Scrutinizing Vacuum Stability in IDM with Type-III Inverse seesaw

We consider the extension of the Standard Model (SM) with an inert Higgs doublet that also contains two or three sets of $SU(2)_L$ triplet fermions with hypercharge zero and analyze the stability of electroweak vacuum for the scenarios. The model represents a Type-III inverse seesaw mechanism for neutrino mass generation with a Dark matter candidate. An effective potential approach calculation with two-loop beta function have been carried out in deciding the fate of the electroweak vacuum. Weak gauge coupling $g_2$ shows a different behaviour as compared to the Standard Model. The modified running of $g_2$, along with the Higgs quartic coupling and Type-III Yukawa couplings become crucial in determining the stability of electroweak vacuum. The interplay between two and three generations of such triplet fermions reveals that extensions with two generations is favoured if we aspire for Planck scale stability. Bounds on the Higgs quartic couplings, Type-III Yukawa and number of triplet fermion generations are drawn for different mass scale of Type-III fermions. The phenomenologies of inert doublet and Type-III fermions at the LHC and other experiments are commented upon.


Introduction
The last element of the Standard Model (SM) was the Higgs boson which was discovered at the CMS and ATLAS detectors of the Large Hadron Collider (LHC) [1,2]. The spin, parity measurements and the combined analysis show the SM-like behaviour of the Higgs boson [3][4][5]. However it has been shown that Standard Model electroweak (EW) vacuum on its own can run into metastability due to quantum corrections [6][7][8][9]. It is well known that the addition of scalars enhance the stability of the EW vacuum via positive loop contributions to the Higgs quartic coupling. Various models, which include scalars from different gauge representations have been proposed [10][11][12][13][14][15][16][17][18][19][20][21] to enhance the stability of EW vacuum. On the contrary an extension with fermion often gives negative contributions to the Higgs quartic couplings that it couples to. Such negative contributions then tend to pull such Higgs quartic couplings toward instability much faster. Thus models with extra fermions and where Majorana masses of the fermions are spontaneously generated are constrained from the vacuum stability .
Apart from the problem of vacuum metastability in SM which depends on the top quark and Higgs boson masses [45][46][47], the theory also fails to provide a stable dark matter (DM) candidate, as well as to give successful explanation for the very tiny eV scale neutrino masses, and their mixings. In this work, we focus on these two aspects by extending the SM with SU (2) L triplet fermions, and SU (2) L inert doublet scalar. The triplet fermion generates the eV light neutrino mass via Type-III seesaw mechanism, while the inert Higgs doublet provides a dark matter candidate, as well as stabilizes the EW vacuum.
The minimal Type-III extensions have one to three generations of SU (2) L fermions with hypercharge zero, which mix with the SM charged and neutral fermions, and also generates tiny eV neutrino mass via electroweak symmetry breaking [48][49][50]. Different extensions of Type-III seesaw and their collider signatures have been studied in [51]- [57], including their spin measurement at the LHC [58]. The stability of EW vacuum in some of these scenarios are studied in [59][60][61][62]. In this article, we consider the inverse seesaw mechanism of neutrino mass generation with two generations of triplet fermions. One among them couples with the SM Higgs boson via Type-III Yukawa coupling, and generates the Dirac mass term. The other triplet fermion generates the Majorana mass term for the triplet fermion.
As discussed earlier an extension with scalar enhances stability of EW vacuum and if the scalar is in the form of SU (2) L inert (Z 2 -odd ) doublet then it also provides the much needed dark matter candidate [14]. SM extension with such inert doublet in the context of vacuum stability have been studied extensively [14][15][16][17]22]. Fields in inert doublet have very interesting phenomenology due to their compressed spectrum and the possibility of real and pseudoscalar dark matter particle [63]- [74]. In this article we will investigate the effect of inert doublet in the context of Type-III fermions.
In our model we have both SU (2) L triplet fermion and SU (2) L doublet scalar. Being in the triplet representation of SU (2) L the new fermions contribute in the evolution of weak gauge coupling g 2 such that g 2 now increases with running scale. This behaviour substantially changes the dynamics of couplings responsible for the EW vacuum stability. We shall see how an enhanced g 2 causes a much lower perturbative scale compared to Type-I [22,23] or only IDM case [16]. Specially we see that with three generations of Type-III fermions it is difficult to attain Planck scale perturbativity and thus two generations are more favoured.
The paper is organised as follows. In section 2, we describe the model, and present the EW symmetry breaking conditions for this model. We discuss the perturbativity and the interplay of two and three generations of SU (2) L triplet fermions in section 3. The EW vacuum stability with all three posibilites are covered in section 4. In section 5, we discuss the phenomenological consequences and present our conclusion. The expressions of the two-loop beta functions used in our analysis are presented in Appendix A.

The Model
The SM is augmented with an inert doublet (ID) and three Type-III fermions with ISS mechanism. We first discuss the scalar sector of the model in Section 2.1. We consider two different scenarios, viz., a canonical Type-III seesaw with small Yukawa couplings and an inverse seesaw (ISS) with large Yukawa couplings to study effects in perturbativity of dimensionless couplings and vacuum stability. The fermionic sector with Type-III seesaw and inverse seesaw is discussed in Section 2.2.

The Scalar Sector
The scalar part consists of two SU (2) L -doublets Φ 1 and Φ 2 , both with hypercharge 1/2: The tree-level SM gauge invariant scalar potential i.e., invariant under SU (2) L × U (1) Y is given by [75,76] V scalar = m 2 11 where we have chosen all the Higgs quartic couplings λ 1,2,3,4 and mass terms m 2 11 , m 2 22 as real. While m 2 12 and the λ 5,6,7 couplings are in general complex but for this study we also have taken them as real number. A Z 2 symmetry is imposed to prohibit the flavor changing neutral currents at tree-level. Under this Z 2 symmetry Φ 2 is odd and Φ 1 is even. This choice also make Φ 2 as inert which can be DM candidate. The λ 6 , λ 7 and m 12 also get removed and Eq. (2.2) reduces to The Φ 1 takes a real vacuum expectation value (VEV) which breaks the EW symmetry as follows where the respective physical mass eigenvalues can be written as This is to note that, being Z 2 odd Φ 2 is inert, which prohibits any mixing between Φ 2 and Φ 1 . This also implies that the gauge eigenstates and the mass eigenstates are the same for the Higgs bosons from both Z 2 odd or even multiplets. In this scenario, Φ 2 being Z 2 -odd does not talk to the fermions. Moreover, we get two CP even neutral Higgs bosons h and H 0 . Here we choose h as the SM-like Higgs boson with mass around 125 GeV discovered at the LHC. The spectrum has one pseudoscalar Higgs boson A and a pair of charged Higgs bosons H ± . It is evident from Eq. (2.6) that the heavy Higgs bosons H 0 , A and H ± are from Φ 2 so they are nearly degenerate in mass spectrum with possible splitting by the help of λ 5 . The sign of λ 5 is crucial in making one of the scalars between A and H 0 as the lightest, and possible DM candidate. The Z 2 -odd symmetry prohibits some of the decays of Φ 2 -type Higgs bosons..

The Type-III and Inverse Seesaw Lagrangians
In addition to the SM particle contents, the Type-III seesaw model contains SU (2) L fermionic triplets Σ with zero hypercharge. Being in the adjoint representation of the SU (2) L group, the Majorana mass term M N of such triplets is gauge invariant. In terms of the usual two-by two notation for triplets, the beyond SM interactions are described by the Lagrangian: where L ≡ (ν, ) L corresponds to the SM lepton doublet, while Φ 1 = iσ 2 Φ 1 (with σ 2 is the second Pauli matrix), Σ c ≡ CΣ T for each fermionic triplet as shown below.
We drop the generation indices here, but this is to remind that there are three set of fermionic triplets for three leptonic doublets. The covariant derivative generates the coupling between the W bosons and the triplet fermions and they are proportional to g 2 as shown below, Without loss of generality, we start from the basis, where M N is real and diagonal. In order to consider the mixing of fermionic triplets with the charged leptons, it is convenient to express the four degrees of freedom of each charged triplet in terms of a single Dirac spinor: On the other hand the neutral fermionic triplet components can be left in twocomponent notation, since they have only two degrees of freedom and mix with neutrinos, which are also described by two-component fields. This leads to the Lagrangian as follows The mass term of the charged sector shows the usual aspect for Dirac particles: where m = y < φ 0 > with y is leptonic Yukawa coupling and v = √ 2 < φ 0 >= 246 GeV. On the other hand the symmetric mass matrix for the neutral states is given by The neutrino mass matrix in this case can be written as Thus the light neutrino mass can be written as which mixes the left-handed neutrinos and the neutral fermionic triplet components (right-handed neutrinos). This leads to the full mass matrix for the neutral states as: Diagonalizing the mass matrix we obtain one small neutrino eigenvalue in the limit of ||M D || ||M N || as : where the triplet fermions take the mass values around M N . From Eq. (2.13), it is evident that to have the light neutrino mass m ν 0.1 eV, we need Y N O(10 −6 ) for M N ∼ O(100) GeV. However, such choices of small couplings cannot affect the RG evolution of other couplings [22], a coupling motivated by inverse seesaw can be relevant here.
The collider signatures of heavy neutrinos and leptons rely upon larger Yukawa couplings. These are further restricted from electroweak precision data [77][78][79][80][81]. In the inverse seesaw frame work [82][83][84][85], we introduce another set of fermions which are SU (2) L triplet, Σ 2i (with i = 1, 2, 3) accompanying the Σ 1i . The inverse seesaw Lagrangian is given by, where M D is a 3×3 Dirac mass mixing matrix, M N is the mixing mass term between the two triplets and µ Σ is the small lepton number violating mass term for the Σ 2fields. In the basis of {ν c L , Σ 0 1 , Σ 0 2 }, the full 9 × 9 neutral components mass matrix can be written as (2.15) one obtains the three light neutrinos diagonalizating the mass matrix Eq. (2.15) as described below other neutrino masses are heavy and they are given by M N ± µ Σ /2. Here the small µ Σ share the load of seesaw making Y N ∼ O(0.1). Such large Yukawa couplings give significant negative contributions in the stability of EW vacuum [86].

Perturbativity
To illustrate the theoretical bounds from perturbativity behaviour of the dimensionless couplings, we impose the following conditions on the dimensionless couplings as perturbative limit at a given scale µ, where g j with j = 1, 2 are the EW gauge couplings 1 , λ i with i = 1, 2, 3, 4, 5 are the quartic couplings corresponding to the scalars and Y k with k = u, d, are the Yukawa couplings for the quarks and leptons. The extension of SM with a SU (2) L inert doublet as well as by SU (2) L triplet fermions can change the running g 2 (µ), which in turn affects the progression of other couplings namely the λ i relevant for the vacuum stability and perturbativity. Below we discuss that how g 2 (µ) gets affected via the extra scalar and fermions of this model.

Running of Gauge couplings:
Eqs. 3.2-3.4 and Figure 1(c) describe the evolution of the SM gauge couplings at a given scale µ (not explicitly mentioned in those Eqs.) at two-loop level. Both g 2 and g 3 decrease with the increase in the running scale µ and remain perturbative in the high scale limit. However, such behaviour can change substantially with the inclusion of other SU (2) L fields such as ID and Type-III Seesaw fermions, which are in the triplet of SU (2) L representation.
In Eq. 3.5-Eq. 3.9, we show the modified evolution of the gauge couplings in the presence of the ID and triplet fermions. Eq. 3.5 shows that an inclusion of inert SU (2) L doublet makes β g 2 less negative at one-loop and so as at two-loop. The behaviours can be verified from the Figure 1(a), where the blue curve implies SM and green line represents SM with ID. For the comparison we also show that an addition of a Y = 0 SU (2) L triplet scalar also reduces the negative impact in β g 2 , as can be read from Eq. 3.6, which is also evident from Figure 1(a) pink curve. However, from Eq. 3.7 we can see that even at one-loop β g 2 has become positive with the factor changed to 5g 3 2 6 if we extend the SM with three generations of Type-III fermions. This behaviour continues even at two-loop. The running of g 2 in this case has been depicted by red curve of Figure 1(a). Unlike the SM, or ID scenario, the coupling g 2 in this case increases with increasing µ. In addition to the three generations of Type-III fermions if we add a SU (2) L inert doublet which gives the much needed DM candidate, the factor at one-loop is enhanced to 5g 3 2 with further enhancement at two-loop and also visible by the sky-blue curve in Figure 1(a). Certainly we can see such models looses perturbativity of the gauge coupling around much lower value of the running scale, i.e., µ 10 10 GeV. However, restricting to two generations of triplet fermions, and one ID, this scale pushes even after the GUT scale µ ∼ 10 17 GeV. This prompt us to choose only two generations of Type-III fermions along with ID, rather than the three generations. The running of the gauge coupling g 2 for the two generation scenario is evident from the purple curve of Figure 1(a), and also from Eq. 3.9. Figure. 1(b) represents running of all the three gauge couplings for the two generation scenario. Therefore, SM extension with an ID and two generations of Type-III fermions with ISS mechanism are more motivated, as this accommodates perturbativity of the gauge couplings above the GUT scale, as well as, can explain two small light neutrino masses, and provide a dark matter candidate from the ID.
The perturbative nature of the scalar quartic couplings are also modified by the inclusion of ID and triplet fermions. Below, we first focus on the analysis of the perturbative behaviour of the scalar quartic couplings with respect to λ 3 in the scenario with SM associated with an ID, and three generations of fermionic triplet. The perturbative limit is calculated if at least one of the coupling crosses the perturbativity or hits Landau pole. Figure 2 describes the perturbative behaviour of λ 3 with the scale µ for Y N = 0.02, where the other quartic couplings λ (i=2, 4,5) are kept at different values at the electroweak scale. For the coupling λ 1 , we choose λ 1 = 0.1264, which gives SM like Higgs boson mass around 125.5 GeV. Here λ i , i = 2, 4, 5 at the EW scale are chosen in very weak (λ i = 0.01 in red ), weak (λ i = 0.10 in green ), moderate (λ i = 0.40 in blue ) and strong (λ i = 0.80 in purple) coupling limits respectively.
In ID+Type-III+ ISS scenario we have three generations of fermionic triplet Σ 1i which generates the Dirac term and three generations of additional fermionic triplet Σ 2j instrumental for inverse seesaw mechanism. Having SU(2) L charge they contribute to the beta functions of the SU(2) L gauge coupling, i.e. β g 2 positively; which is somewhat different than normal Type-I + ISS case [22]. Additionally Y N also contributes positively to the beta functions of λ 3,4,5 at one-loop and negative effects only comes at two-loop. Both Y N and β g 2 push the λ 3,4,5 towards non-perturbative limit. The Higgs quartic couplings λ 1 get negative corrections from the Yukawa coupling Y N , pushing the Higgs potential toward instability at one-loop and two-loop. The detailed two-loop beta functions are given in Appendix A, where only two generation effects are shown. However, in Figure 2 we have considered all three generations of SU(2) L triplet fermions along with ID which makes the theory more stable, but simultaneously becomes non-perturbative below Planck scale for all corresponding values of λ i and Y N . Thus for the Planck scale perturbativity we should restrict ourselves to two generations of fermionic triplets. For λ i = 0.01, 0.10 which is less than λ 1 = 0.1264, λ 1 hits Landau pole before going into instability around 10 15 GeV (red and green lines). This happens due to positive effect of g 2 , which is different than the Type-I case [22]. The bending happens due to further positive effects of λ 1 Tr(Y † N Y N ) and other quadratic terms involving λ i s. Such bending effects grows from λ i = 0.01 (brown line) to λ i = 0.10 (green line). For λ i ≥ 0.2 the perturbative limits come at much smaller case as other λ i s hit the Landau pole before λ 1 .
In Figure 3, we present the perturbative behaviour of the Higgs quartic couplings for SM extension with ID + Type-III + ISS with two generations of fermionic triplets. The perturbativity behaviour of the scalar quartic couplings λ 3, 4, 5 are studied in Figure 3    Planck scale Type I ISS   ID Type III ISS   10 3,4,5) . For Y N values up to 0.1, the effect of triplet fermions on the perturbativity of λ i is hardly noticeable as shown in Figure 4. The exact value of Y N where it starts affecting the perturbativity depends on the initial value of λ i at EW scale. However, due to enhanced g 2 in the case of Type-III, λ i attains perturbation limit before than Type-I case and the effect is more prominent for lower values of λ i (EW). For λ i (EW) = 0.1 in ID+Type-III+ISS, the λ 1 hits Landau pole before other λs as λ 1 (EW) = 0.1264 and gets large positive contribution form the enhanced g 2 in this case. Effects of other λ i s are negligible in this case until a particular value of Y N . Large g 2 effect brings the perturbative scale to 10 28.07 GeV with prolonged effect till Y N 0.23 after which λ 1 Tr(Y † N Y N ) effect takes over bringing perturbative scale further down as can be seen from the green line with the bending effect. Compared to that in ID+Type-I+ISS case, for λ i = 0.1, the perturbativity scale is ∼ 10 33.29 GeV and the effect starts showing up for Y N > 0.15 in ID+Type-I+ISS as can be seen in red line. Here perturbative limits are obtained via the Landau poles of other λ i as g 2 decreases with the scale in this case and large Y N pushes λ 1 towards negative values of instability [22]. However, other λ i gets positive contributions λ i Tr(Y † N Y N ) towards their Landau pole. Like the Type-III case here also being very small λ i effects are negligible.
For higher values of λ i (EW), the perturbativity limits are obtained when other λs hit Landau pole before λ 1 and mostly controlled by λ i (see Appendix A) and λ 1 runs towards instability by large Y N effect. This also results in lesser splitting in the perturbative scale between Type-III and Type-I until λ i Tr(Y † Y N ) effects creep in with larger factor for Type-III as compared to Type-I. For λ i = 0.2, the perturbativity limit comes around the GUT scale 10 16−15 GeV and the fermion effect starts for Y N > ∼ 0.3 in both scenarios but the effect of Y N is much stronger in ID+Type-III+ISS. For further higher values of λ i =0.8, the perturbativity scales are almost the same ∼ 10 5 GeV for both Type-I and Type-III cases mostly governed by the λ i effects. The effect of new fermion comes much later for Y N > 0.60. Higher values of λ i s can accommodate higher values of Y N for vacuum stability in λ 1 direction. On the other hand, the theory becomes non-perturbative at much lower scale. One can read the upper bounds on perturbativity from Figure 4 for λ i and Y N values. We see that for the λ i ≤ 0.15 and Y N ≤ 0.25 , Y N ≤ 0.3 the theory remains perturbative till the Planck scale for Type-III and Type-I respectively.

Stability Bound
In this section we analyse the stability of Higgs potential via two different approaches. Firstly via calculating two-loop scalar quartic couplings and checking if the SM Higgs like quartic coupling λ h becomes negative at some higher scale. In this case λ h = λ 1 at tree-level, but at one-loop and two-loop levels λ h gets contribution from SM fields as well as the ID and the Type-III fermions. The details are provided in subsection 4.1. For simplicity, we only present the expressions of the corresponding beta functions at one-loop in the next subsection. The expressions for the two-loop beta functions are given in the Appendix A. Secondly we follow the effective potential approach as described in detail in section 4.3.

RG Evolution of the Scalar Quartic Couplings
To study the evolutions of dimensionless couplings we have implemented the ID + Type-III Seesaw + ISS scenario in SARAH 4.13.0 [87] for two generations SU (2) L triplet fermions. The corresponding β-functions for various gauge, quartic and Yukawa couplings are calculated at one-and two-loop. The full two-loop β-functions can be found in Appendix A, and used for our numerical analysis of vacuum stability which will be described here. We first look at the one-loop β-function of λ h to observe the effect of the Yukawa and other scalar quartic couplings. λ h = λ 1 at tree-level and effects of other particles start entering at one-loop level. The β-function of the SM-like Higgs quartic coupling λ h in this model receives three different types of contributions: one from the SM gauge, Yukawa, quartic interactions, the second from the Type-III Seesaw Yukawa couplings, and the third from the inert scalar sector (ID): Here Y u , Y d , Y e represent the up, down and electron-type Yukawa couplings and other Yukawa couplings are neglected [9] and g 1 , g 2 are the U (1) Y , SU (2) L gauge couplings respectively their values at the EW scale [45] are: λ 1 = 0.1264, g 1 = 0.3583, g 2 = 0.6478, y t = 0.9369. In this analysis the ISS contribution to the RG evolution of λ 1 starts only at the scale above M N . In Figure 5(a)-5(b), we show the respective λ h running with the scale µ. The plot in the left panel represents the three generation scenario, and the plot in the right panel represents two generations scenario. The two plots describe the behaviour of Type III ISS  ID Type III  Type III ISS   Type III   5   only. The green curve shows the evolution using β SM λ 1 + β Type−III λ 1 + β ISS λ 1 , blue curve describes the evolution using β SM λ 1 + β Type−III λ 1 + β ID λ 1 , and finally the purple curve shows the full evolution using β λ 1 ≡ β SM λ 1 + β Type−III In Figure 5(a), the three generations of fermionic triplet make the g 2 contribution too large (see Eq. 3.8) for both Type-III+ISS (green) and ID+ Type-III +ISS (purple), such that β λ 1 hits the Landau pole at ∼ 10 15.4 GeV and ∼ 10 15.1 GeV respectively before hitting the instability scale (at which λ h ≤ 0 ). This makes the theory non-perturbative below Planck scale. Without ISS the β g 2 is relatively smaller (see subsection 3.1) which restrains λ h from hitting the Landau pole. Thus for Type-III Seesaw scenario β λ h becomes unstable ∼ 10 10.7 GeV but bounce back to stability (where λ h ≥ 0) at ∼ 10 14 GeV. For ID+ Type-III scenario, β λ h remains stable till Planck scale for λ i = 0.01 and Y N = 0.06 respectively. Figure 5 (b) describes the behaviour of two generations of fermionic triplet with reduced positive g 2 effect, which prohibits the Landau pole of λ 1 . We can see the ID+Type-III+ ISS (purple curve) is more stable than Type-III+ISS (green curve). Again without ISS, the g 2 contribution is less so Type-III (red) and ID+Type-III (blue) have gone more negative; especially Type-III (red) is even more negative due to the lack of positive effects of ID scalars. Type III ISS  ID Type III  Type III ISS   Type III   5       In Figure 6 we describe the behaviour of two-loop running of the SM-like Higgs quartic coupling λ h as a function of scale (µ) for six benchmark points. We follow the same colour code as Figure 5 and λ 1 (EW ) = 0.1264 chosen for all the graphs. Figure 6(a) describes for the benchmark points of λ i = 0.01, Y N = 0.01 and M N = 100 GeV where i = 2, 3, 4, 5 with two generations of Type-III fermions. We can clearly see that the stability scales for ID+Type-III+ISS (purple curve) and Type-III+ISS (green) are enhanced to Planck scale. Type-III (red) and ID+ Type-III (blue) hit instability around 10 9.5 GeV. Now if we increase Y N = 0.23 as shown in Figure 6(b), we can see that red and blue curve hit instability earlier around 10 8.5 GeV due to larger negative effects of Y N . Even green and purple curves also hit λ h = 0 due to this negative effect. For a comparative study with Figure 6(a) we plot the running of the λ h for three generations of Type-III fermions in Figure 6(c). We see that the purple and the green curve hits Landau pole again due to larger positive g 2 contributions as explained earlier. Blue and red curve also move towards stability in this case. Figure 6(e) we restrict the fermion generation only to two and with the reduced positive effect the Landau poles are gone with overall shift towards the left. Figure 6(d) shows the comparison with Figure 6(b) with two Type-III fermion generation with Y N enhanced to 0.334. We notice the overall negative effect that shifts all the curves towards the left reducing the instability scale with red curve hitting instability at first at 10 7 GeV. If we increase the number of fermion generation to three in Figure 6(f) the effect will be enhanced as red curve crosses zero at around 10 6 GeV before acquiring the Landau pole at 10 15 GeV. The shapes of all the curves becomes more steeper as compared to Figure 6(d). Compared to Figure 6(c) where Y N = 0.02, if we increase Y N = 0.334 in Figure 6(f), we see that the negative effect creeps in and the Type-III + ISS i.e. green the green curve does not hit the Landau pole. However, in ID+Type-III +ISS case due to the positive contributions from λ i the purple curve hits the Landau pole at 10 15 GeV. Figure 7 describes the variation of the stability scale with Y N and λ i . We notice that for the smaller values of λ i ∼ 0.1 (green circles), the Planck scale stability can be achieved till Y N ≤ 0.32. In this case λ 1 strikes the perturbative bounds before other quartic couplings due to the strong g 2 effect and as λ 1 (EW ) = 0.1264. The other quartic coupling effects are negligible in this region as explained earlier. After Y N ≥ 0.32 (green line) the negative contribution from the new fermions become very strong and push λ h towards negative values at around 10 9 GeV.

Variation perturbativity and stability with Y N
For example as we enhance the λ i at EW scale the effect of quartic couplings in their beta functions increase as λ i Tr(Y † N Y N ) along with the enhanced g 2 effect. These inflate β λ 1 towards the higher scale stability compared to the SM but leads to non-perturbative limit for the other λ i s at lower scale. The point to be noted here is that for the choices of λ i > λ 1 at the EW scale, other λ i ( i = 2, 3, 4, 5) are most likely to hit the Landau pole before λ 1 . For example as we enhance the λ i = 0.2 with λ 1 = 0.1264 at the EW scale, one of the λ i s afflicts the landau pole (shown by effect for the instability respectively. For λ i = 0.3, one of the λ i s hits the Landau pole at much lower scale around 10 10 GeV (cyan colour star) due to large quartic coupling contributions along with λ i Tr(Y † N Y N ). In this case the negative effect of Y N in λ 1 starts much later due to larger value of λ i and at Y N ≥ 0.57 the effect of fermions make the λ h ≤ 0 for 10 5.23 GeV (cyan line).

Vacuum Stability: RG-improved potential Approach
We study the stability of the EW vacuum via the Coleman and Weinberg [88] prescription of RG-improved effective potential at one-loop. The effective potential at one-loop has been calcualted for SM + Type-III-ISS + ID with two generations of fermions. The potential then analysed for the stability, metastability and instability by observing the behaviour of the effective Higgs quartic coupling. For this purpose we scan the parameter space of the model and then segregate them as above mentioned three regions.
Eq. (2.3) describes the tree-level Higgs potential for the model. The tree-level stability conditions of the potential are given by [75] (4.5) In SM λ h receives negative effects from the top quark loop which makes the potential unstable at around 10 10−11 GeV. There is also a possibility of second minima in the h direction due to quantum fluctuation. In this article we will see how different beyond SM fields can contribute to that possibility of second minima. However, such possibility can only occur at higher field values which justifies the choice of effective potential as given below: where λ eff (h, µ) is the effective Higgs quartic coupling and the calculation of which is described below . Then the stability of the vacuum then be guaranteed for the scale µ by assuring that λ eff (h, µ) ≥ 0.

Effective Potential
In our model the one-loop RG-improved effective potential can be written as where V 0 is the tree-level potential given by Eq. represent the corresponding one-loop effective potential terms from the inert scalar doublet, and fermionic triplet, respectively. V 1 describes one-loop Coleman and Weinberg effective potential which can be written as where the summation is over all the fields interacting with the h-field. F = 0, 1 for bosons and fermions respectively, n i is the total of degrees of freedom of the particle and M 2 i are the field-dependent mass terms with the corresponing coefficients are shown in Table 1 in the last column. Note that the massless fileds do not contribute to Eq. (4.9), and to Eq. (4.10) In the case of inverse seesaw in the limit µ Σ → 0, the triplet fermion masses are double-degenerate per generation. Thus we take an extra factor of two for each new Type-III fermion contribution in Eq. (4.10). We then study the λ eff (h, µ) in our model to distinguish the stability, metastability and instability regions. h = µ is taken for the numerical analysis as at this scale we have scale-invariant potential [89].

Phase diagrams: Stable, Metastable and Unstable Regions
For λ eff ≥ 0 the potential is bounded from below and the region is termed as the stable region. In this region we expect to have a global EW minimum. On the other hand, for λ eff < 0 the potential is unbounded from below and the poential can tunnel to the global second minima sufficiently faster. But there exists regions when such tunnelling is not fast enough and EW minimum still can survive due to the tunnelling rate which is lower than the age of the Today's universe. Such possibilities are known as metastable vacuua. The tunnelling rate at zero temperature has been calculated as where T 0 is time corresponds to the age of the universe, and µ is the scale with maxium probability, i.e. ∂P ∂µ = 0. The λ values at different scales can be obtained as: where v 246 GeV. If we substitute P = 1, T = 10 10 years and µ = v in Eq. (4.11), one finds λ eff (v) =0.0623. Demanding P < 1, for T = 10 10 years old universe would estimate the condition that such tunnelling rates corresponds to the metastability as given by [6]: (4.13) The condition of λ eff < 0, corresponds to the potential unbounded from below as mentioned before and it depicts the unstable region. It is evident from Eq. (4.10), these different solutions are sensitive to the choices of the scale µ and other model parameters such us bare masses and couplings. Figure 8 represents the phase diagrams in terms of Higgs boson mass and top quark pole mass in GeV. Different regions of the solution spaces corresponding to the unstable, metastable and stable regions are depicted in red, yellow and green colours respectively. The contours describe the current experimental 1σ, 2σ, 3σ values with the dot specifying the central value in the (M h , M t ) plane [9,31]. To obtain the regions we vary all the λ i = 0.1 − 0.8 while the λ 1 and y t are varied to attain the Higgs boson mass within 120 − 128 GeV and top quark mass within 168 − 182 GeV respectively. In Figure 8 we fix M N = 100 GeV and vary Y N = 0.1 − 0.4 for two generations of SU (2) triplet fermions. Figure 8(a) and Figure 8(b) present the scenarios with ID+Type-III+ISS for relatively lower values of Y N = 0.1, 0.3. It is realized that the scenarios are stable till Planck scale.
As shown in Figure 8(a), in this scenario, λ eff becomes more positive and the region is fully in the stable region till Planck scale. This occurs, as there is more positive contribution from g 2 compared to negative effect from fermions for lower values of Y N . Additionally, the inert doublet also adds more scalars to the effective potential, leading to the enhanced stability. In Figure 8(b) we depict the scenario for Y N = 0.3, where negative fermionic effect starts showing up which is compensated by the scalar effect of IDM. As is evident, the stability is still more than SM, and hence, the 3σ contour in m h − m t plane just touches the region of metastability. Further   It can be seen that the whole region is unstable. Further addition of ISS SU (2) L triplet fermions which directly do not give negative contributions but enhance g 2 to more positive value as discussed before leads to Type-III seesaw+ISS scenario marginally extending into the metastable region for Y N = 0.4, as depicted in Figure 8(e). Instead of ISS fermions addition of inert doublet also have the similar effect and pushes the potential into the metastable region as shown in Figure 8(f). This further motivates the extension of Type-III seesaw scenario with ISS and IDM to achieve the stability at larger values of Y N as described in Figure 8(d). Figure 9 we increase M N = 10 3 GeV and analyse the phase diagrams as before. Due to this enhancement in SU (2) L triplet fermion mass their negative loop effects will now be reduced. This can be realised from Figure 9(a)-9(b), where λ eff is highly positive and the regions are fully stable as compared to Figure 8(a)-8(b). As we increase Y N = 0.4, the negative fermionic effect starts showing up in Figure 9(c). The region now lies in the metastable region, however touching the stable region, contrary to Figure 8(c) where some part is in unstable region.

Discussions and conclusion
In this article we studied the vacuum stability of the electroweak vacuum in the presence of Type-III fermions along with inverse seesaw fermions. Unlike Type-I seesaw case, Type-III fermions are in the triplet representation of SU (2) which contribute to the beta-function of the gauge coupling g 2 even at one-loop level. This is a positive effect and increases from Type-III to Type-III+ISS+ID case step wise and makes the g 2 grow to higher values as scale increases. g 2 certainly becomes not perturbative below Planck scale or GUT scale for three generations of Type-III fermions. It is only with the two generation that we are able to acquire the Planck or GUT scale stability.
The enhancement of g 2 also has impact on the scalar quartic couplings which makes them non-perturbative much before compared to SM or SM+Type-I+ISS [22]. For lower values of quartic couplings the g 2 effect is the dominant. At larger values of λ i (except the λ 1 which is fixed by the Higgs mass at EW scale) λ i Tr(Y † N Y N ) effect creeps in making the quartic couplings further divergent. However, a further increment of Y N will bring down the stability bound by pushing λ 1 to negative direction which is proportional to For Planck scale perturbativity we can go up to λ 5 ∼ 0.17 with λ 1 = 0.126 and other λ i = 0.10 at the EW scale for Y N = 0.40. The effective potential approach calculations show that even for Y N = 0.30 for m N = 100, 1000 GeV the model ID+Type-III+ISS with two generations of new fermions lies in the stable region for lower values of Y N and draws to metastable region for higher values of Y N . However, only Type-III scenario belongs to unstable regions in both cases, whereas ID+Type-III and Type-III+ISS scenarios can be in between metastable and unstable regions.
IDM is generally motivated to provide the much needed DM to explain the DM relic and other experimental observations. Nevertheless, it is also supported to enhance the stability of electroweak vacuum. Being in Z 2 -odd multiplet it does not couple to the SU (2) triplet fermions which makes their phenomenology more illusive. No two-body decays are allowed for the Type-III fermions into any of the inert Higgs bosons. In [16,22] authors have shown that due to compressed spectrum only some three-and four-body decays are allowed maintaining the Z 2 symmetry of the Lagrangian. In [15,16,[90][91][92] a detailed relic calculations has been carried out including the direct, indirect DM searches and collider phenomenology. It has been found out that the lightest Z 2 -odd particles should be heavier than 1176 GeV to satisfy the DM relic constraints which is in the desired range to explain the AMS-02 positron excess observation [16,93]. The decays of charged Higgs boson can give rise to mono-lepton plus missing energy signatures [16] which can be isolated from displaced mono-leptonic signatures in real scalar and complex triplet scenarios [16,94] and other charged Higgs signatures [85,[95][96][97]. Similar displaced charged leptonic signatures can be observed in the models with Type-I seesaw [85,[98][99][100][101][102][103] and Type-III seesaw [49,104,105].
The triplet fermions are searched a the LHC at 13 TeV centre of mass energy with democratic branching fractions [106] and a lower bound of 620 − 840 GeV has been put at 2σ level. However, due the presence and mixing with other set of SU (2) triplet fermions involving in inverse seesaw and non-democratic branching can substantially reduce the mass limit allowing even smaller triplet fermion mass. Type-III fermions can also be looked via their angular distributions at the LHC [58].