Electroweak Breaking and Higgs Boson Profile in the Simplest Linear Seesaw Model

We examine the simplest realization of the linear seesaw mechanism within the Standard Model gauge structure. Besides the standard scalar doublet, there are two lepton-number-carrying scalars, a nearly inert SU2 doublet and a singlet. Neutrino masses result from the spontaneous violation of lepton number, implying the existence of a Nambu-Goldstone boson. Such"majoron"would be copiously produced in stars, leading to stringent astrophysical constraints. We study the profile of the Higgs bosons in this model, including their effective couplings to the vector bosons and their invisible decay branching ratios. A consistent electroweak symmetry breaking pattern emerges with a compressed spectrum of scalars in which the"Standard Model"Higgs boson can have a sizeable invisible decay into the invisible majorons.


Introduction
Non-zero neutrino masses constitute one of the most robust evidences for new physics.
Ever since the discovery [1,2] and confirmation [3,4] of neutrino oscillations took place, the efforts to underpin the origin of neutrino mass have been fierce. Yet the basic dynamical understanding of the smallness of neutrino mass remains as elusive as ever. We have no clue as to what is the nature of the underlying mechanism and its characteristic energy scale. A popular approach to neutrino mass generation is the type-I seesaw mechanism, in which neutrinos get mass due to the exchange of heavy singlet mediators. Following Refs. [5][6][7] we assume here that the seesaw mechanism is realized using just the Standard Model (SM) gauge structure associated to the SU(3) c ⊗ SU(2) L ⊗ U(1) Y symmetry.
In its standard high-scale realization, the seesaw mechanism hardly leads to any phenomenological implication besides those associated to the neutrino masses themselves.
However, the seesaw can arise from low-scale physics [8,9]. For example, the seesaw mechanism can be realized at low scale in two different pathways, the inverse [10,11] and the linear seesaw [12][13][14]. These low-scale seesaw schemes require the addition of a sequential pair of isosinglet leptons, instead of just a single right-handed neutrino added sequentially.
In this work, we examine the simplest variant of the linear seesaw mechanism. In contrast to the conventional formulations [12][13][14], here left-right symmetry is not imposed.
In our setup the linear seesaw mechanism is realized in terms of the simplest Standard Model SU(3) c ⊗ SU(2) L ⊗ U(1) Y gauge structure, in which lepton number symmetry is ungauged. Spontaneous breaking of the lepton number symmetry hence implies the existence of a Nambu-Golstone boson, a variant of the so-called majoron [6,7]. Such minimally extended scalar boson sector contains, in addition to the Standard Model Higgs doublet, a second Higgs doublet, as well as a complex singlet scalar, both carrying lepton number charges. The singlet is required to ensure consistency with the LEP measurement of the invisible Z decay width. To prevent excessive stellar cooling by majoron emission, their vacuum expectation values (vevs) must obey a stringent astrophysical bound on the vev of the second doublet. This nicely fits the generation of neutrino masses by the linear seesaw mechanism. Our "neutrino-motivated" singlet extension of the two-doublet Higgs sector also leads to a peculiar benchmark for electroweak (EW) breaking studies at collider experiments.
We perform a numerical study of the Higgs sector taking into account consistency with astrophysical bound, EW precision data as well as perturbative unitarity and vacuum stability. These imply that the model nearly realizes the structure of the inert Higgs doublet model [15,16]. Moreover, it also has an impact on the physics of the 125 GeV Standard Model Higgs boson discovered at the LHC. Indeed, the profile of the Higgs sector is modified by the mixing of new CP-even states that affect its couplings and the presence of new CP-odd scalars. For example, it implies the existence of a new invisible Higgs decay channel with majoron emission [8]. This has phenomenomenological implications for collider experiments [17][18][19][20][21][22][23][24][25][26][27] and has indeed been searched by LEP and LHC collaborations [28,29].
The paper is organised as follows. In Sec. 2 we describe the basic theory setup of the model, while the main features of the Higgs potential and EW breaking sector are described in Sec. 3. The theoretical and experimental constraints are described in Sections 4 and 5, respectively. The physical profile of the Higgs boson spectra resulting from our numerical scans are presented in Sec. 6 while results for the invisible Higgs decay branching ratio are given in Sec. 7. Finally, a summary is presented in Sec.8.

Basic Theory Setup
The Yukawa sector contains, besides the three Standard Model lepton doublets with lepton number 1, three lepton singlets ν c i with lepton number −1 and three lepton singlets ψ i with lepton number 1. The resulting Yukawa Lagrangian is given as where h ij and f ij are dimensionless Yukawa couplings, M ij is an arbitrary matrix with dimensions of mass, and Φ and χ L are scalar doublets.
After symmetry breaking it will give the linear seesaw mass matrix, where the 3 × 3 sub-matrices are given as where v φ and v L represent the vevs of Φ and χ L , respectively. Here, the lepton number is broken by the M L νS term. This leads to the effective light neutrino mass matrix given by This matrix scales linearly with respect to the Dirac Yukawa couplings contained in M D , hence giving name to this seesaw mechanism. It is clear that this vanishes as M L → 0, ensuring that the small neutrino masses are "protected" by the lepton number symmetry.
Notice that, in contrast to the original left-right symmetric formulations [12][13][14], here we realize the linear seesaw mechanism just in terms of the standard SU(3) c ⊗ SU(2) L ⊗ U(1) Y gauge structure, having the existence of the majoron as its characteristic feature. In the next section we analyse the dynamical origin of the M L νS term from a scalar doublet Higgs vacuum expectation value, whose smallness is required by astrophysics and consistent with minimization of the potential.
The presence of the heavy TeV-scale neutrinos needed to mediate neutrino mass through the linear seesaw leads to a plethora of phenomenological signatures associated to the heavy neutrinos. These include their signatures such as lepton flavor violation processes in high energy collisions [44,45], as well as lepton flavor violation processes at low energies, such as µ → eγ [46,47]. Many other aspects of this theory have also been discussed in similar contexts, such as those associated with violation of unitarity of the lepton mixing matrix [48][49][50][51] and its possible impact upon neutrino oscillation experiments [52].
In this paper we focus primarily on the profile of the Higgs sector.

The Higgs Potential
We consider two doublets, Φ, χ L and a singlet σ, and choose the following lepton number assignments for the Higgs fields, With these quantum number the most general Higgs potential that we can write that respects all the symmetries is For definiteness we assume all couplings to be real.

Minimization Conditions
First we solve the minimization equations for the mass parameters µ, µ L , µ σ in the potential.
We get

Charged Mass Matrix
Substituting the minimization conditions we obtain the charged scalar mass matrix in the basis (φ + , χ + L ) as which we can easily see that has a zero eigenvalue corresponding to the charged Goldstone boson, where we have defined, as usual, The physical charged Higgs has a mass given by (3.8)

Neutral Scalar Matrix
The neutral scalar mass matrix is given by and we can check that has non-zero determinant, so there are three massive CP-even scalars.

Neutral Pseudo-Scalar Matrix
The neutral pseudo-scalar mass matrix is given by It is easy to verify that it has zero determinant and two zero eigenvalues; their eigenvectors are given by where we have defined The remaining pseudo-scalar is defined as A, and its squared mass is given by (3.14)

Parameters of the Lagrangian
It is useful to write the parameters of the Lagrangian in terms of the physical masses, vevs and the relevant angles of rotation. We have already shown in Eq. 3.4 that the quadratic terms of the potential can be written in terms of the vevs and the other parameters. This has already been done in writing the mass squared matrices.

Neutral Pseudo Scalar Mass Matrix
It is convenient to write the rotation matrix that connects the weak eigenstates to the mass eigenstates. We get  The matrix O I is given by Notice that the only parameters involved are again the vevs of the scalar multiplets in the Higgs potential. We also can invert Eq. (3.14) to obtain the parameter β 4 as follows

Charged Higgs Mass Matrix
The physical mass of the charged scalar m H + and pseudo-scalar mass M A are related through involving, again, the three vevs of the theory.

Neutral Scalar Mass Matrix
The diagonalization of the neutral scalar mass matrix will give us six relations that can be solved to get the parameters λ, λ L , λ σ and β 1 , β 2 , β 3 describing the quartic couplings in terms the physical masses, vevs and rotation angles. We define the scalar mass eigenstates where M 2 1 , M 2 2 and M 2 3 are the squared masses of h 1 , h 2 and h 3 , respectively. The matrix O R can be parameterized in terms of the angles θ i as and c i = cos α i , s i = sin α i . Using this parameterization one obtains, from Eq. (3.21), the following relations for the quartic parameters: (3.29)

Theoretical Constraints
In this section we study the theoretical constraints that must be applied to the model parameters in order to ensure consistency of the electroweak symmetry breaking sector.

Stability Constraints
In order to look at the stability or bounded from below (BFB) conditions, we start by considering only the neutral vacuum. Defining x, y and z such that: we can write the quartic terms of the potential 3.3 as where and where δ is some combination of phases. For the potential of the form V 0 , the conditions for stability have been given in Ref. [53]. The problem is the extra piece V 1 . However, we can always say that Now, note that for any positive x, y, we always have Therefore, we can bound our potential in the following way: which can be joined into V 0 to give Now, from Ref. [53] we get the conditions for the potential to be BFB as follows, These conditions are sufficient, although they might be more restrictive than the necessary and sufficient conditions, because of the method of bounding the potential we have used.

Unitarity Constraints
In order to discuss the unitarity constraints, we follow the procedure developed in Ref. [54].
As explained there, we have to obtain all the coupled channel matrices for the scattering of two scalars into two scalars, and bound the highest of their eigenvalues. Since the electric charge and the hypercharge are conserved in this high energy scattering, we can separate the states according to these quantum numbers. For this purpose, and because we are in the very high-energy limit, it is better to work in the unbroken phase. It is convenient then to use the following notation for the Higgs fields.
The relevant two body states are given in the entries of Table 1, and their complex conjugates. It is important to note that the index α is a compound index; it refers to a set of {i, j} indices for the two body states. Also note that in Table 1 the two body states with equal particles have a normalization of 1/ √ 2 that we have not written here, but must be included in the calculation.

Q Y State
Number of states n 1 n * 1 , n 1 n * 2 , n 2 n * 1 , n 2 n * 2 , s * s, ss, s * s * } We will give the full results in the Appendix A, but let us illustrate with the simplest example, the state S ++ α . With the notation of Eq. (4.13), the quartic part of the potantial will read Now consider the scattering This will proceed through the quartic vertex in Fig. 1. Therefore the amplitude M(w + 1 w + 2 → w + 1 w + 2 ) will be given by β 1 + β 5 . Now consider the scattering The Feynman rule for the quartic term would be 4λ, but remembering the factor of 1/ √ 2 for states with identical particles [54] we would get the amplitude and similarly for w + 2 w + 2 → w + 2 w + 2 . Therefore for the coupled channel states in S ++ α we get, where |a 0 | < 1/2, is the partial wave to be bounded, requiring the eigenvalues of Eq. (4.18) to obey, In the Appendix A we present all the coupled channel matrices for the sates in Table 1 and give their eigenvalues. Then the limits implied in Eq. (4.19) were aplied in the code.

Oblique Parameters S, T, U
In order to discuss the effect of the oblique S, T, U parameters, we use the results of Ref. [55,56]. To apply their expressions we have to find the matrices U and V that we now define explicitly. For the matrix U , we have which gives The matrix V is a 2 × 6 matrix defined by Using the rotation matrices O R and O I we get (4.23) To apply the expressions for S, T, U , we need the following matrices: with c β = cos β, s β = sin β . We also need the diagonal elements of V † V : (4.28) We have implemented a numerical code to take all of the above constraints into account.

Experimental Constraints
In this section we study the constraints that must be applied to the scalar potential parameters and which follow from various experimental considerations.

Astrophysics Constraints
Spontaneous breaking of a global symmetry such as lepton number leads to the existence of a Nambu-Goldstone boson, dubbed "majoron". This would be copiously produced in stars, leading to new mechanisms of stellar cooling. If the majoron is strictly massless (or lighter than typical stellar temperatures), one has an upper bound for the majoron-electron coupling [57,58] |g Jee | < ∼ 10 −13 .
with J|φ denoting the majoron projection into the Standard Model doublet. This can be obtained in a model-independent way by using Noether's theorem [7] and checked explicitly from the form of the pseudo-scalar mass matrix, Eq. (3.12). In our case, it leads to the Note, however, that the majoron can, in certain circumstances, acquire a nonzero mass as a result of interactions explicitly breaking the global lepton number symmetry.
These could arise, say, from quantum gravity effects. Unfortunately, we have no way of providing a reliable estimate of their magnitude. If the majoron is massive, it may play the role of warm [34][35][36][37][38][39] or cold Dark Matter [40,41], in addition to having other potential astrophysical and cosmological implications [42,43]. In what follows, we stick to the validity of Eq. (5.3), which amounts to having a (nearly) massless majoron.

LHC Constraints
We have to enforce the LHC constraints on the 125 GeV scalar Higgs boson. These are given in terms of the so-called signal strength parameters,  is separated by production process. We took the recent results from ATLAS [60] shown in Table 3. Finally, we have also enforced the LHC constraints in the other neutral and charged Higgs. This was done using the HiggBounds-4 package [61].
In practice, in order to optimize our scans, we started by imposing just the simple requirement that the coupling of the 125 GeV Higgs scalar boson with the vector bosons, As in the case of any multi-doublet Higgs model, NHDM [54], the couplings of the CP even Higgs bosons to the vector bosons obey a sum rule, where we have used the properties of the orthogonal matrix O R . We have started by enforcing Eq. (5.5) in our scans and then, to this optimized data set, we applied the constraints on Eq. (5.4) using the results from Table 2 and Table 3.
6 The profile of the Higgs bosons

General numerical scan
In this section we present a study of the impact of the previous constraints on the parameter space of the scalar potential in the linear seesaw model. In all plots we have imposed the theoretical constraints plus the LHC constraints on k V . We fix the h 1 mass to 125 GeV and vary other the model parameters in the following way: 1 v L ∈ [10 −6 , 10 2 ] GeV, v σ ∈ [10 3 , 1.2 × 10 4 ] GeV.
This first general scan is useful to see the impact of the stellar cooling constraint, Eq. (5.3), on the parameter space. In fact, to comply more easily with this constraint we have sampled v σ values above 1 TeV, aware that lower values could be possible. The resulting allowed region in the (v σ , v L ) plane is shown in Fig. 2. The yellow region is the set of values of (v σ , v L ) that satisfy Eq. (5.3). From here we immediately see that the allowed range of v L is much smaller than what we start with. This requires a stringent restriction on v L , i.e.
v L < 0.5 GeV. As mentioned above, this limit may be avoided in the presence of explicit soft lepton number violation terms. We took this in account to optimize the constrained scans below. Note that Eq. (5.3) is a relation among three vevs, but there is another relation coming from the W mass, which explains the boundary in the plane (v σ , v L ). In Fig. 2

Constrained Scan
Using the above result, we have performed a constrained scan where all points satisfy the astrophysical constraint, Eq. (5.3). We have also required that the points satisfy the LHC constraints on the signal strengths given in Table 2 and Table 3, at the 3σ level. As We have further restricted the scan to scalar masses values below 600 GeV, which might be explored in the next generation of collider experiments. How the allowed model parameters are constrained is shown in Fig. 3 and Fig. 4. The mixing angles in the neutral scalar rotation matrix are shown in Fig. 3. Their values are restricted by the LHC constraint on the signal strengths, which we imposed at the 3σ level. The color code is defined in Eq. (6.3). We see from Eq. (3.25) that smaller values of v L require the the numerator of that equation to be small, otherwise perturbative unitarity on λ L would be violated.
One can verify that the smallness of the numerator occurs for α 1 close to zero and for a compressed spectrum, as we will discuss below. We should also note that, as the allowed values of v L are quite small, β in Eq. 3.13, is a very small angle, as shown in Fig. 4; here, we show in the left panel the relation of β with v L , and in the right panel the correlation between k 2 V (h 1 ) and α 1 . The color code is the same. We see that the lower limit on k 2 V (h 1 ), set in Eq. (5.5), corresponds indeed to the signal strengths at 3σ, as we have points for all values of k 2 V (h 1 ) in that range. If we enforced the signal strength constraints at 2σ, the range would be reduced as we will show below.

Compressed Higgs Boson Spectra
We have found that, in order to ensure very small values of the lepton number breaking vev v L , the spectrum of Higgs bosons tends to be compressed. The reason can be traced to the perturbative unitarity constraints on the quartic coulings, in particular those on λ L . This can be easily understood by considering Eq. (3.25). As λ L is inversely proportional to the third power of v L , for small values of v L the numerator must be very small, to prevent λ L violating the perturbative unitarity constraints. It turns out that this is achieved by a compressed spectrum and a value of the mixing angle α 1 close to zero. In fact, one can show that, for v L v φ , the numerator of Eq. (3.25) vanishes for α 1 = α 3 = 0 and for equal masses. We did not impose these last conditions on our scan, it just turned out the that the good points have this profile. This is shown in the plots in Fig. 5. Notice that this compression is much stronger than that usually imposed by the oblique parameters.
One can see that, for larger values of v L , the points satisfying all the constraints, including those from the oblique parameters, can have a sizeable splitting.
In our model, the new decay channels of the CP-even scalars that contribute to the invisible decays are just h i → JJ and h i → 2h j (when M i > 2M j ). The latter also contributes as h i → 2h j → 4J. The Higgs-majoron coupling is given by (no summation on a), where O I ij are the elements of the rotation matrix in Eq. (3.17), and the decay width is given by Following our conventions, the trilinear coupling h 2 h 1 h 1 are giving by: 3) and hence, for example when 2M 1 < M 2 , we have the decay width h 2 → h 1 h 1 given by We have computed all the decay channels of the neutral and charged scalars in order to obtain their branching fractions. For finding the new Feynman rules and computing the amplitudes and decay rates we used the new software FeynMaster [63], that makes use of FeynRules [64] QGRAF [65], and FeynCalc [66,67]. For the computation of the decay widths h → γγ and h → Zγ, we used the expressions and conventions given in Ref. [62].  We see that an invisible branching ratio around 20%, close to the present upper bound [28,29], is possible within this model. This is consistent with all the LHC constraints including those on the signal strengths at the 3σ level. However, if 2σ limits are applied, the ratio reduces to a maximum of 10%. This is better illustrated on the left panel of  If it is found at the LHC that this signal strength becomes closer to one, then the invisible branching ratio in this model will be smaller, which can be important for LHC searches. On the right panel of Fig.(7) we show the correlation between the invisible branching ratios of the two lightest CP even Higgs bosons. We see that the invisible branching ratio of the Higgs boson lying above the one with m H = 125GeV can have a wide range of values, from very small (hence visible) to close to an 100% invisible branching ratio. To better understand the different possibilities, we have also plotted in the left panel of Fig. 8 the correlation between the invisible branching ratios of the second and third CP even Higgs bosons. On the right panel of Fig. 8 we show the absolute value of their couplings to vector bosons, which gives a measure of the probability that they are produced and observed. We notice that the effective couplings of the second and third CP even Higgs boson to vector bosons are limited to a maximum value below 1, due to the sum rule in Eq. (5.7) and to the fact that the coupling of the lightest Higgs boson is bounded from below to be in agreement with LHC data, as shown in Eq. (5.5). From these figures we see that the model has a very rich structure with many possibilities. In order to better illustrate this we selected three benchmark points with quite different characteristics.
These points typically have 2M H + < M h 2 , M h 3 , thus allowing the second and third CP even Higgs bosons to decay visibly into the charged one. One such example is given in Table 4.
We see that all the CP even Higgs bosons have non-negligible couplings to vector bosons, although these are bounded due to the sum rule of Eq. (5.7). In contrast, only the  lightest one (the 125 GeV SM Higgs) has an observable invisible branching ratio.
These points typically have 2M H + > M h 2 , M h 3 , so that the second and third CP even Higgs bosons do not decay visibly into the charged one. One such example is given in Table 5.   Table 6.  The first one (P3a) has a very compressed spectrum as a result of a low value of v L , as we have discussed before (see Fig. 5). For the second point (P3b), with a larger v L , it has a broader spectrum, but the couplings to vector bosons are smaller (see Eq. (5.7)).

Summary and conclusions
In this work we have examined the simplest realization of the linear seesaw mechanism within the context of the Standard Model SU(3) c ⊗ SU(2) L ⊗ U(1) Y gauge symmetry structure. In addition to the standard scalar doublet, we employ two scalar multiplets charged under lepton-number. One of these is a nearly inert doublet, while the other is a sin- In contrast, no sum rule holds concerning their visible decay branching, of course, so much so that, with current experimental precision, values of an invisible branching ratio up to 20% are allowed. However, future lepton colliders may play a decisive role here; in fact, it is expected that BR inv may be measured with precision better than 1% level [31], which will impose severe constraints on these decay modes. All in all, the model provides interesting and peculiar benchmarks for electroweak breaking studies at collider experiments. We think these deserve a dedicated experimental analysis that lies beyond the scope of this paper.
We have implemented all the constraints that the eigenvalues should satisfy [54]: