched window for f in terms of the majoron mass.
Given the forecast sensitivities of future SO and CMB-S4 experiments, we showed the sandwiched windows in Fig. 4. It can be seen that null signals of ∆N eff in future CMB experiments will push the low-scale seesaw scenario into a narrow sandwiched parameter space.In particular, if the primordial majoron abundance is inherited from relativistic freeze-out and later decays nonrelativistically into neutrinos, null signals of ∆N eff from CMB-S4 is able to fully close such a low-scale seesaw window.
Finally, although we have mainly focused on the majoron abundance in this work, it is worthwhile to emphasize that any new light particle coupled to neutrinos or photons can also be expected to result in a sandwiched window in the parameter space, as long as the particle has abundances from both the UV sources and the freeze-in production and only decays after the neutrino decoupling.Such a general phenomenon deserves careful investigation in the future.
. Feynman rules for the majoron-neutrino interactions.The arc arrows denote the orientations of fermion flow [63].We have identified ν i ≡ n i and N i ≡ n i+3 (for i = 1, 2, 3).Note that when i = j, there is a factor of 2! due to the two indistinguishable neutrinos.This is why there is an additional factor of (1 + δ ij ) compared with Eq. (A.14).
• For i, j = 1, 2, 3, from Eq. (A.12) we have Therefore, the interaction between the majoron and the active neutrinos is approximately diagonal and is suppressed by m i /f .
• For i, j = 4, 5, 6, we have .16)which means the interaction between the majoron and the sterile neutrinos is also approximately diagonal.
• For i = 1, 2, 3 and j = 4 This describes the interaction among the majoron, the active neutrinos and the sterile neutrinos.
The corresponding Feynman rules for majoron-neutrino interactions are summarized in Fig. 5.

B Neutrino coalescence rate
The collision term of the neutrino coalescence 2ν → J in Eq. (3.11) is given by where δ 4 ≡ δ 4 (p 1 +p 2 −p), and the quantum statistics for J is neglected.f i (E i ) = (e E i /Tν + 1) −1 are distribution functions of incoming neutrinos, with T ν the neutrino temperature, and the squared amplitude is given by To calculate Eq. (B.1), we can first go to the rest frame of J. Suppose in the rest plasma frame with 4-velovity u µ = (1, 0), the 4-momentum of the majoron is denoted by p µ = (E, ⃗ p).Then changing to the rest frame of J with p ′µ = (m J , ⃗ 0) is equivalent to boosting the rest plasma frame with u ′µ = (E/m J , −⃗ p/m J ).In the rest frame of J, we denote the neutrino 4-momenta by p ′µ 1,2 = (ω 1,2 , ⃗ q 1,2 ), where ω 1 = ω 2 = m J /2, |⃗ q 1 | = |⃗ q 2 | = β ν m J /2 are determined by the momentum conservation, with β ν ≡ 1 − 4m 2 i /m 2 J the neutrino velocity.Then, in the rest frame of J, the collision term is given by where the distribution function of the thermalized neutrinos in the rest frame of J reads for i = 1, 2. Integrating out |⃗ q 2 | and |⃗ q 1 | via δ-functions, we are led to Taking the Boltzmann approximation, we have where in the last step we have assumed m i ≪ m J so that β ν ≃ 1, and K 1 is the modified Bessel function.

Figure 2 .
Figure 2.∆N eff calculated from relativistic majoron decay at the epochs of BBN or CMB with different freeze-out temperature T fo (black short-dashed line).The regions above different horizontal lines correspond to the 2σ excluded regions of ∆N eff from the constraints of BBN+Y p +D (green dotted line), Planck (red short-dashed line), as well as the future projected sensitivities of SO (magenta dot-dashed line) and CMB-S4 (blue dashed line).The gray solid line denotes the minimal ∆N eff ≃ 0.027, which corresponds to the freeze-out temperature T fo ≳ O(100) GeV.

Figure 4 .
Figure 4.The sandwiched window for the lepton-number breaking scale from the constraints of the future SO (upper panel: ∆N eff < 0.1) and CMB-S4 (lower panel: ∆N eff < 0.06) experiments.In each subfigure, the green-shaded regions denote the excluded regions from the FONR case with different primordial majoron abundances (characterized by ∆N BBN eff ).The shaded regions in the lower left corner denote the regime of relativistic decay (yellow) and the regions excluded from the FINR cases (purple).The white bands labeled by "low-scale seesaw window" are the viable parameter space where the contributions from FONR and FINR cases coexist.

Figure 6 .
Figure 6.Feynman rules for the majoron self interaction and majoron-Higgs interactions, where θ is the mixing angle between two CP-even scalars ρ and h.

Table 1 .
Correspondence among the freeze-out temperatures T fo , the primordial majoron abundances Y n J,ini , and the contributions to ∆N eff at the BBN epoch.
.6) Combining Eqs.(4.3)-(4.6)and given a freeze-out temperature T fo , one can obtain the neutrino energy density Y ρ ν that comes from the majoron decay.The excess of N eff at the CMB epoch is then given by