NNMSM type-II and -III

We suggest two types of extension of the standard model, which are the so-called next to new minimal standard model type-II and -III. They can achieve gauge coupling unification as well as suitable dark matter abundance, small neutrino masses, baryon asymmetry of the universe, inflation, and dark energy. The gauge coupling unification can be realized by introducing two or three extra new fields, and they could explain charge quantization. We also show that there are regions in which the vacuum stability, coupling perturbativity, and correct dark matter abundance can be realized with current experimental data at the same time.


Introduction
The discovery of the Higgs particle at the Large Hadron Collider (LHC) experiment [1,2] filled the last piece of the standard model (SM). Furthermore, the results from the LHC experiment are almost consistent with the SM, and no signatures of the supersymmetry (SUSY) have been discovered. However, there are some unsolved problems in the SM, e.g., the SM does not have a dark matter (DM) candidate although a SUSY model can give it. The SUSY is one of the excellent candidates beyond the SM, because it can also solve the gauge hierarchy problem in addition to the DM. Moreover, gauge coupling unification (GCU) can be realized in a minimal supersymmetric standard model (MSSM). But the discovery of the Higgs boson with a 126 GeV mass and no signatures of the SUSY may disfavor the SUSY at low energy.
Actually, various extensions of the SM without the SUSY have been proposed. One minimal extension is the new minimal standard model (NMSM) [3]. The NMSM contains a gauge singlet real scalar, two right-handed neutrinos, an inflaton, and a small cosmological constant, which can explain a e-mail: ryo.takahasi88@gmail.com the DM, small neutrino masses, and baryon asymmetry of the universe (BAU), inflation, and dark energy (DE), respectively. 1 The next to new minimal SM (NNMSM) has been suggested in Ref. [10]. The NNMSM adds new particle contents to the NMSM to explain the gauge coupling unification (GCU). These are two adjoint fermions and four vector-like fermions. 2 In the NNMSM, the stability and triviality conditions have also been analyzed by use of recent experimental data of Higgs and top masses [11]. As a result, it could be found that there are parameter regions in which the correct abundance of DM can also be realized at the same time. One is the lighter DM mass, region: 63.5 GeV m S 64.0 GeV, and the other is the heavier one: 708 GeV m S 2040 GeV (writing m S for the lighter DM mass) with the center value of top pole mass.
Reference [10] has introduced six new fields, which were assumed to be at the same mass scale, to realize GCU. In this work, we also extend the NMSM and leave the condition of the same mass scale for the new fields adopted in the NNMSM. As a result, we can reduce the particle contents of the NNMSM while realizing GCU and vacuum stability, and satisfying phenomenological constraints such as DM, small neutrino masses, BAU, inflation, DE, and the proton decay. We suggest two types of model, the first model includes two adjoint fermions for the GCU and the second one has three adjoint fermions. The degrees of freedom of additional fermions in the models decrease compared to the NNMSM. We call the models NNMSM type-II (two adjoint fermions) and NNMSM type-III (three adjoint fermions), respectively. The NNMSM type-II also includes two righthanded neutrinos like the NNMSM, and the type-III does not have them. Both models have a gauge singlet scalar, 1 See also [4][5][6][7][8][9] for other extensions of the SM. inflaton, and a small cosmological constant. We also analyze the stability and triviality bounds with the 126 GeV Higgs mass, the recent updated limits on the DM particle [12], and the latest experimental value of the top pole mass, 173.5 GeV in both models. We will point out that there are parameter regions in both models where the stability and triviality bounds, the correct abundance of DM, and the Higgs and top masses can be realized at the same time.
This paper is organized as follows: In Sect. 2, we suggest the NNMSM type-II and show that the model can realize GCU. Then, the vacuum stability and DM are investigated. Some related arguments in the model such as the inflation, neutrino, and baryogenesis are also given. In Sect. 3, we suggest the NNMSM type-III. In particular, we focus on two simple models in this type of model. Similar phenomenological arguments to those in type-II are also presented. Section 4 is devoted to the summary. We give other setups for the realization of GCU in the Appendix.

Model
We suggest a next to new minimal standard model type-II (NNMSM-II) by reducing the particle contents of the NNMSM [10], which has two adjoint fermions λ a (a = 2, 3), four vector-like fermions L i (L i ) (i = 1, 2), the gauge singlet real scalar boson S, two right-handed neutrinos N i , the inflaton ϕ, and a small cosmological constant in addition to the SM. Our model removes four vector-like fermions from the NNMSM, but adds a new energy scale to the model. The quantum numbers of these particles are given in Table 1. Only the gauge singlet scalar particle has odd parity under an additional Z 2 symmetry, while the other additional particles have even parity. We will show that the singlet scalar becomes DM as in the NNMSM. The runnings of the gauge couplings are changed from the SM due to new particles with charges. The realization of GCU is one of the important results of this work as we will show later.
We consider NNMSM-II as a renormalizable theory, and thus the most general form of the Lagrangian allowed by the symmetries and renormalizability is given by Table 1 Quantum numbers of the additional particles in NNMSM-II (i = 1, 2) with α = e, μ, τ andH = iσ 2 H * where L SM is the Lagrangian of the SM, which includes the Higgs potential. v is the vacuum expectation value (VEV) of the Higgs boson: v = 246 GeV. L S,N ,ϕ, are Lagrangians for the DM, righthanded neutrinos, inflaton, and the cosmological constant, respectively. L SM + L S,N , are the same as those of the NNMSM. 3 L is a new Lagrangian in NNMSM-II where L α is for left-handed lepton doublets in the SM. 4

Gauge coupling unification
First, we investigate the runnings of the gauge couplings in the NNMSM-II. Since we introduce two adjoint fermions, λ 3 and λ 2 , listed in Table 1, the beta functions of the RGEs for the gauge couplings become where t ≡ ln(μ/1GeV), μ is the renormalization scale, and  Fig. 1 The runnings of the gauge couplings in NNMSM-II. The horizontal axis is the renormalization scale and the vertical axis is for the values of α −1 i . The runnings of α −1 1 , α −2 2 , and α −1 3 are described by black, blue, and red solid curves, respectively. We take M 3 7.44 × 10 9 GeV and M 2 = 300 GeV, and the coupling unification is realized at  Table 1 is given by The NNMSM has assumed the same mass scale for the new particles, M NP = M 3 = M 2 = M L i , where M L i are the masses of the vector-like fermions. On the other hand, we allow different mass scales for the two adjoint fermions in NNMSM-II. According to the numerical analyses, taking for the two masses M 3 7.44 × 10 9 GeV and M 2 = 300 GeV can realize GCU with a good precision at 1-loop level as shown in Fig. 1. 5 The beta functions in NNMSM-II are We show the thresholds of new particles with M 3 7.44 × 10 9 GeV and M 2 = 300 GeV by black solid lines. The realization of GCU by adding these adjoint fermions was pointed 5 In this analysis, we take the following values [11]: sin 2 θ W (M Z ) = 0.231, α −1 em (M Z ) = 128, and α s (M Z ) = 0.118, for the parameters in the EW theory, where θ W is the Weinberg angle, α em is the fine structure constant, and α s is the strong coupling, respectively. out in Ref. [4]. The NNMSM-II suggests the GCU at GCU 2.41 × 10 15 GeV, (11) with the unified coupling α −1 38.8.
A constraint from the proton decay experiments is τ ( p → π 0 e + ) > 8.2 × 10 33 years [11]. When we suppose the minimal SU (5) GUT at GCU , the protons decay of p → π 0 e + occurs by exchanging heavy gauge bosons of the GUT gauge group. The partial decay width of the proton for p → π 0 e + is estimated as where α 2 H is the hadronic matrix element, m p is the proton mass, f π is the pion decay constant, D and F are the chiral Lagrangian parameters, A R is the renormalization factor, and V ud is an element of the CKM matrix (e.g., see [4,13,14]). In our analysis, we take these parameters as m p = 0.94 GeV, f π = 0.13 GeV, A R 1.02, D = 0.80, and F = 0.47. The theoretical uncertainty on the proton lifetime comes from the hadron matrix elements, α H = −0.0112 ± 0.0034 GeV 3 [15]. When α H is taken as a smaller value, which is α H = −0.0146 GeV 3 , the proton lifetime is predicted as small, and this case gives a conservative limit. At the point determined by (11) and (12), the proton lifetime can be evaluated as τ 5.19 × 10 33 (1.82 × 10 34 ) years for α H = −0.0146 (−0.0078) GeV 3 . Thus, the value of α H = −0.0078 GeV 3 can satisfy the experimental bound from the proton decay although the conservative case (α H = −0.0146 GeV 3 ) cannot. For the center value of α H = −0.0112 GeV 3 , the proton lifetime is evaluated as τ 8.55 × 10 33 years, which also satisfies the experimental limit. Since the future Hyper-Kamiokande experiment is expected to exceed the lifetime O(10 35 ) years [16], which corresponds to GCU 4.42 +0.63 −0.73 × 10 15 GeV for α H = −0.0112 ± 0.0034 GeV 3 , proton decay is observed if NNMSM-II is correct.
When we take a larger value of M 2 , such as 800 GeV, GCU can be realized with a larger value of M 3 , 2.03 × 10 10 GeV. However, GCU with a larger value of M 2 leads to a smaller scale of GCU , and thus the constraint from the proton decay becomes stronger. In fact, GCU with M 2 = 800 GeV leads to a proton lifetime of τ (2.14 − 7.48) × 10 33 years for α H = −0.0112 ± 0.0034 GeV 3 , which is ruled out by the proton decay experiment. Therefore, the upper bounds on the adjoint fermions masses are (M 3 , M 2 ) (2 × 10 10 , 800) GeV. On the other hand, the LHC experiment gives a lower limit to the SU (2) L triplet mass M 2 of 245 GeV ≤ M 2 by the ATLAS examining the channel of four lepton final states (and (180 − 210) GeV ≤ M 2 by the CMS for the three lepton final states) [17,18]. As a result, the allowed region for M 2 in NNMSM-II is 245 GeV ≤ M 2 800 GeV. The corresponding region for M 3 is 7 × 10 9 GeV M 3 2 × 10 10 GeV to realize GCU. Since the SU (2) L triplet fermion can be discovered up to M 2 ≤ 750 GeV by the LHC with √ s = 14 TeV [19], most of the mass region of M 2 in the NNMSM-II can be checked by the experiment.

Abundance and stability of new fermions
Next, we discuss the abundance and stability of new fermions, λ 3 and λ 2 . λ 3 is expected to be long lived since it cannot decay into the SM sector. A stable colored particle is severely constrained by experiments with heavy isotopes, since it is bounded in nuclei and appears as anomalously heavy isotopes (e.g., see [20]). The number of the stable colored particles per nucleon should be smaller than 10 −28 (10 −20 ) for its mass up to 1 (10) TeV [21,22]. But the calculation of the relic abundance of the stable colored particle is uncertain because of the dependence on the mechanism of hadronization and nuclear binding [23].
In this paper, we apply the same scenario to avoid the problem of the presence of the stable colored particle as in the NNMSM. It is to consider few production scenario for the stable particle, i.e., if the stable particle were rarely produced in the thermal history of the universe and clear the constraints of the colored particle. In fact, a particle with a mass of M is very rarely produced thermally if the reheating temperature after the inflation is lower than M/(35 ∼ 40). 6 Therefore, we consider a reheating temperature T RH of since M 3 = 7.44 × 10 9 GeV. On the other hand, λ 2 is thermally produced but it can decay into the SM particles through the Yukawa interaction in Eq. (7) before the Big Bang nucleosynthesis (BBN). The condition for the decay before the BBN is O(10 −13 ) |y ν α |, which can also be consistent with a constraint from neutrino mass generation as we will discuss later. Therefore, the presence of two new adjoint fermions in NNMSM-II for GCU is not problematic.

Stability, triviality, and dark matter
In this section, we investigate the parameter region where not only stability and triviality bounds but also correct abundance of the DM are achieved. The ingredients of Higgs and DM sector in NNMSM-II are the same as the NNMSM [10], which are given by L SM and L S in Eqs. (2) and (3) but the runnings of the gauge couplings are different from those of the NNMSM. The singlet scalar S becomes the DM also in NNMSM-II. Reference [10] has pointed out that there are two typical regions in which the vacuum stability, the correct abundance of DM, the 126 GeV Higgs mass, and the latest experimental value of the top mass can be realized at the same time. One is the lighter mass region of DM and the other is the heavier one. It has also been shown that the top mass dependence is quite strong even within the experimental error of the top pole mass, M t = 173.5 ± 1.4 GeV. The future XENON100 experiment with 20 times higher sensitivity will be able to rule out the lighter mass region completely.
On the other hand, the heavier mass region can currently be allowed by all experiments searching for DM. The region will be ruled out or checked by the future direct experiments of XENON 100×20, XENON1T and/or combined data from indirect detections of Fermi+CTA+Planck at 1σ CL.
In NNMSM-II, the runnings of gauge couplings are slightly changed with those of the NNMSM. Therefore, we reanalyze the vacuum stability, triviality, and the correct abundance of DM in NNMSM-II. The RGEs for three quartic couplings of the scalars [3] and the DM mass are given by and respectively. We should also use the present limits for the singlet DM model. We comment on Eq. (16): the right-hand side of the equation is proportional to k itself. Thus, if we take a small value of k(M Z ), the evolution of k tends to be slow and remaining at a small value, and the running of λ is close to that of the SM. In our analysis, the boundary conditions of the Higgs self-coupling and top Yukawa coupling are given by for the RGEs, where the VEV of the Higgs field is v = 246 GeV. Let us solve the RGEs, Eqs. (15)- (17), and obtain the stable solutions, i.e., the scalar quartic couplings are within the range of 0 < (λ, k, λ S ) < 4π up to the Planck scale M pl = 10 18 GeV. Figure 2 shows the case of a central value The solutions of the RGEs are described by gray plots in Fig. 2, where the horizontal and vertical axes are log 10 (m S /1 GeV) and log 10 k at the M Z scale, respectively. The boundaries of the plotted region are determined by the stability and triviality conditions. The words 'stability' and 'triviality' in both figures mean the corresponding conditions. We also show the contour satisfying S / DM = 1 with DM = 0.115, where S and DM are density parameters of the singlet DM and the observed value of the parameter [24], respectively. The contour is calculated by micrOMEGAs [25]. Since there is no DM candidate except for the S to compensate for S / DM < 1, which is above the contour, we focus only on the contour. The relic density depends on k and m S but not on λ S , meanwhile λ S affects the stability and triviality bounds. In the figure, λ S (M Z ) is randomly varied from 0 to 4π , where the λ S -dependence of the stability and triviality bounds is not stringent, and most of λ S (M Z ) ∈ [0, 1] as the boundary condition can satisfy the bounds. A direct DM search experiment, XENON100 (2012), gives an exclu-sion limit [12], which is described by the (red) dashed line in Fig. 2. 7 There are two regions, R 1,2 , which satisfy both the correct DM abundance and the triviality bound simultaneously, The future XENON100 experiment with 20 times higher sensitivity, which is described by the (blue) dotted lines in Fig. 2, will be able to completely rule out the lighter m S region R 1 . On the other hand, the heavier m S region, R 2 , can be currently allowed by all experiments searching for DM. It is seen that the future XENON100 × 20 can check up to m S 1,000 GeV (log 10 (m S /1 GeV) 3). The future XENON1T experiment and combined data from indirect detections of Fermi+CTA+Planck at 1σ CL may be able to reach up to m S 5 TeV [12]. The lower and upper bounds on m S in the region R 1 come from the triviality bound on λ and the XENON100 (2012) experiment, respectively. On the other hand, the lower and upper bounds on m S in the region R 2 are given by the stability and triviality bounds on λ, respectively. Since k in the R.H.S of Eq. (15) is effective only above the energy scale of m S , the triviality bound on λ becomes severe as the m S becomes small. We also find that the two regions R 1 and R 2 are almost the same as in the NNMSM [10]. This means that the differences in runnings of the gauge couplings do not affect the stability, triviality, and correct abundance of DM in this class of model. The favored region still depends on the top mass rather than the runnings of the gauge couplings as pointed out in Ref. [10]. In fact, the heavier mass region becomes narrow as the top mass becomes larger due to the stability bound, while the region R 1 does not depend on the top mass, because the triviality bound on λ does not depend on the top Yukawa coupling (see Ref. [10] for detailed discussions of the top mass dependence of the region).

Inflation, neutrinos, and baryogenesis
In this section, realizations of inflation, suitable tiny active neutrino masses, and baryogenesis are discussed. The relevant Lagrangian for the inflaton is given by L ϕ in Eq. (5). The WMAP [24,26] and the Planck [27] measurements of the cosmic microwave background (CMB) constrain the cosmological parameters related with the inflation in the early universe. In particular, the first results based on the Planck measurement with a WMAP polarization low-multipole likelihood at ≤ 23 (WP) [24,26] and high-resolution (highL) CMB data give n s = 0.959 ± 0.007 (68 %; Planck + WP + highL), for the scalar spectrum power-law index, the ratio of tensor primordial power to curvature power, and the running of the spectral index, respectively, in the context of the CDM model. Regarding r 0.002 , the constraints are given for the case of both no running and including running of the spectral indices.
We also adopt the same inflation model in NNMSM-II as in the NNMSM. The inflaton potential is the Coleman-Weinberg (CW) type [28][29][30][31][32]. In this potential Eq. (5), the VEV of ϕ becomes σ . When we take (φ, σ, B) (6.60 × 10 19 GeV, 9.57 × 10 19 GeV, 10 −15 ), the model can lead to n s = 0.96, r = 0.1, dn s /dlnk 8.19 × 10 −4 , and (δρ/ρ) ∼ O(10 −5 ), which are consistent with the cosmological data. The values of the couplings of the infla-ton with the Higgs, DM, right-handed neutrinos, and new adjoint fermions are also constrained, because there is an upper bound on the reheating temperature after the inflation as T RH 1.86 × 10 8 GeV. This upper bound leads to μ 1,2 6.86 × 10 7 GeV and (y i j N , y 3 , y 2 ) 1.79 × 10 −6 . Since κ H,S should be almost vanishing at low energy for the realizations of the EW symmetry breaking and the DM mass, we take the values of κ H,S as very tiny at the epoch of inflation. The smallness of κ H,S also does not spoil the stability and triviality bounds. The lower bound of the reheating temperature depends on the baryogenesis mechanism. When the baryogenesis works through the sphaleron process, the reheating temperature must be at least higher than O(10 2 ) GeV.
The neutrino sector is shown in Eq. (4), where tiny active neutrino masses are obtained through the type-I and -III seesaw mechanisms. Since there are two right-handed neutrinos and one adjoint (SU (2) L triplet) fermion, three active neutrinos are predicted to be massive in NNMSM-II. The Yukawa coupling of the triplet fermion should be |y ν α | O(10 −6 ) so as not to exceed the typical neutrino mass scale of m ν ∼ 0.1 eV. Thus, the region of O(10 −13 ) |y ν α | O(10 −6 ) is allowed in which the lower bound comes from the discussion of the BBN as mentioned above. Recalling the reheating temperature in NNMSM-II, the masses of the right-handed neutrino must be lighter than 1.86 × 10 8 GeV. What mechanism can induce the suitable baryon asymmetry at such a low reheating temperature? One possibility is resonant leptogenesis [33] in which the righthanded neutrinos may be light, up to 1 TeV. Thus, the reheating temperature, 1 TeV T RH 1.86 × 10 8 GeV, can realize resonant leptogenesis, which means for the couplings of the inflaton 369 GeV μ 1,2 6.86 × 10 7 GeV and 9.63 × 10 −12 (y i j N , y 3 , y 2 ) 1.79 × 10 −6 in Eq. (5).

Models
We also discuss other possibilities, which are alternatives to the NNMSM and NNMSM-II, for realizing GCU by different particle contents. Here, we suggest a class of model with several generations of the adjoint fermions and without the right-handed neutrinos. We refer this class of models to NNMSM-III. We focus on two simple models, NNMSM-III-A and -B, in this class of models. Both NNMSM-III-A and -B introduces three new fields such as one λ 3 and two generations of λ 2,i (i = 1, 2) in addition to the singlet DM, inflaton, and the cosmological constant. Then, NNMSM-III-A requires that the mass scales of the three new fields are the same and NNMSM-III-B allows different mass scales between λ 3 and λ 2,i . The quantum numbers of these particles for both models are given in Table 2.
Only the singlet scalar DM has odd parity under an additional Z 2 symmetry like NNMSM-II. Since the runnings of the GCU are still changed from the previous models, we will focus on them later.
NNMSM-III-A and -B are also presented as renormalizable theory. The most general form of the Lagrangian allowed by the symmetries and renormalizability is given by with where the inflaton potential in L ϕ is the same as those of the other NNMSMs, but the interactions are different from them. Then, L is a new Lagrangian for the models. The mass matrix M 2 is assumed to be diagonal for simplicity. The Lagrangians L SM + L S + L are the same as those of the NNMSMs. 8 One of the advantages of NNMSM-III-A is that we do not need two right-handed neutrinos and different mass scales between λ 3 and λ 2,i for the realization of the GCU, unlike NNMSM-II as we will show later. Thus, we introduce one mass scale for the new adjoint fermions and define it as M NP ≡ M 3 = M 2,i for NNMSM-III-A. NNMSM-III-B is a generalization of NNMSM-III-A.

Gauge coupling unification
First, we investigate the runnings of the gauge couplings in the NNMSM-III-A and B. Since we introduce three adjoint 8 We also simply assume that the origin of DE is the tiny cosmological constant as in the other NNMSMs.  Table 2, the beta functions for the gauge couplings are modified to

NNMSM-III-A case
According to the numerical analyses, taking the free parameter M NP as 2.26 × 10 8 GeV in NNMSM-III-A can realize GCU with a good precision at 1-loop level, as shown in with the unified coupling as The model also predicts the proton lifetime as τ = 1.94 +2.06 −0.80 × 10 35 years for α H = −0.0112 ± 0.0034 GeV 3 . Therefore, this model can satisfy the constraint on the proton decay even with the most conservative value of α H , but it There are other initial setups for realizing GCU with the same mass scales for the new adjoint fermions, i.e., more generations of adjoint fermions can also lead to GCU. Some examples are given in the Appendix.

NNMSM-III-B case
Since one can generally take different mass scales between M 3 and M 2,i , we consider the simplest case in the generalization, which is NNMSM-III-B. According to the numerical analyses, when we take M 3 4 × 10 9 GeV and M 2,i 6.73 × 10 8 GeV, GCU can be realized as shown in Fig. 4.
The beta functions in NNMSM-II are with Eq. (29). We show the thresholds of the new particles with masses M 3 4×10 9 GeV and M 2,i 6.73×10 8 GeV by black solid lines. This case suggests the GCU to occur at GCU 2.77 × 10 15 GeV with the unified coupling as This mass spectrum interestingly predicts the proton lifetime as τ = 1.50 +1.60 −0.62 × 10 34 years for α H = −0.0112 ± 0.0034 GeV 3 . Therefore, the model with this mass spectrum can satisfy the constraint on the proton decay even with the most conservative value of α H and it can be checked by the future Hyper-Kamiokande experiment if the model is correct. This is one advantage compared to NNMSM-III-A. In fact, there is an upper bound on the mass scale of M 3 (or M 2,i ), which just comes from the proton decay. Therefore, in a broad region of M 3 4 × 10 9 GeV (or M 2,1 6.73 × 10 8 GeV), there are solutions for the realization of GCU. There are also initial setups for the GCU with different mass scales between λ 3 and λ 2,i and more generations of them. Some simple examples are given in the Appendix.

Abundance and stability of new fermions
We also adopt the few production scenario for the colored particle λ 3 in NNMSM-III to avoid the problem of the presence of the colored particle. The reheating temperature should be T RH < 5.65 × 10 6 GeV for NNMSM-III-A Note that λ 2,i are not also thermally produced because of M λ 2,i > T RH in the models.

Stability, triviality, and dark matter
Regarding the stability, triviality, and DM, since the differences in the runnings of the gauge couplings almost do not affect the stability and triviality bounds, the favored regions in NNMSM-III-A and -B are also almost the same as in NNMSM-II. Thus, NNMSM-III-A and -B predict the same mass region of DM and k as in the other NNMSMs.

Inflation, neutrinos, and baryogenesis
Realizations of inflation, a suitable tiny active neutrino mass and baryogenesis in NNMSM-III are discussed in this section. Regarding the inflation model, the same CW type inflaton potential as in the previous models is utilized in the models, but the upper bounds on the inflaton couplings are changed due to different constraints on the reheating temperature. The upper bounds on the inflaton couplings are Since NNMSM-III does not include the right-handed neutrinos, the neutrino sector is changed from NNMSM-II. In NNMSM-III, the tiny active neutrino mass can be realized by SU (2) L adjoint fermions through the type-III seesaw mechanism. The relevant Lagrangian is given by Eq. (27). Since both NNMSM-III-A and -B have only two generations of λ 2,i , one of the active neutrinos is predicted to be massless, m 1 = 0 (m 3 = 0) for the normal (inverted) mass hierar-chy. Recalling the condition M λ 2,i > T RH in NNMSM-III-A and -B for the few production scenario, λ 2,i do not play a role for generating the baryon asymmetry of the universe. What mechanism can induce the BAU? One possibility is baryogenesis from the dark sector [34,35], which presupposes an asymmetry between DM and anti-DM in which the asymmetry of the dark matter sector including a new dark matter number can be converted into the lepton number. 10 As a result, the baryon number can be generated through the sphaleron process. This means that the reheating temperature should be typically O(10 2 ) GeV T RH for both NNMSM-III-A and -B, which leads to O(10) GeV μ 1,2 and O(10 −13 ) y 3 , y 2 .

Summary
There are some unsolved problems in the SM. These are, for instance, explanations for DM, the gauge hierarchy problem, tiny neutrino mass scales, baryogenesis, inflation, and DE. The extended SM without the SUSY, the so-called NNMSM, could explain the above problems except for the gauge problem by adding two adjoint fermions, four vector-like fermions, two gauge singlet real scalars, two right-handed neutrinos, and a small cosmological constant. In this paper, we suggested two types of alternatives (NNMSM-II and NNMSM-III) to the NNMSM by reducing additional fields while keeping the above merits of the NNMSM.
First, we have taken a setup where the new fermions have a different mass scale of new physics. Under this condition, GCU with the proton stability determines the field contents of NNMSM-II, i.e., two adjoint fermions are added to the SM in addition to two gauge singlet real scalars, two righthanded neutrinos, and a small cosmological constant. The GCU can occur at GCU 2.41 × 10 15 GeV with two mass scales of new particles as M 3 7.44 × 10 9 GeV and M 2 = 300 GeV. We consider the reheating temperature as T RH 1.86 × 10 8 GeV in order not to produce stable adjoint fermions in the early universe. This reheating temperature requires the following issues. The masses of the right-handed neutrino should be smaller than 1.86×10 8 GeV, so that a tiny neutrino mass is realized through the type-I seesaw mechanism with relatively small neutrino Yukawa couplings. The BAU should be achieved, for example, through resonant leptogenesis. We have also analyzed the stability and triviality conditions by use of recent experimental data of the Higgs and top masses. We found the parameter regions in which the correct abundance of DM can also be realized at the same time. One is the lighter m S region: 63.5 GeV m S 64.0 GeV, and the other is the heavier region: 708 GeV m S 2040 GeV with the center value of top pole mass. Both regions are almost the same as in the NNMSM. This means that the differences in runnings of the gauge couplings between NNMSM and NNMSM-II do not affect the stability, triviality, and correct abundance of DM in these classes of model. Therefore, the favored region for the stability and triviality still depends on the top mass rather than the running of the gauge couplings. The future XENON100 experiment with 20 times higher sensitivity will completely check out the lighter mass region. On the other hand, the heavier mass region will also be completely checked by the future direct experiments of XENON100×20, XENON1T and/or combined data from indirect detections of Fermi+CTA+Planck at 1σ CL.
Second, we have also taken a different setup (NNMSM-III), which includes three adjoint fermions (λ 3 and λ 2,i (i = 1, 2)) in addition to two gauge singlet real scalars and a small cosmological constant, but it does not have righthanded neutrinos. Removing the right-handed neutrino is one of the advantages of this setup. Then, we have considered two simple cases in this setup. One is that all masses of adjoint fermions are the same. The other is that masses of the SU (3) C and SU (2) L adjoint fermions are different. The first and the second cases are named NNMSM-III-A and -B, respectively. The GCU can occur at GCU 5.20 (2.77) × 10 15 GeV with M 3 = M 2,i 2.26 × 10 8 (M 3 4 × 10 9 and M 2,i 6.73 × 10 8 ) GeV in NNMSM-III-A (-B). Thus, the reheating temperature should be T RH 5.65 × 10 6 (10 8 ) GeV for model A (B). The tiny neutrino mass can be realized by two adjoint fermions under SU (2) L through the type-III seesaw mechanism, and the BAU can be achieved, e.g., baryogenesis from the dark sector in both models. We have also investigated other initial setups for realizing GCU in the appendix. These have several generations of λ 3 and/or λ 2 . Regarding the stability, triviality, and DM in NNMSM-III, since the differences in the runnings of the gauge couplings almost do not affect the stability and triviality bounds, the favored regions in NNMSM-III are almost the same as in the other NNMSMs. Therefore, NNMSM-III predicts the same mass region of DM and k as the previous models.
Finally, we briefly compare the NNMSMs to any other minimal extensions of the SM such as the minimal leftright model (e.g., see [7]), neutrino minimal standard model (νMSM) [8,9], and some supersymmetric extensions. In the context of the left-right model, GCU, DM, tiny neutrino mass, and BAU can be explained, but there is no complete analysis for the vacuum stability and triviality with the latest center values of the Higgs and top mass. Such a discussion might be interesting, although the RGEs for the Higgs sector are more complicated than that in the NNMSMs. The νMSM with three right-handed neutrinos is one simple extension of the SM, and can explain some problems (DM, tiny neutrino mass, and BAU) at the same time. To achieve GCU in the νMSM, some additional particles are still needed. In addition, the Higgs sector of the νMSM is the same as in the SM, and thus the vacuum in the model becomes unstable before the Planck scale. The degrees of freedom in the NNMSMs are fewer than general extensions of the SM with the left-right symmetry and supersymmetry. These are the advantages of the NNMSMs. Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited. Funded by SCOAP 3 / License Version CC BY 4.0. • Upper bounds on M 3,i and M 2,i are given for each case. We take a conservative limit of τ 10 34 years for the proton decay to obtain the upper bounds. Therefore, all cases listed in Table 4 can satisfy the proton decay constraint. • (N λ 3 , N λ 2 ) = (1, 1) case is just NNMSM-II. • (N λ 3 , N λ 2 ) = (1, 2) case with M 3 = M 2,i (M 3 = M 2,i ) is NNMSM-III-A (-B). • Larger number of N λ 3 leads to larger upper bound on M λ 3,i . • The cases with N λ 2,i = 1 can satisfy the constraint from the proton decay but these are excluded by the LHC experiment searching for the SU (2) L triplet particle. The cases are labeled by ' '.
Hierarchical mass spectra for M 3,i and M 2,i such as M 3,1 < M 3,2 might also realize GCU but we do not consider the hierarchical mass spectra case because of the minimality of the models in this work.