Distinguishing Inert Higgs Doublet and Inert Triplet Scenarios

In this article we consider a comparative study between Type-I 2HDM and $Y=0$, $SU(2)$ triplet extensions having one $Z_2$-odd doublet and triplet that render the desired dark matter(DM). For the inert doublet model (IDM) either a neutral scalar or pseudoscalar can be the DM, whereas for inert triplet model (ITM) it is a CP-even scalar. The bounds from perturbativity and vacuum stability are studied for both the scenarios by calculating the two-loop beta functions. While the quartic couplings are restricted to $0.1-0.2$ for a Planck scale perturbativity for IDM, these are much relaxed ($0.8$ ) for ITM. The RG-improved potentials by Coleman-Weinberg show the regions of stability, meta-stability and instability of the electroweak vacuum. The constraints coming from DM relic, the direct and indirect experiments like XENON1T, LUX and H.E.S.S., Fermi-LAT allow the DM mass $\gsim 700, \,1176$ GeV for IDM, ITM respectively. Though mass-splitting among $Z_2$-odd particles in IDM is a possibility for ITM we have to rely on loop-corrections. The phenomenological signatures at the LHC show that the mono-lepton plus missing energy with prompt and displaced decays in the case of IDM and ITM can distinguish such scenarios at the LHC along with other complementary modes.


Introduction
Higgs boson was the last key stone predicted by Standard Model (SM), which was discovered at the LHC [1,2]. So far five decay modes of the SM Higgs boson are discovered at the LHC [3,4] and they fall nearly by SM prediction. In spite of immense success, SM cannot resolve many theoretical and experimental anomalies; like existence of dark matter (DM), explanation of very light neutrinos, Higgs mass hierarchy, vacuum stability, muon g − 2, etc. Though discovery of Higgs boson was a direct proof of the role of a scalar in electro-weak symmetry breaking (EWSB) the existence of other Higgs multiplets cannot be ruled out. Recent studies also show that SM stands in a metastable state [5] and need other scalar to make the electro-weak (EW) vacuum stable till Planck scale. This motivates to extend the SM by other Higgs multiplets.
The simplest extension could be via a singlet [6][7][8] but there could be a possibility of extension with another SU (2) Higgs doublet, i.e. two Higgs doublet model (2HDM) [9][10][11][12][13] or with a SU (2) triplet [14] which can enhance the vacuum stability. The extensions of SM with fermions motivated by Seesaw mechanisms often suffers from vacuum instability and one needs some extra scalar to compensate the negative effects [15]. Many of these extensions have a Z2-odd particle, i.e. inert particle which is stable and being lightest among them, can be a dark matter candidate.
Supersymmetric sector in its minimal framework has 2HMD of Type-II [16]. However, the minimal scenario is often challenged by fine-tuning of ∼ 125 GeV light SM-like Higgs boson mass. One of the remedies of this problem is also to extend the Higgs sector beyond its minimal form. This can be achieved by extension by a SM gauge singlet [17], SU (2) triplet [18] or via singlet and triplet superfields [19]. In this case the DM particles is generated by R-parity and it is a supersymmetric particle with odd R-parity. The extended Higgs superfields mix at the superpotential level causing the mixing of Higgs bosons after EWSB among different representations, i.e. doublet-singlet, doublet-triplet, etc [20]. However, we see the situation is very different for non-SUSY Higgs extensions, especially for the inert models. There are no mixing among these extra Higgs states and the SM particles, making them more illusive to produce and detect at the colliders. Nevertheless, they can provide the much needed dark matter candidate and also make the EW vacuum more stable.
In this article we consider two different extensions of SM to attain the dark sector. In the first one we extend SM to Type-I 2HDM with Z2-odd SU (2) doublet that constitutes the dark sector and the scenario is known as inert Higgs doublet (IDM). In the second case we consider the dark sector as Y = 0 SU (2) triplet which is again Z2-odd and the scenario is known as inert Higgs Triplet scenario (ITM). Both the scenarios help in extending the vacuum stability [9,14]; however, we will see that they differ in various constraints coming from perturbativity, vacuum stability, DM relic abundance, direct detection and collider searches. IDM has more scalar with relatively larger mass splitting among the Z2-odd states whereas the ITM has only two Z2-odd states mass degenerate at the tree-level.
Another aspect extended Higgs sector is the search for Higgs quartic coupling. The SM Higgs quartic coupling is till to be measured precisely and only bounds are obtained from the di-Higgs production constraints at the LHC [21,22]. Extended Higgs sectors have many such quartic couplings and they differ from IDM to ITM and are very crucial in determining the fate of the Higgs potential. One or few such quartic couplings can provide the much needed Higgs-DM coupling [7,11]. In this case we focus our region where the DM mass is greater than discovered Higgs mass, i.e. 125.5 GeV. Considering the bounds from vacuum stability, perturbativity, DM relic and direct DM searches we estimate the allowed parameter space and try to distinguish IDM and ITM at the LHC via the compressed spectrum and less number Z2-odd states for the later.
Higgs sector dark matter also has appeal as the quartic coupling between SM-like Higgs boson and dark sector is crucial in measuring such scenario experimentally as well as theoretically. There have been lots of work done in measuring Higgs-DM coupling [7,10,11,23,24]; nevertheless a comprehensive study including bounds from vacuum stability, perturbativity, DM relic and direct DM is expected and which is the topic of this article.
This article is arranged as follows. In section 2 and section 3 we discuss the IDM and ITM briefly along with electro-weak symmetry breaking conditions and the tree-level Higgs boson masses. The comparative study of tree-level mass spectra between IDM and ITM is detailed in section 4. The perturbativity and vacuum stability bounds are discussed in section 5 and section 6 respectively. The DM relic and direct dark matter constraints are calculated in section 7 and section 8 respectively. Indirect bounds are discussed in section 9. In section 10 we dispense the parameter space verses the validity scale and in section 11 we discuss the LHC phenomenology briefly. Finally we conclude in section 12.

Inert Doublet Model (IDM)
The inert 2HDM is a minimalist (apart from SM singlet) extension of the SM with a second SU (2) Higgs doublet Φ2 with the same quantum numbers as the SM Higgs doublet Φ1. The Lagrangian is invariant under the Z2 parity transformation where Φ2 → −Φ2, Φ1 → Φ1 and all the SM fields are even under this symmetry. Such discrete symmetry guarantees the absence of Yukawa couplings between fermions and the inert doublet Φ2 and prohibits any tree-level flavor changing neutral currents. The most general renormalizable, CP conserving potential for inert doublet model [10,25]- [32] is given by where, and m 2 11 , m 2 22 and λ1−5 are real parameters. Electro-weak symmetry breaking is achieved by giving real vev to the first Higgs doublet i.e. Φ1 and the second Higgs doublet does not take part in EWSB. At EW minima, with v 246 GeV, whereas the second Higgs doublet, being Z2-odd, does not take part in symmetry breaking; hence the name is'inert 2HDM'.
Using minimization conditions, we express the mass parameter m 2 11 in terms of other parameters as follows: Except for the SM Higgs boson, h, four new physical scalar states are present: one charged Higgs boson pair H ± , one CP-even neutral Higgs boson H0 and one CP-odd neutral Higgs boson A. Lightest of the the two neutral Higgs bosons can be a candidate of cold dark matter that would be discussed later. After electroweak symmetry breaking, the masses of the scalar particles are given by: Since, Φ2 is inert, there is no mixing between Φ1 and Φ2 and the gauges eigenstates are same as the mass eigenstates for the Higgs bosons. The Z2 symmetry prevents any such mass mixing through Higgs portal and it also prevents the second Higgs doublet to couple to fermions. In this case we get two CP-even neutral Higgs h and H0, where h is likely to be the discovered Higgs boson around 125 GeV at the LHC [1,2] and the other is yet to be found out. Similarly we are also looking for the pseudoscalar Higgs boson A and the charged Higgs boson H ± at the collider. It can be seen from Eq. 2.4 that H0, A and H ± are nearly degenerate. Depending upon the sign of λ5 one of scalar between H0 and A can be lighter and a cold dark matter candidate [25]- [32]. Unlike [12,13] here we concentrate of MH 0 , MA > m h and the corresponding couplings.

Inert Triplet Model (ITM)
In completing SM with a dark sector we can have DM in the SU (2) triplet representation which does not take part in the EWSB. This can be simply achieved by adding a SU (2) real triplet scalar with Y = 0 hypercharge and again making it Z2-odd to provide to take part in EWSB [14]. Here we introduce in addition to SM Higgs doublet i.e. Φ, another SU (2)L triplet scalar with Y=0, i.e. T and due to Z2-odd nature, the triplet field does not take part in EWSB, i.e. the vev of the triplet, vT = 0.
The Higgs Lagrangian for ITM case can be written as, where the covariant derivatives involving the gauge-fields are given by, Now we impose an additional Z2 symmetry under which triplet is assigned to be odd and other fields are even. The Lagrangian is invariant under the Z2 parity transformation where T → −T and all the SM fields are even. A Z2 symmetric potential for ITM can be written as: In ITM the triplet field does not get vev i.e., vT = 0 and only doublet gets vev as given by, Here v = 246 GeV and the model in known as 'inert triplet model'. In minimization conditions, we express the mass parameter m 2 h in terms of other parameters as follows: Triplet field does not contribute to mass of any of the SM particle and the gauge bososn masses solely get contribution from Φ as shown below: Thus in this case ρ = M 2 w cos 2 θw M 2 z stays in SM value at the tree-level. Except for the SM Higgs boson, h, three new physical scalar particle states are present: one charged Higgs boson pair T ± and one CP-even neutral Higgs boson T0. After EWSB the physical Higgs boson masses can be read as: where mT and λ ht are the parameters as shown in the Higgs potential Eq. 3.4. Note that at the tree-level from Eq.3.7, masses of neutral and charged components are the same, but loop corrections tend to make the charged components, T ± slightly heavier than the neutral one T0 with a mass gap of δM (M ± T , MT 0 ) = 166 MeV [33]. Hence, T0 turns out to be lightest component of triplet scalar and a suitable DM candidate.
Next we compare both the models after EWSB by their physical mass eigenstates, mass spectrum and perturbativity, stability bounds. We mentioned earlier that for IDM we have one extra excitation as CP-odd Higgs boson i.e. A which can be a DM candidate. Whereas in case of ITM the DM is always a purely CP-even scalar. In sections below we categorically address the issues regarding the mass spectrum, bounds from perturbativity and vacuum stability, DM relic and direct dark matter detection.

Mass spectrum of IDM and ITM
In Figure 1 we describe the mass correlations among the heavier Higgs states for both IDM. Figure 1(a) depicts the mass correlation between MH 0 and MA in GeV and the green colour corresponds to the masssplitting greater than MW and red colour describes the mass-splitting less than MW . In this case the tree-level mass splitting is generated by the λ5 term. Such mass splitting is greater in the lower mass range and as the mass spectrum increases, m22 term dominates over the λ3−5 which makes all Z2 odd states almost degenerate. We find that the mass splitting between MH and MA is greater than W boson mass till MH 0 = 600GeV. This mass-splitting between M + H and MA keeps below MW for M H ± 400 GeV as can be seen from Figure 1(b). We also note that the mass splitting between M + H and MA is lower than the corresponding splitting between MH 0 and MA.        T vs MT 0 in GeV at tree-level. We see that at the tree-level there is no mass-splitting between triplet states. One has to rely on the loop-contributions for O(166) MeV mass splitting between T ± and T0 which will be crucial for the phenomenological studies [33].

Perturbativity Bound
To emulate the theoretical bounds from perturbative unitarity of the dimensionless couplings, we impose that all dimensionless couplings of the model must remain perturbative for a given value of the energy scale µ, i.e. the couplings must satisfy the following constraints: where λi with i = 1, 2, 3, 4, 5 are the scalar quartic couplings; gj with j = 1, 2 are EW gauge couplings; 1 and Y k with k = u, d, are all Yukawa couplings for the up, down types quarks and leptons respectively. The two-loop beta functions generated by SARAH 4.13.0 [34], given in Appendix A and Appendix B are used to check the variations of the dimensionless couplings with the scale of the variation (µ in GeV).  The perturbativity behaviour of the scalar quartic couplings λ3,4,5 is studied in Figure 4(a)-4(c) respectively where the other quartic couplings λ (i=2, 3,4,5) are fixed at some values. Here red, green, blue and purple curves in each plot correspond to different initial conditions for other λi at the EW scale, representative of very weak (λi = 0.01), weak (λi = 0.10), moderate (λi = 0.40) and strong (λi = 0.80) coupling limits respectively. The dashed black line corresponds to Planck scale (10 19 GeV). Higgs quartic coupling λ3 remains perturbative till Planck scale for λ3 0.51, 0.32 for λi(EW) = 0.01, 0.10 respectively as shown in Figure 4(a). For λi(EW) = 0.40, 0.80 theory becomes non-perturbative at much lower scale ∼ 10 8.9 , 10 5. 6 GeV respectively for almost all initial values of λ3.  Here red, green, blue and purple curves in each plot correspond to different initial conditions for λ t and λ ht at the EW scale, representative of very weak (λ t = 0.01), weak (λ t = 0.1), moderate (λ t = 0.4) and strong (λ t = 0.8) coupling limits respectively.

Stability Bound
In this section we discuss the stability of Higgs potential via two different approaches. Firstly via calculating two-loop scalar quartic couplings and checking if the SM-like Higgs quartic coupling λ h is getting negative at some scale. In this case λ h = λ1 at tree-level but at one-loop and two-loop levels λ h gets contribution from SM fields as well as the BSM scalars as we describe in the subsection 6.1. For the simplicity in subsection 6.1 we give the expressions of the corresponding beta functions at one-loop level and in the Appendix A, B the two-loop beta functions are given.

RG Evolution of the Scalar Quartic Couplings
To study the evaluations of dimensionless couplings we implemented both the IDM and the ITM scenarios in SARAH 4.13.0 [34] and the corresponding β-functions for various gauge, quartic and Yukawa couplings are calculated at one-and two-loop levels. The explicit expressions for the two-loop β-functions can be found in Appendix A, B and they are used in our numerical analysis of vacuum stability in this section. To illustrate the effect of the Yukawa and additional scalar quartic couplings on the RG evolution of the SM-like Higgs quartic coupling λ1 in the scalar potential (2.1) and (3.4), let us first look at the one-loop βfunctions. λ h = λ1 at tree-level and at the one-loop level, the β-function for the SM Higgs quartic coupling in this model receives two different contributions: one from the SM gauge, Yukawa, quartic interactions and the second from the inert scalar sectors of IDM/ITM as shown below: where, Here g1, g2, g3 are respectively the U (1)Y , SU (2)L and SU (3)c gauge couplings, and Yu, Y d , Y are respectively the up, down and lepton-Yukawa coupling matrices of SM. We use the SM input values for these parameters at the EW scale: λ1 = 0.1264, g1 = 0.3583, g2 = 0.6478, yt = 0.9369 and other Yukawa couplings are neglected [35,36]. Figure 6 depicts the running of SM-like Higgs quartic coupling at two-loop level for four benchmark points with (λ2,3,4,5) for IDM and (λ ht,t ) for ITM to be 0.010, 0.060, 0.068 and 0.100 respectively. For both the cases λ1 = 0.1264 is kept at two-loop level for the SM-like Higgs boson mass at 125.5 GeV. Here the red curve corresponds to the IDM and the green curve corresponds to the ITM. For λi(EW)=0.010, in Figure 6(a), the effect of scalars on stability is less and both IDM and ITM becomes unstable at same scale ∼ 10 9.7 . In Figure 6(b) for λi(EW) = 0.060 we see that the λ h becomes negative around 10 12 GeV but λ h turns upward at 10 16 GeV and touches zero value for 10 20 GeV in the case of IDM while for ITM it still stays negative. As λi(EW) enhances to 0.068 in Figure 6(c), the stability scale increases to ∼ 10 13.5 in ITM while IDM becomes completely stable. Since, there are more number of scalars in IDM than ITM, the theory becomes stable at much lower values of λi. Further enhancement of λi(EW ) to 0.100, Figure 6(d) makes both IDM and ITM stable till Planck scale.

Vacuum Stability from RG-improved potential Approach
In this section, we investigate the vacuum stability via RG-improved effective potential approach by Coleman and Weinberg [37], and calculate the effective potential at one-loop for IDM/ITM. The parameter space of the models are then scanned for the stability, metastability and instability of the potential by calculating the effective Higgs quartic coupling and implementing the constraints as discussed in the paragraph follows.
Before going to quantum corrected potential lets look at the stability conditions of the tree-level potential of IDM/ITM. The tree-level potential of IDM is given in Eq. (2.1) and the potential is bounded from below in all the directions is ensured by the tree-level stability conditions given by [38] Similarly, the tree-level potential of ITM is given in Eq. (3.4) and the corresponding tree-level stability conditions are given by [23] Considering the running of couplings with the energy scale in the SM, we know that the Higgs quartic coupling λ h gets a negative contribution from top Yukawa coupling yt, which makes it negative around      minimum exists at much higher scale than the EW minimum, we can safely consider the effective potential in the h-direction to be where λ eff (h, µ) is the effective quartic coupling which can be calculated from the RG-improved potential. The stability of the vacuum can then be guaranteed at a given scale µ by demanding that λ eff (h, µ) ≥ 0. We follow the same strategy as in the SM in order to calculate λ eff (h, µ) in our model, as described below. The one-loop RG-improved effective potential in our model can be written as where V0 is the tree-level potential given by Eq. are the corresponding one-loop effective potential terms from the IDM and the ITM loops. In general, V1 can be written as where the sum runs over all the particles that couple to the h-field, F = 1 and 0 for fermions and the bosons in the loop, ni is the number of degrees of freedom of each particle, M 2 i are the tree-level field-dependent masses given by with the coefficients given in Table 1 and m 2 corresponds to Higgs mass parameter. Note that the massless particles do not contribute to Eq. (6.10), and so to Eq. (6.9). Therefore, for the SM fermions, we only include the dominant contribution from top quarks, and neglect the other quarks. We take h = µ for the numerical analysis as at that scale the potential remains scale invariant [40]. Table 1. Coefficients entering in the Coleman-Weinberg effective potential, cf. Eq. (6.9).
Using Eq. (6.9) for the one-loop potentials, the full effective potential in Eq. (6.8) can be written in terms of an effective quartic coupling as in Eq. (6.7). This effective coupling can be written as follows: where the corresponding coefficients for all the required fields are given in the Table 1. The nature of λ eff in the models thus guides us to identify the possible instability and metastability regions, as discussed below.

Stable, Metastable and Unstable Regions
The parameter space where λ eff > 0 is termed as the stable region, since the EW vacuum is the global minimum in this region. For λ eff < 0, there exists a second minimum deeper than the EW vacuum. In this case, the EW vacuum could be either unstable or metastable, depending on the tunnelling probability from the EW vacuum to the true vacuum. The parameter space with λ eff < 0, but with the tunnelling lifetime longer than the age of the universe is termed as the metastable region. The expression for the tunnelling probability to the deeper vacuum at zero temperature is given by where T0 is the age of the universe and µ denotes the scale where the probability is maximized, i.e. ∂P ∂µ = 0. This gives us a relation between the λ values at different scales: , (6.13) where v 246 GeV is the EW VEV. Setting P = 1, T = 10 10 years and µ = v in Eq. (6.12), we find λ eff (v) =0.0623. The condition P < 1, for a universe about T = 10 10 years old is equivalent to the requirement that the tunnelling lifetime from the EW vacuum to the deeper one is larger than T0 and we obtain the following condition for metastability [5]: (6.14) The remaining parameter space with λ eff < 0, where the condition (6.14) is not satisfied is termed as the unstable region. As can be seen from Eq. (6.11), these regions depend on the energy scale µ, as well as the model parameters, including the gauge, scalar quartic and Yukawa couplings.    [36,41]. To obtain the regions we vary all the λi(EW) = 0.01 − 0.80 for random values maintaining the Planck scale perturbativity and also maintain the m h and mt within limits shown in Figure 7. Figure 7(a) shows the scenario where λ1 = 0 and all other λi = 0 and clearly the region is in metastbale state as expected for SM [36]. Introduction of inert doublet adds more scalars to the effective potential so the λ eff becomes more positive and the region is fully in the stable region as can be seen from Figure 7(b). In Figure 7(c) we depicts the scenario for ITM, where such extra scalar degrees of freedoms are lesser than IDM but more than SM, so the 3σ contour in m h − mt plane includes some region of metastability. In this context we also want to mention that the extra scalars are necessary and come as saviour for the models with right-handed neutrino with O(1) neutrino Yukawa coupling [15].

Calculation of Relic Density in freeze out scenario for IDM and ITM
After the theoretical constraints from perturbativity and vacuum stability we focus on the constraints coming from the measurement of the relic density of dark matter by WMAP and Planck experiments [42] and the current value is given by where h = 0.67 ± 0.012 is the scaled current Hubble parameter in units of 100km/s.Mpc. Here, we use this value as upper bound on the contribution on dark matter production for the models IDM [43] and ITM [14]. Before the onset of freeze-out, the universe was hot and dense and as the universe expands, the temperature falls down. In this scenario the respective dark matter particles will not be able to find each other fast enough to maintain the equilibrium abundance. For the case of IDM lightest of H0 and A can become the dark matter candidate and for this study we focus on the A as the DM candidate and for ITM it is T0. So when the equilibrium ends and the freeze-out starts, Inert particles T0 and A, can contribute in the relic density of DM through freeze-out mechanism [44]. We now examine the thermal relic abundance of DM particle. φDM (A/T0 for IDM/ITM). The evolution of the number density of DM is obtained by solving the Boltzmann equation [45] where H is the Hubble parameter, v φ DM stands for the relative velocity of the dark matter particles, ... represents the thermal average of a function in brackets, n φ DM , n φ DM ,eq and σ are the number density of DM particle, the number density in thermal equilibrium and the total annihilation cross-section of φDM respectively. All the particles in the Z2-odd multiplets for both IDM/ITM will eventually contribute with σv φ DM . For IDM for lower mass splitting among A, H0, H ± both the annihilation AA → SM SM and co-annihilation AH ± → SM SM should be included while estimating σv φ DM as shown in Figure 8. AA → ZZ/W + W − are the dominant channels in getting the DM relic for IDM but co-annihilation channel H ± A → γW ± also contribute. Unlike IDM, the mass splitting between dark matter (T0) and charged components (T ± ) is much smaller for ITM, O(166) MeV. Thus the co-annihilation T0T ± → ZW ± contribution is substantial along with the annihilation T0T0 → W ± W ± as shown in Figure 9. Below we scan the parameter space for both IDM and ITM to find out the regions with correct DM relic as given in Eq. (7.1).
For this scan we take the allowed parameter space from perturbativity and stability till Planck scale for the analysis of correct DM relic density by Micromegas 5.0.8 [46][47][48] . Figure 10 describes the variation of relic density with the masses of charged Higgs boson and DM (A/T0 for IDM/ITM). The colour code of DM relic (Ωh 2 ) is shown from blue to red for 0.0 − 0.4 for both IDM and ITM respectively. The correct values of Ωh 2 = 0.1199 ± 0.0027 is specified by a star in both the cases. We can read from Figure 10(a) that for IDM MA > ∼ 700 GeV corresponds to correct DM relic value. However, for ITM the correct relic value corresponds to MT 0 > ∼ 1176 GeV as shown in Figure 10(b). The presence of one extra Z2-odd scalar in IDM compared to ITM, results into higher the DM number density in IDM case and thus requires more

Constrains from Direct Dark Matter experiments
In this section, we discuss the direct detection prospects of DM candidate for both IDM and ITM scenarios. Dark matter can be detected via elastic scattering with terrestrial detectors, the so-called direct detection method. From the particle physics point of view, the quantity that determines the direct detection rate is the dark matter-nucleon (DM − N) scattering cross-section. In the IDM, the DM − N scattering process relevant for direct detection is Higgs-mediated. The tree-level spin-independent DM-nucleon interaction cross section, in IDM scenario [49,50] is given by Eq. (8.1) were M h is the mass of the SM-like Higgs boson, MA is the mass of the DM candidate, MN is the nucleon mass that we took equal to the average of proton and neutron masses, fN is the nucleon form factor, taken equal to 0.3 for the subsequent analysis and λ345 = λ3 + λ4 − 2λ5 with λ5 > 0, is the combined coupling that is responsible for the scattering. we have used Micromegas 5.0.8 [46][47][48] to calculate the direct spinindependent scattering cross-sections and DM relic density for the parameter space and later compare with the experimental bounds from different direct detection experiments as discussed later.
In the case of ITM, the T0 DM candidate can interact with nucleon by exchanging Higgs boson and the DM-nucleon scattering cross section is given by [14] Eq 8.2 where the coupling constant fN is given by nuclear matrix elements and MN = 0.939 GeV is nucleon mass which is average of the proton and neutron masses, M h is the SM-like Higgs boson mass, MT 0 is the dark matter mass and λ ht is only responsible Higgs coupling here.  Figure 11. SI cross-section verses dark matter mass in GeV. Here we have shown XENON100, LUX and XENON1T data in red, green and blue regions respectively. Widths in green and blue regions are to make them transparent such that other bounds are visible.
There are several experiments to detect DM particles directly through the elastic DM-nucleon scattering. The strong bounds on the DM-nucleon cross section are obtained from XENON100 [51], LUX [52] and XENON1T [53] experiments. The minimum upper limits on the spin independent cross sections are:  Figure 11 describes the variation of spin independent (SI) DM-nucleon scattering cross-section with DM mass for both IDM and ITM. The red colour corresponds to the cross-section bound satisfied by XENON100 experiment [51], green colour satisfies the LUX experimental bound [52] and the blue colour corresponds to the experimental bound of XENON1T experiment [53] for both IDM and ITM. The crosssection varies with the DM mass and the Higgs quartic coupling λ345 for IDM and λ ht for ITM. If the Higgs quartic coupling is chosen to be small enough λ345 = 0.01 for IDM, the minimum DM mass satisfying the XENON1T bound is 420 GeV 11(a). Unfortunately this value of quartic coupling in ITM i.e. λ ht =0.01 is not allowed by the vacuum stability. The enhancement in Higgs quartic coupling λ 345/ht = 0.2 increases the lower bound of DM mass to 2770 GeV by XENON1T data11(b).
The variation of DM mass with Higgs quartic coupling λ3 in IDM and λ ht in ITM is depicted in Figure 12. The light purple and blue colour describe the allowed regions by stability and perturbativity till Planck scale for IDM and ITM respectively. The black vertical lines correspond to the relic density bound satisfied by DM mass 700, 1200 GeV for IDM, ITM respectively. The green and red colour points describe the minimum values of MDM for a given λ345/λ ht for IDM and ITM respectively that satisfy the direct Dark matter constraint of XENON1T [53]. In IDM the effective quartic coupling λ345 allows to choose maximum allowed value of λ3 satisfying the direct DM constraints, while in the case of ITM the minimum value of MDM increases with increase in λ ht .    16 GeV where as the SM-like Higgs stays with mass 125.5 GeV for both the cases. One more number of Z2-odd field in IDM as compared to ITM which contributes to the number density of the dark matter. Thus IDM requires more annihilation cross-sections than ITM in getting the correct DM relic, which results in lower DM mass (∼ 700) GeV for IDM as compared to ∼ 1.2 TeV for ITM.

Constraints from H.E.S,S and Fermi-Lat experiemtns
Since both the cases (IDM and ITM) the dark matter annihilate to W ± W ∓ directly, the bounds on < σv > in W ± W ∓ mode from H.E.S.S [54] and Fermi-LAT [55] would be very evident. We impose such bounds on our parameter space as shown in Figure 14 describes < σv > in W ± W ∓ mode verses the DM mass by pink lines: Figure 14(a) for IDM and Figure 14(b) for ITM respectively. The Blue line corresponds to the H.E.S.S bounds [54] and the green line corresponds to Fermi-LAT bounds [55] in W ± W ∓ mode. As expected due to triplet coupling to W ± is larger (See Eq. 3.2) in comparison with the doublets, the cross-section in W ± W ∓ mode is larger for a given mass. The start ( ) points are the chosen benchmark points as discussed in Table 2 are allowed by both H.E.S.S [54] and Fermi-LAT [55] data in W ± W ∓ mode. In the context of IDM other indirect bounds are discussed in the literature [56].  [54] and the green line corresponds to Fermi-LAT bounds [55] in W ± W ∓ mode. The start ( ) the points are chosen benchmark points as discussed in Table 2.

Dependence on the validity scale
In this section we discuss how the parameter space depends on the validity scale of perturbativity and vacuum stability along with the relic and direct DM constraints. While implementing that we consider three different scales; namely the Planck scale (10 19 GeV), the GUT scale (10 15 GeV) and the 10 4 GeV scale as the upper limit of the theory. It would be interesting to see how two different DM models differ in such different requirements.

Validity till Planck scale
Here we consider that all the dimensionless couplings remain perturbative and the EW vacuum remains stable till Planck scale (µ 10 19 GeV). In Figure 15 we present the parameter points in DM mass verses DM relic density for both IDM and ITM. The Red coloured points are allowed by the electroweak symmetry breaking. Among those points, the Green coloured points correspond to the points which are allowed by both perturbativity and stability till Planck scale (µ 10 19 GeV). The black and blue lines correspond to those points which are allowed by direct detection cross-section bound of XENON1T [53] for two different benchmark scenarios chosen for IDM and ITM. The benchmark points chosen for direct detection are λ345 = 0.050 (λ3 = 0.200, λ4 = 0.100, λ5 = 0.125) and λ345 = 0.09 (λ3 = 0.200, λ4 = 0.200, λ5 = 0.155) for IDM as shown in Figure 15(a) described by black and blue lines. We see that the similar constraints for ITM are presented in Figure 15(b) for λ ht = 0.05 and λ ht = 0.09 respectively. In the case of ITM, the quartic coupling value λ ht = 0.05 is allowed by perturbativity till Planck scale but only to µ 10 9 GeV by vacuum stability, while λ ht = 0.09 is allowed by both till Planck scale. The dashed horizontal line defines the correct DM relic density as given in Eq: 7.1.   Figure 16 shows the DM mass verses relic density variation in IDM and ITM. Simialr to previous case here also green colour corresponds to the points which are allowed by both perturbativity and vacuum stability till GUT scale (10 15 GeV). For IDM and ITM, the allowed parameter space by both perturbativity and vacuum stability remain same as Planck scale. The black and blue lines again correspond to those points which are allowed by the direct detection cross-section bound of XENON1T [53]. The corresponding benchmark points are chosen λ345/λ ht = 0.05, 0.09 for IDM/ITM respectively as shown in Figure 16(a) and Figure 16(b). As discussed earlier for ITM, the EW vacuum is stable till µ ∼ 10 9 GeV for λ ht = 0.05. x

Validity till 10 4 GeV
The above analysis is repeated for the benchmark points which are allowed by perturbativuty, vacuum stability, DM relic bound and direct detection cross-section bound till scale µ ∼ 10 4 GeV as shown in Figure 17. In this scenario, green colour corresponds to points which are allowed by both perturbativity and vacuum stability till 10 4 GeV scale. The allowed parameter space by vacuum stability and perturbativity increases for both IDM and ITM as we see more green points as compared to Figure 15 and Figure 16. The corresponding benchmark points are chosen λ345/λ ht = 0.05, 0.09 for IDM/ITM respectively as shown in Figure 17(a) and Figure 17(b) and all the points are allowed by the perturbativity and vacuum stability constraints till µ ∼ 10 4 GeV.

LHC Phenomenology
LHC is looking for the heavier states specially for the another Higgs bosons for both CP-even and CP-odd but so far no new resonances are found out and only cross-section bounds have been given by both CMS and ATLAS [57,58]. In this article we consider the extension of SM with a inert SU (2) doublet or inert Y = 0 SU (2) triplet. In both the cases the extra scalar gives rise to a lightest Z2-odd particle which does not decay and can contribute as missing energy in the collider [59,60]. IDM has one pseudoscalar Higgs boson (A), one CP-even Higgs boson (H0) and the charged Higgs boson (H ± ) and all are from the inert doublet Φ2, which is Z2 odd and their mass splittings are mostly MW in allowed mass range, making a quasi-degenerate mass spectra. Contary to IDM, ITM has only a CP-even real Higgs boson (T0) and a charged Higgs boson (T ± ). In this case their tree-level masses are identical unlike IDM case and only mass splitting of 166 MeV comes from loop-corrections.
In ITM the triplet does not take part in EWSB and so there is no mass mixing between the doublet and triplet which is very different from the supersymmetric triplet case [18,19] where such mixing occur from the superpotential. Moreover, Y = 0 triplet nature does not allow it to couple to fermion in both SUSY and non-SUSY cases disparate from Y = 2 triplet case of Type-II seesaw. The normal Y = 0 triplet which takes part in EWSB, breaks the custodial symmetry (vT = 0) which implies g W ∓ −Z−H ± = 0 at tree-level.  Table 2. Dominant 3-body decay modes and corresponding branching ratios, decay width and decay length for the benchmark points of IDM and ITM.

Model Masses in GeV
This makes ρ > 1, which strongly constrains vT 5 GeV [61]. In case of ITM, we have vT = 0 as triplet stays in Z2-odd, which certainly ceases the g W ∓ −Z−H ± coupling to exist. Thus the charged Higgs boson decays to mono-lepton or di-jet plus / ET via off-shell W ± and DM unlike tri-lepton plus missing energy in case of triplets that gets vev and breaks custodial symmetry at tree-level [62][63][64][65].
Associated production of charged Higgs boson with another triplet neutral scalar in ITM scenario thus gives rise to mono-lepton or di-jet plus missing energy signature. A pair of charged Higgs boson will give rise to di-lepton plus missing energy [66,67]. The signatures of ITM and IDM [68][69][70] are very similar and the only difference is that in case of IDM we have additional neutral scalar (CP-even or CP-odd) which gives rise to distinguishing signature and thus can be separated from the ITM. Due to Z2-odd, both inert Higgs bosons do not couple to fermions and their decay only happen via gauge mode on-or off-shell.
In Table 2 we present the benchmark points for the future collider study which are allowed by the vacuum stability, perturbativity bounds till Planck scale, dark matter relic and DM constraints. The heavy Higgs boson and charged Higgs boson mass stay around 912 GeV and 903 GeV respectively with the pseudoscalar boson mass around 899 GeV. In this allowed mass range, the mass gap among the other heavier Higgs bosons are of the order of O(1) GeV, giving rise to naturally soft decay products for the associated Higgs productions. Here the decays of a Z2 odd Higgs boson is only possible via three-body decays to quarks and leptons via off-shell gauge boson and DM particle. In Table 2 we also show the dominant three-body decay modes for the heavy CP-even Higgs boson in IDM with branching fractions of BR(H0 → Add) ∼ 12.21% and BR(H0 → Ass) ∼ 12.20% respectively with a total decay width of ∼ 5.80 × 10 −7 GeV. This corresponds to decay length of ∼ 10 −9 meter, which essentially give rise to a prompt decay. The other subdominant decay modes are with BR(H0 → i=2,3 Aνiνi) ∼ 10.75% and (H0 → Aūu) ∼ 9.58% respectively.
Similarly lower panel of Table 2 shows the benchmark point for the ITM scenario. Here the charged Higgs bosons and the triplet neutral scalar stay almost mass degenerate with nMT 0 =1178.60 GeV and M T ± =1178.76 GeV respectively. Such spectrum only allows the three body decays with branching ratios of BR (T ± → T0du) ∼ 72.72% and BR(T ± → T0ν ± ) ∼ 24.30% respectively. A very small decay width of 1.51 × 10 −16 GeV easily gives rise to O(33) meter displaced charged Higgs boson decay. [8,[71][72][73][74][75][76] Next we focus on the production cross-sections of the chosen benchmark points at the LHC with centre of mass energy of 14,100 TeV [77]. In Table 3 present the cross-sections of various associated Energy IDM ITM  Table 3. Production cross-section at LHC for 14 TeV and 100 TeV center of mass energy.
Higgs production modes at the LHC with centre of mass energy of 14 and 100 TeV. Here we used CalcHEP 3.7.5 [78] for calculating the tree-level cross sections and decay branching fraction for the chosen benchmark points. For the cross-sections NNPDF 3.0 QED LO [79] is used as parton distribution function and √ŝ is used as scale, where s = E 2 cm is the known Mandelmstam variable. The associated Higgs productions include the production modes of H ± H ∓ , H ± H0, H ± A in IDM and T ± T ∓ , T ± T0 in ITM as shown in Table 3. The charged Higgs pair production and associated productions cross-sections at tree-level are σ(H ± H ∓ ) = 1.88 × 10 −2 fb, σ(H ± H0) = 3.49 × 10 −2 fb and σ(H ± A) = 3.64 × 10 −2 fb respectively for IDM. Similar cross-sections for ITM are given by σ(T ± T ∓ ) = 3.07 × 10 −3 fb, σ(T ± T0) = 6.82 × 10 −3 fb respectively at the LHC with 14 TeV centre of mass energy. It is evident that the cross-sections are very low due to electro-weak nature of the process and around TeV mass of the particles. Nevertheless the situation improves at 100 TeV with σ(H ± H ∓ ) = 1.87 fb, σ(H ± H0) = 3.29 fb, σ(H ± A) = 3.30 fb for IDM and σ(T ± T ∓ ) = 6.16 × 10 −1 fb, σ(T ± T0) = 1.23 fb for ITM respectively. At 100 TeV LHC and with sufficiently large integrated luminosity studying the mono-lepton plus missing energy with prompt and displaced leptons one can distinguish such scenarios. IDM has one more massive mode compared to ITM which could also be instrumental in distinguishing such scenarios [80].

Conclusions
In this article we consider two possible extensions of SM which give rise to a potential DM candidate and further extensions of which can address many other phenomenological issues [13,14]. For this purpose Z2-odd SU (2) doublet extension, IDM and Y = 0 SU (2) triplet extension, ITM are analysed. The EWSB conditions in case of IDM give rise to extra CP-even(H0) and CP-odd(A) Higgs bosons along with a charged Higgs boson H ± . Here lightest of the two neutral Higgs boson can be the DM candidate. However, for ITM there is only one CP-even(T0) neutral Higgs boson and one charged Higgs boson (T ± ) that come from the Z2 odd triplet multiplet. The EW mass gap among these Z2-odd particles varies between O(MW ) to O(1) GeV in case of IDM at the tree-level. In comparison the Z2 odd particles in ITM are all mass degenerate at the tree-level and only O(1) MeV mass splitting comes from loop correction.
After EWSB we checked the perturbative unitarity of all the dimensionless couplings for both IDM and ITM scenarios. Due to existence of large numbers of scalars IDM scenario gets perturbative bounds below Planck scale even with relatively smaller values of one of the Higgs quartic couplings at the EW scale i.e. λi 0.1 − 0.2. On the other hand, ITM scenario remains perturbative till Planck scale for higher values of Higgs quartic coupling, i.e. λ t,ht 0.8 and λ ht 0.5, λt=1.3. Similar to perturbativity, the stability of EW vacuum gives bounds on the parameter space by requiring that SM direction of the Higgs potential is stable and for SM such validity scale is µ ∼ 10 9−10 GeV [36]. Introduction of the Z2 scalar in both the cases i.e. IDM and ITM moves the region to greater stability. Thus models with right-handed neutrinos with large Yukawa can be in the stable region by the help of these scalars [15].
After checking the perturbative unitarity and stability we move to calculate the DM relic abundance for both the scenarios. The dominant mode of annihilation for the both the cases are into W ± W ∓ and co-annihilation is in association with the charged Higgs boson into W ± Z. However, due to presence of one extra Z2 scalar in IDM compared to ITM, the DM number density is relatively on higher side than ITM. This requires more annihilation or co-annihilation to obtain the correct relic compared to the ITM case, leading to lower mass bound on DM mass i.e. mDM > ∼ 700 GeV in IDM compared to ITM, where it is mDM > ∼ 1176 GeV. Later we also considered the direct-DM bounds from DM-nucleon scattering crosssection from XENON100, LUX and XENON1T [51][52][53]. The corresponding indirect bounds on < σv > in W ± W ∓ mode from H.E.S.S [54] and Fermi-LAT [55] are also taken into account.
At the end we studied their decay modes by calculating their decay branching fractions for the allowed benchmark points. We also estimate their production cross-sections for various associated Higgs-DM production modes at the LHC for the centre of mass energy of 14, 100 TeV respectively. Compressed spectrum for ITM will easily lead to displaced mono-or di-charged leptonic or displaced jet final states along with missing energy. Such displaced case however not so natural in case of IDM. Nevertheless, such inert scenarios can easily be distinguished from the normal Type-I 2HDM and Y = 0 real scalar triplet, where both of them take part in EWSB as their decay products are not so restrictive [80].   .