Dynamical inverse seesaw mechanism as a simple benchmark for electroweak breaking and Higgs boson studies

The Standard Model(SM) vacuum is unstable for the measured values of the top Yukawa coupling and Higgs mass. Here we study the issue of vacuum stability when neutrino masses are generated through spontaneous low-scale lepton number violation. In the simplest dynamical inverse seesaw, the SM Higgs has two siblings: a massive CP-even scalar plus a massless Nambu-Goldstone boson, called majoron. For TeV scale breaking of lepton number, Higgs bosons can have a sizeable decay into the invisible majorons. We examine the interplay and complementarity of vacuum stability and perturbativity restrictions, with collider constraints on visible and invisible Higgs boson decay channels. This simple framework may help guiding further studies, for example, at the proposed FCC facility.


INTRODUCTION
The main leitmotiv of the LHC has been to elucidate the mechanism of spontaneous symmetry breaking in the SM. With the discovery of a scalar particle with properties closely resembling those of the SM Higgs, the ATLAS [1] and CMS [2] experiments have achieved this goal, although only partially. The Higgs discovery provides motivation for further studies, for example, at the upcoming FCC facility [3,4] and complementary lepton machines. If this scalar particle is indeed the SM Higgs boson, its mass measurement allows us to study the stability of the vacuum up to high energies through the renormalization group equations (RGEs). Given the measured values of the top quark and Higgs boson masses, the SM Higgs quartic coupling remains perturbative all the way up to the Planck scale (M P ), but goes negative well below, Fig. 1. Thus, the Higgs vacuum in the SM is not stable.
The discovery of neutrino masses [5][6][7] provides us with another important milestone in particle physics. It brings to surface one of the most important SM shortcomings, i.e. the absence of neutrino masses, which stands out as a key problem. Therefore, despite its outstanding achievements, it is now widely expected that the SM cannot be the final theory of nature up to the Planck scale. It is therefore important to analyze the problem of vacuum instability within neutrino mass extensions of the SM.
The purpose of this paper is to re-examine the consistency, i.e. the stability-perturbativity of the electroweak vacuum within neutrino mass extensions of the SM and also to confront the resulting restrictions with information available from collider experiments LEP and LHC.
We adopt the seesaw paradigm realized within the minimal SU(3) c ⊗ SU(2) L ⊗ U(1) Y gauge structure. There are two versions, "explicit" [8] and "dynamical" [9,10]. In the former case lepton number is broken explicitly, while in the second, the breaking occurs via the vacuum expectation value (vev) of a SU(3) c ⊗ SU(2) L ⊗ U(1) Y singlet scalar σ. This "dynamical" variant harbors a physical Nambu-Goldstone boson, called majoron J [9,10]. The most obvious way to account for the small neutrino masses is to assume very heavy right-handed neutrinos as mediators. As an interesting alternative to such high-scale type-I seesaw, we have the low-scale extensions such as the inverse seesaw mechanism [11]. For sizeable Diractype Yukawa couplings one finds that the Higgs vacuum stability problem can become worse than in the SM [12][13][14][15][16][17][18][19][20].
One of the attractive features of low-scale seesaw models is that we can have large Diractype Yukawa coupling even with light mediators, e.g. at the O(TeV) scale. In this case, the Yukawa couplings will evolve for a much longer range, compared to the high scale type-I seesaw [21]. As a result the Higgs quartic coupling can become negative much sooner than in SM. Thus, it will have larger negative effect upon vacuum stability. However, we will see how in dynamical low-scale seesaw scenarios [22] electroweak vacuum stability can be substantially improved [14,21].
The prototype model is characterized by a very simple set of scalar bosons: in addition to the SM-like Higgs boson H 125 found at the LHC, there is another CP -even scalar H . The mixing angle sin θ between the CP-even scalars plays a key role for our study. Moreover, there is a massless CP-odd boson, the majoron J, the physical Nambu-Goldstone associated to spontaneous breaking of lepton number symmetry. In such low-scale seesaw the majoron can couple substantially to the Higgs boson [23], leading to potentially large invisible decays, e.g. H 125 → JJ and H → JJ. A sizeable mixing between the two CP -even scalars can have important phenomenological consequences, particularly for collider experiments like the LHC. As a result of this mixing, the couplings of the SM-like Higgs scalar H 125 can deviate appreciably from the SM values. These can modify the so-called signal "strength parameter" µ f associated to a given "visible" final-state f , which can be tested at the LHC [24,25]. The modified couplings and the existence of invisible decays are constrained by Higgs measurements at the LEP and LHC experiments [26][27][28]. Here we adopt the conservative range, while for the invisible Higgs boson decays we take bound coming from the CMS experiment [29] 1 , Collider bounds and vacuum stability conditions lead to complementary constraints on the allowed range of the mixing angle sin θ. The goal of this work is to exploit this complementarity to test the simplest dynamical inverse seesaw scenario. We confront the collider limits with the consistency restrictions arising from the stability-perturbativity of the electroweak vacuum.
The work is organized as follows: in Sec. 2 we briefly summarize the issue of vacuum stability in the SM. In Secs. 3 and 4, we discuss vacuum stability in the inverse seesaw mechanism with explicit lepton number breaking as well as in the dynamical inverse seesaw mechanism. We use the two-loop RGEs given in Ref. [21] to evolve all the SM parameters as well as the new ones. We derive the full two-loop RGEs of the relevant parameters of the dynamical inverse seesaw, and list them in Appendix A. In Sec. 5, we discuss the production and decays of the two CP-even scalars H 125 and H , including both visible as well as invisible decay modes. In Sec. 6 and 7, we study in detail the sensitivities of Higgs boson searches at LEP and LHC. In Sec. 8, we address the issue of vacuum stability taking into account the collider constraints. Finally in Sec. 9 we conclude.

VACUUM STABILITY IN THE STANDARD MODEL
Before starting in earnest it is useful to briefly sum up the lessons from previous vacuum stability studies in the SM. If the 125 GeV scalar discovered at LHC is indeed the SM Higgs boson then we can determine its quartic coupling at the electroweak scale. This measurement can subsequently be used to study the stability of the fundamental vacuum at high energies, all the way up to Planck scale. In Fig. 1, we summarize the status of the electroweak vacuum within the SM, following the discussion of Ref. [20,21]. Here we adopt the MS scheme, taking the parameter values at low scale as input, see [20] for details. The top quark mass scale is set as m t = 173 ± 0.4. We have used two-loop renormalization group equations (RGEs) for the quartic coupling λ SM , the Yukawa coupling Y ν , as well as for SU(3) c ⊗ SU(2) L ⊗ U(1) Y gauge couplings g 1 , g 2 and g 3 . Fig. 1 clearly shows that SM Higgs quartic coupling λ SM goes negative around µ ∼ 10 10 GeV. As a result, the potential is not bounded from below, indicating an unstable vacuum. Note that the SM vacuum stability is very sensitive to the input value of top-quark mass. A dedicated analysis shows that SM Higgs vacuum is not absolutely stable, but rather metastable with very long lifetime [31][32][33]. In what follows we will examine the implications of vacuum stability requirements within seesaw models of neutrino mass generation.

INVERSE SEESAW AND VACUUM STABILITY
The seesaw mechanism based on the SU(3) c ⊗ SU(2) L ⊗ U(1) Y gauge group can be realized either in "high-scale" or in "low-scale" regimes. The vacuum stability issue has been examined in the high-scale type-I seesaw mechanism with large Yukawa couplings in Ref. [20].
It has been found that majoron extensions of these schemes can restore the vacuum stability all the way up to Planck scale once one takes into account the scalar threshold corrections.
However, such high scale seesaw schemes typically involves mediator masses much larger than the electroweak scale, unaccessible at collider experiments. Here we revisit the issue, but in the context of type-I "low-scale" seesaw mechanism, in which mediators would be accessible to high energy colliders. The simplest prototype is the inverse seesaw mechanism [11,22,34]. In inverse seesaw, lepton number is violated by introducing extra SM gauge singlet fermions S i with small Majorana mass terms, associated to the conventional "right-handed" neutrinos ν c i . The relevant part of the Lagrangian is given by where L i = (ν, ) T ; i = 1, 2, 3 are the lepton doublets, Φ is the Higgs doublet and ν c i , S i are SM gauge singlet fermions. The ν c i , S i transform under the lepton number symmetry U (1) L as ν c i ∼ −1 and S i ∼ +1, respectively. The smallness of light neutrino masses is controlled by the lepton number violating Majorana mass parameter µ S . This allows the Yukawa couplings Y ν to be sizeable, even when the messenger mass scale M lies in the TeV scale, without conflicting with the observed smallness of the neutrino masses.
Thanks to the potentially large Dirac neutrino Yukawa coupling Y ν required to generate adequate neutrino masses in such schemes the vacuum stability problem aggravates. To examine the effect of the new fermions ν c and S upon the stability of the electroweak Higgs vacuum we need to take into account the effect of the threshold corrections. To begin with, below the threshold scale Λ ≈ M , we need to integrate out the new fermions, so the theory is just the SM plus an effective dimension-five Weinberg operator. This affects the running of Higgs quartic coupling λ κ below the scale Λ, though the correction is negligibly small.
As a result, in the effective theory, the running of λ κ below the scale Λ is almost same as in the SM. Above the threshold scale Λ we have the full Ultra-Violet (UV) complete theory. Now the Yukawa coupling Y ν will affect the running of the Higgs quartic coupling which we now call λ so as to distinguish it from the quartic coupling below the threshold scale.
The two-loop system of RGEs governing the evolution of λ, Y ν and the SM couplings are listed in Ref. [21]. Integrating out the heavy neutrinos also introduces threshold corrections to the SM Higgs quartic coupling λ at the scale Λ [20,21,35]. As a result we also need to consider the shift due to threshold corrections in λ at Λ when solving RGEs. The threshold corrections imply that where n is the number of singlet fermions ν c . Having set up our basic scheme, let us now look at the impact of the new Yukawa coupling Y ν on the stability of the Higgs vacuum. As shown in Fig. 2, above the threshold scale Λ, Yukawa coupling Y ν can completely dominate the RGEs behaviour of quartic coupling λ. In Fig. 2 we compare the evolution of the Higgs quartic coupling λ within the SM (dashed, red) with the (3, n, n) inverse seesaw completion. Here n denotes the number of ν c and S species. We show the results for n = 1 (solid, blue), n = 2 (dot-dash, magenta) and n = 3 (dot, green). For this comparison, we have fixed the Yukawa coupling |Y ν | = 0.4 for the (3,1,1) case. For (3, n, n) with n ≥ 2, we took the diagonal entries of Y ν as Y ii ν = 0.4, while all the off-diagonal ones are neglected. We have fixed the threshold scale, which also sets the mass scale of the singlet neutrinos, as Λ = 10 TeV. One sees how, the larger the value of n, the more strongly the (3, n, n) inverse seesaw scenarios aggravate the Higgs vacuum stability problem. This destabilizing effect of the neutrino Yukawas can be potentially cured in the presence of other particles that can revert the trend found above.

MAJORON COMPLETION AND VACUUM STABILITY
As a well-motivated completion of the above scheme, we now turn to the dynamical version of the inverse seesaw mechanism [22]. Lepton number is now promoted to a spontaneously broken symmetry within the SU(3) c ⊗ SU(2) L ⊗ U(1) Y gauge framework. To do this, in addition to the SM singlets ν c i and S i , we add a complex scalar singlet σ carrying two units of lepton number. This symmetry is then broken by the vev of this complex singlet σ.

The relevant Lagrangian is given by
The neutral component of the doublet Φ and the singlet σ acquire vevs v Φ √ 2 and vσ √ 2 , respectively leading to the light neutrino masses given by For m ν ∼ 0.1 eV, we can have Yukawa couplings Y ν of order one, for TeV scale v σ and M .

Scalar Potential
The SU(3) c ⊗ SU(2) L ⊗ U(1) Y as well as the global lepton number symmetry invariant scalar potential is given by Consistency conditions: boundedness and perturbativity The above scalar potential must be bounded from below. This implies that at any given energy scale µ, the quartic couplings should satisfy where λ i (µ) are the values of the quartic couplings at the running scale µ. To have an absolutely stable vacuum, one needs to satisfy the condition given in Eq. 8 at each and every energy scale.
To ensure perturbativity, we take a conservative approach of simply requiring that There will be additional constraints from unitarity or from electroweak precision data through the S, T and U parameters. However, for our parameter range of interest they lead to rather loose constraints [26] compared to the LHC bounds which we will shortly discuss in detail.

Mass spectrum
In order to obtain the mass spectrum for the scalars after SU(3) c ⊗ SU(2) L ⊗ U(1) Y and lepton-number symmetry breaking, we expand the scalar fields as Using this expansion, the potential in (7) leads to a physical massless Goldstone boson, namely the majoron J = Im σ plus two massive neutral CP-even scalars H i (i = 1, 2). The mass matrix of CP-even Higgs scalars in the basis (R 1 , R 2 ) reads as with the mass eigenvalues given by where the scalars H 1 and H 2 have masses m H 1 and m H 2 respectively, and by convention where θ is the mixing angle. The rotation matrix satisfies We can use Eq. (15) and (16) to solve for the potential parameters λ Φ , λ σ , λ Φσ in terms of mixing angle θ and the scalar masses m H i as Vacuum stability and spontaneous lepton number violation We now look at the stability of the electroweak vacuum in more detail. To see how the couplings evolve with energy we use the full two-loop RGEs governing the evolution of the Higgs quartic coupling [14], which are listed in Appendix A. However, to understand the main features, its enough to look at the one-loop β functions for the quartic couplings, which are given as follows where β f = µ df dµ . Notice that the one-loop contributions of the new fermions ν c i and scalars σ to the beta-function of λ Φ shown in Fig Vacuum stability in this model can be studied in two different regimes . Due to its potential testability at LHC, here we focus on the second      GeV, θ = 0.12, v σ = 1 TeV, whereas the right panel has m H = 800 GeV, θ = 0.08, v σ = 3 TeV. We have taken the Yukawa coupling |Y ν | = 0.4 for (3,1,1) case, while for the (3, n, n) with n ≥ 2 we took Y ii ν = 0.4 with zero off-diagonal entries. To avoid overcrowding the plot, only the evolution of the quartic scalar coupling λ Φ has been shown in Fig. 4.
In Fig. 4  in the RGEs making the other quartic couplings, e.g. λ σ , nonperturbative at scales far below M P . This can be clearly seen in the approximation Y S ≈ 0, where β λσ is always positive and hence λ σ can only increase. For relatively large input values of λ σ and small λ Φσ , the running of λ σ can be approximated as β λσ ∝ λ 2 σ . Hence, it can encounter a Landau pole at a scale far below the Planck scale. Similarly, with a very large starting value of λ Φ and small λ Φσ , the running of λ Φ will be dominated by the term +24λ 2 Φ . As a result one can hit a Landau pole at a scale far below M P . In Appendix. B we have discussed in detail each of these scenarios. Thus λ Φσ can neither be taken too small, nor too large, only a small optimal parameter range will satisfy both stability and perturbativity constraints. However, this small optimal range can be probed in an important way by colliders such as the LHC, as we will discuss in the next section.

COLLIDER CONSTRAINTS AND INVISIBLE HIGGS BOSON DECAYS
In this section we examine how Higgs measurements at LEP and LHC can constrain the parameter space of the Majoron inverse seesaw model. For this we have considered the following scenarios: We should remind the reader that, by definition, we always take m H 2 > m H 1 .
Rather than discussing collider constraints in terms of quartic couplings, its more convenient to use the mass basis quantities, e.g. scalar masses and mixing angles, since experimental results are quoted in terms of these quantities. In our simple model, the mixing angle θ, the mass m 2 H and the ratio of the two vevs tan β = v Φ vσ (with v Φ = 2m W g ) can be taken as free parameters, in terms of which all others can be fixed.
Before discussing the collider constraints, notice that in the Majoron inverse seesaw ex- where J denotes the Majoron and The decay widths for H i → JJ are given by If m H 2 > 2m H 1 , H 2 can also decay to H 1 H 1 with the decay width given by These new decay widths will lead to invisible decays as Turning to the lighter scalar boson decays to final state f of SM particles, one finds the branching fractions If Γ inv (H 1 ) = 0, the branching fraction would be same as that of SM. The lighter H 1 may decay predominantly into pair of majorons depending upon the mass m H 1 and mixing angle θ. The invisible branching ratio for H 1 is given by For the heavier state H 2 , the branching fraction into SM final state f is Similarly, the invisible branching ratio for H 2 is given by The coupling of H 1 and H 2 to other SM fermions and gauge bosons are suppressed relative to the standard values by sin θ (cos θ) and cos θ (sin θ) for Case I and Case II respectively.
Hence the single H 1 or H 2 production cross-sections are given as, where σ SM (pp → H 1 ) and σ SM (pp → H 2 ) are the SM cross-sections for Higgs production at m H 1 and m H 2 . Note that they are modified by factors sin 2 θ (cos 2 θ) or cos 2 θ (sin 2 θ) with respect to the conventional ones.
In the following sections, we discuss the constraints on the relevant parameter space of where σ SM hZ is the standard cross section, and R H Z is the suppression factor related to the coupling of the Higgs boson to the gauge boson Z. Of course we have R SM hZ = 1 in the SM. In our case, BR(H → bb) is modified with respect to SM due to the presence of invisible reduced with respect to that of the SM, lighter masses become allowed.
In Fig. 5, in addition to the LEP constraints, we have also showed in magenta color the constraints on the invisible decay of H 125 coming from the LHC. As one can see, these constraints supersede those from LEP, and severely restrict the allowed parameter space.
They come from the current upper bound on the branching ratio to invisible decay modes BR(H 125 → Inv) ≤ 19% given by the CMS collaboration [41] and a similar one from AT-LAS [30]. The results are shown in Fig. 5  We now turn to invisible Higgs decays at hadron colliders [42]. Apart from the above LHC limit, we also have the LHC measurements of several visible decay modes of the 125 GeV Higgs boson. These are given in terms of the so-called signal strength parameters, where σ is the cross-section for Higgs production, NP and SM stand for new physics and SM respectively.
For the 8 TeV data, we list the results for signal strength parameters from combined ATLAS and CMS analysis [24] in  For the 13 TeV Run-2, there is no combined final data so far, and the available data is separated by production processes. Table II compiles the recent results from ATLAS [25].
We note that in our model, the expected signal strength parameter µ f for any SM final state f can only be less than unity, as shown in the left panel of Fig. 6. The left panel of In the right panel of Fig. 6, we show the correlation between µ f and µ f where f = f .  into account the limits in Tab. I and Tab. II, we assume that the LHC allows for deviations in the range given in Eq. 1. GeV (bottom panel). As before, the blue region is excluded from LEP, while the magenta region is excluded from the LHC constraint in Eq. (1). The green region is allowed by the LHC limit. As before, the kink is associated with the opening of the decay channel BR(H → Inv) for v σ = 1 TeV. The color code is the same as in Fig. 7. The left panel of Fig. 8 shows that in our model, due to the LHC limit Eq. (1), the maximum invisible branching ratio of H 125 is about 20%. Whereas for the lighter H , the LHC limit Eq. (1) imples that it should decay mainly via the invisible mode, as shown in the right panel of In Table III  for v σ > 1 TeV the limit from Eq. (2) gets relaxed since, the larger the v σ , the smaller the   invisible decay mode H 125 → JJ. As a result, for larger v σ values the Higgs invisible decay gives a weaker exclusion limit on | sin θ| than that coming from µ f .
Notice that the Higgs invisible branching ratio changes with the scale of dynamical breaking of lepton number, i.e. value of v σ that triggers neutrino mass generation. Fig. 11 shows how one can get information on this fundamental scale by Higgs boson measurements. These plots show indeed that the Higgs invisible branching ratio can be used to probe the scale of spontaneous lepton number breaking v σ . As can be seen from Fig. 11 for a fixed value of the mixing angle sin θ the Higgs invisible branching ratio varies in a monotonic fashion with v σ . Thus, one can use other LHC results i.e. signal strength measurements to obtain limits on sin θ and then use Fig. 11 to constrain the scale of dynamical lepton number breaking. For example, for | sin θ| = 0.1 we find that v σ cannot be less than 500 GeV while for | sin θ| = 0.2 v σ cannot be less than 900 GeV. Future improvement on the Higgs invisible branching ratio measurement can be used to further constrain the scale of dynamical lepton number breaking, as depicted in Fig. 11. Finally, as in Case I, one can use Eqs. (1) and (2) to obtain correlations between different observables, analogous to the Fig. 8 and Fig. 9. However, we will not show these plots explicitly. Instead, we will make use of such constraints and correlations in conjunction with the vacuum stability constraints in order to obtain complementary limits in Section. 8.

PERTURBATIVITY AND VACUUM STABILITY
We now examine the combined implications of collider constraints in conjunction with the restrictions that follow from vacuum stability and perturbativity of the theory. As we will see, these two sets of constraints give complementary information on the Majoron inverse seesaw model. In most cases, vacuum stability is threatened by the violation of the condition λ Φ > 0 and λ σ > 0. In order to have a stable vacuum one also needs relatively large values of λ Φσ , which means non-negligible mixing parameter | sin θ| between the two CP even scalars.
On the other hand the LEP and LHC provide stringent bounds on the mixing between the two CP even scalars i.e. they require small values of | sin θ|. We now dicuss this interplay in more detail for both Case I and Case II. In subsequent sections we show stable, unstable and non-perturbative m H −sin θ regions, associated to green, red and brown colors, respectively.
We have categorized these regions using the following criteria: • Green Region: This is the region where we can have stable vacuum all the way up to the Planck scale, and all the couplings are within their perturbative regime. In our numerical scans these conditions are implemented by requiring: where µ is the running scale. All other couplings e.g. the gauge and Yukawa couplings are also required to be perturbative till the Planck scale.
• Red Region: In this region the vacuum is unstable, as the potential becomes unbounded from below at some high energy scale before Planck scale. This means that any one or more than one of these conditions are realised: Note that inside the red region there can be parameters for which the potential is unbounded, and also some of the quartic couplings are non-perturbative as well, although we are excluding Landau poles.
• Brown Region: This region implies the existence of non-perturbative couplings at some energy scale before the Planck scale. This happens if any one of the following Note that the gauge coupling running here is similar to the SM running and hence they always remain perturbative. We are also including Landau poles inside the non-perturbative regions, since as one approaches the Landau pole the perturbative approach is no longer reliable. In Appendix. B, we discuss how Landau poles can arise in our the RGEs.
There we have also discussed other scenarios leading to non-perturbative couplings, such as the continuous growth of a coupling or the saturation of some coupling with respect to the energy scale µ.
Let us now look at the combined results of the collider constraints and stabilityperturbativity constraints of our model. We start with Case I first. The region allowed by collider constraints is the one in between the black contours in the left panel of Fig. 12. One sees that in this case the collider constraints are very stringent and rule out almost all the parameter space, except for a very thin small region very close to | sin θ| ≈ 0. However, even this small allowed region is ruled out by the combination of vacuum stability and perturbativity constraints. Finally, we stress that Fig. 12 corresponds to the (3, 1, 1) Majoron inverse seesaw, taking Y ν = 0. A non-zero Yukawa coupling will have an effect on the evolution of the quartic coupling λ Φ , aggravating the vacuum instability problem. Moreover, higher (3, n, n) realizations with n ≥ 2 also will only aggravate the stability problem. Thus, we can safely say that at least for v σ = 100 GeV, there is no viable parameter space within the Majoron inverse seesaw approach to have a stable and perturbative vacuum up to the Planck scale.
Since higher dynamical lepton number breaking scales v σ relax the collider limits, will that help us find a viable parameter space? Although the collider restrictions clearly disappear Before closing this section, we comment on the possibility that the lepton number symmetry is spontaneously broken at a very low scale, i.e. v σ ∼ O(KeV) [43]. This case unfortunately requires extreme fine tuning to be viable. From (19) one sees that it is only allowed if the masses of the CP even scalars is fine-tuned to be nearly degenerate i.e.
. This case would require a separate analysis, properly taking into account the threshold corrections coming from integrating out the SM particles as well as the QCD corrections. For large m H and | sin θ| ∼ 0, λ σ (M Z ) is large but λ Φσ (M Z ) is small. Hence, the one-loop λ σ evolution can be approximated as β λσ ≈ 20λ 2 σ (see Eq. 22) and following Eq. (B1) the coupling becomes non-perturbative before reaching the Planck scale.
Keeping all other parameters the same, we now switch on the neutrino Yukawa coupling Y ν = 0.6 to analyze its effect 4 , see the lower panel in Fig. 15. As expected, upon switching on the neutrino Yukawa coupling, the vacuum will be unstable over a larger parameter space. One sees from the lower panel in Fig. 15, that the region with unstable vacuum increases appreciably. Correspondingly, the green region where the vacuum is stable and all couplings are perturbative decreases in size. After imposing the LEP-LHC constraints (region between the black lines), we find that no viable allowed region remains, as all the green region falls within the collider-forbidden region. Thus, stability-perturbativity in conjunction with LEP-LHC constraints, completely rule out this benchmark.
The main conclusions drawn for the case of v σ = 700 GeV hold for larger v σ values. In The different regions in these figures can be explained in a way similar to the case of v σ = 700 GeV. The most important change now is that the collider constraints from Eqs. (1) and (2) get relaxed with increasing value of v σ . Indeed, from Table. III and Fig. 10, one sees that, the larger the v σ , the more relaxed are the LHC constraints. The fact that these constraints are weaker for v σ = 3 TeV than for v σ = 1 TeV explains why the green region in such relatively large v σ , we obtain a stable vacuum consistent with LHC constraints even for non-zero Yukawa couplings.
We remark that the above analysis has been performed for the missing partner (3,1,1) inverse seesaw mechamism, where there is only one Yukawa coupling. For higher (3, n, n) inverse seesaw schemes with n ≥ 2, the results will be similar, but more Yukawa couplings means that larger regions will be ruled out by the stability-perturbativity requirement. We have also considered these cases in detail. In Fig. 18 and 19 we compare the stability properties of the (3, 1, 1) and (3,2,2) missing partner seesaw with the standard sequential (3,3,3) inverse seesaw.
We have taken the Yukawa coupling |Y ν | = 0.4 for the (3,1,1) case, while for the (3, n, n) with n ≥ 2 and we assumed Y ii ν = 0.4 and the off-diagonal entries as Y ij ν = 0. One sees from these figures that when the dynamical lepton number breaking scale v σ is high i.e. v σ > ∼ 1 TeV, thanks to the relaxed LHC constraints (Fig. 10  perturbativity regions will enlarge considerably.

SUMMARY AND OUTLOOK
We have examined the dynamical inverse seesaw mechanism as a simple benchmark for electroweak breaking and Higgs boson physics. We first briefly summarized the issue of vacuum stability in the context of inverse seesaw mechanism with explicit lepton number violation and compared with the SM, Fig. 1. The addition of fermion singlets (ν c and S) has a destabilizing effect on the running of the Higgs quartic coupling λ and we found that with sizeable Yukawa coupling Y ν the quartic coupling λ becomes negative before the SM instability scale µ ∼ 10 10 GeV, Fig. 2 that of the SM, adding a singlet σ whose vacuum expectation value v σ drives the spontaneous violation of lepton number and neutrino mass generation. It has two characteristic features: i) the existence of a massless Nambu-Goldstone boson, J, dubbed majoron, and ii) two CP -even neutral Higgs H 1 and H 2 given as a simple admixture of the doublet and singlet scalar given by sin θ. This leads to potentially large Higgs decays into the invisible majorons, in Eq. (2), modifying also the rates for SM decay channels in a simple manner, Moreover, the stability properties of the electroweak vacuum can be substatially improved, see Figs. 3 and 4. We further study how the vacuum stability can further restrict the parameter space, as illustrated in Figs. 12, 13, 14, 15, 16, 17, 18 and 19. The take-home message implied by our findings is that, with appreciable mixing angle between CP -even neutral Higgs H 1 and H 2 , we can have stable vacuum even for relatively large Yukawa coupling. Therefore further experimental improvements, either at the LHC or at the future FCC [3,4], will further restrict the allowed regions. Needless to say, an open conflict between stability-perturbativity and experiment could mean the breakdown of the theory, and the presence of new physics below the Planck scale. We are still far from this long term goal and, in the meatime, our dynamical inverse seesaw picture offers a very simple benchmark for electroweak breaking and precision Higgs boson studies at upcoming collider facilities. We see from Eq. 22 that if Y S ≈ 0, one-loop evolution of λ σ can be approximated as
• Eq.B1 can be generalized in the following way: From this we see that for n > 1 we can have a Landau pole.

Continuous growth
If β c = Ac n with n ≤ 1, we see from Eq. B3 that in this case we will not have pole but c(µ) can grows continuosly. For example with n = 1 2 , and with n = 0, Although running of c(µ) will not encounter any Landau pole but it can go to nonperturbative region with relatively large values of c(M Z ) and A.

Saturation
If β c has a zero at the finite value c(µ * ) then the growth of c will be saturated at c(µ * ) for µ → ∞. To illustrate this let us consider the following form of β c , β c = (A − Bc(µ)).
This can have a zero at c(µ * ) = A B . Now using c(µ * ) = A B in the above equation, we find c(µ) = A B for any µ. Note that this will happen for any β c which has a zero at the finite value c(µ * ).
One loop and two-loop RGEs for the quartic couplings are in general very complex.