Two Component Singlet-Triplet Scalar Dark Matter and Electroweak Vacuum Stability

We propose a two component dark matter set-up by extending the Standard Model with a singlet and a hypercharge-less triplet scalars, each of them being odd under different $Z_2$ symmetries. We observe that the inter-conversion between the two dark matter components allow a viable parameter space where masses of both the dark matter candidates can be below TeV, even though their individual contribution to single component dark matter rules out any such sub-TeV dark matter. We find that a lighter mass of the neutral component of the scalar triplet, playing the role of one dark matter component, compared to the scalar one is favored. In addition, the set-up is shown to make the electroweak vacuum absolutely stable till the Planck scale, thanks to Higgs portal coupling with the scalar dark matter components.


I. INTRODUCTION
The Standard Model (SM) of particle physics undoubtedly emerges as the fundamental theory of interactions after discovery of the Higgs boson at the Large Hadron Collider (LHC) [1,2]. However there still remains some issues, confirmed by experimental observations and can't be resolved within SM. For example, observation of cosmic microwave background radiation by Planck [3] reveals that about 26.5% of Universe is made up of mysterious dark matter (DM). The SM of particle physics, however, can not account for a dark matter candidate. Dark matter direct search experiments such as LUX [4], XENON-1T [5], PandaX-II [6,7] search for the evidences of DM-nucleon interaction. Till date no such direct detection signal of DM has been detected which limits the DM-nucelon scattering cross-section. Apart from dark matter there also exists problem with the stability of electroweak (EW) vacuum within the Standard Model as the electroweak vacuum becomes unstable at large scale Λ I ∼ 10 10 GeV [8][9][10][11][12] for top quark mass m t = 173.2 GeV [13]. This instability of EW vacuum at large scale can be restored in presence of additional scalars.
In order to address the above mentioned issues, we need to go beyond the SM. In this work, we will include new scalar particles which can serve as dark matter candidate and also stabilize the EW vacuum simultaneously. It is also to be noted that the null detection of DM in direct detection (DD) experiments also triggers the possibility of dark sector to be multicomponent which is explored in many literatures in recent . In multi-component dark matter scenario DM-DM conversion plays a significant role to determine the observables such as direct detection and relic density and also helps to stabilize EW vacuum with increased number of scalars. In this work, we consider a multi-component dark matter with scalar singlet and scalar triplet with zero hypercharge.
Study of scalar singlet dark matter and its effects on electroweak vacuum is done extensively in earlier works [39][40][41][42][43][44][45][46][47][48]. In a pure scalar singlet scenario, due to the presence of the quartic coupling between Higgs and dark matter can help Higgs quartic coupling become positive making the EW vacuum stable till Planck scale, M P l . It is found that singlet scalar with mass heavier than 900 GeV can satisfy the constrains coming from relic density, direct detection and vacuum stability [32]. Introducing an inert doublet as a possible dark matter component attracts a great amount of attention in recent days. It is found that there exists an intermediate region (80 -500) GeV, beyond which the neutral component of the inert Higgs satisfies the relic and DD constraints [49][50][51][52][53][54][55][56][57][58][59][60]. Recently it has been shown that in multi-component DM scenarios involving inert Higgs doublet(s) and/or singlet scalar, the region can be revived [31,32].
Moving toward a further higher multiplet, it is found that an inert triplet can also be a possible dark matter candidate. A hypercharge-less (Y =0) inert triplet scalar can serve as a feasible dark matter candidate similar to inert Higgs doublet [49][50][51][52][53][54][55][56][57][58][59][60]. However, the allowed mass ranges of inert triplet dark matter is very much different from that of the inert doublet. Similar to the case of inert doublet, annihilation of triplet scalar is mostly gauge dominated which leaves a larger desert region compared to inert doublet. Also, due to small mass splitting between charged and neutral triplet scalar, co-annihilation channels into SM particles becomes relevant. Earlier studies [15,[61][62][63] reported that a pure inert triplet (with Y =0) dark matter, consistent with relic density, direct detection and vacuum stability constrains can be achieved with triplet mass ∼ 1.9 TeV and triplet Higgs quartic coupling The other possibility is to have Y = 2 triplet scalar which is also investigated. It was shown in [61] that with Y = 2, dark matter mass M DM ≥ 2.8 TeV is allowed. For Y = 2 possibility, things are further restricted, mostly from the direct detection bounds. It is to be noted that unlike Y = 2 inert triplet scalar, neutral particles of Y = 0 inert triplet scalar does not have direct interaction with the Z boson which arise from the kinetic term in case of Y = 2 triplet. As a result, additional quark nucleon scattering via Z boson exchange occurs for Y = 2 inert triplet. This interaction term contributes to dark matter direct detection significantly and because of large scattering cross-section, most of the available parameter space is ruled out [61]. In this work we concentrate on Y = 0 triplet scalar.
As mentioned above, due to large gauge dominated annihilation, the relic density of Y = 0 inert triplet dark matter remains under-abundant up to ∼1.8 TeV. Therefore, this leaves a great opportunity to explore the phenomenology of multi-component dark matter set-up involving the inert triplet and a singlet scalars. Similar to the case of scalar singlet, the triplet Higgs quartic coupling also helps to stabilize the EW vacuum. In a work [15], the authors explored a multi-component DM scenario with an inert triplet and a singlet scalar, both are odd under different Z 2 symmetries in which the conversion of one to other dark matter did not play any significant role in the parameter space considered. In this work however, we want to explore the below-TeV regime of both the dark matters in the two-component framework as this sub-TeV region is of great importance from the collider and dark matter experiments. In our study, we aim to show the importance of the conversion coupling in realizing the relic density. At the same time we also emphasise on the Higgs portal coupling of both the dark matters as they play a significant role in dark matter phenomenology and in making the EW vacuum absolutely stable till M P l . We therefore search for a viable parameter space in this multi-component dark matter scenario that satisfies constraints from dark matter observables as well as electroweak vacuum stability can also be achieved.
The paper is organized as follows. The model is introduced in section II and the various theoretical and experimental constraints deemed relevant are detailed in section III. Sections IV sheds light on the DM phenomenology. We then discuss the status of vacuum stability in V in this scenario and finally conclude in section VI.

II. MODEL
In the present set-up, we extend the Standard Model particle content by introducing a SU (2) L triplet scalar T having hypercharge Y = 0 and a SU (2) L singlet scalar S. In addition, we include discrete symmetries Z 2 × Z 2 under which all the SM fields are even while additional fields transform differently. In Table I we provide the charge assignments of these additional fields under the SM gauge symmetry and the additional discrete symmetries imposed on the framework. Both the scalar singlet S and the neutral component of T can play the role of the dark matter candidates as they are charged odd under different Z 2 and hence stable. Therefore the present set-up can accommodate a two-component dark matter scenario. The most general renormalisable scalar potential of our model, V (H, T, S), consistent where and In the above expression of Eq. The scalar fields can be parametrised as and after the EWSB, the masses of the physical scalars are given as In Eq. (5), m h = 125.09 GeV [65], is the mass of SM Higgs. It is to be noted that although mass of neutral and charged triplet scalar are degenerate, a small mass difference of ∆m is generated via one loop correction [66,67] and therefore T 0 can be treated as a stable DM candidate. This mass difference is expressed as where α is the fine structure constant, M W , M Z are the masses of the W and Z bosons, ∆m can be expressed as [66] ∆m = α 2 M W sin 2 θ W 2 166 MeV.
The couplings λ HS and λ HT denote the individual Higgs portal couplings of two DM candidates S and T 0 respectively whereas the coupling κ provides a portal which helps in converting one dark matter into another (depending on their mass hierarchy). For our analysis purpose, we first implement this model in LanHEP [68], choosing the independent parameters in the scalar sector as: (m T 0 , m S , λ HS , λ HT , κ).

A. Theoretical constraints
The parameter space of this model is constrained by the theoretical consideration like the vacuum stability, perturbativity and unitarity of the scattering matrix. These constraints are as follows: (i) Stability: Due to the presence of extra scalars (T and S) in our model, the SM scalar potential gets modified which can be seen from Eq (1). In order to ensure that the potential is bounded from below, the quartic couplings in the potential must satisfy the following co-positivity conditions. Following [69,70] we have derived the copositivity conditions for our present set-up: where µ is the running scale. These condition should be satisfied at all the energy scales till M P l in order to ensure the stability of the entire scalar potential in any direction.
where λ i and κ represents the scalar quartic couplings involved in the present setup whereas g i and y αβ denotes the SM gauge and Yukawa couplings respectively. We will ensure the perturbativity of the couplings present in the model till the M P l energy scale by employing the renormalisation group equations (RGE).
(iii) Tree level unitarity: One should also look for the constraints coming from perturbative unitarity associated with the S matrix corresponding to scattering processes involving all two particle initial and final states [71,72]. In the present set-up, there are 13 neutral and 8 singly charged combination of two particle initial/final states. All the details are provided in the Appendix A. The constraints imposed by the tree level unitarity of the theory are as follows: where |x 1,2,3 | are the roots of the following cubic equation:

B. Experimental constraints
The important experimental constraints follow from electroweak precision experiments and collider searches.
(i) Electroweak precision parameters: A common approach to study beyond the SM is considering the electroweak precision test. The presence of an additional scalar triplet in the setup may contribute to the oblique parameters. These extra contributions to the oblique parameters coming from the present setup are given as [63,73] S 0, (11a) The contribution to the S parameter from the triplet scalar fields is negligible. It is invisible Higgs decay width of the SM Higgs boson as [13]: In the present set up we focus mostly in the parameter space where m T 0 , m S > m h 2 so the above constraint is not applicable.
(ii) LHC diphoton signal strength: Due to the presence of the interaction between the SM Higgs h and the triplet scalar T in Eq. (3), the charged triplet scalar T ± can contribute significantly to h → γγ at one loop. The Higgs to diphoton signal strength can be written as Now when triplet is heavier than m h /2, we can further write The analytic expression of Γ(h → γγ) triplet can be expressed as [74] Γ where G f , is the Fermi constant. The form factors induced by top quark, W gauge boson and T ± loop respectively. The formula for the form factors are listed below.
In order to ensure that µ γγ lies within the experimental uncertainties, the analysis should respect the latest signal strength from ATLAS [75] and CMS [76]. The measured value of µ γγ are given by µ γγ = 0.99 ± 0.14 from ATLAS and µ γγ = 1.17 ± 0.10 from CMS. (iv) Relic density and Direct detection of DM: The parameter space of the present model is to be constrained by the measured value of the DM relic abundance from the Planck experiment [3]. One can further restrict the parameter space by applying bounds on the DM direct detection cross-section coming from the experiments like LUX [4], XENON-1T [5], PandaX-II [6,7]. Detailed discussions on the dark matter phenomenology are presented in section IV.

IV. DARK MATTER PHENOMENOLOGY
The present set-up contains two dark matter candidates T 0 and S, both are odd under different discrete symmetries Z 2 and Z 2 which remain unbroken. To obtain the correct relic densities of the dark matter candidates one needs to solve the coupled Boltzmann equations. In order to do that we first identify all the relevant annihilation channels of both the dark matter candidates. In Fig.1 2. Annihilation channels for triplet scalar dark matter T 0 .
Here one can relate y i (i = S, T 0 ) to Y i by y i = 0.264M P l √ g * µ dm Y i whereas one can redefine ) are now written in terms of µ dm as Here M Pl = 1.22 × 10 19 GeV, g * = 106.7, m i = m S , m T 0 , X represents all the SM particles and finally, the thermally averaged annihilation cross-section can be expressed as and is evaluated at T f . The freeze-out temperature T f can be derived by equating the DM interaction rate Γ = n DM σv with the expansion rate of the universe H(T ) (20), K 1,2 (x) represents the modified Bessel functions.
We use Θ function in Eq. (18) to explain the conversion process (corresponding to Fig.4) of one dark matter to another which strictly depends on the mass hierarchy of DM particles.
These coupled equations can be solved numerically to find the asymptotic abundance of the DM particles, y i µ dm m i x ∞ , which can be further used to calculate the relic: where x ∞ indicates a very large value of x after decoupling. Total DM relic abundance is then given as It is to be noted that total relic abundance must satisfy the DM relic density obtained from Planck [3] Ω Total h 2 = 0.1199±0.0027 .

B. Direct detection
Direct detection (DD) experiments like LUX [4], PandaX-II [6,7] and Xenon1T [5,77] look for the indication of the dark matter-nucleon scattering and provide bounds on the DM-nucleon scattering cross-section. In the present model, dark sector contains two dark matter particles. Therefore, both the dark matter can appear in direct search experiments.
However, one should take into account the fact that direct detection of both triplet and singlet DM are to be rescaled by factor f 0 Ω Total with j = T 0 , S. Therefore, the effective direct detection cross-section of triplet scalar DM T 0 is given as [78] and similarly the effective direct detection cross-section of scalar singlet is expressed as [79] where m N is the nucleon mass, λ HT and λ HS are the quartic couplings involved in the DM-Higgs interaction. A recent estimate of the Higgs-nucleon coupling f gives f = 0.32 [80].
Below we provide the Feynman diagrams for the spin independent elastic scattering of DM with nucleon.

C. Results
To study the proposed two component DM scenario, we first write the model in LanHEP [68] and then extract the model files to use in micrOMEGAs 4.3.5 [81]. In doing this analysis, all the relevant constraints as mentioned in section III are considered. For the sake of better understanding, we divide our analysis in two parts: Note that our aim is to have mass of both the DM candidates below TeV which is an interesting regime for experimental studies. Here we mostly rely on two facts to satisfy our goal: (i) single component of DM does not require to produce the entire relic contribution and (ii) conversion involving two DMs is expected to contribute non-trivially. Below we proceed one after other cases. As we observe above that the triplet contribution to the relic is essentially under-abundant (irrespective of the choice of portal coupling λ HT ) in this region, we expect that the singlet scalar can make up the rest of relic while an important contribution to be contributed by the DM-DM conversion. As stated before, the relevant parameters that would control the study are m T 0 , m S , λ HS , λ HT , and κ and we find below their importance. In the left panel of Fig.7, we show the variation of the individual contributions toward relic abundances from triplet (Ω T 0 h 2 ) and singlet (Ω S h 2 ) with their respective masses, m T 0 and m S respectively, such that the total relic abundance Ω Total h 2 satisfies the Planck limit Below we discuss implications of this plot in detail.
In order to understand the importance of conversion coupling κ, we begin with κ = 0 case. It is to be noted that even when the conversion coupling κ is set at 0, conversion between DM candidates (SS → T 0 T 0 ) can take place via s-channel diagram as shown in Hence Ω T 0 h 2 has a limitation, it can't provide more than ∼ 10 percent contribution as seen from Fig. 6(b). However once the κ has a sizeable magnitude, the T 0 contribution to the relic is enhanced to some extent due to the DM-DM conversion as can be seen from the black patch (paired with purple) for κ = 0.3 and brown patch (paired with red) for κ = 0.5.
In Fig.7  DM mass is kept fixed at 680 GeV while κ is considered to be 0.3 (one of the two benchmark values of Fig. 8(a)). The total relic is shown here by the orange line. We observe that for the singlet scalar contribution, it exactly follows the pattern of its sole contribution (below 680 GeV) as shown in Fig. 6  In Fig.9, we repeat the plots of Fig.8 for a smaller value of λ HS = 0.05, though keeping λ HT fixed at 0.2. As λ HS is decreased, the annihilation of S into the SM particles is also decreased which in turn enhances the relic density of S for a given mass. Hence, a relatively smaller contribution from T 0 (compared to Fig. 8(a)) is required and as a result, lower mass of m T 0 is allowed. In other words, a shift of the black patch (of parabolic nature) toward left (i.e. shift toward lowered masses) is observed. For example, with the same value of κ = 0.3 is in Fig. 8 also, while a pair of DM masses m S , m T 0 = (690, 680) GeV satisfies the total relic in case with λ HS = 0.2, a lower set of masses (407, 400) GeV can satisfy the relic in case with λ HS = 0.05. At this point, we can recall our finding from Fig. 6 also. The relic contribution from T 0 is essentially governed by the T 0 T 0 annihilations to finals state gauge bosons, and being almost insensitive to λ HT value, the maximum contribution of Ω T 0 h 2 incorporating a sizeable κ can be around 30 percent of the total relic (provided we stick to the low mass regime of DMs, i.e. ∼ below TeV) with appropriate κ. Therefore the significant relic has to be obtained from S. Hence, the above conclusion that a smaller λ HS allows for a lighter DM pair remains valid for any choice of λ HT . With a similar line of consideration as in Fig. 8(a), here also we use the conservative bound on m T 0 as m T 0 > 287 GeV. Finally in view of constraints on the mass of the triplet DM as stated in section III, we put a vertical dashed line at m T 0 = 287 GeV such that the right side of it can be recognized as the allowed parameter space. As a result, some of the parameter space becomes disallowed for κ = 0.3.

Case II: m S < m T 0
We now study the DM phenomenology considering the mass hierarchy among DM components as m S < m T 0 . Note that in this case the DM-DM conversion can take place having the form: T 0 T 0 → SS and hence contribution from the singlet scalar would be more than that of the case-I. Following Fig. 6(b), we know that the maximum contribution of Ω T 0 h 2 is less than 30 percent only provided we restrict m T 0 to be in sub-TeV regime. Furthermore due to T 0 T 0 → SS conversion in this case, contribution to relic by Ω T 0 h 2 would be even less.
This particular case is therefore not very promising from the perspective of two component DM. Hence in this case, we extend the mass range of T 0 to be more than TeV (though less than 1.8 TeV) while m S is kept below 1 TeV.
In the present set-up, presence of these new scalar fields i.e. S and T provides a positive contribution to the beta function of λ H through their Higgs portal interactions as which (for details, see Appendix B) helps in making the EW vacuum stable.
While λ H > 0 till M P l ensures the absolute stability of the EW vacuum, violation of this at a scale below M P l could be problematic. In case λ H (µ) becomes negative at some scale (as happens for SM at Λ I ), there may exist another deeper minimum other than the EW one. Then the estimate of the tunneling probability P T of the EW vacuum to the second minimum is essential to confirm the metastability of the Higgs vacuum. The Universe will be in a metastable state, provided the decay time of the EW vacuum is longer than the age of the Universe. The tunneling probability is given by [8,82], where T U is the age of the Universe, µ B is the scale at which the tunneling probability is maximized, determined from β λ H (µ B ) = 0. Solving the above equation, the metastability requires: GeV and α S (m Z ) = 0.1184.  portal couplings, λ HS and λ HT . Therefore we would like to explore here whether the same parameter space can make the EW vacuum stable. As stated above, both the couplings λ HS and λ HT (with different pre-factors) play a significant role in the running of effective Higgs quartic coupling λ eff H . Presence of these Higgs portal couplings are therefore expected to make the EW vacuum stable. For the analysis purpose, we have chosen two benchmark points BP-I and BP-II as shown in Table   III. Both these points satisfy the total relic density, the direct detection bounds and are also allowed by the constraints coming from ATLAS [75] on the Higgs singnal strength µ γγ ( as discussed in section III). While choosing the benchmark points we have kept κ fixed at 0.3 so that the conversion of the heavier dark matter to the lighter one remains effective. We In Fig. 11 we show the running of the effective Higgs quartic coupling λ eff H in our model for the two benchmark points as mentioned in Table III, BP-I (solid red lines) as well as BP-II (dotted red lines) and compare it with that of the SM (dashed red lines). As expected, we observe in Fig. 11 that due to the presence of additional scalar couplings like λ HS and λ HT , the β λ H gets affected and hence make λ eff H positive till M P l . The conclusion remains valid for both the benchmark points, BP-I and BP-II. Increase in the value of λ eff H for BP-II is due of the involvement of larger λ HS = 0.5. In Fig. 12, we plot the running of all the scalar quartic couplings in our model for both the BPs. We observe in Fig. 12 that all the couplings remain positive and perturbative till the Planck scale M P l for both the BPs. It is interesting to note that the self quartic coupling of the scalar singlet S in Fig. 12 (b) shoots up, this happens because of the specific choice of λ HS = 0.5 made in BP-II as shown in Table III. This rapid increase in the evolution of λ S for the large value of λ HS is dictated by the presence of 12λ 2 HS term in the β λ S .

VI. CONCLUSIONS
In this work, we explore a two-component DM scenario made out of one singlet scalar and the neutral component of a hypercharge-less triplet scalar. As a single component dark matter, none of these candidates satisfies the relic density and the DD constraints having mass below TeV. While the singlet scalar starts to satisfy the relic and DD with its mass close to 1 TeV alone, the Y = 0 triplet can do so with its mass close to 2 TeV. Hence we particularly focus in this sub-TeV region as this regime is otherwise an interesting one from the perspective of collider and dark matter experiments. We are able to show that the DM-DM conversions becomes helpful so as to realize our goal of restricting both the dark matters in sub-TeV regime for m S > m T 0 . In case of reverse mass hierarchy, such a realization turns out to be not that favorable though. In this case where m S < m T 0 , triplet mass beyond 1 TeV (but much less than 2 TeV) with m S below 1 TeV can do the job.
In this entire analysis, the conversion coupling κ plays a pivotal role. We observe that though it is mostly the scalar singlet contribution which contributes dominantly to the relic, the parameter space with κ = 0 is completely disallowed. This is due to the fact that the relic density then would be mainly followed from S only and hence the effective cross-section in DD can not have adequate suppression which is otherwise expected via Eq.(23) with a sizeable κ. The parameter space that satisfy the relic and DD constraints is also consistent in making the electroweak vacuum absolutely stable. This is mainly achieved through the contributions of the Higgs portal couplings of the dark matters. The set-up also bears an interesting discovery potential at LHC. Due to its multi-component nature, the present setup can accommodate smaller value of triplet scalar mass (below TeV) which provides a possibility of probing the charged scalar more proficiently at LHC via the disappearing charge track at the detector. A detailed study in this direction remain an interesting possibility to explore in future. and After determining the eigenvalues of Eq.(A6) we conclude that the tree level unitarity constraints in this set up are the following: |λ HT | < 8π, |λ HS | < 8π, |κ| < 8π, and |x 1,2,3 | < 16π (A11) where |x 1,2,3 | are the roots of the following cubic equation: x 3 + x 2 (−36λ H − 3λ S − 5λ T ) + x(−27κ 2 − 36λ 2 HS − 108λ 2 HT + 108λ H λ S + 180λ H λ T +15λ S λ T ) + 972κ 2 λ H − 648κλ HS λ HT + 324λ 2 HT + 180λ 2 HS λ T λ S − 540λ H λ T λ S = 0.