Realizing the supersymmetric inverse seesaw model in the framework of R-parity violation

If, on one hand, the inverse seesaw is the paradigm of TeV scale seesaw mechanism, on the other it is a challenge to find scenarios capable of realizing it. In this work we propose a scenario, based on the framework of R-parity violation, that realizes minimally the supersymmetric inverse seesaw mechanism. In it the energy scale parameters involved in the mechanism are recognized as the vacuum expectation values of the scalars that compose the singlet superfields $\hat N^C$ and $\hat S$. We develop also the scalar sector of the model and show that the Higgs mass receives a new tree-level contribution that, when combined with the standard contribution plus loop correction, is capable of attaining $125$GeV without resort to heavy stops.


I. INTRODUCTION
A current exciting challenge in particle physics is the explanation of the smallness of the neutrino masses through new physics at TeV scale. In this regard, the inverse seesaw mechanism(ISS) [1] became the paradigm of successful TeV scale seesaw mechanism. Its minimal implementation requires the introduction to the electroweak standard model (SM) of two sets of three neutral fermion singlets , N = (N 1 , N 2 , N 3 ) and S = (S 1 , S 2 , S 3 ), composing the following mass terms in the flavor basis, where ν = (ν 1 , ν 2 , ν 3 ) is the set of standard neutrinos. In the basis (ν , N C , S) the neutrino mass may be put in the following 9 × 9 matrix form, The challenge concerning the ISS mechanism is to find scenarios that realize it. This means to propose models that generate the mass terms in Eq. (1). In this regard, as the ISS mechanism works in the TeV scale, it seems to be natural to look for realization of the ISS mechanism in the framework of theories that we expect will manifest at TeV scale [2,3], which is the case of supersymmetry ( SUSY). Thus it seems to be interesting to look for scenarios that realize the ISS mechanism in the context of SUSY [4][5][6].
We know already that a natural way of obtaining small neutrino mass in the context of the MSSM is to consider that R-parity, R ≡ (−1) 2S+3(B−L) , is violated through bilinear terms like µ iLiĤu in the superpotential [7]. Thus we wonder if R-parity violation (RPV) is an interesting framework for the realization of the SUSYISS mechanism. For this, we implement the SUSYISS mechanism in a framework where R-parity and lepton number are violated explicitly but baryon number is conserved in a way that we call the minimal realization of the SUSYISS mechanism once the necessary set of superfields required to realize it is the original one,N C i andŜ i , only. Moreover, it has been extensively discussed that the minimal supersymmetric standard model (MSSM) faces difficulties in accommodating a Higgs of mass of 125 GeV, as discovered by ATLAS and CMS [8] while keeping the principle of naturalness [9]. This is so because, at tree level, the MSSM predicts a Higgs with a mass whose value cannot exceed 91 GeV. Thus robust loop corrections are necessary in order to lift this value to 125 GeV. Consequently stops with mass far above 1TeV are required. To accept this is to put the naturalness principle aside. We show that the SUSYISS mechanism developed here accommodates a 125 GeV Higgs mass without resort to robust loop corrections.

II. THE MECHANISM
The supersymmetric version of the ISS (SUSYISS) mechanism [4] requires the assumption of two sets of three singlet superfieldsN C i ,Ŝ i (i = 1, 2, 3) composing, with the MSSM superfields, the following extra terms in the superpotential, W ⊃LĤ uN C +ŜM NN C + 1 2Ŝ µ NŜ . A successful extension of the MSSM that realizes the SUSYISS mechanism must generate these terms. This would be an interesting result in particle physics since we would be providing an origin for the energy scales M N and µ N [5].
The mechanism we propose here is minimal in the sense that it requires the addition to the MSSM of the two canonical singlet superfieldsN C i andŜ i , only. Moreover, we impose that the superpotential be invariant under the following set of discrete symmetries, Z 3 ⊗ Z 2 , according to the following transformation: under Z 3 the transformations are, Now we make an important assumption. We assume that the scalars that compose the super- respectively. This assumption provides the source of the canonical mass terms M N and µ N of the SUSYISS mechanism. Note that, from the last two terms in the superpotential above, we have that the VEV of the scalarS becomes the source of the mass scale M N while the VEV of the scalarÑ C becomes the source of the mass scale µ N . In other words, the superpotential above together with the assumption that the scalarsN C i andŜ i develop non zero VEVs has the required ingredients to realize the SUSYISS mechanism.
Another important point of the model is to discuss the possible values v S i and v N i may take. For this we have to obtain the potential of the model. The soft breaking sector will play an important role in the form of the potential.
The most general soft breaking sector of our interest involves the following terms, Note that the last two trilinear terms violate explicitly lepton number and the energy scale parameters A s and A v characterize such violation.
A common assumption in developing ISS mechanisms it to assume that the new neutral singlet fermions are degenerated in masses and self-couplings. However, for our case here, it seems to be more convenient, instead of considering the degenerated case, to consider the case of only one generation of superfields. The extension for the case of three generations is straightforward and the results are practically the same.
The potential of the model is composed by the terms is given above in Eq. (5). The relevant contributions to V D are, In what concerns the F-term, the relevant contributions are given by the following terms, With the potential of the model in hand, we are ready to obtain the set of constraint equations Let us first focus on the third relation in the equation above. Observe that the dominant term inside the parenthesis is M 2 ν . Outside the parenthesis, on considering for while that v N < v S , the In view of this, from the third relation above we have that, For Mν > v S , we have v ν < v u,d,S , as expected.
Let us now focus on the fifth relation of the Eq. (8). The dominant term inside the parenthesis is This expression for v N is similar to the v ν case and suggests that v N is also small.
Let us now focus on the forth relation. Taking v ν , v N v S , we have that the dominant terms in that relation are, Perceive that M S dictates the value of v S . As the neutral singlet scalarS belongs to an extension of the MSSM, then it is reasonable to expect that its soft mass term M S lies at TeV scale.
Consequently v S must assume values around TeV. In regard to the first and second relations they control the standard VEVs v u and v d .
Let us return to the expression to v N in Eq. (10). As the neutral singlet scalarÑ also belongs to an extension of the MSSM, then it is reasonable to expect that its soft mass term MÑ lies at TeV scale, too. In this case perceive that the value of v N get dictated by the soft trilinear term A s . Thus a small v N means a tiny A s . As A s is a trilinear soft breaking term, then it must be generated by some spontaneous SUSY breaking scheme. The problem is that we do not know how SUSY is spontaneously broken. Thus there is no way to infer exactly the value of A s . Moreover, note that A s is a soft trilinear term involving only neutral scalar singlets by MSSM which turns its estimation even more complex. We argue here that it is somehow natural to expect that such terms be small. and evaluate A s in such a scenario is out of the scope of this paper. However, whatever be the case, in the framework of GMSB scheme A s must be naturally small and consequently v N , too. In this point we call the attention to the fact that the idea behind the ISS mechanism is that lepton number is explicitly violated at low energy scale. This suggests that the GMSB seems to be the adequate spontaneous SUSY breaking scheme to be adopted in realizing SUSYISS mechanism.
Let us discuss the case of gravity mediated supersymmetry breaking. As in the ISS mechanism lepton number is assumed to be explicitly violated at low energy scale, it is expected that There is still an issue to consider in respect to the scalar potential. As can be easily verified, the value of the potential at origin of the fields is zero. In order to guarantee that electroweak symmetry will be broken, we need the potential in the minimum to be negative. Taking the constraints in Eq. (8) to eliminate the soft masses in the scalar potential, we have, For the magnitudes of VEVs discussed above, the dominant term is − 1 4 λ 2 s v 4 S , which is negative. For the case of one generation considered here this is a strong evidence of the stability of the potential.
After all these considerations, we are ready to go to the central part of this work that is to develop the neutrino sector. For this we have, first, to obtain the mass matrix that involves the neutrinos. Due to the RPV the gauginos and Higgsinos mix with the neutrinos ν, N and S.
Considering the basis (λ 0 , λ 3 , ψ h 0 u , ψ h 0 d , ν, N c , S), we obtain the following mass matrix for these neutral fermions, where M 1 e M 2 are the standard soft breaking terms of the gauginos. We remark that on considering For this decoupling to be effective we must have v ν of order MeV or less. Diagonalization of this mass matrix implies that the lightest neutrino, which is predominantly the standard one, ν, get the following mass expression, This is exactly the mass expression of the ISS mechanism. For v S around TeV and v N around keV we obtain neutrinos at eV scale for v u at electroweak scale. In the case of three generations the pattern of the neutrino masses will be determined by Y ij ν .
To demonstrate the validity of these aproximations we can compute the mass eigenvalues from the full matrix in (13). For typical values of the supersymmetric parameters and v S ∼ 10 TeV, v N ∼ 10 keV, v ν ∼ 1 MeV and Y ν ∼ λ s = 0.21, we have the following order of magnitude for the mass eigenvalues (∼ TeV, ∼ TeV, ∼ 10 2 GeV, ∼ 10 2 GeV, ∼ 10 −1 eV, ∼ TeV, ∼ TeV), where the lightest particle is exclusively the standard neutrino. This result is encouraging and indicates that RPV is an interesting framework to realize the SUSYISS mechanism.
We end this section making a comparison of the SUSYISS developed here with the µνSSM in Ref. [11]. This model resorts to R-parity violation to solve the µ problem. However neutrino masses at sub-eV scale require considerable amount of fine tuning of the Yukawa couplings. We stress that, in spite of the fact that the SUSYISS model contains the particle content of the µνSSM, unfortunately it does not realize the µνSSM. This is so because if we allow a term likeŜĤ uĤd in the superpotential in Eq. (4), as consequence the entries ψ h 0 d S and ψ h 0 u S in the mass matrix in Eq. (13) would develop robust values which jeopardize the realization of the ISS mechanism.

III. THE MASS OF THE HIGGS
Now, let us focus on the scalar sector of the model. We restrict our interest in checking if the model may accommodates a 125 GeV Higgs mass without resorting to tight loop contributions.
For the case of one generation the model involves five neutral scalars whose mass terms compose a 5 × 5 mass matrix that we consider in the basis (H u , H d ,ν ,Ñ ,S). We do not show such a mass matrix here because of the complexity of their entries. Instead of dealing with a 5 × 5 mass matrix, which is very difficult to handle analytically, we make use of a result that says that an upper bound on the mass of the lightest scalar, which we consider as the Higgs, can be obtained by computing the eigenvalues of the 2 × 2 submatrix in the upper left corner of this 5 × 5 mass matrix [12]. This is a common procedure adopted in such cases which give us an idea of the potential of the model to generate the 125 GeV Higgs mass.
The dominant terms of this 2 × 2 submatrix are given by, We made use of the hierarchy among the VEVs, as discussed above, to obtain such a 2×2 submatrix.
On diagonalizing this 2 × 2 submatrix we obtain the following upper bound on the mass of the Higgs, Note also that Eq. (11) imposes that either M 2 S or v 2 S is negative. In order to the second term in Eq. (17) gives a positive contribution to the Higgs mass we take M 2 S negative and Y ν and λ s with opposite sign.
What is remarkable in the mass expression in Eq. (17) is that the second term provides a robust correction to the Higgs mass even involving the parameters that dictate the neutrino masses as the couplings Y ν and λ s and the VEV v S . This suggest an interesting connection between neutrino and Higgs mass. For illustrative proposals, perceive that for Y ν of the same order of λ s , v ν around MeV, v u around 10 2 GeV and v S of order tens of TeV, the second term provides a contribution of tens of GeV to the Higgs mass. This contribution is enough to alleviate the pressure on the stop masses and their mixing in order to keep valid the principle of naturalness.
In order to check the range of values the stop mass and the A t term may develop in this model, we add to m 2 h given above the leading 1-loop corrections coming from the MSSM stop terms [13], GeV is the VEV of the standard model, X t ≡ A t − µcot(β) is the stop mixing parameter and m s ≡ (mt 1 mt 2 ) 1/2 is the SUSY scale (scale of superpartners masses) where mt is the stop mass. In the analysis done below, we work with degenerated stops and, in all plots, we take v ν = 1 MeV and v S = 4 × 10 4 GeV.  Let us discuss a little some phenomenological aspects of the SUSYISS mechanism developed here. First of all, observe that the aspects of RPV concerning the mixing among neutralinos and neutrinos, as well as charginos and charged leptons, are dictated by the VEVs v ν and v N and the couplings Y ν and λ s , which are both small. The squarks sector is practically unaffected. Thus, with relation to these sectors, the phenomenology of the SUSYISS mechanism is practically similar to the case of the supersymmetric version of the ISS mechanism [4,14]. The signature of the SUSYISS mechanism developed here should manifest mainly in the scalar sector of the model due to the mixing of the neutral scalars with the sneutrinos which will generate Higgs decay channel with lepton flavor violation h → l i l j .
In general, as far as we know, this is the first time the ISS mechanism is developed in the framework of RPV. Thus many theoretical, as well phenomenological aspects of the model proposed here must be addressed such as experimental constraints from RPV, accelerator physics, analysis of the renormalization group equation, spontaneously SUSY breaking schemas, etc., which we postpone to a future paper [15]. Moreover, needless to say that in SUSY models with RPV the lightest supersymmetric particle is not stable which means that neither the neutralino nor sneutrino are candidates for dark matter [16] any longer. We would like to remark that because of the Z 3 symmetry used in the superpotential above cosmological domain wall problems are expected [17].
However, the solution of this problem in the NMSSM as well in the µνSSM [11] cases may be applied to our case, too [18].
Finally, concerning the stability of the vacuum, we have to impose that the potential be bounded from below when the scalar fields become large in any direction of the space fields and that the potential does not present charge and color breaking minima. Concerning the latter condition, we do not have to worry about this condition here because the new scalar fields associated to the superfield singlets,Ŝ andN C , are neutrals under electric and color charges. Concerning the former issue, the worry arises because at large values of the fields the quartic terms dominate the potential.
Thus we have to guarantee that at large values of the fields the potential be positive. Thus we have to worry with the quartic couplings, only. The negative value of λ s leads to two negative quartic terms. Considering this, on analyzing the potential above, we did not find any direction in the field space in which λ s negative leads to a negative potential. All direction we find involves a set of condition where it is always possible to guarantee that the potential be positive at large value of the fields [19]. Moreover, we took λ s negative for convenience. We may arrange the things such that all couplings be positive. For example, on taking λ s positive, v ν in Eq. (9) get negative, which guarantee a positive contribution to the Higgs masses and that all quartic couplings be positive.
However, a complete analysis of the stability of the potential is necessary. This will be done in [15].

IV. CONCLUSIONS
In this work we proposed the realization of the SUSYISS model in the framework of RPV. The main advantage of such framework is that it allows the realization of the SUSYISS model with a minimal set of superfield content where the superfieldsŜ andN C of the minimal implementation are sufficient to realize the model. To grasp the important features of the SUSYISS, we restricted our work to the case of one generation of superfields. As nice result, the canonical mass parameters M N and µ N of the SUSYISS mechanism are recognized as the VEVs of the scalarsS andÑ that compose the superfieldsŜ andN C . There is no way to fix the values of the VEVs v S and v N .
However, it seems plausible that v S and v N develop values around TeV and keV scale, respectively.
Thus, we conclude that RPV seems to be an interesting framework for the realization of the SUSYISS mechanism. We recognize that in order to establish the model a lot of work have to be done, yet. For example, we have to find the spontaneous SUSY breaking scheme that better accommodates the mechanism, develop the phenomenology of the model and its embedding in GUT schemes. We end by saying that the main results of this work are that the model proposed here realizes minimally the SUSYISS mechanism and provides a 125 GeV Higgs mass respecting the naturalness principle.   (blue dotted X t = 600GeV), (red dashed X t = 700GeV) and (red solid X t = 800GeV).