Type II neutrino seesaw mechanism extension of NMSSM from SUSY breaking mechanisms

We propose to accommodate Type II neutrino seesaw mechanism in (G)NMSSM from GMSB and AMSB, respectively. The scale of the heavy triplet scalar is identified to be the messenger scale. General features related to the Type II seesaw mechanism in NMSSM are discussed. The Type II neutrino seesaw-specific interactions can give additional Yukawa deflection contributions to the soft SUSY breaking parameters of NMSSM, which are indispensable to realize successful EWSB and accommodate the 125 GeV Higgs. Relevant numerical results, including the constraints of DM etc in AMSB-type scenarios, are also given. We find that Type II neutrino seesaw mechanism extension of NMSSM from both AMSB and GMSB can lead to realistic low energy NMSSM spectrum, both admitting the 125 GeV Higgs as the lightest CP-even scalar. The possibility of the 125 GeV Higgs being the next-to-lightest CP-even scalar in AMSB-type scenarios is totally ruled out by DM direct detection experiments.


Introduction
TeV scale supersymmetry(SUSY) is one of the most promising candidates for new physics beyond the Standard Model(SM). It can prevent the Higgs boson mass from acquiring dangerous quadratic divergence corrections, realize successful gauge coupling unification and provide viable dark matter(DM) candidates, such as the lightest neutralino assuming exact R-parity. Besides, the discovered 125 GeV Higgs [1,2] lies miraculously in the narrow 115-135 GeV window predicted by MSSM, which can also be seen as a triumph of low scale SUSY. However, low energy SUSY confronts many challenges from LHC experiments, the foremost of which is the null search results of superpartners at LHC. Recent analyses based on Run 2 of 13 TeV LHC and 36f b −1 of integrated luminosity constrain the gluino mass mg to upon 2 TeV [3] and the top squark mass mt 1 to upon 1 TeV [4] in some simplified models.
It is known that the low energy SUSY spectrum can be determined by the SUSY breaking mechanism. So we need to survey which SUSY breaking mechanism can give the desired low energy spectrum. Depending on the way the visible sector f eels the SUSY breaking effects in the hidden sector, the SUSY breaking mechanisms can be classified into gravity mediation [5], gauge mediation [6], anomaly mediation [7] scenarios, etc. Both GMSB and AMSB are calculable, predictive, and phenomenologically distinctive. Especially, they will not cause flavor and CP problems that bothers gravity mediation models.
The nature of tiny neutrino masses, which is discovered by neutrino oscillation experiments, is one of the major unresolved problems of particle physics now. It is known that Weinberg's effective dimension-5 operator is the lowest one which can generate tiny Majorana neutrino masses. Such an operator can be ultraviolet(UV)-completed to obtain three types of tree-level seesaw mechanism: type I seesaw [8], involving the exchange of righthanded neutrinos; type II seesaw [9], involving the exchange of scalar triplet; type III [10], involving the exchange of fermion triplet. The SUSY extension of seesaw mechanism can provide an unified framework to solve all the remaining puzzles of SM together.
GMSB can hardly explain the 125 GeV Higgs with TeV scale soft SUSY breaking parameters because of the vanishing trilinear terms at the messenger scale. Although nonvanishing A t at the messenger scale can be obtained in GMSB with additional messengermatter interactions [11,12,13], it is rather ad hoc to include such interactions in the superpotential. So it is interesting to see if certain types of messenger-matter interactions can arise naturally in an UV-completed model. Yukawa mediation contributions from messenger-matter interactions can also possibly be present [14] in deflected AMSB [15,16], which can elegantly solve the tachyonic slepton problem of minimal AMSB through the deflection of the renormalization group equation (RGE) trajectory [17]. In this paper, we can see that such messenger-matter interactions can be naturally present if we combine seesaw mechanism with the SUSY breaking mechanisms. Some of the messengers can act as the heavy fields integrated out in the neutrino seesaw mechanism. Although similar ideas had been proposed in [18,19] for MSSM, many new features related to the realization of next-to-minimal supersymmetric standard model(NMSSM) had not been discussed in previous works.
We will discuss the realization of NMSSM through the combination of neutrino seesaw mechanism with GMSB and deflected AMSB. NMSSM is the simplest gauge singlet extension of MSSM [20]. It can elegantly solve the µ-problem in MSSM by generating an effective µ-term after the singlet scalar acquires a vacuum expectation value (VEV). Furthermore, due to possible new tree level contributions to the Higgs mass, NMSSM can easily accommodate the 125 GeV Higgs boson, ameliorating the fine-tuning involved. Therefore, the Type II neutrino seesaw extension of NMSSM is a unified frame, which is phenomenological interesting.
This paper is organized as follows. In Sec 2, we discuss the realization of Type II seesaw mechanism in NMSSM from GMSB and AMSB, respectively. The relevant soft SUSY breaking parameters are given. New features related to realization of the Type II seesaw mechanism are discussed. Relevant numerical results are studied in Sec 3. Sec 4 contains our conclusions.

Type II neutrino seesaw mechanism in SUSY
In the ordinary Type II seesaw mechanism [9], tiny neutrino masses can arise through the Yukawa interaction of the left-handed SU (2) L doublet lepton to a very heavy SU (2) L triplet scalar with lepton number L = −2, as well as the coupling between the scalar triplet to the Higgs The third term plays a key role in determining the minimum of the full scalar potential so as to give a tiny vacuum expectation value(VEV) of ∆ L . Such a tiny VEV can in turn induce a Majorana mass for left-handed neutrinos The extension of Type II neutrino seesaw mechanism in MSSM is non-trivial. There are two SU (2) L doublet Higgs in the MSSM, so the Type II seesaw extension in MSSM can be seen as a special case of Type II neutrino seesaw extension in two-Higgs doublet model, which contains interactions between both scalar Higgs doublets to the heavy scalar triplet.
We can further extend the Type II seesaw mechanism to NMSSM. We need to introduce vector-like SU (2) L triplet superfields with U (1) Y quantum number Y = ±2 in the superpotential with general NMSSM superpotential From the superpotential (2.3), we can obtain the F-term of the triplets We require the SUSY to be unbroken at the triplet scale M T . So the F-flat conditions The neutrinos will acquire tiny Majorana masses through the type II seesaw mechanism This result can be understood to arise from the scalar potential The m T y u ∆ H * u H * u ∆ T term plays the role of the third term in (2.1). We should note that F-flat conditions for H u and H d can not be satisfied for solutions in eqn (2.6) with non-negligible µ term. Therefore, tiny SUSY breaking effects of order |F Hµ | 2 ∼ µ 2 v 2 will appear. In fact, the minimum conditions for H u and H d should also involve the soft SUSY breaking terms. The term involving µ with also gives a subleading contributions to neutrino masses. Such a SUSY extension of Type II neutrino seesaw mechanism can be nontrivially embedded into SUSY breaking mechanisms. In SUSY extension of neutrino Type II seesaw mechanism, there is an alternative contribution to neutrino mass in addition to the eqn (2.7). The trilinear soft term will be generated after SUSY breaking. From the minimum condition of the total scalar potential, including the soft SUSY breaking terms, the triplet VEV can be approximately given by So the resulting neutrino masses are given by The three terms can be destructive if or similarly for µ, tiny neutrino masses can be generated by fine tuning even if either terms in eqn (2.12) are not very small. In this paper, the messenger threshold can be identified to be the heavy triplet scalar threshold in Type II seesaw mechanism. This possibility provide an economic unified framework to taking into account both SUSY extension and neutrino masses. So m T is always much larger than the A H d H d ∆ T , which is typically the soft SUSY breaking scale. Successful EWSB requires µ to lie at the soft SUSY breaking scale. Therefore, the second and third terms in eqn (2.12) are always subleading unless the messenger scale is very low.

NMSSM with Type II neutrino seesaw mechanism from GMSB
The discovered 125 GeV Higgs have already set stringent constraints on the SUSY breaking mechanisms. To accommodate the 125 GeV Higgs in MSSM, TeV scale stop masses with near-maximal stop mixing are needed [21]. For O(10) TeV stops with small A t , although still possible to interpret the 125 GeV Higgs, exacerbate the little hierarchy problem arising from the large mass gap between the measured value of the weak scale and the sparticle mass scale. As ordinary GMSB predicts vanishing A t at the messenger scale, it necessitates the introduction of additional Yukawa deflection contributions if we would like to reduce the fine tuning involved. In the SUSY extension of Type II neutrino seesaw mechanism, the Higgs sector can participate in new interactions with the triplets, which leads to additional Yukawa mediation contributions to trilinear couplings at the messenger scale. Large A t can lead to reduced electroweak fine-tuning [22] even with TeV scale stops. Most discussions can be generalized to NMSSM for small λ or large tan β.
In GMSB, the VEV of spurion X is given by As emphasized in [20], successful electroweak symmetry breaking(EWSB) in NMSSM necessitates non-vanishing soft SUSY masses for S and A κ . As the soft mass of the gauge singlet S receives no contributions from ordinary GMSB, additional Yukawa mediation contributions should be included. It was noted in [23] that double species of SU (2) L triplet superfields with SU (3) c × SU (2) L × U (1) Y quantum number ∆ i (1, 3, 1) and ∆ i (1, 3, −1) are needed to avoid possible mixing between the spurion X and the gauge singlet S if we couple the messengers to S. The superpotential can take the form To preserve gauge coupling unification, the ∆ i (1, 3, 1) and ∆ i (1, 3, −1) messengers should be embedded into complete SU(5) representations So the superpotential (2.14) in terms of SU(5) representation is given by So the messenger scale need to satisfy M mess 10 11 GeV for n = 1 and M mess 10 14 GeV for n = 2. Although such a double-messenger-species choice of superpotential is phenomenological viable, it had been utilized in our previous model buildings, see [25] for an example. In this work, we choose an alternative possibility to realize NMSSM spectrum.
Couplings of the form in the superpotential will trigger mixing between X and S via messenger loops, generating the following Kahler potential after integrating out the messengers which will give a tadpole term for S after SUSY breaking Such a tadpole term can generate a suitable VEV for S . Therefore, we adopt this possibility for GMSB. The superpotential (2.19) can be embedded into the following form with complete SU(5) multiplets So the whole GMSB superpotential is given by with W SB (24 H , · · ·) the SU(5) symmetry breaking sector, which possibly involving 24 H Higgs etc. Besides, proper doublet-triplet(D-T) splitting mechanism is assumed so that the Higgs triplets in 5 H and5 H will be very heavy and be absent from the low energy spectrum at the messenger scale M mess . After we integrating out the messengers, tiny Majorana neutrino masses will be generated by GNMSSM extension of Type II seesaw mechanism, the superpotential (2.3). We should note that Z 3 -invariant NMSSM can be generated if we adopt the superpotential (2.16) instead of (2.22). From the superpotential (2.3), the general expressions for soft SUSY breaking parameters can be calculated with the wavefunction renormalization approach [26].
• The expressions for gaugino masses (2.24) So we have • The expressions for trilinear couplings In our convention, the anomalous dimension are expressed in the holomorphic basis [12] G with ∆G ≡ G + −G − the discontinuity across the messenger threshold. Here G + (G − ) denote respectively the value above (below) the messenger threshold.
So we have the soft SUSY breaking trilinear couplings Here we neglect possible RGE effects of y HuHu∆ etc between the GUT scale and the messenger scale.
• The soft SUSY masses are given as For later convenience, we list the discontinuity of various Yukawa beta functions and define We can calculate the soft scalar masses and (2.35)

SUSY breaking from anomaly mediation
To generate realistic EWSB in NMSSM, soft scalar masses for singlet S is necessarily present. As the gauge singlet receives no contributions from pure gauge mediation, additional Yukawa mediation contributions should be present in addition to pure GMSB contributions. To evade possible Landau pole below the GUT scale, less messengers in 15 representation are preferable, which otherwise may introduce the mixing between X and S, leading to generalized version of NMSSM.
To simplify the previous problems in GMSB, we can move to the predictive AMSB scenario, in which AMSB contribution to m 2 S is naturally present. Unfortunately, the minimal AMSB predicts negative slepton square masses and must be extended. The most elegant solution to tachyonic slepton is the deflected AMSB [15] scenario in which additional messenger sectors are introduced to deflect the AMSB trajectory and lead to positive slepton mass by additional gauge mediation contribution. The triplets in type II seesaw can naturally be fitted into the messenger sector.
In AMSB, the GMSB contributions of the messengers will cancel the change of AMSB contributions if simple mass thresholds for messengers are present. To evade such a difficulty, a pseudo-moduli field X can be introduced with its lowest component VEV X = M + θ 2 F X determining the messenger threshold. The deflection parameter, which characterizes the deviation from the ordinary AMSB trajectory, will depend on the form of the pseudo-moduli superpotential W (X). So, the W mess;B in eqn.(2.23) changes into 36) within which the coupling between S and 15, 15 can be absent. So Z 3 -invariant NMSSM can be adopted here. Expression of W (X) can be fairly generic and leads to a deflection parameter d of either sign given by After integrating out the messengers, the superpotential of form 2.3 can be obtained with W N M SSM taking the Z 3 invariant form. We can calculate the soft SUSY breaking parameters following the approach in [14].
• The soft gaugino mass is given at the messenger scale by (2.38) So the gaugino masses are given as • The trilinear soft terms will be determined by the superpotential after replacing canonical normalized superfields and given by 1 Similarly, we can obtain the m 2 S and ξ S terms. The trilinear soft terms etc are given by • The soft scalar masses are given by Details of the expression involving the derivative of ln X can be found in our previous works [27,28,29].
Expressions for scalars can be parameterized as the sum of each contributions with δ A the anomaly mediation contributions, δ G the general gauge mediation contributions and δ I the interference contributions, respectively. The expressions of δ G can be obtained by the following replacement in eqn(2.33).
The expressions of δ A are given by ordinary AMSB The expressions of the interference term are with ∆G y b , ∆G y b , ∆G λ , ∆G κ given in eqn (2.32).

Soft SUSY breaking contributions to Type II neutrino seesaw
As discussed in subsection 2.1, the soft SUSY breaking trilinear term can give subleading contribution to Majorana neutrino mass via Type II seesaw mechanism. In order to give the complete expressions for neutrino masses, we list the second subleading contribution here.
We require the knowledge of trilinear scalar coupling∆ T − H d − H d in both SUSY breaking mechanism. Before we integrate out the messengers involving ∆ T , AMSB already gives a trilinear coupling

Numerical results
Standard GMSB or AMSB can not give viable NMSSM spectrum. Lacking gauge interactions for S, the A λ , A κ -terms are typically very suppressed in GMSB [30]. In AMSB, large A λ , A κ needs large λ and κ so as to induce large positive m 2 S , suppressing the singlet VEV [31]. In our scenario, new interactions involving H u , H d and triplets can possibly give phenomenological viable parameters.
In ordinary setting, the spurion X is normalized so that y X = 1.
• Due to possible mixing between X and S through messengers in 15 representation of SU(5), tadpole terms in the scalar potential of S can be generated In our numerical study, κ is a free parameter while tan β is not. This choice is different to ordinary numerical setting in NMSSM in which tan β is free while κ is a derived quantity [32]. Such a choice can be convenient for those predictable NMSSM models from top-down approach. A guess of tan β is made to obtain the relevant Yukawa couplings y t , y b at the EW scale. After RGE evolving up to the messenger scale, the whole soft SUSY breaking parameters at the messenger scale can be determined. Low energy tan β can be obtained iteratively with such a spectrum from the minimization conditions of the Higgs potential. Obtaining an iteratively stable tan β indicates that the EWSB conditions are satisfied by the model input.
We use NMSSMTools 5.5.0 [33] to scan the whole parameter space. We interest in relatively large values of λ in order to increase the tree-level mass of the 125 GeV CP-even Higgs boson. Besides, the couplings λ 0 , λ 1 , λ 2 should be pertubative and λ, κ should satisfy the perturbative bound λ 2 + κ 2 0.7. The parameters are chosen as In our scan, we impose the following constraints: • (I) The lower bounds from current LHC constraints on SUSY particles [34,35]: -Light stop mass: mt 1 0.85 TeV.
• (II) We impose the following lower bounds for neutralinos and charginos, including the invisible decay bounds for Z-boson. The most stringent constraints of LEP [36] require mχ± > 103.5GeV and the invisible decay width Γ(Z →χ 0χ0 ) < 1.71 MeV, which is consistent with the 2σ precision EW measurement constraints Γ non−SM  [38], which is not included in the NMSSMTools.
• (V) The relic density of cosmic DM should satisfy the Planck data Ω DM = 0.1199 ± 0.0027 [39] in combination with the WMAP data [40](with a 10% theoretical uncertainty). We impose only the upper bound of Ω DM in our numerical studies because other DM species can also possibly contribute to the relic abundance of DM.
It is known that gravitino will be much lighter in GMSB than that in mSUGRA and in general will be the LSP. Such a light gravitino is also motivated by cosmology since it can evade the gravitino problem. The interaction of goldstino component of gravitino is 1/F X instead of 1/M P l . If gravitinos are in thermal equilibrium at early times and freeze out at the temperature T f , their relic density is [41] Ωgh 2 = mg keV In order to obtain the required DM relic density, one needs to adjust the reheating temperature as a function of the gravitino mass. Besides, it is shown in [42] that the late decay of the lightest messenger to visible sector particles can induce a substantial amount of entropy production which would result in the dilution of the predicted gravitino abundance. As a result, one would obtain suitable gravitino dark matter for arbitrarily high reheating temperatures. Due to the flexibility of the theory, we do not impose the DM relic density constraints in our GMSB scenario.
We have the following discussions related to our numerical results • Type II neutrino seesaw in NMSSM from AMSB In NMSSM, the 125 GeV Higgs boson in general can be either the lightest or the next-to-lightest CP-even scalar. Depending on the nature of the 125 GeV Higgs, we have the following discussions -A) 125 GeV Higgs is the lightest CP-even scalar.
As noted previously, successful EWSB in AMSB-type scenarios are fairly nontrivial. In our Type II neutrino seesaw extension scenario, interactions between Higgs and messengers can give additional contributions to A κ and A λ while not for m 2 S , making the EWSB condition a bit easier to realize. Our numerical scan indicates that EWSB conditions alone can already ruled out a large portion of the total parameter space. Combing with the constraints from (I) to (V), we can obtain the survived points that lead to realistic SUSY Figure 1: Survived points that can satisfy the EWSB conditions and the constraints from (I) to (V) in case the 125 GeV Higgs is the lightest CP-even scalar in AMSB-type scenario. In the upper left(right) panels, the allowed values of λ versus κ with the corresponding tan β (the messenger scale M mess versus F φ ) are given, respectively. In the middle panels, the gluino mass mg versus the Higgs mass m h (left) and the couplings λ 0 , λ 1 as well the deflection parameter d versus M mess (right) are shown, respectively. The BGFT measures are also shown in different colors. In the lower panels, the Higgs mass m h versus A t (or lighter stop masst 1 ) with the corresponding BGFT are shown. spectrum at low energy, which are shown in fig.1. In the left panel of fig.1, the allowed κ versus λ regions are given. We can see that κ should lie between 0.1 to 0.24 while λ should lie between 0.19 to 0.29. The tan β, which is obtained iteratively from the minimization condition of the Higgs potential for EWSB, are constrained to lie between 4 and 14 for 60 TeV ≤ F φ ≤ 160 TeV.
It is interesting to note that the messenger scale, which is just the heavy triplet scalar scale in Type II seesaw mechanism, are constrained to lie in a small band, from 1.0 × 10 14 GeV to 1.0 × 10 15 GeV. In general, light M mess is possible for tiny y u ∆ . However, successful EWSB in NMSSM as well as non-tachyonic slepton requirements etc forbid too small y u ∆ , as relatively large couplings are needed to give non-negligible Yukawa mediation contributions to the soft SUSY breaking parameters.
The values of F φ can determine the whole scales of the soft spectrum. We can see from the middle left panel of fig.1 that the gluino are constrained to lie upon 3.5 TeV. Such a heavy gluino can evade the current LHC bounds. The upper bounds for gluino is 12 TeV, which is not accessible in the near future experiment.
From the middle left panel of fig.1, it is also clear that our scenario can successfully account for the 125 GeV Higgs boson. The Higgs mass in NMSSM can be approximately given by [20] with v ≈ 174 GeV, m 2 T = m 2 U 3 and A t the stop trilinear coupling. From the allowed values of λ and tan β, it can be seen that the NMSSM specific λ 2 v 2 sin 2 2β contributions to the Higgs mass m 2 h is small, which is estimate to be 7 2 for tan β = 10. So large A t or heavy stop is necessary to give the 125 GeV Higgs. Fortunately, A t receives additional contributions in our scenario, which will increase A t for negative deflection parameters. The middle right panel of fig.1 shows that the preferred deflection parameters lie near −1.5, which indeed increase the value of A t . It can also be seen from this panel that the allowed regions require non-vanishing λ 0 ,λ 1 couplings, which means that Yukawa deflection in AMSB by the triplet messengers etc is indispensable to obtain realistic low energy SUSY spectrum. The plot of Higgs mass m h versus A t or lighter stop masst 1 are shown in the lower panels of fig.1. It can be seen that larger A t ort 1 can predict larger Higgs mass as expected.
The Barbieri-Giudice fine-tuning(FT) measure with respect to certain input parameter a is defined as [43] ∆ a ≡ ∂ ln M 2 Z ∂ ln a , (3.9) while the total fine-tuning is defined to be ∆ = max a (∆ a ) with {a} the set of parameters defined at the input scale.
The Barbieri-Giudice FT measures of our scenario are shown in the middle left panel of fig.1. In the allowed region, the BGFT satisfies ∆ 500. The FT can be as low as 100 in low gluino mass regions. As the gluino mass is determined by F φ , which set all the soft SUSY mass scale, lighter mg in general indicates lighter stop, reducing the FT involved. We can see from the lower panels of fig.1 that lighter A t ort 1 will lead to smaller BGFT for fixed Higgs mass. On the other hand, increasingt 1 while at the same time increasing A t can possibly make the involved FT unchanged [22]. In NMSSM, the lightest neutralino can act as the DM candidate. We can see from the upper left panel of fig.2 that in most of the previous allowed parameter space, the neutralino will lead to under-abundance of DM, although full abundance of DM is still possible for a small portion of the parameter space. This can be understood from the ingredients of the neutralino, which is shown in the upper right panel. We can see that DM is singlino-like in most of the parameter space. The almost pure singlino-like DM is a distinctive feature of NMSSM.
Its relic density can be compatible with WMAP bounds if it can annihilate via s-channel CP-even (or CP-odd) Higgs exchange when such Higgs has sufficient large singlet component for not too small κ. Under abundance of DM will not cause a problem as other specie of DM, such as axion or axino, can possibly contribute to the remaining abundances of DM.
The direct detection experiments, such as LUX [44],Xenon [45],PandaX [46], will set upper limits on the WIMP-nucleon scattering cross section. The spinindependent(SI) and spin-dependent(SD) scattering cross section of the neutralino DM is displayed in the left and right lower panels in fig.2, respectively. As the SI neutralino-nucleon interaction arises from s-channel squark, t-channel Higgs (or Z) exchange at the tree level and neutralino-gluon interactions from the one-loop level involving quark loops, singlino-like DM can evade the direct detection constraints if the t-channel exchanged Higgs is not too light for heavy squarks. The SD neutralino-nucleon interaction is dominated by Z 0 exchange for heavy squarks with the corresponding cross section proportional to the difference of the Higgsino components σ SD ∝ |N 2 13 − N 2 14 |. If the two Higgsino components are large but similar, the SD cross section can become small, which however will lead to large σ SI as σ SI ∝ |N 2 13 + N 2 14 |. We can see from the low panels that although some portion of the allowed parameter space is ruled out by DM direct detection experiments, especially by σ SI in case the singlino-like DM provides full abundance of DM(the green points), a large portion of parameter space is still not reached by current experiments if there are other DM components other than the lightest neutralino.
-B) The 125 GeV Higgs is the next-to-lightest CP-even scalar.
It can be seen in the panels of fig.3 that the nature of SM Higgs as the next-tolightest CP-even scalar in addition to EWSB conditions and bounds from (I) to (V) can rule out most of the points in the parameter space. We can give similar discussions as Case A.
Numerical results indicate that the non-trivial deflection parameter d and couplings λ 0 , λ 1 are absolutely necessary to obtain realistic low energy NMSSM spectrum. From the upper left panel of fig.3, we can see that the central value of d is −1.8 and the couplings λ 0 , λ 1 are constrained to take non-vanishing values. From the upper and middle panels of fig.3, we can see that the allowed κ should lie between 0.1 to 0.17 while λ should lie between 0.21 to 0.26 with the iteratively obtained tan β lying between 4 and 13 for 50 TeV ≤ F φ ≤ 130 TeV.
From the left panel in the second row of fig.3, the gluino can be seen to be constrained to lie between 3.5 TeV to 5 TeV, which maybe accessible in the HE-LHC. It is also obvious from this panel that the 125 GeV Higgs mass can readily act as the next-to-lightest CP-even scalar. As the width of the SMlike Higgs boson is quite narrow, the masses of the lightest CP-even scalar and the lightest CP-odd scalar can not be too light so as that the 125 GeV Higgs decaying into h 1 h 1 and a 1 a 1 are kinetically suppressed. Otherwise, such exotic decay modes may have sizable branching ratios and in turn suppress greatly the visible signals of the SM-like Higgs boson at the LHC. We show the masses of the lightest CP-even scalar versus the lightest CP-odd scalar in the middle right panel of fig.3. All the survived points can pass the constraints from the package HiggsBounds 5.3.2 [47]. The lightest neutralino can be the DM candidate, which however can provide only under abundance of cosmic DM. Even though the singlino-like DM can not account for the full DM relic abundance, direct DM detection bounds from spin-independent cross section σ SI can rule out the majority of the survived points (see the panels in the bottom of fig.3). Besides, the spin-dependent cross section σ SD can fully rule out the whole parameter space of this scenario. This can be understood from the ingredients of neutralino (shown in the right panel of the third row), in which the difference of the Higgsino components can be sizable.
• Type II neutrino seesaw in NMSSM from GMSB Ordinary GMSB predicts vanishing trilinear couplings A κ , A λ and vanishing m 2 S , therefore it can not predict realistic low energy NMSSM spectrum. Additional Yukawa deflection contributions in addition to GMSB, which just due to the terms related to Type II neutrino seesaw, can lead to new contributions to trilinear couplings and soft scalar masses.
The survived points after imposing the EWSB constraints and the bounds from (I) to (IV) are shown in fig.4. Again, as shown in upper left panel of fig.4, numerical results indicate that non-trivial couplings λ 0 , λ 1 , λ 2 , especially λ 2 which contributes to the tadpole term, are required to obtain realistic low energy GNMSSM spectrum. Those couplings can contribute to the trilinear couplings A κ , A λ and m 2 S , which are indispensable in realizing successful EWSB.
The allowed values of κ, which lie between 0.54 to 0.66, are much larger than that of the AMSB cases. The allowed ranges of λ and the iteratively obtained (from EWSB conditions) tan β, are found to lie between [0.1,0.2] and [10,20], respectively. As the survived points require large κ, small λ and intermediate tan β, the NMSSM specific tree-level contribution λ 2 v 2 sin 2 2β are always small. Besides, the mixing with the singlet scalar will provide destructive contributions to Higgs mass, which can be seen in eqn (3.8). Therefore, large A t or heavy stop masses are still needed in this scenario to accommodate 125 GeV Higgs. Fortunately, due to the new contributions to A t from Type II seesaw specific interactions, the 125 GeV Higgs can be successfully obtained.
The scale of the triplets in GMSB scenario are bounded loosely to lie between 10 13 GeV to 10 15 GeV, as tachyonic slepton problems etc that obsessing AMSB are absent in GMSB.
From the lower right panel of fig.4, the gluino can be seen to be constrained to lie between 4 TeV to 10 TeV, which can be accessible only in the future VLHC with √ s = 100 TeV. The BGFT involved in obtaining the 125 GeV Higgs can be as low as 100. Although the 125 GeV Higgs boson can be either the lightest or the next-tolightest CP-even scalar, our numerical results indicate that it can only be the lightest CP-even scalar in this scenario.
We checked that the NLSP in our GMSB scenario will always be the lightest neutralino. The dominant decay mode isχ 0 1 → γ +G with its LHC signal being multi-photon+missing energy. We know that the triplet messenger scale should be very high (of order 10 15 GeV) to accommodate the Type II seesaw mechanism. Therefore, to obtain TeV scale SUSY particle, the SUSY breaking √ F X should be of order 10 8 ∼ 10 9 GeV. As the parameter 1/F X determines the lifetime of the NLSP decaying into gravitino, the average distance traveled by neutralino NLSP can be large so as that it decays outside the detector and therefore behaves like a stable particle.

Conclusions
We propose to accommodate Type II neutrino seesaw mechanism in (G)NMSSM from GMSB and AMSB, respectively. The scale of the heavy triplet scalar is identified to be the messenger scale. General features related to the Type II seesaw mechanism in NMSSM are discussed. The Type II neutrino seesaw-specific interactions can give additional Yukawa deflection contributions to the soft SUSY breaking parameters of NMSSM, which are indispensable to realize successful EWSB and accommodate the 125 GeV Higgs. Relevant numerical results, including the constraints of DM etc in AMSB-type scenarios, are also given. We find that Type II neutrino seesaw mechanism extension of NMSSM from both AMSB and GMSB can lead to realistic low energy NMSSM spectrum, both admitting the 125 GeV Higgs as the lightest CP-even scalar. The possibility of the 125 GeV Higgs being the next-to-lightest CP-even scalar in AMSB-type scenarios is totally ruled out by DM direct detection experiments. Figure 3: Survived points that can satisfy the EWSB conditions and the constraints from (I) to (V) in case the 125 GeV Higgs is the next-to-lightest CP-even scalar in AMSB-type scenario. Other notations are the same as that in Fig.1 except the right panel in the second row, which shows the lightest CP-even scalar mass m h1 versus the lightest CP-odd scalar mass m a1 . Figure 4: Survived points that can satisfy the EWSB conditions and the constraints from (I) to (V) in GMSB-type scenario. The 125 GeV Higgs is found to be the lightest CP-even scalar for all the survived points. Other notations of the panels are the same as that in Fig.1.