Vacuum stability and spontaneous violation of the lepton number at low energy scale in a model for light sterile neutrinos

It is well known that the Standard Model of the Electroweak interactions rests on a metastable vacuum. This can only be fixed by means of new physics. Presently neutrino physics provides the most intriguing framework to formulate new physics. This is so because, in addition to the problem of the lightness of the active standard neutrinos, currently MiniBooNE experimental result may be indicating that sterile neutrinos exist and are light, too. In this case, it is reasonable to expect that the framework that yields light active and sterile neutrinos could stabilize the vacuum, too. In order to achieve this goal, we consider an extension of the standard model which involves new fermions in the form of right-handed neutrinos ($\nu_R$) and new scalars in the form of triplet ($\Delta$) and singlet ($\sigma$). Within this framework, tiny masses are obtained when we consider that lepton number is spontaneously broken at low energy scale which means that $\Delta$ and $\sigma$, both, develop very small vacuum expectation values. We investigate if this setting leads to a stable vacuum. For this we obtain the whole set of conditions over the Quartic Terms of the Potential that ensures that the model is Bounded From Below(BFB) and evaluate the RGE-evolution of the self coupling of the Higgs. We show that in such a scenario the Quartic Coupling $\Phi^T \Delta \Phi \sigma$, where $\Phi$ is the standard Higgs doublet, is responsible for the stability of the Electroweak Vacuum up to Planck scale. We also extract constraints over the parameters of the Potential by means of Lepton Flavor Violating(LFV) processes and from invisible decay of the standard-like Higgs.


I. INTRODUCTION
Although people has devoted considerable attention to the study of extensions of the standard Higgs sector[1], [2], [3], relative few attention has been given to a Higgs sector involving triplet (∆), doublet (Φ) and singlet (σ) of scalars [4], [5], [6]. From now on we refer to this case as 3-2-1 model. This model is interesting by its own right. However it get even more interesting when right-handed neutrinos are introduced, too. This is so because in this case the 3-2-1 model yields the most general neutrino mass matrix involving Majorana and Dirac mass terms for both neutrinos. Hence, when we assume that lepton number is spontaneously violated at low energy scale, right-handed neutrinos acquire light masses and may explain the recent MiniBooNE experimental result [7] by means of neutrino oscillation.
In this work we derive the complete set of conditions that guarantee the Potential of the 3-2-1 model to be BFB. For the specific case when lepton number is spontaneously broken at low energy scale, we obtain the spectrum of scalars of the model and discuss the stability of the vacuum by evaluating the RGE-evolution of the self-coupling of the standard-like Higgs up to Planck scales . This case is particularly interesting because it encompasses a Majoron and a light CP-even scalar in their spectrum of scalars. We discuss the contributions of these scalars for the invisible decay channels of the standard-like Higgs and of the neutral gauge boson Z. We also obtain the constraints that LFV put over the parameters of the Potential.
In what concern neutrino physics, we provide a solution, i.e., a set of values for the Yukawa couplings, that recovers the standard neutrino sector and provides at least one right-handed neutrino with mass resting on eV scale and robustly mixed with the standard neutrinos in such a way that accommodates MiniBooNE current results by means of neutrino oscillation and is in agreement with cosmological data. This work is organized as follows. In Sec. II we develop the main aspect of the model including neutrino masses, while in Sec. III we develop the scalar sector. In Sec. IV we discuss the stability of the vacuum. In Sec. V we present our final remarks.

II. THE 3-2-1 MODEL
The leptonic sector of the model is composed by the standard doublet L plus right-handed neutrinos in the singlet form, (1) where i = e , µ , τ , while the standard scalar sector is composed by one triplet, one doublet and one singlet of scalars, The quark sector is the standard one.
The most general potential involving this scalar content and that conserves lepton number is composed by the following terms With such lepton and scalar content, the Yukawa interactions that generate mass for all neutrinos of the model is given by The Yukawa interactions of the charged fermions are the standard ones.
When the neutral scalars of the model develop vacuum expectation values (VEV) different , the Yukawa interactions in Eq. (4) provide the following mass terms for the neutrinos, Considering the basis ν = (ν L ν C R ) T , we can simplify Eq. (5) to and the 6×6 symmetric mass matrix is given by where The relation among the flavor basis, ν, with the physical ones, N = where M 1 = diag(m 1 m 2 m 3 ) T and M 2 = diag(m 4 m 5 m 6 ) T .
In order to go further we need to obtain information about the VEVs v 1 , v 2 and v 3 . For this we have to develop the scalar sector of the model.
Firstly, we expand the neutral scalar fields around their respective VEVs, and obtain the set of minimum conditions required by the potential above to allow spontaneous breaking of the symmetries of the model which include the global B − L symmetry, On analyzing this set of constraint, observe that the first and third relations provide This relation is interesting because it relates v 3 , which has an upper bound of 2 GeV, with v 1 that is free to develop any value. According to this relation, if we assume that µ 3 ∼ µ 1 than we have v 1 ∼ v 3 . Any hierarchy among v 3 and v 1 translates in hierarchy among the energy mass scale µ 3 and µ 1 . For example: if we assume v 1 at TeV and v 3 at eV scale, the relation above implies µ 3 = 10 12 µ 1 which sounds very weird. Thus, it seems that the potential above prefers scenarios where both v 1 and v 3 are not so distant one from another. Since v 3 must be small to accommodate standard neutrino masses, then v 1 must be small, too. We can conclude that this model prefers that right-handed neutrinos are light particles. The most strong reason to the existence of light right-handed neutrinos is the explanation of shortbaseline neutrino results (LSND and MiniBooNE) [7][8] by means of neutrino oscillation. In this case, the natural value for v 1 is one such that accommodates at least one right-handed neutrino with mass around eV with robust mixing with the standard neutrinos and is in conciliation with cosmology. We follow this scenario.
Such a scenario may be realized for the following set of values for the VEVs, v 1 = 10 5 eV; v 2 = 246 GeV; v 3 = 1 eV, (12) and the following set of values for the Yukawa couplings, The mixing matrix, U , responsible by the diagonalization of M D+M and that relates the basis ν with N , as in Eq. (8), is given by The values of m 1 , m 2 and m 3 given in Eq. (16) and the upper left 3 × 3 submatrix of U accommodate the current solar and atmospheric neutrino oscillations data. A nice thing to observe is that the mixing angles between N 4 , ν µ and ν e , together with the mass value of m 4 , are in such a way that they allow the explanation of neutrino anomalies suggested by the data from SBL neutrino experiments by means of neutrino oscillation. Finally, observe in U that N 5 and N 6 practically decouple from the other neutrinos. In other words, this case recovers the 3 + 1 sterile neutrino scenario.
A problem with models involving eV sterile neutrino is that they present a tension with current cosmological data [9]. We discuss this point later.

III. SCALAR SECTOR
We saw in the previous section that the scenario we are developing here is capable of accommodating neutrino physics including short-baselibe (SBL) anomalies as LSND and MiniBooNE. This provides a strong reason for we go deep into the development of such case. Thus, in this section we perform a careful analysis of the spectrum of the scalars of the model.

A. Spectrum of scalars
Here we are interested in the spectrum of scalars for the specific case when v 3 << v 1 << v 2 . We start developing the CP-even sector. Considering the basis (R 1 R 3 R 2 ), the potential above together with the minimum conditions provide, The complexity of this mass matrix does not allow us to obtain neither the eigenvalues or the eigenvectors. However, according to the hierarchy of the VEVs we assumed here, this matrix may be approximated by This means that R 2 decouple from the other ones, while R 1 and R 3 mix among themselves to form H 1 and H 3 according to the following relation where The masses are given by, Observe that, for the hierarchy of the VEVs assumed here, we have that H 2 will play the role of the standard Higgs while H 3 is a heavy Higgs, with mass around TeV scale, and H 1 is a light one with mass at eV scale.
In the CP-odd sector, things are much simple and the mass matrix in the basis (I 1 , I 2 , I 3 ) is given by Its diagonalization leads to a Goldstone boson, G, that is dominantly I 2 and will be eaten by the standard gauge boson Z; a massless pseudo-scalar, J, which we call the Majoron and a heavy pseudo-scalar, A, which is dominantly I 3 . The relation among these pseudo-scalars with the basis is given by where U I is given by For the case of interest here, the Majoron is related to the basis in the following way, which allow we conclude that it is dominantly singlet.
The mass of the pseudo-scalar A take the following expression, which allow we conclude that it is a heavy particle even for the set of VEVs considered here.
In what concern the charged scalars, in considering the basis (∆ + , φ + ), we have the following mass matrix for these scalars We can easily diagonalize this matrix and find the physical fields We see that there are not any relevant mixing between the charged fields. G ± is the Goldstone while H ± is the simply charged scalar whose mass expression is given by Observe that it must be heavy for the choice of the VEVs used here.
The doubly-charged scalar acquires the following mass expression which must be heavy, too.
Thus, we see have that, although the VEVs v 1 and v 3 are much smaller than v 2 , we have that the scalars that belong to the triplet ∆ are heavier than the standard-like Higgs and their masses are practically determined by the parameter κ. This is a consequence of the hierarchy of the VEVs. It is curious that the same hierarchy among the VEVs does the opposite with regard to the scalars belonging to the singlet σ. The scenario predicts a light scalar H 1 . The heavy scalars may be probed at the LHC, while the massless J and light H 1 will contribute to the invisible decay channels of the Higgs and Z.

B. Some constraints
The coupling constants κ, β 2 , λ 3,5 will play an important role in the RGE-evolution of the quartic coupling of the standard-like Higgs λ 1 . Thus, information on these parameters In what concern the invisible decay of Z, the Lagrangian of interest is given by Because R 3 mix with R 1 to compose H 1 and I 3 mix with I 1 to compose J, we have that this Lagrangian generates an interaction among Z , H 1 and J modulated by the following vertex where g is the SU (2) coupling constant and c W = cos(θ W ) with θ W being the Weinberg angle. is given in Eq. (21). The current data gives Γ(Z) inv = 500.1 ± 1.9 MeV [10].
Because M H 1 << M Z , the vertex above provides the following expression for the decay The expression for the decay width of Z in two neutrinos is given by Observe that Eqs. (35) and (36) provide According to this we have that Γ Z→JH 1 must be smaller than 2.1 MeV. Once v 3 v 1 = , at the end of the day we get This result confirms the hierarchy among the VEVs we are considering here.
In order to check that our scenario obeys the constraint put by the invisible decay of Z as discussed above, see that for v 1 = 10 5 eV and v 3 = 1 eV, we get Γ(Z → JH 1 ) = 124.5×10 −20 MeV which is much smaller than 2.1 MeV. The other possible contribution to Γ(Z) inv is Γ(Z 0 → JJJ). However we must have that Γ(Z 0 → JH 1 ) > Γ(Z → JJJ) because the later decay is obtained from the first by means of the decay H 1 → JJ. Thus, we conclude here that the invisible Z decay is not a threat to our model. Now let us extract constraints over the parameters of the potential by means of the invisible Higgs decay channels and the LFV process µ → eγ.
Let us consider the contributions that our case give to the invisible decay of the standardlike Higgs H 2 . We consider the following contributions Γ(H 2 → H 1 H 1 ) and Γ(H 2 → JJ).
Their decay widths take the expression [12] Γ( The prediction for the total decay width of the standard Higgs is around 4 MeV with ∼20% being invisible decay rates( BR(H 2 → inv) = 0, 26±0, 17). All this allows we conclude that β 2 , λ 3 and λ 5 are constrained to lie around 10 −2 or smaller.
Thus we conclude here that the 3-2-1 model in the regime of low energy scale, although has a Majoron, which is a massless pseudo-scalar, and a light CP-even scalar it is a safe model in what concern the invisible decay of the standard neutral gauge boson Z. As a nice fact we have that our particular case gives reasonable contribution to the invisible decay of the standard Higgs through the channels Γ(H 2 → H 1 H 1 ) and Γ(H 2 → JJ). In other words, our case may be constrained by future improvement of the data concerning Higgs physics.
In what concern LFV processes, the muon decay channel µ → eγ may provide strong constraints on the parameters of the Potential. In one-loop order we have the following expression for the branching ratio of this process [11] where α is the fine structure constant and G F = 1.1663787 × 10 −5 GeV −2 .
On substituting the expression of the mass of the doubly charged scalar given in Eq.
(32), we have that for the fixed values of Y L 's given in Eq. (14) and of the VEVs given in Eq. (12), the upper bound BR(µ → γe) < 5.7 × 10 −13 [15]translates in the following lower bound over κ With this set of constraints in hand, we are ready to analysis the RGE-evolution of the quartic coupling of the standard-like Higgs λ 1 .

IV. VACUUM STABILITY
Now that we have developed the scalar sector by finding the spectrum of scalars for a particular set of values of the VEVs and obtained some constraints over the parameters of the potential due to Higgs invisible decay and lepton flavor violation, it is the moment to investigate the stability of the vacuum by finding the bound from below conditions and calculating the running of the self coupling of the Higgs.

A. Bound from Below conditions
In order to assure that the scalar Potential of the 3-2-1 model is bounded from below at large field strength, where the potential is generically dominated by the Quartic terms, we need to find the set of conditions that guarantee that the parameters of the Quartic Couplings of the Potential are positive when the fields go to infinity. We find the whole set of conditions and paved the way for similar models. We follow the techniques employed in [16].
We also need to develop the following parameters, where 1 2 ≤ ζ ≤ 1, 0 ≤ ξ ≤ 1 and −1 ≤ α ≤ 1. Two of them are already knew in the literature. The third one is a new parameter. We can see in detail in Appendix A how we can limit this parameter.
Let also define new variables x and y that must vary between 0 and 1 in the following way: Replacing Eq. (45) in Eq. ( 42) we get, We manage things such that we can express these quartic terms in the following way, We can fix y = 0 or y = 1 to obtain the cases when the quartic couplings of the potential is positive. When we do this, we obtain the following conditions For A x > 0 we need to use the same argument as before. Fixing x = 0 and x = 1 we have similar conditions for the inequalities λ 1 > 0, These new conditions depends of the parameters in Eq. (47). They vary in different ranges, but we only need to study the boundary values of these intervals. In this case the new conditions are: Using the same argument for the condition in Eq. ( 49), it turns easy to see that Using the condition in Eq. (50) and the same fact that C x can have x = 0 or x = 1, we obtain β 2 − 2κα + 2 β 1 λ 1 > 0, The first inequality has two different solutions while the second has four ones. At the end of the day, we have  The main contributions for the beta function of λ 1 involve the following terms where g and g Y are the gauge couplings of the standard gauge group SU (2) and U (1) Y while y t is the Yukawa coupling of the quark top.
Observe that the couplings β 2 , λ 3,5 and κ give positive contributions to the running of λ 1 .
However, as showed above, the invisible Higgs decay requires β 2 , λ 3,5 be minor then 10 −2 which turns insignificant they contributions to the RGE-evolution. Rest us the contribution of the parameter κ. In Fig. 1  In such a scenario, we obtained the whole set of conditions that guarantee the model is Bounded From Below and studied the RGE-evolution of the self-coupling of the standard-like Higgs. As main result we have that the quartic coupling κΦ T ∆Φσ plays a central role in the process and stability of the vacuum requires κ > 0.3.
As interesting consequence, we remark that the model has one Majoron (J) and one light Higgs (H 1 ) composing the spectrum of scalar of the model. Their contributions to the invisible decay rate of the standard-like Higgs, H 2 → JJ and H 2 → H 1 H 1 , were considered and the results are the bounds β 2 , λ 3 , λ 5 ≤ 10 −2 over the couplings of the potential.
In what concern the neutrino sector, the scenario recovers the 3+1 sterile neutrino model which explain MiniBooNE experiment by means of neutrino oscillation. However, we know that light sterile neutrinos are strongly disfavored by current cosmological data involving Big Bang Nucleosynthesis(BBN) , Cosmic Microwave Background(CMB) anisotropies and Large Scale Structure(LSS) [9]. This is so because, in face of the large mixing required by MiniBooNE, neutrino oscillation may conduct sterile neutrino to thermal equilibrium with the active neutrino even before neutrinos decouple from the primordial plasma. A possible solution for this tension requires the suppression of the production of these neutrinos in the early universe. This avoids that they thermalize with the active ones at high temperature.
This may be achieved by means of secret interactions [19] which is nothing more than the interaction of the sterile neutrino with a pseudo-scalar, I, ∼ g sν C S γ 5 ν S I.
The solution to the tension requires I be lighter than the lightest sterile neutrino and g s take values in the range 10 −6 − 10 −5 . Observe that our scenario recover this solution. For this, recognize that g is Y R 11 whose value in the matrix in Eq. (15) is 1, 4 × 10 −5 and I is the Majoron J. In order to generate a small mass to J we just need to consider a term like: M σσσ in the potential. This term will generate a mass term to J proportional to M . On assuming that M < m N 4 we have a secret sector that reconciliates eV sterile neutrino with cosmology as done in [20].

VI. APPENDIX A
Here we will give a hint for the proof of the limitation of the parameter α. The definition of this parameter is Then, we can study term by term to see what is the behavior of this parameter, e.g., to see if it is limited or not. As an example, we choose the first term of the Numerator and expand the fields in the real and imaginary parts. Using the following expansion we will obtain the Denominator terms (only the real part) The idea here is to look closely in each real function and study their limitation range.
For the first term, R 2 2 R 3 R 1 , we have the following relation (for R 2 = 0 ) We can see easily that this last term is limited in the range [-1,1] with polar coordinates.
We use similar arguments for next terms and find that α lies in the range [-1,1].