Scalar assisted singlet doublet fermion dark matter model and electroweak vacuum stability

We extend the so-called singlet doublet dark matter model, where the dark matter is an admixture of a Standard Model singlet and a pair of electroweak doublet fermions, by a singlet scalar field. The new portal coupling of it with the dark sector not only contributes to the dark matter phenomenology (involving relic density and direct detection limits), but also becomes important for generation of dark matter mass through its vacuum expectation value. While the presence of dark sector fermions affects the stability of the electroweak vacuum adversely, we find this additional singlet is capable of making the electroweak vacuum absolutely stable upto the Planck scale. A combined study of dark matter phenomenology and Higgs vacuum stability issue reflects that the scalar sector mixing angle can be significantly constrained in this scenario.


Introduction
Although the discovery of the 125 GeV Higgs boson at Large Hadron Collider (LHC) [1,2] undoubtedly marks the ultimate success of the Standard Model (SM), there are issues in particle physics and cosmology, supported by observations, which can not be explained in the SM framework. For example, SM justifies only 5% of the total matter content of the Universe preferably known as visible matter. Compelling evidences from astrophysical and cosmological observations of cosmic microwave background radiation (CMBR), spiral galaxy rotation curve, colliding clusters etc. indicate the presence of unknown matter, called dark matter (DM) which constitutes 25% of the Universe. There are some other theoretical issues for which SM can not provide clear answer.
In particular, it is well known that the Higgs quartic coupling (λ H ) turns negative at energy scale (Λ SM I ∼ 10 10 GeV [3][4][5][6][7] with m t = 173.2 GeV [8]) leading to a possible instability of the electroweak (EW) minimum [9]. However the conclusion crucially depends on precise value of the top quark and Higgs mass. In presence of a deeper minimum compared to the EW one, question will also arise why the Universe has chosen the EW vacuum over the global minimum [10][11][12][13][14][15].
In order to circumvent these shortcomings of the SM, one has to introduce new physics beyond the Standard Model. In an earlier attempt [16], the SM is extended with two SM singlet scalars, one is with zero and other has non-zero vacuum expectation value (vev). It is shown in [16] that while the singlet scalar with zero vev plays the role of the DM, the other scalar with non-zero vev mixes with SM Higgs (Higgs portal) and affect the dark matter phenomenology in such a way that the scalar DM having mass ∼ 200 GeV and onward can satisfy the relic density and direct search constraints from LUX [17], XENON-1T [18], Panda 2018 [19] and XENON-nT [20]. On the other hand, it turns out that the interaction of the scalar fields with SM Higgs can modify the instability scale (Λ I ) larger than Λ SM I even by several order of magnitude. In fact the scalar with non-zero vacuum expectation value having mass smaller than Λ SM I can indeed make the electroweak vacuum absolutely stable [16] with the help of threshold effect [21,22]. Other works involving DM and EW vacuum stability can be found in [23][24][25][26][27][28]. The extra scalar field(s) could also be connected to several other unresolved physics of the Universe involving inflation [29,30] or neutrinos [31][32][33][34][35][36][37] etc.
In this work we consider the singlet doublet dark matter (SDDM) scenario and explore how it can be extended minimally (if required) so as to achieve the EW vacuum stability till M P . In a typical SDDM model [38][39][40][41][42][43][44][45][46][47][48][49][50][51], the dark sector is made up with two Weyl fermion doublets and one Weyl singlet fermion. The Yukawa interactions of them with SM Higgs result three neutral fermion states, the lightest of which becomes a viable candidate for DM provided the stability is guaranteed by some symmetry argument. Unlike Higgs portal dark matter models, singlet doublet dark matter scenario directly couples the mass and dynamics of dark sector with the SM gauge sector. This is analogous to the case of supersymmetric extensions [52] where the supersymmetry breaking scale provides mass of dark matter [44]. Singlet doublet dark matter models also induce considerable coannihilation effect which is absent in usual Higgs portal DM scenarios. Another interesting feature of SDDM model is related to evading the direct detection bound with some specified "blind spots" of the model [44].
The SDDM carries different phenomenology from the usual extension of dark sector with vectorlike fermion doublet and singlet [53][54][55][56][57] due to the involvement of three neutral Majorana fermions in SDDM as compared to two vector like neutral fermions in [53][54][55]57]. In case of vector like singlet doublet models, it is possible to have interaction with Z boson which can enhance the spin independent dark matter nucleon cross-section considerably. On the other hand, in case of SDDM, such interaction is suppressed [44]. Although in SDDM, spin dependent interaction (i.e. axial vector interaction) survives, the bounds on spin dependent dark matter nucleon cross-section [58] is not that stringent compared to spin independent limits and hence remain well below the projected upper limits. Therefore it relaxes the bounds on model parameters in the singlet doublet model allowing the model to encompass a large range of parameter space.
Although the SDDM has many promising features as mentioned above, it also has some serious issues with the Higgs vacuum stability. The model involves new fermions, which can affect the running of Higgs quartic coupling leading to instability at high energy scale. In an attempt to solve the Higgs vacuum stability where the DM is part of the SDDM model, we propose an extension of the SDDM with a SM singlet scalar. We employ a Z 4 symmetry under which all the beyond SM fields carry non-trivial charges while SM fields are not transforming. The salient features of our model are the followings: • There exists a coupling between the additional scalar and the singlet Weyl fermion which eventually contributes to the mass matrix involving three neutral Weyl fermions. After the SM Higgs doublet and the scalar get vevs, mixing between neutral singlet and doublet Weyl fermions occur and the lightest neutral fermion can serve as a stable Majorana dark matter protected by the residual Z 2 symmetry. In this way, the vev of the additional scalar contributes to the mass of the DM as well as the mixing.
• Due to the mixing between this new scalar and the SM Higgs doublet, two physical Higgses will result in this set-up. One of these is identified as the Higgs discovered at LHC and we consider the other Higgs to be heavier. This introduces a rich DM phenomenology (and different as compared to usual SDDM model) as the second Higgs would also contribute to DM annihilation and the direct detection cross section.
• The presence of the singlet scalar with non-zero vev helps in achieving the absolute stability of the EW vacuum. Here the mixing between singlet doublet scalars (we call it scalar mixing) plays an important role. Hence the combined analysis of DM phenomenology (where this scalar mixing also participates) and vacuum stability results in constraining this scalar mixing at a level which is even stronger than the existing limits on it from experiments.
The paper is organised as follows. In Sec. 2 we describe the singlet scalar extended SDDM model.
Various theoretical and observational limits on the specified model under are presented in Sec. 3.
In the next Sec. 4 we present our strategy, the related expressions including Feynman diagrams for studying dark matter phenomenology of this model. The discussion on the allowed parameter space of the model in terms of satisfying the DM relic density and direct detection limits are also in Sec. 4. In Sec. 5, the strategy to achieve vacuum stability of the scalar enhanced singlet doublet model is presented. In Sec. 6, we elaborate on how to constrain parameters of the model while having a successful DM candidate with absolute vacuum stability within the framework. Finally the work is concluded with conclusive remarks in Sec. 7.

The Model
Like the usual singlet doublet dark matter model [39][40][41][42], here also we extend the SM framework by introducing two doublet Weyl fermions, ψ D 1 , ψ D 2 and a singlet Weyl fermion field ψ S . The doublets are carrying equal and opposite hypercharges (Y = 1 2 (− 1 2 ) for ψ D 1 (D 2 ) ) as required from involves vev of φ but also the dark matter phenomenology becomes rich due to the involvement of two physical Higgs (as a result of mixing between φ and the SM Higgs doublet H). Apart from these, φ is playing a crucial role in achieving electroweak vacuum stability. The purpose of φ will be unfolded as we proceed. For the moment, we split our discussion into two parts as: extended fermion and scalar sectors of the model.

Extended fermion sector
The dark sector fermions ψ D 1 , ψ D 2 and ψ S are represented as, Here field transformation properties under SM (SU (2) L × U (1) Y ) are represented within square brackets. The additional fermionic Lagrangian in the present framework is therefore given as where D µ is the gauge covariant derivative in the Standard Model, (2) it can be easily observed that after φ gets a vev, the singlet fermion ψ S in the present model receives a Majorana mass, m ψ S = c φ .
Apart from the interaction with the singlet scalar φ, the dark sector doublet ψ D 1 and singlet ψ S can also have Yukawa interactions with the Standard Model Higgs doublet, H. This Yukawa interaction term is given as The matrix is constructed with the basis X T = (ψ S , ψ 0 1 , ψ 0 2 ). On the other hand, the charged components have a Dirac mass term, m ψ ψ − 1 ψ + 2 + h.c.. In general the mass matrix M could be complex. However for simplicity, we consider the parameters m ψ S , λ and m ψ to be real. By diagonalising this neutral fermion mass matrix, we obtain , where the three physical states P T = (χ 1 , χ 2 , χ 3 ) are related to X by, where V is diagonalizing matrix of M. Then the corresponding real eigenvalues obtained are given as [41,59] where A = 1, B = −m ψ S , C = −(m 2 ψ + λ 2 v 2 2 ), D = m 2 ψ m ψ S (provided the discriminant (∆) of M is positive). Now R and the angle θ m can be expressed as where P = 2B 3 −9ABC+27A 2 D and Q = 3 is the discriminant of the matrix M. The lightest neutral fermion, protected by the unbroken Z 2 , can serve as a potential candidate for dark matter.
Below we define the Dirac fermion (F + ) and three neutral Majorana fermions (F i=1,2,3 ) as, where ψ − 1 and ψ + 2 are identified with ψ − and ψ + respectively. In the above expressions of F + and F i , α(α) = 1, 2 refers to upper (lower) two components of the Dirac spinor that distinguishes the left handed Weyl spinor from the right handed Weyl spinor [60]. Hence m F + = −m ψ corresponds to the Dirac mass term for the charged fermion. Masses of the neutral fermions are then denoted as m F i = m χ i . As we have discussed earlier, although the Z 4 symmetry is broken by φ , a remnant Z 2 symmetry prevails in the dark sector which prevents dark sector fermions to have direct interaction with SM fermions. This can be understood later from the Lagrangian of Eq. (22) which remains invariant if the dark sector fermions are odd under the remnant Z 2 symmetry. Now we need to proceed for finding out various interaction terms involving these fields which will be crucial in evaluating DM relic density and finding direct detection cross-sections. However as in our model, there exists an extra singlet scalar, φ with non-zero vev, its mixing with SM Higgs doublet also requires to be included. For that purpose, we now discuss the scalar sector of our framework.

Scalar sector
As mentioned earlier, we introduce an additional real singlet scalar φ that carries Z 4 charge as given in Table 1. The most general potential involving the SM Higgs doublet and the newly introduced scalar is given as After electroweak symmetry is broken and φ gets vev, these scalar fields can be expressed as Minimization of the scalar potential leads to the following vevs of φ and H given by Therefore, after φ gets the vev and electroweak symmetry is broken, the mixing between the neutral component of H and φ will take place (the mixing is parametrized by angle θ) and new mass or physical eigenstates will be formed. The two physical eigenstates (H 1 and H 2 ) can be obtained in terms of H 0 and φ 0 as where θ is the scalar mixing angle defined by Similarly the mass eigenvalues of these physical scalars are found to be Using Eqs. (16)(17)(18), the couplings λ H , λ φ and λ φH can be expressed in terms of the masses of the physical eigenstates H 1 and H 2 , the vevs (v, v φ ) and the mixing angle θ as One of these H 1 , H 2 would be the Higgs discovered at LHC. The other Higgs can be heavier or lighter than the SM Higgs. Before proceeding for discussion of how this model works in order to provide a successful DM scenario and the status of electroweak vacuum stability, we first summarize relevant part of the interaction Lagrangian and the various vertices relevant for DM phenomenology and study of our model.

Interactions in the model
Substituting the singlet and doublet fermion fields of Eqs. (2)(3) in terms of their mass eigenstates following Eq.(5) and using the redefinition of fields given in Eq.(10), gauge and Yukawa interaction terms can be obtained as Here the expressions of different couplings are given as With the consideration that all the couplings involved in M are real, the elements of diagonalising matrix V in Eq.(5) become real [59] and hence the interactions proportional to the imaginary parts in Eq.(22) will disappear. Only real parts of X ij , Y ij will survive. From Eq. (22) we observe that the Largrangian remains invariant if a Z 2 symmetry is imposed on the fermions. Therefore this residual Z 2 stabilises the lightest fermion that serves as our dark matter candidate.
The various vertex factors involved in DM phenomenology, generated from the scalar Lagrangian, are We consider the lightest scalar state H 1 as the Higgs with mass m H 1 = 125.09 GeV [61]. Therefore we are left with only three independent parameters: m H 2 , sin θ and v φ ; in the scalar sector.
These parameters can be constrained using the limits from perturbativity, perturbative unitarity, electroweak (EW) precision data and the single induced NLO correction to the W boson mass as has been extensively studied in earlier works [62][63][64]. Below we mention these for completeness purpose.

Constraints
In this section we illustrate the important theoretical and experimental bounds that can constrain the parameter space of the proposed model.

Theoretical Constraints
• The scalar potential should be bounded from below in any field direction. This poses some constraints [65,66] on the scalar couplings of the model which we will discuss in Sec. 5 in detail. The conditions must be satisfied at any energy scales till M P in order to ensure the stability of the entire scalar potential in any field direction.
• One should also consider the the perturbative unitarity bound associated with the S matrix corresponding to scattering processes involving all two-particle initial and final states. In The unitarity constraints are obtained as [67,68] λ H < 4π, λ φH < 8π, and • In addition, all relevant couplings in the framework should maintain the perturbativity limit.
The perturbative conditions turn out to be We will ensure the perturbativity of the couplings present in the model till M P energy scale by employing the renormalization group equations.

Experimental Constraints
where the couplings, λ H 1 χ 1 χ 1 and λ Zχ 1 χ 1 can be obtained from Eq. (22). The bound on Z invisible decay width from LEP is Γ inv Z ≤ 2 MeV at 95% C.L. [69] while LHC provides bound on Higgs invisible decay and invisible decay branching fraction Γ inv • Apart from the invisible decay, the Higgs production cross-section also gets modified in the present model due to mixing with the real scalar singlet. As a result, Higgs production crosssection at LHC is scaled by a factor cos 2 θ and the corresponding Higgs signal strength is , where σ SM is the SM Higgs production cross-section and Br SM is the measure of SM Higgs branching ratio to final state particles X. The simplified expression for the signal strength is given as [64,[71][72][73][74][75][76][77]] where Γ 1 is the decay width of H 1 in SM. In absence of any invisible decay (when m χ 1 > m H 1 /2), the signal strength is simply given R = cos 2 θ. Since H 1 is the SM like Higgs with mass 125.09 GeV, R 1. Hence, this restricts the mixing between the scalars. The

ATLAS [69] and CMS [70] combined result provides
This can be translated into an upper bound on sin θ 0.36 at 3σ.
Similarly, one can also obtain signal strength of the other scalar involved in the model , where Γ 2 being the decay with of H 2 with mass m H 2 in SM and Γ T ot H 2 is the total decay width of the scalar H 2 given as Γ T ot H 2 = sin 2 θ Γ 2 + Γ inv H 2 . However due to small mixing with the SM Higgs H 1 , R is very small to provide any significant signal to be detected at LHC [64].
• The presence of fermions in the dark sector and the additional scalar φ will affect the oblique parameters [78] S, T and U through changes in gauge boson propagators. However only T parameter could have a relevant contributions from the newly introduced fields. Contributions to the T parameter by the additional scalar field φ can be found in [79]. However in the small mixing case, this turns out to be negligible [80] and can be safely ignored [41]. When we consider fermions, the corresponding T parameter in our model is obtained as [40,81] where and Λ is the cutoff of the loop integral which vanishes during the numerical estimation.
• The mass of the SM gauge boson W also gets correction from the scalar induced one loop diagram [82]. This poses stronger limit on the scalar mixing angle sin θ 0.3 for m H 2 < 1 TeV [64].

Dark matter phenomenology
In , our DM candidate remains singlet dominated and for m ψ < m ψ S , this becomes doublet like [48]. In the present work we will investigate the characteristics of the dark matter candidate irrespective of its singlet or doublet like nature.

Dark Matter relic Density
Dark matter relic density is obtained by solving the Boltzmann equation. The expression for dark matter relic density is given as [83,84] where M P denotes the reduced Planck mass (2.4 × 10 18 GeV) and the factor J(x f ) is expressed as where x f = m χ 1 /T f , with T f denoting freeze out temperature and g is the total number of degrees of freedom of particles. In the above expression, σ|v| is the measure of thermally averaged annihilation cross-section of dark matter χ 1 into different SM final state particles. It is to be noted that annihilation of dark matter in the present model also includes co-annihilation channels due to the presence of other dark sector particles. Different Feynmann diagrams for dark matter annihilations and co-annihilations are shown in Fig. 1   The thermally averaged dark matter annihilation cross-section σ|v| is expressed as are the corresponding mass splitting ratios. Therefore it can be easily concluded that for smaller values of mass splitting co-annihilation effects will enhance the final dark matter annihilation cross-section significantly. The effective degrees of freedom g ef f is denoted as In the above expression g i , i = 1 − 3 are spin degrees of freedom of particles. Using Eqs. (33)(34)(35)(36), relic density of the dark matter χ 1 can be obtained for the model parameters. The relic density of the dark matter candidate must satisfy the bounds from Planck [85] with 1σ uncertainty is given   Figure 3: The dominant co-annihilation channels of DM (χ 1 ) with charged fermion ψ − . Figure 4: The dominant co-annihilation channels of the charged fermion pair ψ + and ψ − . The expression for spin independent direct detection cross-section in the present singlet doublet model is given as [86] σ SI m 2

Direct searches for dark matter
where λ H i χ 1 χ 1 , i = 1, 2 denotes the coupling of dark matter χ 1 with the scalar H 1 and H 2 as given in the Eq.(22). In the above expression of direct detection cross-section, m r is the reduced mass for the dark matter-nucleon scattering, m r = mχ 1 mp mχ 1 +mp , m p being the proton mass. The scattering factor λ p is expressed as [87] where f q is the atomic form factor [88,89].
As we have mentioned earlier, following the interaction Lagrangian described in Eq. (22), we have an axial vector interaction of the neutral Majorana fermions with the SM gauge boson Z. This will infer spin dependent dark matter nucleon scattering with the detector nuclei. The expression for the spin dependent cross-section is given as [90] σ SD = 16m 2 where d q ∼ g 2 2c W m 2 Z ReX 11 (following Eq. (22)) and λ q depends on the nucleus considering χ 1 as the dark matter candidate.

Results
In this section we present the dark matter phenomenology involving different model parameters and constrain the parameter space with theoretical and experimentally observed bounds discussed in Sec. 4. As mentioned earlier, the dark matter candidate is a thermal WIMP (Weakly Interacting Massive Particle) in nature. The dark matter phenomenology is controlled by the following parameters, We have implemented the model in MicrOmegas [91] and using these above mentioned parameters, we solve the Boltzmann equation for the dark matter candidate in order to obtain the dark matter relic density. The direct detection cross-section obtained are also taken into account. The model in general consists of three neutral fermions χ i , i = 1 − 3 and one charged fermion ψ + which take part in this analysis. The lightest fermion χ 1 is the dark matter candidate that annihilates into SM particles and freeze out to provide the required dark matter relic density. The heavier neutral particles in the dark sector χ 2,3 and the charged particle ψ annihilates into the lightest particle χ 1 .
Also χ 2,3 co-annihilation contributes to the dark matter relic abundance (when the mass differences are small). Different possible annihilation and co-annihilation channels of the dark matter particle is shown in Figs. 1, 2, 3, 4 .
We have kept the mass of the heavier Higgs m H 2 below 1 TeV from viewpoint of future experimental search at LHC. Now in this regime, sin θ is bounded by sin θ 0.3 [64]. In the small sin θ approximation, λ φ almost coincides with the second term in Eq. (20). Now it is quite natural to keep the magnitude of a coupling below unity to maintain the perturbativity at all energy scales (including its running). Hence with the demand λ φ < 1, from Eq.
Unless otherwise stated, for discussion purpose we have kept the heavy Higgs at 300 GeV and v φ , while varied, is kept between 500 GeV and 10 TeV. the regions that satisfy dark matter relic density has spin dependent cross-section ∼ 10 −42 − 10 −44 cm 2 which is well below the present limit obtained from spin dependent bounds (for the specific mass range of dark matter we are interested in) from direct search experiments [58]. Therefore, it turns out that the spin independent scattering of dark matter candidate is mostly applicable in restricting the parameter space of the present model.
In Fig. 7, we depict the effect of scalar mixing in dark matter phenomenology keeping parameters c and λ both fixed at 0.1 along with the same values of m ψ and m H 2 used in Fig. 6. The vev v φ is varied within the range 500 GeV ≤ v φ ≤ 10 TeV. Similar to Fig. 6 (there with λ), here also we notice a scaling with respect to different values of sin θ as the dark matter annihilations depend upon it and there exist two resonances. However beyond m χ 1 ∼ 250 GeV, dependence on sin θ mostly disappears as seen from the Fig. 7 as we observe all three lines merge into a single one.
Note that this is also the region where co-annihilations start to become effective as explained in the context of Fig. 6. It turns out that due to the presence of axial type of coupling in the Lagrangian  From Fig. 7, it is observed that scalar mixing has not much role to play in the co-annihilation region. However the scalar mixing has significant effect in the direct detection (DD) of dark matter.
To investigate the impact of sin θ on DD cross section of DM, we choose few benchmark points (set of λ, m χ 1 values) in our model that satisfy DM relic density excluding the resonance regions (m χ 1 m H 1 /2 resonance regime is highly constrained from invisible Higgs decay limits from LHC).
In Fig. 11 (top left panel) we plot the values of DM mass against dark matter spin independent cross-section for the above mentioned ranges of parameters with sin θ = 0.1 which already satisfy DM relic abundance obtained from Planck [85].  The bounds on DM mass and SI direct detection scattering cross-section from LUX [17], XENON-1T [18], Panda 2018 [19] and XENON-nT [20] are also shown for comparison. The spin dependent scattering cross-section for the allowed parameter space is found to be in agreement with the present limits from Panda 2018 [19] and does not provide any new constraint on the present phenomenology. From Fig. 11 (top left panel) it can also be observed that increasing λ reduces the region allowed by the most stringent Panda 2018 limit. This is due to the fact that an increase in λ enhances the dark matter direct detection cross-section as we have clearly seen from previous plots (see Fig. 6). Here we observe that with the specified set of parameters, dark matter with mass above 100 GeV is consistent with DD limits with λ = 0.01 − 0.15 (see the blue shaded region). For the brown region, we conclude that with λ = 0.15 − 0.30, DM mass above 400 GeV is allowed and with high λ = 0.30 − 0.50, DM with mass 600 GeV or more is only allowed. We also note that a large region of the allowed parameter space is ruled out when XENON-nT [20] direct detection limit is taken into account.
Similar plots for the same range of parameters given in Eq.(41) for sin θ = 0.2 and 0.3 are shown in top right panel and bottom panel of Fig. 11 respectively. These plots depict the same nature as observed in top left panel of Fig. 11. In all these plots, the low mass region (m χ 1 62.5 GeV) is excluded due to invisible decay bounds on Higgs and Z. It can be observed comparing all three plots in Fig. 11, that the allowed region of DM satisfying relic density and DD limits by Panda 2018 becomes shortened with the increase sin θ. In other words, it prefers a larger value of DM mass with the increase of sin θ. This is also expected as the increase of sin θ is associated with larger DD cross-section (due to H 1 , H 2 mediated diagram). Hence overall we conclude from this DM phenomenology that increase of both λ and sin θ push the allowed value of DM mass toward a high value. In terms of vacuum stability, these two parameters, the Yukawa coupling λ and the scalar mixing sin θ, affect the Higgs vacuum stability differently. The Yukawa coupling destabilizes the Higgs vacuum while the scalar mixing sin θ makes the vacuum more stable. Detailed discussion on the Higgs vacuum stability is presented in the next section.
A general feature of the singlet doublet model is the existence of two other neutral fermions, χ 2,3 and a charged fermion, ψ + . All these participate in the co-annihilation process which contributes to the relic density of the dark matter candidate, χ 1 . ψ + can decay into W + and χ 1 . For mass splitting ∆m = m ψ − m χ 1 between χ 1 and ψ + smaller than the mass of gauge boson W + , the three body decay of charged fermion, ψ + into χ 1 associated with lepton and neutrino becomes plausible.
This three body decay must occur before χ 1 freezes out, otherwise it would contribute to the relic.
Therefore, the decay lifetime of ψ + should be smaller compared to the freeze out time of χ 1 . The freeze out of the dark matter candidate χ 1 takes place at temperature T f = m χ 1 /20. Therefore, the corresponding freeze out time can be expressed as where M P is the Planck mass M P = 1.22 × 10 19 GeV and g − 1 2 * effective number of degrees of freedom. The decay lifetime of the charged fermion ψ + is given as τ ψ + = 1 Γ ψ + , where Γ ψ + is the decay width for the decay ψ + → χ 1 l +ν l , is of the form In the above expression, G F is the Fermi constant and the terms I 1,2,3 is expressed as where x = E χ 1 and a = m χ 1 , E χ 1 being the total energy of χ 1 . In order to satisfy the condition that ψ + decays before the freeze out of χ 1 , one must have τ ψ + ≤ t. The integrals I 1,2,3 in Eq. (44) are functions of mass splitting ∆m and so is the total decay width Γ ψ + . To show the dependence on ∆m, we present a correlation plot m χ 1 against ∆m in Fig. 12. Fig. 12 is plotted for the case sin θ = 0.1 (consistent with Fig. 11 having the region allowed by the DD bound from Panda 2018). We use the same color code for λ as shown in Fig. 11. The horizontal red line indicates the the region where ∆m = m W . From Fig. 12 we observe that for smaller values of λ (0.01-0.15), ∆m < m W is satisfied upto m χ 1 ∼ 500 GeV. The mass splitting increases for larger λ values. We find that for the chosen range of model parameters (Eq.(41)), the decay life time τ ψ + is several order of magnitudes smaller than the freeze out time of χ 1 .
We end this section by estimating the value of T parameter in Table 2 . 8).
small and hence it does not pose any stringent constraint on the relic satisfied parameter space.
However with large λ ∼ 1, the situation may alter. In our scenario, we have seen in Fig. 6 that for value of λ larger than 0.4, the direct detection cross-section of dark matter candidate also increases significantly and are thereby excluded by present limits on dark matter direct detection cross-section. To make this clear, here we present a plot, Fig. 13 (left panel), of relic density and GeV, m H 2 = 300 GeV. As before, v φ is varied between 500 GeV and 10 TeV. Similar plot with same set of c, m H 2 but with m ψ = 1000 GeV is depicted in right panel of Fig. 13. Different ranges of dark matter masses are specified with different colors as mentioned in the caption of Fig. 13.
From Fig. 13 we observe that allowed range of λ reduces with the increase of scalar mixing due to the DD bounds. From Fig. 13 (left panel) we get a maximum allowed λ ∼ 0.25 while the same for the m ψ = 1000 GeV (right panel) turns out to be λ = 0.5. Furthermore as we will see the study of vacuum stability, discussed in Sec. 5, indicates that the Yukawa coupling λ should not be large in order to maintain the electroweak vacuum absolutely stable till Planck scale. Therefore, larger values of λ (close to 1) is not favoured in the present scenario.

EW vacuum stability
In the present work consisting of singlet doublet dark matter model with additional scalar, we have already analysed (in previous section) the parameter space of the set-up using the relic density and direct detection bounds. Here we extend the analysis by examining the Higgs vacuum stability within the framework. It is particularly interesting as the framework contains two important parameters, (i) coupling of dark sector fermions with SM Higgs doublet (λ) and (ii) mixing (parametrized by angle θ) between the singlet scalar and SM Higgs doublet. The presence of these two will modify the stability of the EW vacuum. First one makes the situation worse than in the SM by pushing the Higgs quartic coupling λ H negative earlier than Λ SM I . The second one, if sufficiently large, can negate the effect of first and make the Higgs vacuum stable. Thus the stability of Higgs vacuum depends on the interplay between these two. Moreover, as we have seen, the scalar singlet also enriches the dark sector with several new interactions that significantly contribute to DM phenomenology satisfying the observed relic abundance and direct detection constraint. Also the scalar mixing angle is bounded by experimental constraints (sin θ 0.3) as we have discussed in Sec. 3.
The proposed set up has two additional mass scales: the DM mass (m χ 1 ) and heavy Higgs (m H 2 ). Although the dark sector has four physical fermions (three neutral and one charged), we can safely ignore the mass differences between them when we consider our DM to fall outside the two resonance regions (Figs. 6-10). As we have seen in this region (see Fig. 12), co-annihilation becomes dominant, all the masses in the dark sector fermions are close enough (∼ m χ 1 , see Figs. [6][7][8][9][10]. Hence the renormalisation group (RG) equations will be modified accordingly from SM ones with the relevant couplings entering at different mass scales. Here we combine the RG equations (for the relevant couplings only) [93] together in the following (provided µ > m φ , m χ 1 ), where β SM is the SM β function (in three loop) of respective couplings [3,[94][95][96].
In this section our aim is to see whether we can achieve SM Higgs vacuum stability till Planck mass (M P ). However we have two scalars (SM Higgs doublet and one gauge singlet φ) in the model.
Therefore we should ensure the boundedness or stability of the entire scalar potential in any field direction. In that case the following matrix has to be co-positive. The conditions of co-positivity [65,66] of such a matrix is provided by Violation of λ H > 0 could lead to unbounded potential or existence of another deeper minimum along the Higgs direction. The second condition (λ φ (µ) > 0) restricts the scalar potential from having any runway direction along φ. Finally, λ φH (µ) + 2 λ H (µ)λ φ (µ) > 0 ensures the potential to be bounded from below or non-existence of another deeper minimum somewhere between φ or H direction.
On the other hand, if there exists another deeper minimum other than the EW one, the estimate of the tunneling probability P T of the EW vacuum to the second minimum is essential. The Universe will be in metastable state only, provided the decay time of the EW vacuum is longer than the age of the Universe. The tunneling probability is given by [3,9], where T U is the age of the Universe, µ B is the scale at which probability is maximized, determined from β λ H = 0. Hence metastable Universe requires [3,9] λ H (µ B ) > −0.065 In SM, the top quark Yukawa coupling (y t ) drives the Higgs quartic coupling to negative values.
In our set up, the coupling λ has very similar effect on λ H in Eq. (47). So combination of both y t and λ make the situation worse (by driving Higgs vacuum more towards instability) than in SM.   Table 4: Initial values of the relevant mass scales (DM mass m χ 1 and heavy Higgs mass m H 2 ), v φ and the couplings (c and λ) of the dark sector used to study the Higgs vacuum stability.
In Fig.14, we constrain the sin θ − λ parameter space using the absolute stability criteria (λ eff H (µ) > 0 for µ = m t to M P ) for the EW vacuum for BP-I and BP-II as values of parameters given in Table 4. The solid green line in Fig. 14 indicates the boundary line in sin θ − λ plane  In Fig.14 (right panel) we include two black dots (X and Y) which we use as two reference points used to indicate the evolution of λ eff H and the co-positivity conditions explicitly mentioned in Eq. (53). Thus in Fig.15, running of λ eff H is shown against the energy scale µ for those two reference points: X (λ = 0.18 and sin θ = 0.1) and Y (λ = 0.4 and sin θ = 0.2) of Fig. 14 (right panel).
We find for sin θ = 0.2, λ eff H remains positive starting from µ = m t to M P energy scale and for sin θ = 0.16, although λ eff H stays positive throughout its evolution, it marginally reaches zero at M P . Hence the X point appears as the boundary point in sin θ − λ plane Fig. 14 (right panel) beyond which the SM Higgs vacuum becomes unstable. In Fig. 15 (right panel) we show the evolution of all the co-positivity conditions from µ = m t to M P .

Conclusion
We have explored a dark matter model by extending the Standard Model of particle physics with a singlet scalar and a dark sector comprised of two Weyl doublets and a Weyl singlet fermions.
The scalar singlet acquires a vev and contributes to the mass the dark sector particles consisting of three neutral Majorana fermions and one charged Dirac fermion. The lightest Majorana particle is stable due to the presence of a residual Z 2 symmetry and hence we study whether this can account for the dark matter relic density and also satisfy the direct detection bounds. There exists a mixing of the singlet scalar with the SM Higgs doublet in the model which results in two physical scalars, which in turn affect the DM phenomenology. We have found that apart from the region of two resonances, there exists a large available region of parameter space satisfying various theoretical and experimental bounds particularly due to large co-annihilations effects present. On the other hand, inclusion of new fermions in the model affects the Higgs vacuum stability adversely by leading it more toward instability at high scale due to new Yukawa like coupling. This issue however can be resolved by the involvement of extra scalar singlet. We find that with the demand of having a dark matter mass ∼ few hundred GeV to 1 TeV consistent with appropriate relic density and DD limits and simultaneously to make the EW vacuum absolutely stable upto the Planck scale, we can restrict the scalar mixing angle significantly. The result is carrying a strong correlation with the dark sector Yukawa coupling, λ. It turns out that, with higher dark matter mass, the allowed range of sin θ becomes more stringent from this point of view. Hence future limits of sin θ will have the potential to allow or rule out the model under consideration.