Anatomy of Higgs mass in Supersymmetric Inverse Seesaw Models

We compute the one loop corrections to the CP even Higgs mass matrix in the supersymmetric inverse seesaw model to single out the different cases where the radiative corrections from the neutrino sector could become important. It is found that there could be a significant enhancement in the Higgs mass even for Dirac neutrino masses of $\mathcal{O}$(30) GeV if the left-handed sneutrino soft mass is comparable or larger than the right-handed neutrino mass. In the case where right-handed neutrino masses are significantly larger than the supersymmety breaking scale, the corrections can utmost account to an upward shift of 3 GeV. For very heavy multi TeV sneutrinos, the corrections replicate the stop corrections at 1-loop. We further show that general gauge mediation with inverse seesaw model naturally accommodates a 125 GeV Higgs with TeV scale stops.

On the other hand, neutrino masses constitute one of the strongest signatures of physics beyond standard model. It is imperative that any supersymmetric extension of the standard model should also contain an explanation for non-zero neutrino masses. Among many ideas to generate tiny neutrino massse, the inverse seesaw model [20] is interesting as it is applicable at the weak scale with neutrino Yukawa couplinig of order one, and thus testable at colliders like LHC.
In the present work, we revisit the consequences of the inverse seesaw model for the lightest CP even Higgs boson mass [21,22]. We find parameter regions in which the one-loop corrections to the light Higgs mass can be very significant, leading to an increase of O(10) GeV, for the neutrino Yukawa coupling larger than about 0.2. This is in the line of observations of Refs. [23,24] which explored the role of extra vector like matter at TeV scale in increasing the light Higgs mass.
We then apply these corrections to phenomenological minimal supersymmetric standard models (PMSSM) and general gauge mediaed supersymmetry breaking models (GMSB) where the Higgs mass can become 125 GeV for supersymmetry breaking scale around TeV.
The paper is organized as follows. In the next section, we present one loop corrections to the Higgs mass and study the various parameter regimes. In the section 3, we work out two numerical examples in (1) PMSSM (2) General gauge mediated supersymmetric inverse seesaw model. We conclude in the section 4. Appendix A contains the main formulae, whereas appendices B and C contain RGE equations and some ancillary formulae.

II. ONE LOOP CORRECTIONS TO THE HIGGS MASS IN MSSM
The inverse seesaw model is characterized by a small lepton number violating mass, unlike the Type-I seesaw, the right-handed neutrinos can be as light as TeV or even below, with their Yukawa couplings of order one. This is achieved by having an additional singlet field, which we denote by S. The superpotential for this model is given as where W M SSM stands for the standard MSSM superpotential, and N c and S are singlet fields carrying lepton number +1,-1 respectively.
We consider one right-handed neutrino and one singlet S field in the discussion. It can be easily generalized to the case of two/three generations. The mechanism of how neutrino gets mass is well documented in the literature. We revisit it here briefly. In the basis, {ν L , N c , S}, the mass matrix, M ν , for the neutral leptons is given by Where m D = Y N H u . The eigenvalues are given as Since the inverse seesaw model is typically a low scale model, unlike the traditional seesaw mechanisms, one wonders if they can give large enough contribution to the light CP-even Higgs boson mass. This is more important to explore in the regions where m D can be relatively large 10 GeV. It should be noted that the range in m D (0.2 − 0.3)v for has been explored by collider searches [25,26]. There are constraints, however, on the size of m D for a given value of M R from electroweak precision tests [27]: This constraint is strictly for the electron and muon generations. For the third generation, it is slightly weaker, at the level of 0.07. This requires M R 3 TeV for m D close to the top quark mass. To compute the corrections to the light Higgs mass from the neutrino sector, we use the one loop effective potential methods of Coleman-Weinberg [28]. The methods have been used to derive the well-known one-loop corrections from the top-stop sector [29,30] and we extend them to the neutrino sector in the inverse seesaw model.
The scalar potential in this model consists of In the basis, {ν L ,Ñ c ,S}, the mass matrix, M 2 ν , for the sneutrinos is given by In the above matrix, elements with * correspond to symmetric entries of the mass matrix. The eigenvalues of the above mass matrix can be easily derived in the limit µ S m D M R , as required by the inverse seesaw mechanism and the electroweak precision tests. In the leading order of m D M R /d 2 , m D X N /d 1 1, they are given as One-loop corrections for the Higgs mass matrix will be derived from the one-loop effective scalar potential given by the standard form: In the basis Φ T = (Re{H 0 d }, Re{H 0 u }), the corrections to the CP even Higgs mass are given as where M 2 0 stands for the tree level mass matrix, ∆M 2 t and ∆M 2 ν are contributions from the top/stop sector and the neutrino/sneutrino sectors respectively. The full mass matrix has the W sin β 2 2 log In the following we write down the contribution from the neutrino sector in a compact notation as follows: In the above, and B ij and A ijk 's are given in the Appendix. While the above formulae are written for a single generation of right handed and singlet neutrinos, they can be easily generalized to three generations of right-handed and singlet neutrinos. The neutrino contributions to the light Higgs mass, though similar to those from the top/stop sector, have a couple of distinct features: (a) there is no colour factor associated with the neutrino contributions, so they typically lower than the top/stop contributions by a factor three, (b) The fermionic contributions, from the right-handed neutrinos can be significant, reducing the total contribution to the Higgs mass. This is highly dependent on the hierarchies between the relevant parameters: the soft masses and the right handed neutrino masses. To understand the overall relevance of these contributions, we will consider a few interesting cases below. Note that in our numerical analysis, we restrict m D M R /d 2 , m D X N /d 1 to be less than 0.1. We now consider the case where M R is the largest mass scale in the theory. This limit has been earlier considered in Ref. [22]. In this case, the enhancement in the light Higgs mass is much smaller and restricted to a few GeV. This is because the neutrino 1-loop correction, which is negative, significantly suppresses the total contribution from the neutrino sector. This is illustrated in Fig. 3 where

III. APPLICATIONS TO PMSSM AND GMSB
In the present section, we present two numerical examples as an application to the above calculation.

A. PMSSM and Inverse Seesaw
The phenomenological MSSM is low energy parameterisation of the supersymmetry breaking soft terms in terms of 19-22 parameters (See for example, [31]). To study the inverse seesaw model in As we can clearly see from the figure, even if m SU SY is below 1 TeV, there is enough contribution from the neutrino sector to a 125 GeV Higgs mass As right-handed neutrino mass M R increases, we get closer to the case-2 discussed in the previous section where larger right-handed neutrino masses make the contribution of right-handed neutrinos to Higgs mass negative, and thus reducing the Higgs mass. This effect is seen in the right panel of Fig. 6.

B. GMSB and Inverse Seesaw
Minimal gauge mediation models have been strongly constrained by the recent discovery of the Higgs mass of 125 GeV [5]. This is because the stop mixing parameter X t is predicted to be very small in these models. The X t can be made large through renormalisation group corrections, but this would require gluino masses to be greater than 8 TeV. Thus, the only way these models can accommodate a light CP even neutral higgs boson with a mass around 125 GeV is by increasing the masses of the stops beyond 4 TeV. This range for the stop masses is far beyond the LHC reach.
One can then consider modification by including either messenger-matter interactions, new fields or new interactions to achieve the Higgs mass within the required ball park. The feature of small X t also persists in general gauge mediation which is an umbrella of all possible gauge mediations both in the perturbative and non-perturbative regime. A recent analysis of the Higgs mass in general gauge mediation is presented in Ref. [32].
Incorporating the inverse seesaw model in general gauge mediation could generate the Higgs mass in the right ball park, due to the additional corrections induced by the neutrinos. We study this possibility in the present subsection. The set up of general gauge mediation we consider is specified by the following boundary conditions at the messenger scale: This generates a large enough positive contribution to m 2 N at the weak scale as m 2 Hu is negative at the weak scale from the requirement of electroweak symmetry breaking. The question then remains whether with the above boundary conditions it is possible to reproduce either of the conditions m N M R or m L M R to enhance the Higgs mass significantly.
Assuming as before only one right-handed neutrino and one singlet, we find that it is indeed possible to generate a Higgs mass of 125 GeV. We have to choose an appropriate boundary condition for the third generation sleptons such that it is close to the M R mass. In the table I we present An interesting application of this model lies in general gauge mediation where we have shown that implementing inverse seesaw model can enhance the light Higgs mass to the 125 GeV for stops less than a TeV, without resorting to any mechanism to enhance the stop mixing parameter X t .

Appendix A: Appendix
Here we have collected the expressions for B ij and A ijk 's which are used in the calculation of one-loop corrected Higgs mass.
where we have suppressed the generation indices.

Appendix B: RGE equations in SISM
In the last section of the appendix we present the renormalisation group equations for some of the superpotential and soft terms relevant to the analysis of general gauge mediation. To derive the formulae we use the standard formulae available in the literature [33,34]. The notation we use From eq (C5), it is evident that Higgs mass receives large correction from sneutrino masses. As sneutrino masses implicitly depend on m D , increase in m D increases the Higgs mass.