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 SU(2)L 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.

The paper is organised as follows. In section 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 section 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 section 6 while results for the invisible Higgs decay branching ratio are given in section 7. Finally, a summary is presented in section 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

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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, , 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

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

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It is easy to verify that it has zero determinant and two zero eigenvalues; their eigenvectors are given by 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 (3.17)

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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 h 1 , h 2 and h 3 as such that 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 where

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.
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 figure 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 refs. [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

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Using the rotation matrices O R and O I we get 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 , from the form of the pseudo-scalar mass matrix, eq. (3.12). In our case, it leads to the constraint 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]. If the majoron mass exceeds the characteristic temperatures of stellar environments, then the bound in eq. (5.3) need not apply. While the missing energy signature associated to the light majoron would remain, there would be important changes in the phenomenological analysis of the scalar sector.
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,  Table 3. ATLAS results for the 13 TeV data, ref. [60].
Applying this first restriction is useful in order to optimize the size of our data points sample. Notice that our model has the same structure of the Higgs coupling to the vector bosons as any two Higgs doublet model. This means we can, for instance, use the results of ref. [62] to obtain 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  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 figure 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 figure 2 we also plot the allowed points once all the constraints on the model are implemented. The color code is as follows: the points in dark green have v L > 0.1 GeV, those in green have v L ∈ [0.01, 01]GeV and those in yellow green have v L < 0.01 GeV. We have imposed a lower bound on the value of v σ which is visible in the figure.

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  v L < 0.01GeV yellow green points.
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 figure 3 and figure 4. The mixing angles in the neutral scalar rotation matrix are shown in figure 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 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 figure 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 JHEP10(2019)245  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 figure 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 con-

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tributes 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: 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 the computation of the decay widths h → γγ and h → Zγ, we used the expressions and conventions given in ref. [62]. The results obtained for the invisible branching ratios of the 125 GeV Higgs boson are shown in figure 6. In the left panel we show the BR(h 125 → invisible) as a function of v L and in the right panel as a function of the mass of the second scalar Higgs boson. The points in dark green satisfy the LHC constraints on the signals strengths at 3σ level. The points in yellow green satisfy those constraints at 2σ.
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 figure 7, where we plot the invisible branching ratio of the 125 GeV Higgs boson, versus the signal strength of the Higgs boson produced via gluon fusion and decaying in the ZZ final state.
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 figure 7 we show the correlation between the invisible JHEP10(2019)245  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 figure 8 the correlation between the invisible branching ratios of the second and third CP even Higgs bosons. On the right panel of figure 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. Notice that these couplings are bounded due to the sum rule of eq. (5.7). In dark green the LHC constraints are imposed at 3σ, while in yellow green (on top) is shown the result of requiring 2σ LHC constraints.  Table 4. Parameters for point P1.  Table 5. Parameters for point P2.

P1: BR
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.
We see that the second CP even Higgs boson can have a sizable invisible branching ratio, while its coupling to the vector bosons is still large, therefore allowing, in principle, to be searched at the LHC. In this case, due to eq. (5.7), the coupling to vector bosons of the third CP even Higgs boson is very small, making it very difficult to produce. 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 figure 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 singlet. Neutrino mass generation through spontaneous violation of lepton number implies the existence of a Nambu-Goldstone boson. The existence of such "majoron" would lead to stringent astrophysical constraints. A consistent electroweak symmetry breaking pattern requires a compressed mass spectrum of scalar bosons in which the Standard Model Higgs boson can have a large invisible decay into the invisible majorons.
The Higgs boson production rates are similar to what is expected in two-doublet Higgs schemes. Indeed, we saw in eq. (5.7) that the couplings of the CP even Higgs bosons to the vector bosons obeys a simple sum rule, characteristic of two-doublet Higgs boson schemes. 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.

A Unitarity constraints
In this appendix we list all the coupled channel matrices for the states in table 1. As we discussed before, these matrices can be separated by charge Q and hypercharge Y of the initial and final state, as the states with different values of Q, Y will not mix.
A.1 Q = 2, Y = 2 We start with the highest charge combination. We have already discussed this case. We get the following matrix Finally we get the last coupled channel matrix,