Electroweak breaking and neutrino mass:"invisible"Higgs decays at the LHC (Type II seesaw)

Neutrino mass generation through the Higgs mechanism not only suggests the need to reconsider the physics of electroweak symmetry breaking from a new perspective, but also provides a new theoretically consistent and experimentally viable paradigm. We illustrate this by describing the main features of the electroweak symmetry breaking sector of the simplest type-II seesaw model with spontaneous breaking of lepton number. After reviewing the relevant"theoretical"and astrophysical restrictions on the Higgs sector, we perform an analysis of the sensitivities of Higgs boson searches at the ongoing ATLAS and CMS experiments at the LHC, including not only the new contributions to the decay channels present in the Standard Model (SM) but also genuinely non-SM Higgs boson decays, such as"invisible"Higgs boson decays to majorons. We find sensitivities that are likely to be reached at the upcoming Run of the experiments.


I. INTRODUCTION
The electroweak breaking sector is a fundamental ingredient of the Standard Model many of whose detailed properties remain open, even after the historic discovery of the Higgs boson [1,2]. The electroweak breaking sector is subject to many restrictions following from direct experimental searches at colliders [3,4], as well as global fits [5,6] of precision observables [7][8][9]. Moreover, its properties are may also be restricted by theoretical consistency arguments, such as naturalness, perturbativity and stability [10]. The latter have long provided strong motivation for extensions of the Standard Model such as those based on the idea of supersymmetry.
Following the approach recently suggested in Refs. [11,12] we propose to take seriously the hints from the neutrino mass generation scenario to the structure of the scalar sector. In particular, the most accepted scenario of neutrino mass generation associates the smallness of neutrino mass to their charge neutrality which suggests them to be of Majorana nature due to some, currently unknown, mechanism of lepton number violation . The latter requires an extension of the SU(3) c ⊗ SU(2) L ⊗ U(1) Y Higgs sector and hence the need to reconsider the physics of symmetry breaking from a new perspective. In broad terms this would provide an alternative to supersymmetry as paradigm of electroweak breaking. Amongst its other characteristic features is the presence of doubly charged scalar bosons, compressed mass spectra of heavy scalars dictated by stability and perturbativity and the presence of "invisible" decays of Higgs bosons to the Nambu-Goldstone boson associated to spontaneous lepton number violation and neutrino mass generation [13].
In this paper we study the invisible decays of the Higgs bosons in the context of a type-II seesaw majoron model [14] in which the neutrino mass is generated after spontaneous violation of lepton number at some low energy scale, Λ EW Λ ∼ O(TeV) [15,16] 1 . This scheme requires the presence of two lepton number-carrying scalar multiplets in the extended SU(3) c ⊗ SU(2) L ⊗ U(1) Y model, a singlet σ and a triplet ∆ under SU(2) -this seesaw scheme was called "123"-seesaw model in [14] and here we take the "pure" version of this scheme, without right-handed neutrinos. The presence of the new scalars implies the existence of new contributions to "visible" SM Higgs decays, such as the h → γγ decay channel, in addition to intrinsically new Higgs decay channels involving the emission of majorons, such as the "invisible" decays of the CP-even scalar bosons. As a result, one can set upper limits on the invisible decay channel based on the available data which restrict the "visible" channels.
The plan of this paper is as follows. In the next section we describe the main features of 1 The idea of the Majoron was first proposed in [17] though in the framework of the Type I seesaw, not relevant for our current paper. On the other hand the triplet Majoron was suggested in [18] but has been ruled out since the first measurements of the invisible Z width by the LEP experiments. Regarding the idea of invisible Higgs decays was first given in Ref. [19], though the early scenarios have been ruled out.
the symmetry breaking sector of the "123" type II seesaw model. In section III we discuss the "theoretical" and astrophysical constraints relevant for the Higgs sector. Taking these into account, we study the sensitivities of Higgs boson searches at the LHC to Standard Model scalar boson decays in section IV. Section V addresses the non-SM Higgs decays of the model. Section VI summarizes our results and we conclude in section VII.

II. THE TYPE-II SEESAW MODEL
Our basic framework is the "123" seesaw scheme originally proposed in Ref. [14] whose Higgs sector contains, in addition to the SU(3) c ⊗ SU(2) L ⊗ U(1) Y scalar doublet Φ, two lepton-number-carrying scalars: a complex singlet σ and a triplet ∆. All these fields develop non-zero vacuum expectation values (vevs) leading to the breaking of the Standard Model (SM) gauge group as well as the global symmetry U (1) L associated to lepton number. The latter breaking accounts for generation of the small neutrino masses. Therefore, the scalar sector is given by with L = 0 and L = −2, respectively, and the scalar field σ with lepton number L = 2. Below we will consider the required vev hierarchies in the model.

A. Yukawa Sector
Here we consider the simplest version of the seesaw scheme proposed in Ref. [14] in which no right-handed neutrinos are added, and only the SU(3) c ⊗ SU(2) L ⊗ U(1) Y electroweak breaking sector is extended so as to spontaneously break lepton number giving mass to neutrinos. Such "123" majoron-seesaw model is described by the SU In this model the neutrino mass (see Fig. 1) is given by, where v 1 and v 2 are the vevs of the singlet and the doublet, respectively. Here κ is a dimensionless parameter that describes the interaction amongst the three scalar fields (see below), and m ∆ is the mass of the scalar triplet ∆.  is the dimensionful parameter responsible of lepton number violation, see eq. (3). Therefore if y ν ∼ O(1) and the mass M ∆ lies at 1 TeV region then one has that ∆ ∼ m ν and µ ∼ 1 eV. Note that one may consider two situations: v 1 Λ EW (high-scale seesaw mechanism) in whose case the scalar singlet and the invisible decays of the Higgs are decoupled [15]; the second interesting case is when Λ EW v 1 few TeV (low-scale seesaw mechanism). In this case the parameter κ is the range [10 −14 , 10 −16 ] for y ν ∼ O (1). In this case one has new physics at the TeV region including the "invisible" decays of the Higgs bosons. Therefore, led by the smallness of the neutrino mass we can qualitatively determine that the analysis to be carried out is characterized by having a vev hierarchy and the smallness of the coupling κ, that is κ 1.

B. The scalar potential
The scalar potential invariant under the SU(3) c ⊗ SU(2) L ⊗ U(1) Y ⊗ U(1) L symmetry is given by [15,16] As mentioned above the scalar fields σ, φ and ∆ acquire non-zero vacuum expectation values, v 1 , v 2 and v 3 , respectively, so that, they can be shifted as follows, The minimization conditions of eq. (4) are given by, and from these one can derive a vev seesaw relation of the type where κ is the dimensionless coupling that generates the mass parameter associated to the cubic term in the scalar potential of the simplest triplet seesaw scheme with explicit lepton number violation as proposed in [21] and recently revisited in [12].

Neutral Higgs bosons
One can now write the resulting squared mass matrix for the CP-even scalars in the weak basis (R 1 , R 2 , R 3 ) as follows, 2 From now on we follow the notation and conventions used in Ref. [16].
The matrix M 2 R is diagonalized by an orthogonal matrix as follows, We use the standard parameterization O R = R 23 R 13 R 12 where and c ij = cos α ij , s ij = sin α ij , so that the rotation matrix O R is re-expressed in terms of the mixing angles in the following way: On the other hand, the squared mass matrix for the CP-odd scalars in the weak basis (I 1 , I 2 , I 3 ) is given as, The matrix M 2 I is diagonalized as, where the null masses correspond to the would-be Goldstone boson G 0 and the Majoron J, while the squared CP-odd mass is The mass eigenstates are linked with the original ones by the following rotation, where the matrix O I is given by, with

Charged Higgs bosons
The squared mass matrix for the singly-charged scalar bosons in the original weak basis (φ ± , ∆ ± ) is given by, We now define where c ± and s ± are given as The massless state corresponds to the would-be Golstone bosons G ± and the massive state H ± is characterized by, On the other hand, the doubly-charged scalars ∆ ±± has mass

C. Scalar boson mass sum rules
Notice that using the fact that the smallness of the neutrino mass implies that the parameters κ and v 3 are very small one can, to a good approximation, rewrite eq. (6) schematically in the form, and eq. (11) becomes, As a result, the scalar H 3 and the pseudo-scalar A are almost degenerate, In the same way, by using eqs. (11), (18) and (19), one can derive the following mass relations, which can be rewritten in the form, This sum rule is also satisfied in the Type-II seesaw model with explicit breaking of lepton number. Imposing the perturbativity condition one finds that the squared mass difference between, say doubly and singly charged scalar bosons, cannot be too large [12]. Explicit comparison shows that λ 5 in eq. (4) corresponds to λ H∆ in Ref. [12]. Therefore when the couplings of the singlet σ in eq. (4) are small, λ 5 is constrained to be in the range [−0.85, 0.85], so that the remaining couplings are kept small up to the Planck scale and vacuum stability is guaranteed. See Figure 4 in Ref. [12]. Likewise when one decouples the triplet one also recovers the results found in Ref. [11].

III. THEORETICAL CONSTRAINTS
Before analyzing the sensitivities of the searches for Higgs bosons at the LHC experiments, we first discuss the restrictions that follow from the consistency requirements of the Higgs potential. We can rewrite the dimensionless parameters λ 1,2,3 and β 1,2,3 in eq. (4) in terms of the mixing angles, α ij and scalar the masses m H 1,2,3 by solving . Hence one gets, In addition, using eqs. (11), (18) and (19) we can write the dimensionless parameters λ 4,5 and κ as functions of the vevs v 1,2,3 and the masses of the pseudo-, singly-and doublycharged scalar bosons (i.e. m A , m H ± and m ∆ ±± , respectively) as, .
From the theoretical side we have to ensure that the scalar potential in the model is bounded from below (BFB).

A. Boundedness Conditions
In order to ensure that the scalar potential in eq. (4) is bounded from below we have to derive the conditions on the dimensionless parameters such the quartic part of the scalar potential is positive V (4) > 0 as the fields go to infinity. We have that the parameter κ 1 (due to the smallness of the neutrino mass) and non-negative. This follows from where we have used the last expression in eq. (24) and the fact that v 3 v 2 , v 1 . Then κ is neglected with respect to the other dimensionless parameters λ i and β j , i.e. λ i , β j κ. As a result the quartic part of the potential V (4) | κ=0 turns to be a biquadratic form λ ij ϕ 2 i ϕ 2 j of real fields. Therefore, in this strict limit, the copositivity criteria described in [22] may be applied and the boundedness conditions for eq. (4) are the following, where λ 24 ≡ λ 2 + λ 4 . In addition all the dimensionless parameters in the scalar potential are required to be less than √ 4π in order to fulfill the perturbativity condition.

B. Astrophysical constraints
In our type-II seesaw model there are some constraints on the magnitude of SU (2) triplet's vev ∆ = v 3 , that one must take into account. First of all, v 3 is constrained to be smaller than a few GeVs due to the ρ parameter ( ρ = 1.0004 ± 0.00024 [23]).
On the other hand, the presence of the Nambu-Goldstone boson associated to spontaneous lepton number violation and neutrino mass generation implies that there is a most stringent constraint on v 3 coming from astrophysics, due to supernova cooling. If the majoron is a strict Goldstone boson (or lighter than typical stellar temperatures) one has an upper bound for the Majoron-electron coupling This is discussed, for example, in Ref. [24] and references therein. This implies Taking into account the profile of the Majoron [14] 3 one can translate this as a bound on the projection of the Majoron onto the doublet as follows [16] | J|φ | = 2|v 2 |v 2 Notice that this restriction on the triplet's vev is stronger that the one stemming from the ρ parameter. The shaded region in Fig. 2 corresponds to the allowed region of v 3 as function of v 1 . To close this section we mention that our phenomenological analysis remains valid if the Nambu-Goldstone boson picks up a small mass from, say, quantum gravity effects.

IV. TYPE-II SEESAW HIGGS SEARCHES AT THE LHC
We now turn to the study of the experimental sensitivities of the LHC experiments to the parameters characterizing the "123" type-II majoron seesaw Higgs sector, as proposed in [14]. In the following we will assume that m H 1 < m H 2 < m H 3 where 1,2,3 refer to the mass ordering in the CP even Higgs sector. Therefore, there are two possible cases that can be considered 4 : where m H is the mass of the Higgs reported by the ATLAS [2] and CMS [25] collaborations, i.e. m H = 125.09 ± 0.21(stat.) ± 0.11(syst.) GeV [26]. For case (i), we have to enforce the constraints coming from LEP-II data on the lightest CP-even scalar coupling to the SM and those coming from the LHC Run-1 on the heavier scalars. Such situation has been discussed by us in Ref. [13] in the simplest "12-type" seesaw Majoron model. In case (ii), only the constraints coming from the LHC must be taken into account.
The neutral component of the Standard Model Higgs doublet couplings get modified as follows, where we have defined C i ≡ O R i2 and O R ij are the matrix elements of O R in eq. (9).

A. LEP constraints on invisible Higgs decays
The constraints on H 1 , when m H 1 < 125 GeV, stem from the process e + e − → Zh → Zbb which is written as [27] where σ SM hZ is the SM hZ cross section, R hZ is the suppression factor related to the coupling of the Higgs boson 5 to the gauge boson Z.
where C 1 = cos α 13 sin α 12 , eq. (28). Notice that C 1 ≈ sin α 12 for the limit α 13 1 and then one obtains the same exclusion region depicted in Fig. 1 in Ref. [13]. 4 Recall that m H3 ≈ m A , eq. (22), which implies that the mass of H 3 must be close to that of the doublycharged scalar mass. Therefore, as we will see in the next section, the existing bounds on searches of the doubly-charged scalar exclude the case where m H3 is lighter than the other CP-even mass eigenstates. 5 The Feynman rules for the couplings of the Higgs bosons H i to the Z are the following:

B. LHC constraints on the Higgs signal strengths
In addition, we have to enforce the limits coming from the Standard Model decay channels of the Higgs boson. These are given in terms of the signal strength parameters, where σ is the cross section for Higgs production, BR(h → f ) is the branching ratio into the Standard Model final state f , the labels NP and SM stand for New Physics and Standard Model respectively. These can be compared with those given by the experimental collaborations. The most recent results of the signal strengths from a combined ATLAS and CMS analysis [28] are shown in Table I  One can see with ease that the LHC results indicate that µ V V ∼ 1. In our analysis, we assume that the LHC allows deviations up to 20% as follows,

C. LHC bounds on the heavy neutral scalars
In our study we will impose the constraints on the heavy scalars from the recent LHC scalar boson searches. Therefore, we use the bounds set by the search for a heavy Higgs in the H → W W and H → ZZ decay channels in the range [145 − 1000] GeV [29] and in the h → τ τ decay channel in the range range [100 − 1000] GeV [30]. We also adopt the constraints on the process h → γγ in the range [65 − 600] GeV [31] and the range [150, 850] GeV [32]. Besides, we impose the bounds in the A → Zh decay channel in the range [220 − 1000] GeV [33].

D. Summary of the searches of charged scalars
The type-II seesaw model with explicit breaking of lepton number contains seven physical scalars: two CP-even neutral scalars H 1 and H 2 , one CP-odd scalar A and four charged scalars ∆ ±± and H ± . Such a scenario has been widely studied in the literature and turns out to be quite appealing because it could be tested at the LHC [34][35][36][37][38][39][40][41][42][43][44]. For instance, the existence of charged scalar bosons provides additional contributions to the one-loop decays of the Standard Model Higgs boson. Indeed, they could affect the one-loop decays h → γγ [39,40] and h → Zγ [40] in a substantial way. In this case the signal strength µ γγ can set bounds on the mass of the charged scalars, ∆ ±± and/or H ± .
The doubly-charged scalar boson has the following possible decay channels: ± ± , W ± W ± , W ± H ± and H ± H ± . However, it is known that for an approximately degenerate triplet mass spectrum and vev v 3 < ∼ 10 −4 GeV the doubly charged Higgs coupling to W ± is suppressed (because it is proportional to v 3 as can be seen from Table III) and hence ∆ ±± predominantly decays into like-sign dileptons [41,44,45]. In this case, CMS [46] and ATLAS [47] have currently excluded at 95% C.L., depending on the assumptions on the branching ratios into like-sign dileptons, doubly-charged masses between 200 and 460 GeV 6 . For v 3 > ∼ 10 −4 GeV, the Yukawa couplings of triplet to leptons are too small so that ∆ ±± dominantly decays to like-sign dibosons, in which case the collider limits are rather weak [43,[48][49][50].
In the present "123" type-II seesaw model there are two additional physical scalars, a massive CP-even scalar H 3 and the massless majoron J. The latter, associated to the spontaneous breaking of lepton number, provides non-standard decay channels of other Higgs bosons as missing energy in the final state 7 .

V. INVISIBLE HIGGS DECAYS AT THE LHC
We now turn to the case of genuinely non-standard Higgs decays. We focus on investigating the LHC sensitivities on the invisible Higgs decays. In so doing we take into account how they are constrained by the available experimental data. In the previous section we mentioned that in our study the CP-even scalars obey the following mass hierarchy m H 1 < m H 2 < m H 3 . Furthermore, we will also assume that the masses m H 3 , m A , m H + and 6 From doubly-charged scalar boson searches performed by ATLAS and CMS one can also constrain the lepton number violation processes pp → ∆ ±± ∆ ∓∓ → ± ± W ∓ W ∓ and pp → ∆ ±± H ∓ → ± ± W ∓ Z [41]. This may also shed light on the Majorana phases of the lepton mixing matrix [34][35][36]. 7 These include, for example, H i → JJ and H ± → JW ∓ . Here we focus mainly on the first, the decays of H ± deserve further study but it is beyond the scope of this work and will be considered elsewhere. m ∆ ++ are nearly degenerate. As a consequence, the decay of any CP-even Higgs H i into the pseudo-scalar A is not kinematically allowed. Therefore, the new decay channels of the CP-even scalars are just, H i → JJ and H i → 2H j (when m H i < m H j 2 for i = j). The latter contributing also to the invisible decay channel of the Higgs as, The Higgs-Majoron couplings are given by, where O I ij are the elements of the rotation matrix in eq. (13) and the decay width is given by Following our conventions we have that the trilinear coupling H 2 H 1 H 1 turns out to be, ) . (34) and hence, for example when m H 1 < 2m H 2 , the decay width H 2 → H 1 H 1 is given by As we already mentioned, a salient feature of adding an isotriplet to the Standard Model is that some visible decay channels of the Higgs receive further contributions from the charged scalars, namely the one-loop decays h → γγ and h → Zγ. That is, the scalars H ± and ∆ ±± contribute to the one-loop coupling of the Higgs to two-photons and to Z-photon, leading to deviations from the Standard Model expectations for these decay channels. The interactions between CP-even and charged scalars are described by the following vertices, Note that the contributions of H ± and ∆ ±± to the decays h → γγ and h → Zγ are functions of the singlet's vev v 1 , this is in contrast to what happens in the type-II seesaw model with explicit violation of lepton number. According to eq. (26) the dimensionless parameters λ i and β i can change the sign of the couplings of g HaH + H − and g Ha∆ ++ ∆ −− , hence the contribution of the charged scalars to h → γγ and h → Zγ may be either constructive or destructive.
For the computation of the decay widths h → γγ and h → Zγ we use the expressions and conventions given in Ref. [51]. The decay width Γ(H a → γγ) turns out to be where G F is the Fermi constant, α is the fine structure constant and the form factors X j i are given by 8 , where τ x = 4m 2 x /m 2 Z . Here N F c and Q F denote, respectively, the number of colors and electric charge of a given fermion. The one-loop function f (τ ) is defined in appendix B. The parameters C a correspond to the Standard Model Higgs couplings in eq. (28).
The decay width Γ(H a → Zγ), using the notation in Ref. [51], is expressed as follows 8 We have taken into account that v 3 is very small so that any contribution involving the triplet's vev is neglected. Then for instance the Feynman rule for the vertex Table III).
where the form factors X j i are given by 9 , where C b 0 and ∆B b 0 are defined in appendix B.

VI. TYPE-II SEESAW NEUTRAL HIGGS SEARCHES AT THE LHC
We stated above that in our study we are assuming m H 1 < m H 2 < m H 3 and v 1 v 2 . Furthermore, because of the ρ parameter and the astrophysical constraint on the triplet's vev we also have that v 3 v 1 , v 2 . We found that the smallness of v 3 and the perturbativity condition of the potential lead to a very small mixing between the mass eigenstate H 3 and the CP-even components of the fields, σ and Φ, in other words, the angles α 13 and α 23 must lie close to 0 or π. As a result, we obtain the following relation, This extra mass relation is derived from eq. (24), by using eq. (25) and the fact that α 13,23 ∼ 0(π). In addition, also as a result of α 13,23 ∼ 0(π), we find that the coupling of H 3 to the Standard Model states is negligible, In Fig. 12 of appendix A we give a schematic illustration of the mass profile of the Higgs bosons in our model. The mass spectrum and composition are summarized in Table II, and provide a useful picture in our following analyses.

A. Analysis (i)
In this case we have taken the isotriplet vev v 3 = 10 −5 GeV, automatically safe from the constraints stemming from astrophysics and the ρ parameter. We have also considered 9 Here we have also assumed v 3 the following mass spectrum, and varied the parameters as v 1 ∈ [100, 2500] GeV, α 12 ∈ [0, π] and α 13,23 where 0 ≤ δ α < 0.1. As described in section IV we must enforce the LEP constraints on the lightest CP-even Higgs H 1 and LHC constraints on the heavier scalars. The near mass degeneracy of H 3 , A, H ± and ∆ ±± ensures that the oblique parameters are not affected.
In analogy to the type-II seesaw model with explicit lepton number violation we expect that, because of v 3 < 10 −4 GeV, the doubly-charged scalar predominantly decays into same sign dileptons [41,44,45] and that m ∆ ±± = 500 GeV is consistent with current experimental data, see subsection IV D.
We show in Fig. 3 the mass of the lightest CP-even scalar as a function of the absolute value of its coupling to the Standard Model states, |C 1 | in eq. (28). The blue region corresponds to the LEP exclusion region and the green(red) one is the LHC allowed(exclusion) region provided by the signal strengths 0.8 < µ XX < 1.2. The presence of light charged scalars can enhance significantly the diphoton channel of the Higgs [39]. Fig. 4 shows the correlation between µ ZZ and µ γγ (µ Zγ ) on the left(right) with µ γγ 1.2 for charged Higgs bosons of 500 GeV. The correlation between the signal strength µ ZZ and the signal strengths µ γγ and µ Zγ is shown in Fig. 4. Note that the former may exceed one due to the new contributions of the singly and doubly charged Higgs bosons. Figure 4: Analysis (i). On the left, we show correlation between µ ZZ and µ Zγ . On the right, correlation between µ ZZ and µ γγ . The color code as in Fig. 3.
The invisible decays of the Higgs bosons, characteristic of the model, turn out to be correlated to the visible channels, represented in terms of the signal strengths, as shown in Fig. 5. Note that the upper bound on the invisible decays of a Higgs boson with a mass of 125 GeV has been found to be BR(H 2 → Inv) 0.2. This limit is stronger than those provided by the ATLAS [52] and the CMS [53] collaborations 10 .
In Fig. 6 we depict the correlation between the invisible branching ratios of H 2 with the one of the lightest scalar boson H 1 . And, as can be seen, H 1 can decay 100% into the invisible channel (majorons). 10 The ATLAS collaboration has set an upper bound on the BR(H → Inv) at 0.28 while the CMS collaboration reported that the observed (expected) upper limit on the invisible branching ratio is 0.58(0.44), both results at 95% C.L.  Finally, as we have mentioned we obtained that the reduced coupling of H 3 to the Standard Model states is C 3 ∼ O(10 −7 ) so that it is basically decoupled. As a result its invisible branching is essentially unconstrained, 10 −5 BR(H 3 → Inv) ≤ 1. On the other hand we find that the constraint coming from the LHC on the pseudo-scalar A with a mass of 500 GeV is automatically satisfied as well, since from the LHC,

B. Analysis (ii)
We now turn to the other case of interest, namely with v 3 = 10 −5 GeV, as before. Now we scanned over v 1 ∈ [100, 2500] GeV, α 12 ∈ [0, π] and α 13,23 = δ α (π − δ α ) (42) where 0 ≤ δ α < 0.1. As we already mentioned in this case we only have to take into account the constraints coming from Run 1 of the LHC at 8 TeV, see Table I. In practice we assume µ XX = 1.0 +0.2 −0.2 . We show in Fig. 7 the correlation between µ ZZ and µ γγ (µ Zγ ) on the left(right). As before, the allowed region is in green while the forbidden one is in red. We can see that µ γγ 1.2 for m H ± m ∆ ±± = 600 GeV.  In this case we find that eq. 32 (for α 13,23 ∼ 0(π) and v 3 v 1 , v 2 ) at leading order is given by, where m H 1 = 125 GeV. BR(H 1 → Inv) versus the Higgs-majoron coupling g H 1 JJ is shown on the right of Fig. 9. Note also from the left panel in Fig. 9 that BR(H 1 → Inv) is anti-correlated with v 1 , as expected. In Fig. 10 we show the correlation between the invisible branching ratio of H 2 (the Higgs with a mass in the range 150 GeV < m H 2 < 500 GeV) and the one of H 1 . We have verified that the LHC constraints on the heavy scalars (H 2 , H 3 and A) are all satisfied. As an example, the reader can convince her/himself by looking at Fig. 11 that H 2 easily passes the restriction stemming from σ(ggH 2 )BR(H 2 → τ τ ) (top left) and/or σ(bbH 2 )BR(H 2 → τ τ ) (top right). The black continuous lines on those plots represent the experimental results from Run 1 of the CMS experiment [30]. We also found that the square of the reduced coupling of H 2 to the Standard Model states is C 2 2 0.1 for m H 2 = [150, 500] GeV. Then, one finds that the experimental upper bounds set by the search for a heavy Higgs in the H → W W and H → ZZ decay channels in [3,29] are automatically fulfilled. However, improved sensitivities expected from Run 2 may provide a meaningful probe of the theoretically consistent region, depicted in green. Also in this case, H 3 is decoupled, so the restrictions on H 3 and the massive pseudoscalar A are automatically fulfilled.

VII. CONCLUSIONS
In this paper we have presented the main features of the electroweak symmetry breaking sector of the simplest type-II seesaw model with spontaneous violation of lepton number. The Higgs sector has two characteristic features: a) the existence of a (nearly) massless Nambu-Goldstone boson and b) all neutral CP-even and CP-odd, as well as singly and doubly-charged scalar bosons coming mainly from the triplet are very close in mass, as illustrated in Fig. 12 of appendix A. However, one extra CP-even state, namely H 2 coming from a doublet-singlet mixture can be light. After reviewing the "theoretical" and experimental restrictions which apply on the Higgs sector, we have studied the sensitivities of the searches for Higgs bosons at the ongoing ATLAS/CMS experiments, including not only the new contributions to Standard Model decay channels, but also the novel Higgs decays to majorons. For these we have considered two cases, when the 125 GeV state found at CERN is either (i) the second-to-lightest or (ii) the lightest CP-even scalar boson. For case (i), we have enforced the constraints coming from LEP-II data on the lightest CP-even scalar coupling to the Standard Model states and those coming from the LHC Run-1 on the heavier scalars. In case (ii), only the constraints coming from the LHC must be taken into account. Such "invisible" Higgs boson decays give rise to missing momentum events. We have found that the experimental results from Run 1 on the search for a heavy Higgs in the H → W W and H → ZZ decay channels are automatically fulfilled. However, improved sensitivities expected from Run 2 may provide a meaningful probe of this scenario. In short we have discussed how the neutrino mass generation scenario not only suggests the need to reconsider the physics of electroweak symmetry breaking from a new perspective, but also provides a new theoretically consistent and experimentally viable paradigm.